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K

BALTIMORE LECTURES

ON

MOLECULAR DYNAMICS

AKD

THE WAVE THEORY OF LIGHT

aonDon: C. J. OLAY and SONS,

CAMBBID6E UNIVEBSITY PBESS WAREHOUSE,

AVE MABIA LANE.

0Usaoi»: 60, WELLINGTON STBEBT.

BaUinwre: PUBLICATION AGENCY OF THE JOHNS HOPKINS UNIVEESITY.

Bombsg anti eialnitta: MACMILLAN AND CO., Ltd

%tmia: F. A. BBOCKHAUS.

[All lUghti reserved.]

BALTIMORE LECTURES

ON

MOLECULAR DYNAMICS

AND

THE WAVE THEORY OF LIGHT

PBINTBD BT J. AND C. F. OLAT,
AT THS UNIVSBBXTT PRXBS.

PREFACE.

HAVING been invited by President Oilman to deliver a
course of lectures in the Johns Hopkins University after
the meeting of the British Association in Montreal in 1884, on
a subject in Physical Science to be chosen by myself, I gladly
accepted the invitation. I chose as subject the Wave Theory
of Light with the intention of accentuating its failures ; rather
than of setting forth to junior students the admirable success with
which this beautiful theory had explained all that was known
of light before the time of Fresnel and Thomas Young, and
had produced floods of new knowledge splendidly enriching the
whole domain of physical science. My audience was to consist of
Professorial fellow-students in physical science ; and from the
beginning I felt that our meetings were to be conferences of
coefficients, in endeavours to advance science, rather than teach-
ings of my comrades by myself I spoke with absolute freedom,
and had never the slightest fear of undermining their perfect
faith in ether and its light-giving waves : by anything I could
tell them of the imperfection of our mathematics ; of the insuffi-
ciency or faultiness of our views regarding the dynamical qualities
of ether; and of the overwhelmingly great difficulty of finding
a field of action for ether among the atoms of ponderable matter.
We all felt that difficulties were to be faced and not to be
evaded ; were to be taken to heart ivith the hope of solving
them if possible ; but at all events with the certain assurance
that there is an explanation of every difficulty though we may
never succeed in finding it.

vi PREFACE.

It is in some measure satisfactory to me, and I hope it will
be satisfactory to all my Baltimore coeflScients still alive in our
world of science, when this volume reaches their hands; to find
in it dynamical explanations of every one of the difficulties with
which we were concerned from the first to the last of our twenty
lectures of 1884. All of us will, I am sure, feel sjnnpathetically
interested in knowing that two of ourselves, Michelson and Morley,
have by their great experimental work on the motion of ether
relatively to the earth, raised the one and only serious objectiou*
against our dynamical explanations; because they involve the
assumption that ether, in the space traversed by the earth and
other bodies of the solar system, is at rest absolutely except in so
far as it is moved by waves of light or radiant heat or variations
of magnetic force. It is to be hoped that farther experiments
will be made; to answer decisively the great question: — is, or
is not, ether at rest absolutely throughout the universe, except
in so far as it is moved by waves generated by motions of ponder-
able matter? I cannot but feel that the true answer to this
question is in the affirmative, in all probability : and provisionally,
I assume that it is so, but always bear in mind that experimental
proof or disproof is waited for. As far as we can be contented
with this position, we may feel satisfied that all the difficulties
of 1884, set forth in Lectures I, X, and XV, are thoroughly
explained in Lectures XVIII, XIX, and XX, as written afresh
in 1902 and 1903.

It seems to me that the next real advances to be looked for in
the dynamics of ether are : —

(I) An explanation of its condition in the neighbourhood of
a steel magnet or of an electromagnet ; in virtue of which mutual
static force acts between two magnets whether in void ether or in
space occupied also by gaseous, liquid, or solid, ponderable matter.

(II) An investigation of the mutual force between two moving
electrions, modified from purely Boscovichian repulsion; as it must
be by the composition, with that force, of a force due to the inertia

* See Appendix A § 18 and Appendix B § 10.

PBSFACE. yii

of the ether set in motion by the motion of each of the elcctrions.
It seems to me that, of these, (II) may be at present fairly within
our reach ; but that (I) needs a property of ether not included
in the mere elastic-solid-theory worked out in the present volume.
My object in undertaking the Baltimore Lectures was to find how '
much of the phenomena of light can be explained without going
beyond the elastic-solid-theory. We have now our answer : every
thing non-magnetic; nothing magnetic. The so-called "electro-'
magnetic theory of light " has not helped us hitherto : but the
grand object is fully before us of finding a comprehensive dynamics
of ether, electricity, and ponderable matter, which shall include
electrostatic force, magnetostatic force, electromagnetism, electro-
chemistry, and the wave theory of light.

I take this opportunity of expressing the gratitude with which
I remember the hearty and genial cooperation of my coefficients
in our meetings of 19 years ago in Baltimore, and particularly
the active help given me by the late Prof. Rowland, from day to
day all through our work.

I desire also to specially thank one of our number, Mr A. S.
Hathaway, for the care and fidelity with which ho stenograph ically
recorded my lectures, and gave his report to the Johns Hopkins
University in the papyrograph volume published in December 1884.
The first eleven lectures, as they appear in the present volume,
have been printed from the papyrograph, with but little of even
verbal correction ; and with a few short additions duly dated.

Thirteen and a half years after the delivery of the lectures,
some large additions were inserted in Lecture XII. In Lectures
XIII, XIV, XV, freshly written additions supersede larger and
larger portions of the papyrograph report, which still formed
the foundation of each Lecture. Lectures XVI — XX have been
written afresh during 1901, 1902, 1903.

In my work of the last five years for the present volume
I have received valuable assistance successively from Mr W. Craig
Henderson, Mr W. Anderson, and Mr G. A. Witherington ; not

a 6

VUl PREFACE.

only in secretarial affairs, but frequently also in severe mathe-
matical calculation and drawing; and I feel very grateful to them
for all they have done for me.

The printing of the present volume began in August, 1885 ;
and it has gone on at irregular intervals during the 19 years
since that time; in a manner which I am afraid must have
been exceedingly inconvenient to the printers.

I desire to thank Messrs J. and C. F. Clay and the Cambridge
University Press for their never-failing obligingness and efficiency
in working for me in such trying circumstances, and for the
admirable care with which they have done everything that could
be done to secure accuracy and typographical perfection.

KELVIN.

Netherhall,

Jaiiuary^ 1904.

CONTENTS.

LECTURE I.

Introductory.

PAGES

Tho wave theory of light ; molecular trcatDicnt by Frcsnel and

Cauchy ; molar by (jreoii 6-6

Ordinary dispersion. Anomalous disjjcrsion ; hypothc«os of Cauchy
and Helmholtz. Time-element in conutitution of matter.
" Electromagnetic Theory of Light." Nature of ether ; tenta-
tive comparison with shoemakers' wax 6-14

Direction of the vibrations in polarized light. Dynamical theory
of refraction and reflection not comi)lete ; difterences of rigidity
or of density of the ether on the two sides of a reflecting
interface ; exi)eriments of Prof. Rood on reflectivity . . 14-17

Double refraction ; its dilHcultias. Explanations suggested by
liankinc and by Huyleigh tosteil by Stokes ; and Iluyghens'
l^aw verified. Further illustr;ition by mcchaniwil model.
Conclusion 17-21

LECTURE II.

Part I.

lolar. Dynamics of elastic solid. James Thomson's radian. General

e<{uations of energy and force, liquations for isotropic material.
Relation between Green's twenty-one cocfticients and the bulk

and rigidity mod \il uses 22-26

(jreneral equations of motion ; moditication for hcterogeneousnoss ;
case of zero rigidity ; compressible fluid. Lord Riiyleigh's work
on the blue sky 26-28

Part II.

Ifolecular. Dynamics of a row of connected particles. Illustrative mechanical

model. The [)ropagation of light waves through the atmosphere.
Equations of motion 28-33

CONTENTS.

LECTURE III.

Part I.

PAGES

Molar. Dynamics of elastic solid. Illustrative model for twenty-one

coefficients. Equations of motion ; methods of solving. Velo-
cities of the condensational and distortional waves. Leslie's
experiment on soimd of a bell in hydrogen explained by Stokes 34-38

Part IL

Molecular. Variations of complex serial molecule. Continued fraction to aid

solution 38-40

LECTURE IV.

Molar. Equations of motion of elastic solid ; method of obtaining all i^ssible

solutions. Energy of condensational waves of ether negligible.
"Electromagnetic theory of lighf wants dynamical founda-
tion ; propagation of electric and magnetic disturbances to be

brought within the wave theory of light 41-46

Solutions for condensational waves travelling outwards from a point ;
spherical waves ; sphere vibrating in ether ; two spheres
vibrating oppositely in lino of centres ; aerial vibrations round
tuning fork ; cone of silence 46-51

LECTURE V.

Molar. Vibrations of air around a tuning fork continued .... 52

Molecular. Vibrations of serial molecule. Period and ratios of displacement ;

critical periods ; infinite i)eriod 53-58

Fluorescence. Finite sequences of waves in dispersive medium ;

beginnings and endings of light 58-60

LECTURE VI.

Molar. Ratio of rigidity to compressibility. Velocity-potential. Conden-

sational wave ; vibrations at a distance from origin ; circular

bell, timing fork, and elliptic bell 61-65

Vibrations of approximately circular plates ; planes of silence.

Velocity of groups of waves through transparent substances . 65-68

Molecular. Vibrations of serial molecule. Lagrange ; " algorithm of finite

differences " ; number of terms in determinant . . . 69-70

LECTURE VII.

Molecular. Vibrations of serial molecule. Solution in terms of roots ; deter-
mination of roots 71-76

Metallic reflection ; Rayleigh on Cauchy ; Sellmeier, Helmholtz,
Lommel. Lommel on double refraction, interesting but not
satisfactory ; Stokes' experimental disproof of Rankine's theory
[if ether wholly incompressible. See Lecture xix, § 184, below] 76-79

CONTENTS.

XI

LECTURE VIII.

IColAr. Solutions for distortional waves. Rotational oscillation in origin ;

illustration by woodon ball in jelly. Double source with
contrary rotations. Solution for waves generated by to-and-fro
vibrator ; Stokes, Raylcigh, and the blue of the sky

Molecular. Sudden and gradual commencements of vibration ; fluorescence

and phosphorescence ; initial and permanent refraction ;
anomalous dispersion fading after a time ....

LECTURE IX.

Ifolar. Contrary vibrators in one line. Sudden banning and gradual

ending of vibrations in a source of light Observations on
interference ; Sellmeier, Fizeau. Conference with Prof. Row-
land ; conclusion that subsidence of vibrations of molecular
source is small in several hundred thousand vibrations ;

dynamics invoked

Amount of energy in the vibration of an atom. Sellmeier,
Helmholtz, and Lommel, on the loss of energy in progress
of waves. Dynamics of absorption ; first taught by Stokes.
Sellmeier's dynamics of anomaloas dispersion

Molecular. Problem of seven vibrating particles. Dynamical explanation

of ordinary refraction ; mechanical model ; expression for
refractive index

PAGES

80-00

90-93

94-99

99-103

103-107

Molar.

Moleoolar.

Molar.

LECTURE X.

Energy of waves. Name for v^ ; Laplacian preferred. Geometrical
illustrations of Fourier's theorem for representation of
arbitrary functions

Cauchy and Poisson on deep-sea waves; great contest of 1815,
Cauchy or Poisson to rule the waves

Difficulties regarding polarization by reflection, double refraction ;
form of wave-surface. Anomalous disi)ersion, fluorescence,
phosphorescence, and radiant heat, discoverable by dynamics
alone. Radiant heat from Leslie cube. Refractivity through
wide range. Refractivity of rock-salt ; Langley .

LECTURE XL

" Anisotropy" rejected ; aeolotropy suggested by Prof. Lushington.
Aeolotropy in an elastic solid. Father Boscovich judged
obsolete in 1884 [reinstated as guide in 1900. See Lectures xv
to XX and Appendices passim]. Green's twenty-one modu-
luses of elasticity. Navier- Poisson ratio ; disproof, and
Green's full theory illustrated by mechanical models. Most
general plane wave expressed in terms of the twenty-uue
independent moduluses

108-114
114-117

117-121

122-134

CONTENTS.

LECTURE III.

Part I.

PAGES

Molar. Dynamics of elastic solid. Illustrative model for twenty-one

coefficients. Equations of motion ; methods of solving. Velo-
cities of the condensational and distortional waves. Leslie's
experiment on sound of a bell in hydrogen explained by Stokes 34-38

Part II.

Molecular. Variations of complex serial molecule. Continued fraction to aid

solution 38-40

LECTURE IV.

Molar. Equations of motion of elastic solid ; method of obtaining all possible

solutions. Energy of condensational waves of ether negligible.
"Electromagnetic theory of light" wants dynamical founda-
tion ; propagation of electric and magnetic disturbances to be

brought within the wave theory of light 41-46

Solutions for condensational waves travelling outwards from a point ;
spherical waves ; sphere vibrating in ether ; two spheres
vibrating oppositely in line of centres ; aerial vibrations round
tuning fork ; cone of silence . . . . . . 46-51

LECTURE V.

Molar. Vibrations of air aroimd a tuning fork continued .... 52

Molecular. Vibrations of serial molecule. Period and ratios of displacement ;

critical periods ; infinite period 53-58

Fluorescence. Finite sequences of waves in disi)ersive medium ;

beginnings and endings of light 58-60

LECTURE VI.

Molar. Ratio of rigidity to compressibility. Velocity-potential. Conden-

sational wave ; vibrations at a distance from origin ; circular

bell, tuning fork, and elhptic bell 61-65

Vibrations of approximately circular plates ; planes of silence.

Velocity of groups of waves through transparent substances . 65-68

Molecular. Vibrations of serial molecule. Lagrange ; " algorithm of finite

difierences " ; number of terms in determinant . . 69-70

LECTURE VII.

Molecular. Vibrations of serial molecule. Solution in terms of roots ; deter-
mination of roots 71-76

Metallic reflection ; Rayleigh on Cauchy ; Sellmoier, Helmholtz,
Lommel. Lommel on double refraction, interesting but not
satisfactory ; Stokes' experimental disproof of Rankine's theory
[if ether wholly incompressible. See Lecture xix, § 184, below] 76-79

CONTENTS.

XI

LECTURE VIII.

Molar. Solutions for distortional waves. Rotational oscillation in origin ;

illustration by wooden ball in jelly. Double source with
contrary rotations. Solution for waves generated by to-and-fro
vibrator ; Stokes, Rayleigh, and the blue of the sky

Molecolar. Sudden and gradual comniencemcnts of vibration ; fluorescence

and phosphorescence ; initial and (permanent refraction ;
anomalous dispersion fading after a time ....

LECTURE IX.

Molar. Contrary vibrators in one line. Sudden banning and gradual

ending of vibrations in a soiurce of light Observations on
interference ; Sellmeier, Fizeau. Conference with Prof. Row-
land ; conclusion that subsidence of vibrations of molecular
source is small in several hundred thousand vibrations ;

dynamics invoked

Amount of energy in the vibration of an atom. Sellmeier,
Helmholtz, and Lommel, on the loss of energy in progress
of waves. Dynamics of absorption ; first taught by Stokes.
Sellmeier's dynamics of anomalous dispersion

Molecular. Problem of seven vibrating ^mrticles. Dynamical explanation

of ordinary refraction ; mechanical model ; expression for
refractive index

PAGES

80-00

90-93

94-99

99-103

103-107

LECTURE X.

Molar. Energy of waves. Name for V" ; Laplacian preferred. Geometrical

illustrations of Fourier's theorem for representation of

arbitrary functions 108-114

Cauchy and Poisson on deep-sea waves; great contest of 1815,

Cauchy or Poisson to rule the waves 114-117

Molecolar. Difficulties regarding polarization by reflection, double refraction ;

form of wave-surface. Anomalous disi)cr8ion, fluorescence,
phosphorescence, and radiant heat, discoverable by dynamics
alone. Radiant heat from Leslie cube, llefractivity through
wide range. Refractivity of rock-salt ; Langley . . . 117-121

LECTURE XL

Molar. " Aiiisotropy" rejected ; aeolotropy suggested by Prof. Lushington.

Aeolotropy in an elastic solid. Father Boscovich judged
obsolete in 1884 [reinstated as guide in 1900. See Lectures xv
to XX and Appendices passim]. Green's twenty-one modu-
luses of elasticity. Navier- Poisson ratio ; disproof, and
Green's full theory illustrated by mechanical models. Most
general plane wave expressed in terms of the twenty-one
independent moduluses 122-134

Xll CONTENTS.

LECTURE Xn.

Part I. pages

Molar. Three sets of plane waves with fronts parallel to one plane ; wave-

surface with three sheets ; Blanchet ; Poisson, Coriolis, Sturm,
Liouville, and Duhamcl, on Blanchet. Purely distortional
wave; Green. Incomprassibility ; energy of condensational
waves in ether; density of the luminiferous ether . . 135-142

Part II.

Moleoular. Mutual force between atom and ether. Vibrating molecule;

polarized soim;e of light. Dynamics of dispersion ; Sellmeier's
theory ; refractivity formula, and Rubens' exi)erimental results
for refractivity of rock-salt and sylvin. "Mikrom" and

"michron" 142-154

Critical periods, fi^=^l, fi^=Oy fi^ passing from - oo to -|-oo . Con-
tinuity in undulatory theory through wide range of frequencies ;
mechanical, electrical, and electromagnetic vibrations all in-
cluded in sound and light 155-162

LECTURE XIII.

Molecular. Prof. Morloy'a numerical solution for vibrator of seven ixsriods.

Dynamical wave machine 163-165

Molar. iEolotropy resumed. Direction of displacement and return force.

Incompressible aoolotropic elastic solid. French classical
nomenclature. Square oDolotropy. Rankine's nomenclature.

Cubic SBolotropy 165-170

-lEolotropic elasticity without skewness ; equalisation of velocities
for one plane of distortion. Green's dynamics for Fresnel's
wave-surface 170-175

Moleoular. Application of Sellmeier's dynamical theory, to the dark lines

DiD^ ; to a single dark line. Photographs of anomalous
dispersion by prisms of sodium- vajwur by Heiwi Bccquerel 176-184

LECTURE XIV.

Molecolar. Rowland's model vibrator. Motion of ether with embedded mole-
cules 185-190

Molar. Mathematical investigation of spherical waves in elastic solid . 190-219

Stokes' analysis into two constituents, equivoluminal tuid irrota-

tional 193-194

Simplification of waves at groat distance from origin . . . 197

Examples by choice of arbitrary functions 199

Details of motion, in equator, in 45* cone, and in axis of disturbance 200-201
Originating disturbance at a spherical siuface S; fiill solution for

S rigid 202-219

Diagrams to illustrate motion of incompressible elastic solid

tlu*ough infinite sjKice around a vibrating rigid globe . . 206-207

CONTENTS.

Xlll

[olar. Rates of transmitting energy outwards by the two waves .

Examples

LECTURE XV.

iolecnlar. Model vibrator; excitation of synchronous vibrators in molecule

by light

!olar. Thlipeinomic treatment of compressibility. Double refraction;

difficulty ; wave-velocity depends on line of vibration ; Stokes ;
Cauchy and Green's extraneous force ; stress theory
Hypothetical elastic solid. Further development of stress theory ;
influence of ponderable atoms on ether ; Kerr's observations
unfavourable to the stress theory of double refraction ;
Glassebrool^ founds successfully on aeolotropic inertia

LECTURE XVL

olar. Mechanical value of smilight and ^lossible density of ether.

Rigidity of ether; etherial non-resistance to motion of ix>n-
derable matter. Is ether gra^tational ? . . . .
Gravitational matter distributed throughout a very large volume ;
quantity of matter in the stellar system ; velocities of stars ;
great velocity of 1830 (troombridge ; Newcomb's suggestion.
Total apparent area of stars exceedingly small Number of
visible stars i Masses of stars

LECTURE XVIL

olecalar. Molecular dimensions ; est iniatos of Thomas Vouii^, W. Thomson,

Johnstone Stoney, Losc'hmidt. Tensilo strt?ngth in l)lack siM>t
of soap bubble ; Newton ; its thickness niciusured by Reinold
and Riicker. Thinnest film of oil on wattT; Riintgen and
Rayleigh

Kinetic theory of gases; molecular diftusivity ; th<H)ry of Clausius
and Maxwell ; 0. E. Moyc^r. Inter -diffasivity of pairs of gases ;
viscosity of gases; exiHirimental detenninations of inter-
diffusivities by Ijoschmidt and of viscosities by Oberniayer;
Maxwell ...........

Numlxjr of molecules in a cubic centimetre of gas intimated.
Stokes on the polariz<Hi light from a cloudless sky; Ilayleigh's
theory of the blueness of the .sky. Transparency of atmo-
sphere; Bouguer and Aitken. Haze due to larger particles.
Dynamics of the V)lue sky

Sky -light over Etna and Monte Ros«'i ; olxservations of Majorana
and Sella. Definition of I^iimbert's "albedo." Table of
albedos; Becker, Miiller. Prismatic analysis of sky-light;
Rayleigh; researches of (JiusepiK) Zettwuch. Atmosi)heric
absori)tion; Miiller and Becker. Molecular dimensions of
particular ga.seous li([uid and solid substances

PAOIS

211-214
214-219

220-221

221-240

241-259

260-267

267-278

279-286

286-298

298-313

313-322

xiv CONTENTS.

PAGER

Moleonlar. Vibrations of polarized light are perpendicular to plane of polari-
zation ; Stokes' cxi>erimental proofs ; experiments of Holtzraann
and Lorenz of Denmark ; confirmation of Stokes by Rayleigh
and Lorenz 322-323

LECTURE XVIII.

Reflection op Light.

Molar. Refraction in opaque substances. Translucenco of metallic films.

Ideal silver. Molecular structure ; best polish obtainable.
Reflectivity ; definition ; results of Fresnel, Potter, Jamin,
and Conroy. Exi>erimental method for reflectivities . . 324-332
Polarizational analysis of reflected light ; experiments of Fresnel

cos (t"^ 't)

and Brewster; deviations from Fresnel's — ,-; — '-i-. Principal

cos (t — ,i) ^

incidence and principal azimuth ; definition ; results for

different metals by Jamin and Conroy 332-343

FresnePs laws of reflection, and Green's theory, and obsen^ation
graphically represented for water, glass, and diamond ; great
deviations from Green's theory. Tangents of principal
azimuth for seven transparent substances .... 343-351

Suggested correction of Green's theory ; velocity of condensational
wave very small instead of very large ; by assuming negative
compressibility ; voluminal instjibility and stability. Equa-
tions of motion. Definition of wave-plane. Reflection and
refraction at an interface l^etween solids. Solutions for
vibrations peri)endicular to plane of incidence. Total internal
reflection of an impulse and of sinusoidal waves ; forced
clinging wave at interface in less dense mediiun. Differences
of density and rigidity in the two mediums ; equal rigidities
proved by observation 351-363

Solutions for vibrations in plane of incidence. Direct incidence.

Rayleigh on the reflectivity of glass. Grazing incidence . 363-373

Oj>acity and reflectivity of mettds ; metallic films ; electromagnetic
screening. Continuity of dynamical wave-theory between
light and slow vibrations of magnets to be discovered. Green's
solution for vibrations in the plane of incidence realised ; case
of infinite density 373-379

Quasi wave-motion in medium of negative or imaginary density ;
ideal silver. Phase difterencc between components of reflected
light. Princii>al incidence. Adamantine proi>erty . . 379 -386

Theory of Fresnel's rhomb. P}ia.sal changes by internal reflection.
Reflection of circularly jwkrized light. Tables of phasal back-
set by single reflection, and of virtual advance by two reflec-
tions 386-393

CONTENTS.

XV

PAOKB

Errors in construction of Fresnel's rhomb determined by
MacCullagh ; errors of two Fresnel's rhombs determined by
observation. An eighty years' error regarding change of
phase in total reflection

Orbital direction of light from a mirror. Rules for the orbital
direction of light; circularly or elliptically polarized light
emerging from Fresnel's rhomb with illustrative diagram

Dynamics of loss of light in reflection ; principal azimuth is 45* if
reflectivity perfect

393-403

403-406
406-407

LECTURE XIX.
Reconciliation bbtwesn Fresnbl and Orebn.

ieooUur. Pruf. Morle/s numerical solutions of the problem of seven

mutually interacting particles 408-409

lar. Navier-Poisson doctrine disproved for ordinary matter by Stokes

considering deviations in direction of greater resistance to
compression, and by Thomson and Tait in the opposite direc-
tion. Bulk-modulus of void ether practically infinite positive ;
but n^ative just short of ^n in space occupied by ponderable
matter ; to reconcile Fresnel and Green. Velocity of conden-
sational-rarefactional waves practically infinite in void ether,
nearly zero in space occupied by ponderable matter ; return
to hypothesis of 1887 with modification .... 409-412

ileoular. Theory of interactions between atoms of ponderable matter and

ether, giving dynamical foundation for these assumptions . 412-415

>l*r. Dynamical theory of adamantinism ; imaginary velocity of con-
densational, rarefactional waves. Numerical reckoning of
adamantinism defined (o-). Adamantine reflection ; phasal
lag ; diagrams of phasal lag for diamond. Principal incidence,
and principal azimuth in adamantine reflection. Difierenccs
of princijMil incidence from Brewsterian angle very small ;
values of the adamantinism of different substances . . 415-424
Double refraction. Huyghens construction confirmed experimen-
tally by Stokes. Fresnel's wave-surface verified in a biaxal
crystal by Glazebrook. iEolotropic inertia, with the assump-
tion of very small velocity for the condensational wave in
transparent matter, gives dynamical explanation of Fresnel's
laws of polarized light in a crystal. Archibald Smith's in-
vestigation of Fresnel's wave- surface 424-430

Vibrational line proved not in wave-plane but i>erpendicular to
ray by Glazebrook ; Glazebrook and Basset's construction for
drawing the vibrational line ; calculation and diagram for
arragonite. Theory wanted for vibrational amplitude in any
part of wave-surface 430-436

XVI CONTENTS.

LECTURE XX.

PAGES

Molecular. Chiral rotation of the plane of polarization. Electro-etherial ex-
planation of eaolotropic inertia. Dynamical explanation of
chiral molecule, called a chiroid 436-440

Molar. Formulas expressing chiral inertia in wave-motion ; given also by

Boussinesq in 1903. Imaginary terms and absorption ; all
terms real ; velocity of circularly polarized light in chiral
medium. Coexistent waves of contrary circular motions;
sign of angular velocities about three mutually perpendicular
axes ; FresneFs kinematics of coexistent waves of circular
motions. Left-handed molecules make a right-handed medium 440-449
Solution for wave-plane perpendicular to a principal axis. Optic
chirality not imperceptible in a biaxal crystal ; experiment-
ally determined in sugar by Dr H. C. Pocklington. Optic
chirality in quartz crystal ; two trial assimiptions compared ;
illustrative diagrams. Modification of double refraction by

chirality in quartz 450-458

Mathematical theories tested experimentally by McConnel. Amount
of rotation of vibrational line of polarized light travelling
through various substances. Chirality in crystals and in
crystalline molecules ; Sir George Stokes. Faraday's mag-
neto-optic rotation 458-461

Moleonlar. Electro-etherial theory of the velocity of light through transparent

bodies ; explanation of uniform rigidity ; theory of refraction.
Energy of luminous waves in transparent substances . 461-467

APPENDIX A.

On the Motion produced in an Infinite Elastic Solid by
THE Motion through the Space occupied by it of a
Body acting on it only by Attraction or Repulsion.

Suggestions regarding the distribution of ether within an atom,

and regarding force between atoms and ether .... 468-475

Absolute orbits of ten particles of ether disturbed by a moving
atom. Stream lines of relative motion of 21 particles of
ether • . . 475-481

Kinetic energy of the ether within a moving atom ; extra inertia
of ether moving through a fixed atom. Velocity of light
through an assemblage of atoms. Difficulty raised by Michel-
Hon's experiment ; suggested explanation by Fitzgerald and
by Lorentz of Leyden 481-485

CONTENTS.

XVU

APPENDIX B.

NiNSTSENTH CeNTURY ClOUDS OYKB THR DtNAMICAL

Thrort of Hrat and Light.

Cloud I. The motion of ponderable matter through ether, an
elastic solid. Early views of Fresnel and Young. Hypo-
thesis of atoms and ether occupying the same space ; uniform
motion of atom imresisted if less than the velocity of light
Hypothesis sufficient to explain the production of light by
vibrating atom, and consistent with aberration. Difficulty
raised by Michelson's experiment. Cloud I not wholly dis-
sipated

Cloud II. Waterstonian-Maxwellian distribution of energies ;
demonstration by Tait for hard Boscovich atoms. The
Boltzmann-Maxwell doctrine of the partition of energy ; its
application to the kinetic theory of gases ; ratio of thermal
capacities ; Clausius' theorem and the B.M. doctrine ; the
latter not reconcilable with truth regarding the specific heats
of gases

Dynamical test-cases for the B.M. doctrine ; reflections of ball
from elliptic boundary; kinematic construction for finding
the paths within boundary of any shape ; scalene triangle
with rounded comers ; deviations from B.M. doctrine .

Statistics of reflections from corrugated boundary ; marlin spike-
shaped boundary ; chattering impacts of rotators on a fixed
plane ; particle constrained to remain near a closed surface ;
ball bounding from corrugated floor ; caged atom struck by
others outside ; non- agreement with B.M. doctrine

B.M. doctrine applied to the equilibrium of a tall column of gas
under gravity. Boltzmann and Maxwell and Hayleigh on the
difficulties regarding the application of the B.M. doctrine to
actual gases. Simplest way to get rid of the difficulties is to
abandon the doctrine, and so dissolve Cloud II

PA0B8

486-492

493-504

504-512

512-524

524-527

APPENDIX C.

On the Disturbance produced by two particular forms op
Initial Displacement in an infinitely long Material
System for which the velocity of Periodic Waves

DEPENDS ON THE WaVE-LENGTH.

Solution in finite tenns for elastic rod. Solution by highly con-
vergent definite integral for all times and places in Poisson
and Cauchy's deep-water wave problem ; four examples of
results calculated by quadratures ; finite solution for deep-
water waves due to initial disturbance ....

628-531

XVUl

CONTENTS.

APPENDIX D.

On the Clustbrino of Gravitational Matter in
any part of the universe.

Ether is gravitationless matter filling all space. Total amount
of ponderable matter in the space known to astronomers.
Total of bright and dark star-disc apparent areas. Vast
sphere, of perfectly compressible liquid, or of uniformly
distributed atoms, given at rest; the subsequent motion
under gravity. Clustering of gravitational matter distri-
buted throughout infinite space. Atomic origin of all
things; Democritus, Epicurus, Lucretius . . . .

PAGES

632-540

APPENDIX E.

AePINUB ATOMIZED.

Electrion ; definition. Assumptions regarding atoms of ponderable
matter and electrions and their interactions; neutralizing
quantum. Resinous and vitreous, instead of positive and
negative. Overlapping of two mono-electrionic atoms ; case
(1) each atom initially neutralized ; case (2) one atom void,
the other neutralized ; in each case, after separation, the larger
atom is left void ; probably true explanation of frictional
and separational electricity ; commerce of energies involved 541-550

Stable equilibrium of several electrions in an atom. Exhaustion
of eneigy in stable groups of electrions. Electrionic explana-
tion of Faraday's electro-inductive capacity, of BecquerePs
radio-activity, and of electric conductivity .... 550-559

Electrionic explanation of pyro-electricity and piezo-electricity ;
di-polar electric crystal Discoveries of Hatiy and the
brothers Curie. Octo-polar electric aeolotropy explained in
a homogeneous assemblage of pairs of atoms, and in a
homogeneous assemblage of single atoms. Voigt's piezo-
electricity of a cubic crystal 559-568

APPENDIX F.

Dynamical illustrations of the magnetic and the heli^oidal
rotatory etkcta of transparent bodies on polarized light.
Mounting of two-period pendulum rotated ; problem of the
motion solved ; solution examined ; vibrations in rotating
long straight rod of elliptic cross-section ....

Interior melting of ice; James Thomson's physical theory of
the plasticity of ice ; illustrated by Tyndall's experiments.
Stratification of vesicular ice by pressure observed in

569-577

CONTENTS. XIX

PAGX8

glaciers, and experimentally produced bj Tyndall ; explained
by the physical theory. Forbes explains blue veins in
glaciers by pressure, and the resultant plasticity of a glacier
by the weakness of the vesicular white strata. Tyndall's
vesicles produced in clear ice by focus of radiant heat ; all
explained by the physical theory 577-583

§

APPENDIX G.
Htdbokinetic Solutions and Observations.

Part I. On the motion of free solids through a liquid. Eulerian
equations of the motion, lines of reference ; axes fixed rela-
tively to the body ; lines of reference fixed in space more
convenient ; isotropic heli^oid (now called chiroid) . . . 584-588

Part II. Equations of motion when liquid circulates through

holes in movable body 588-590

Part III. The influence of frictionless wind on waves in A-iction-
less water ; particular wind velocities ; maximum wind velocity
for level surface of water to be stable 590-593

Part IV. Waves and ripples. Minimum wave-velocity. Moving
force of ripples chiefly cohesion. The least velocity of fric-
tionless air that can raise ripples on quiescent frictionless
water. Effect of viscosity ; Stokes on the work against vis-
cosity required to maintain a wave. Scott Russell's Report
on Waves ; capillary waves 593-598

Part V. Waves imder motive i)ower of gravity and cohesion
jointly without wind. Cohesive force of water. Fringe of
ripples in front of vessel ; ship-waves abaft on each quarter ;
influence of cohesion on short waves. An ex|)eri mental deter-
mination of nunimum velocity of solid to produce ripples or
waves in water 598-601

APPENDIX H.
On the Molecular Tactics op a Crystal.

Homogeneous assemblage of bodies ; theorem of BravaiH. Thirteen
nearest neighbours in acute-angled homogeneous assemblage.
Equilateral assemblage. Stereoscopic representation of cluster
of thirteen globes in contact. Barlow's model of homogeneous
assemblages. Homogeneous assemblage implies homogeneous
orientations. Homogeneous assemblage of symmetrical
doublets 602-609

How to draw partitional boundary in three dimensions ; parti -
tional boundary for homogeneous plane figures. Homogeneous
assemblage of tetrakaidekahedronal cells. Stereoscopic view
of equilateral tetrakaidekahedron. Equivoluminal cells with
minimum partitional area 609-617

CONTENTS.

PAGES

Different qualities on two parallel sides of a crystal ; oppositely
electrified by same rubber ? oppositely electrified by changes
of temperature ; and by lateral pressures. Homogeneous
packing of molecules or of continuous solids, as bowls or
spoons or forks ; case of single contact. Orientation for
closest packing in this case. Definition of " chiraL" Scalene
convex solids homogeneously assembled. Ratio of void to
whole space in heaps of gravel, sand, and broken stones.
Orientation of molecules for closest packing. Change of
configuration of assemblage in stable equilibrium causes
expansion. Osborne Reynolds on the dilatation of wet sand
when pressed by the foot ; seen when glass-plate is sub-
stituted for foot. Assemblage of globular molecules (or of
ellipsoids if all similarly oriented) is rigid and constant in
bulk, provided each molecule remains in coherent contact
with twelve neighbours 617-626

G^metrical problem of each solid touched by eighteen neighbours.
Stereoscopic view of tetrahedron of convex scalene solids.
Suggestion for a twin-crystaL Miller's definition of twin-
crystal. Twin-plane as defined by Miller, and as defined
by Stokes and Rayleigh. Successive twin-planes regular or
irregular. Madau's change of tactics by heat in chlorate of
potash ; Raylcigh's experiment. Stokes and Rayleigh show
that the brilliant colours of chlorate of potash crystals are
due to successive twinnings at regular intervals ; cause of
twinnings and counter-twinnings at regular intervals undis-
covered 626-637

Ternary tactics in lateral and terminal faces of quartz. The
triple anti-symmetry required for the piezo-electricity in
quartz investigated by the brothers Curie. Optic and piezo-
electric chirality in quartz and tourmaline ; Voigt and Riecke.
Orientational macling, and chiral macling, of quartz. Geo-
metrical theory of chirality 637-642

APPENDIX I.
On the Elasticity of a Crystal according to Boscovich.

Crystalline molecule and chemical molecule ; crystalline molecule
probably has a chiral configuration in some or all chiral
crystals ; Marbach, Stokes ; certainly however many chemical
molecules have chirality ; magnetic rotation has no chirality.
Forcive required to keep a crystal homogeneously strained.
Boscovich's theory. Only 15 coefficients for simplest Bosco-
vichian crystal 643-646

Displacement ratios and their squares and products. Work
against mutual Boscovichian forces. Elastic energy of strained

CONTENTS. XXI

PAGES

simple homogeneous assemblage. Navier-Poisson relation
proved for simple assemblage of Boscovich atoms. In an
equilateral assemblage, when each point experiences force
only from its twelve nearest neighbours, the diagonal rigidity
is half the facial rigidity, relatively to the principal cube . 646-655
Single assemblage in simple cubic order. Equilateral assemblage ;
elasticity moduluses ; conditions for elastic isotropy ; forces
between points for equilibrium. Scalene homogeneous as-
semblage of points; scalene tetrahedron in which perpen-
diculars from the four comers to opposite faces intersect in
one point ; scalene assemblage giving an incompressible solid
according'to Boscovich. Twenty-one coefficients for elasticity
of double assemblage 655-661

APPENDIX J.
Molecular Dynamics of a Crtstal.

Work done in separating a homogeneous assemblage of molecules
to infinite distances from one another. Dynamics of double
homogeneous assemblage. Distances of neighbours : in single
equilateral assemblage: in double assemblage. Scalene as-
semblage in equilibrium if infinite. Work-curves for two
assemblages 662-671

Stabilities of monatomic and diatomic assemblages ; stability not
secfiired by positive moduluses; stability of infinite row of
similar particles ; Boscovich's curve ; equilibrium in the neigh-
l)ourhood of ends of a long row of atoms. Alternating varia-
tions of density seven or eight nets from surface of crystal . 671-680

APPENDIX K.
On Variational Electric and Magnetic Screening.

Imperfectly conducting material acts as screen, only against steady
or slowly varying electrostatic force. Blackened i)aper trans-
jmrcnt for Hertz waves of high frequency; but opaque to
light waves. Electro-conductive plate acts as screen against
magnetic changes. Rotating conductor screens steady mag-
netic force. Experimental illustration 681-687

APPENDIX L.

Electric Waves and Vibrations in a Submarine

Telegraph Wire.

ERRATA.

Page 14, line 19, for ** refracted " read *< reflected."
15, delete footnote.

39, line 7 from foot, for ** vibrations " read ** vibration.*'
46, line 7 from foot, for " K" read "r."

48, in denominator of last term of equation (18), for *' x' " read ** r*."
70, in heading, for «* Part I." read ** Part II."

73, line 5, before '* the " insert *' the reciprocals of the squares of.*'

74, lines 2 and 6, for *« c *' read " t."

75, in footnote, for *< Appendix C " insert "Lect. XIX., pp. 408, 409," and
delete ** with illustrative carves/'

86, line 10 from foot, for " dx* " read " dt»."

87, line 10, delete ** that."

4t3 4r>

87, line 3 from foot, for "yS^-'lT^" read "vV= "ij^"

88, line 15, for ''words painting " read " word-painting."
90 and 92, in heading, for " Part I.'* read ** Part D."

103, last line, for -;; read -. .

106, line 14, for '• X" read " x ."

239, for marginal " molecular" read *' molar."

NOTES OF LECTURES

ON

MOLECULAR DYNAMICS

...^_^_^___ AND

ERRATUM.
Page 116, line 13 from foot, for " t," read •* iri.

>f

1884.

COPYRIGHT BY THE JOHNS HOPKINS UNIVERSITY,

BALTIMORE, MD.

> .

»»

ERRATA.

Page 14, line 19, for "refracted" read ** reflected."
„ 15, delete footnote.

„ 89, Une 7 from foot, for ** vibrations " read ** vibration."
„ 46, line 7 from foot, for " T" read "r."

„ 48, in denominator of last term of equation (18), for *' x' " read ** r*."
„ 70, in heading, for «• Part I." read ** Part U.'*

73, line 5, before ** the " insert ** the reciprocals of the squares of."

74, lines 2 and 6, for " c " read •* t."

75, in footnote, for **Appendix C" insert **Lect. XIX., pp. 408, 409," and
delete " with illastrative carves."

86, Une 10 from foot, for *« dx^ " read ** dtV

87, line 10, delete ** that."

4t> 4ir3

„ 87, line 8 from foot, for «« y«^ _ ^ ^ " read ** vV = - -rr ^•"

„ 88, line 15, for ** words painting " read ** word-painting."
„ 90 and 92, in heading, for ** Part I." read '* Part H."

„ 108, last line, for -r read

it
If

ft

ft

106, line 14, for "X" read " x .»»
„ 239, for marginal " molecular " read ** molar."

ff

NOTES OF LECTURES

ON

MOLECULAR DYNAMICS

AND

ERRATUM.
Page 116. line 13 ftrom foot, for " t," read •* ni**

1884.

COPYRIGHT BY THE JOHNS HOPKINS UNIVERSITY.

BALTIMORE. MD.

ERRATA.

Page 14, line 19, for ** refracted" read *' reflected."
" ^g' delete footnote.

„ 100, last line, for ^ read ^ .

„ 106, line 14, for "X" read " x .'»

„ 239, for marginal *< molecular ** read '* molar."

NOTES OF LECTURES

ON

MOLECULAR DYNAMICS

AND

THE WAVE THEORY OF LIGHT.

DELIVERED AT THE JOHNS HOPKINS UNIVERSITY, BALTIMORE,

BT

SIR WILLIAM THOMSON,

PROFESSOR IN THE UNIVERSITY OF GLAROOW.

STENOGRAPHIC ALLY REPORTED BY

A. S. HATHAWAY,

LATELY FELLOW IN MATHEMATICS OF THE JOHNS HOPKINS UNIVERSITY.

1884.

COPYRIGHT BY THE JOHNS HOPKINS UNIVERSITY,

BALTIMORE, MD.

ADVERTISEMENT.

In the month of October, 1884, Sir William Thomson of
Glasgow, at the request of the Trustees of the Johns Hopkins
University in Baltimore, delivered a course of twenty lectures
before a company of physicists, many of whom were teachers of
this subject in other institutions. As the lectures were not written
out in advance and as there was no immediate prospect that they
would be published in the ordinary form of a book, arrangements
were made, with the concurrence of the lecturer, for taking down
what he said by short-hand.

Sir William Thomson returned to Glasgow as soon as these
lectures were concluded, and has since sent from time to time
additional notes which have been added to those which were
taken when he spoke. It is to be regretted that under these
circumstances he has had no opportunity to revise the reports.
In fact, he will see for the first time simultaneously with the
public this repetition of thoughts and opinions which were freely
expressed in familiar conference with his class. The "papyro-
graph" process which for the sake of economy has been employed
in the reproduction of the lectures does not readily admit of cor-
rections, and some obvious slips, such as Canchy for Cauchy, have
been allowed to pass without emendation ; but the stenographer
has given particular attention to mathematical formulas, and
he believes that the work now submitted to the public may be
accepted, on the whole, as an accurate report of what the lecturer
said.

A S. HATHAWAY.

Dec, 1884.

LECTURE I.

5 p.m. Wednesday, Oct 1, 1884.

The most important branch of physics which at present makes
demands upon molecular dynamics seems to me to be the wave
theory of light. When I say this, I do not forget the one great
branch of physics which at present is reduced to molecular dyna-
mics, the kinetic theory of gases. In saying that the wave theory
of light seems to be that branch of physics which is most in want,
which most imperatively demands, applications, of molecular dyna-
mics just now, I mean that, while the kinetic theoiy of gases is a
part of molecular dynamics, is founded upon molecular dynamics,
works wholly within molecular dynamics, to it molecular dynamics
is everything, and it can be advanced solely by molecular dynamics;
the wave theory of light is only beginning to demand imperatively
applications of that kind of dynamical science.

The dynamics of the wave theory of light began very molecu-
larly in the hands of Fresnel, was continued so by Cauchy, and to
some degree, though much less so, in the hands of Green. It was
wholly molecular dynamics, but of an imperfect kind in the hands
of Fresnel. Cauchy attempted to found his mathematical investi-
gations on a rigorous molecular treatment of the subject. Green
almost wholly shook off the molecular treatment, and worked out
all that was to be worked out for the wave theory of light, by the
dynamics of continuous matter. Indeed, I do not know that it is
possible to add substantially to what Green has done in this sub-
ject. Substantial additions are scarcely to be made to a thing
perfect and circumscribed as Green's work is, on the explanation
of the propagation of light, of the refraction and the reflection of
light at the bounding surface of two different mediums, and of the

6 LECTURE I.

propagatiou ol' light through crystals, by a strict mathematical
treatment, founded on the consideration of homogeneous elastic
matter. Oreen's treatment is really complete in this respect, and
there is nothing essential to be added to it. But there is a great
deal of exposition wanting to let us make it our own. We must
study it; we must try to see what there is in the very concise €uid
sharp treatment, with some very long formulas, which we find in
Green's papers.

The wave theory of light, treated on the assumption that the
medium through which the light is propagated is continuous and
homogeneous, except where distinctly separated by a bounding
inter-face between two different mediums, is really completed by
Green. But there is a great deal to be learned from that kind of
treatment that perhaps scarcely has yet been learned, because the
subject has not been much studied nor reduced to a very popular
form hitherto.

Cauchy seemed unable to help beginning with the considera-
tion of discreet particles mutually acting upon one another.
But, except in his theory of dispersion, he virtually came to the
same thing somewhat soon in his treatment every time he began
it afresh, as if he had commenced right away with the considera-
tion of a homogeneous, elastic solid. Green preceded him, I believe,
in this subject. I have read a statement of Lord Rayleigh that there
seems to have been an attributing to Cauchy of that which Green
had actually done before. Green had exhausted the subject ; but
there is I believe no doubt that Cauchy worked in a wholly inde-
pendent way.

What I propose in this first Lecture — we must have a little
mathematics, and I therefore must not be too long with any kind
of preliminary remarks — is to call your attention to the outstand-
ing difficulties. The first difficulty that meets us in the dynamics
of light is the explanation of dispersion; that is to say, of the fact
that the velocity of propagation of light is different for different
wave lengths or for light of different periods, in one and the same
medium. Treat it as we will, vary the fundamental suppositions
as much as we can, as much as the very fundamental idea allows
us to vary them, and we cannot force from the dynamics of a
homogeneous elastic solid a difference of velocity of wave propa-
gation for different periods.

Cauchy pointed out that if the sphere of action of individual

INTRODUCTORY. 7

molecules be comparable with the wave lengths, the fact of the
difference of velocities for different periods or for different wave
lengths in the same medium is explained. The best way, perhaps,
of putting Cauchy's fundamental explanation is to say that there is
heterogeneousness through space comparable with the wave length
in the medium, — that is, if we are to explain dispersion by
Cauchy's unmodified supposition. We shall consider that, a little
later. I have no doubt the truth is perfectly familiar already to
many of you that it is essentially insufficient to explain the facts.

Another idea for explaining dispersion has come forward more
recently, and that is the assumption of molecules loading the
luminiferous ether and somehow or other elastically connected with
it The first distinct statement that I have seen of this view is in
Helmholtz's little paper on anomalous dispersion. I shall have
occasion to speak of that a good deal and to mention other names
whom Helmholtz quotes in this respect, so that I shall say
nothing about it historically, except that there we have in Helra-
holtz's paper, and by some Oerman mathematicians who preceded
him, quite another departure in respect to the explanation of
dispersion. The Cauchy hypothesis gives us something comparable
with the wave length in the geometrical dimensions of the body.
Or, to take a crude matter of fact view of it, let us say the ratio of
the distance from molecule to molecule (from the centre of one
molecule to the centre of the next nearest molecule) to the wave
length of light is the fundamental characteristic, as it were, to
which we must look for the explanation of dispersion upon
Cauchy's theory.

We may take this fundamental idea in connection with the two
hypotheses for accounting for dispersion : that we must have, in the
very essence of the ponderable medium, some relation either to
wave length or to period, and it seems at first sight (although this
is a proposition that may require modification) that with very long
waves the velocity of propagation should be independent of the
period or wave length. That, at all events, seems to be the case
when the subject is only looked upon according to Cauchy's view.
We are led to say then that it seems that for very long waves
there should be a constant velocity of propagation. Experiment
and observation now seem to be falling in very distinctly to affirm
the conclusions that follow from the second hypothesis that I
alluded to to account for dispersion. In this second hypothesis,

H LKCTUBK I.

itmU^i tA having a geometrical dimension in the ponderable
fitsUUit which in crjmparable with the wave length, we have a
tuufbLtntsnibi time-relation — a certain definite interval of time
mmthUow ingrained in the constitution of the ponderable matter,
which in c^miiiarable with the period. So that, instead of a relation
^4 U^ui(ih U) length, we have a relation of time to time.

Now, how are we to get our time element ingrained in the
f'^ftmiiintiou of matter? We need scarcely put that question
uoW'i%r4hiyH. W<5 are all familiar with the time of vibration of the
tif/tiiinn At/ftn, and the great wonders revealed by the spectroscope
ar^; all full #if indications showing a relation to absolute intervals of
tirru; jfi ihfi f/rofK^'ities of matter. This is now so well understood,
tl«at it in no n<;w idea to propose to adopt as our unit of time one
*4 ihit fMndarri<;ntal periods — for instance, the period of vibration of
light in on<; or (ither of the sodium D lines. You all have a
AyuMttirjil i«l<;a of this already. You all know something about
th^' time of vibration of a molecule, and how, if the time of
vihration of light [lassiug through any substance is uearly the
MAffM; tm the natural time of vibration of the molecules of the
ntt\^Uiut'Js, this approximate coincidence gives rise to absorption.
W<; all know of (course, according to this idea, the old dynamical
i;xp|iination, first proposed by Stokes, of the dark lines of the solar

Wa havi! now this interesting point to consider, that if we
would work out the idea of dispersion at all, we must look definitely
Ut tmntn of vibration, in connection with all ponderable matter. To
g«;t a firm hyixithesis that will allow us to work on the subject, let
us iirMgine space, otherwise full of the luminiferous ether, to be
Initially rjccupied by something different from the general lumini-
ferous ether. That s<iniething might be a portion of denser ether,
or u |Kirtion of more rigid ether ; or we might suppose a portion of
ether U} have greater density and greater rigidity, or different
density and different rigidity from the surrounding ether. We will
rvnne t^ck to that subject in connection with the explanation
(A the blue sky, and, particularly. Lord Rayleigh's dynamics of the
blue sky. In the meantime, I want to give something that will
allow us to bring out a very crude mechanical model of dis-
persion.

In the first place, we must not listen to any suggestion that we
are to look upon the luminiferous ether as an ideal way of putting

INTRODUCTORY. 9

the thing. A real matter between us and the remotest stars
I believe there is, and that light consists of real motions of that
matter, motions just such as are described by Fresnel and
Young, motions in the way of transverse vibrations. If I knew
what the electro-magnetic theoiy of light is, I might be able
to think of it in relation to the fundamental principles of the wave
theory of Ught. But it seems to me that it is rather a backward
step from an absolutely definite mechanical notion that is put
before us by Fresnel and his followers to take up the so-called
Electro-magnetic theory of light in the way it has been taken up
by several writers of late. In passing, I may say that the one
thing about it that seems intelligible to me, I do not think is
admissible. What I mean is, that there should be an electric dis-
placement perpendicular to the line of propagation and a magnetic
disturbance perpendicular to both. It seems to me that when we
have an electro-magnetic theory of light, we shall see electric dis-
placement as in the direction of propagation, and simple vibrations
as described by Fresnel with lines of vibration perpendicular to the
Une of propagation, for the motion actually constituting light. I
merely say this in passing, as perhaps some apology is necessary
for my insisting upon the plain matter-of-fact dynaniics and the
true elastic solid as giving what seems to me the only tenable
foundation for the wave theory of light in the present state of our
knowledge.

The luminiferous ether we must imagine to be a substance
which so far as luminiferous vibrations are concerned mgves as if
it were an elastic solid. I do not say it is an elastic solid.
That it moves as if it were an elastic solid in respect to the lumini-
ferous vibrations, is the fundamental assumption of the wave theory
of light.

An initial difficulty that might be considered insuperable is,
how can we have an elastic solid, with a certain degree of rigidity
pervading all space, and the earth moving through it at the
rate the earth moves around the sun, and the sun and solar system
moving through it at the rate in which they move through space,
at all events relatively to the other stars ?

That difficulty does not seem to me so very insuperable.
Suppose you take a piece of Burgundy pitch, or Trinidad pitch, or
what I know best for this particular subject, Scottish shoemaker s
wax. This is the substance I used iu the illustration I intend

10 LECTURE I.

to refer to. I do not know how far the others would succeed
in the experiment. Suppose you take one of these substances, the
shoemaker s wax, for instance. It is brittle, but you could form
it into the shape of a tuning fork and make it vibrate. Take a
long rod of it, and you can make it vibrate as if it were a piece of
glass. But leave it lying upon its side and it will flatten down
gradually. The weight of a letter, sealed with sealing-wax in the
old-fashioned way, used to flatten it, will flatten it. Experiments
have not been made as to the absolute fluidity or non-fluidity of
such a substance as shoemaker's wax ; but that time is all that is
necessary to allow it to yield absolutely as a fluid, is not an im-
probable supposition with reference to any one of the substances
I have mentioned. Scottish shoemaker's wax I have used in this
way : I took a large slab of it, perhaps a couple of inches thick,
and placed it in a glass jar ten or twelve inches in diameter. I filled
the glass jar with water and laid the slab of wax in it with a
({uantity of corks underneath and two or three lead bullets on the
upper side. This was at the beginning of an Academic year. Six
months passed away and the lead bullets had all disappeared, and
I suppose the corks were half way through. Before the year had
passed, on looking at the slab I found that the corks were floating
in the water at the top, and the bullets of lead were tumbled about
on the bottom of the jar.

Now, if a piece of cork, in virtue of the greater specific gravity
of the shoemaker's wax would float upwards through that solid
material and a piece of lead, in virtue of its greater specific gravity
would move downwards through the same material, though only at
the rate of an inch per six months, we have an illustration, it seems
to me, quite suflGicient to do away with the fundamental difficulty
from the wave theory of light. Let the luminiferous ether be
looked upon as a wax which is elastic and I was going to say
brittle, (we will think of that yet — of what the meaning of brittle
would be) and capable of executing vibrations like a tuning fork
when times and forces are suitable — when the times in which the
forces tending to produce distortion act, are very small indeed, and
the forces are not too great to produce rupture. When the forces
are long continued then very small forces suffice to produce un-
limited change of shape. Whether infinitesimally small forces
produce unlimited change of shape or not we do not know; but
very small forces suffice to do so. All we need with respect to the

INTRODUCTORY. 11

luminiferous ether is that the exceedingly small forces brought
into play in the luminiferous vibrations do not, in the times during
which they act, suffice to produce any transgression of limits of
distortional elasticity. The come-and-go effects taking place in
the period of the luminiferous vibrations do not give rise to the
consumption of any large amount of energy: not large enough an
amount to cause the light to be wholly absorbed in say its pro-
pagation from the remotest visible star to the earth.

If we have time, we shall try a little later to think of some of
the magnitudes concerned, and think of, in the first place, the mag-
nitude of the shearing force in luminiferous vibrations of some
assumed amplitude, on the one hand, and the magnitude of the
shearing force concerned, when the earth, say, moves through the
luminiferous ether, on the other hand. The subject has not been
•gone into very fully; so that we do not know at this moment
whether the earth moves dragging the luminiferous ether altogether
with it, or whether it moves more nearly as if it were through
a frictionless fluid. It is conceivable that it is not impossible that
the earth moves through the luminiferous ether almost as if it were
moving through a frictionless fluid and yet that the luminiferous
ether has the rigidity necessary for the performance of the lumini-
ferous vibration in periods of from the four hundred million
millionth of a second to the eight hundred million millionth of
a second corresponding to the visible rays, or from the periods
which we now know in the low rays of radiant heat as recently
experimented on and measured for wave length by Abney and
Langley, to the high uJtra-violet rays of light, known chiefly by
their chemical actions. If we consider the exceeding smallness of
the period from the 100 million millionth of a second to the 1600
million millionth of a second through the known range of radiant
heat and light, we need not fully despair of understanding the
property of the luminiferous ether. It is no greater mystery at
all events than the shoemaker's wax or Burgundy pitch. That is
a mystery, as all matter is ; the luminiferous ether is no greater
mystery.

We know the luminiferous ether better than we know any
other kind of matter in some particulars. We know it for its
elasticity ; we know it in respect to the constancy of the velocity
of propagation of light for diCFerent periods. Take the eclipses of
Jupiter's satellites or something far more telling yet, the waxings

12 LECTURE I.

and wanings of self-luminous stars as referred to by Prof. Newoomb
in a recent discussion at Montreal on the subject of the velocity of
propagation of light in the luminiferous ether. These phenomena
prove to us with tremendously searching test, to an excessively
minute degree of accuracy, the constancy of the velocity of propa-
gation of all the rays of visible light through the luminiferous
ether.

Luminiferous ether must be a substance of most extreme
simplicity. We might imagine it to be a material whose ultimate
property is to be incompressible ; to have a definite rigidity for
vibrations in times less than a certain limit, and yet to have the
absolutely yielding character that we recognize in wax-like bodies
when the force is continued for a sufficient time.

It seems to me that we must come to know a great deal more
of the luminiferous ether than we now kuow. But instead of
beginning with saying that we know nothing about it, I say that
we know more about it than we know about air or water, glass or
irou — it is far simpler, there is far less to know. That is to say,
the natural history of the luminiferous ether is an infinitely
simpler subject than the natural history of any other body. It
seems probable that the molecular theory of matter may be so far
advanced sometime or other that we can understand an excessively
fine-grained structure and understand the luminiferous ether as
differing from glass and water and metals in being very much more
finely grained in its structure. We must not attempt, however, to
jump too far in the inquiry, but take it as it is, and take the great
facts of the wave theory of light as giving us strong foundations
for our convictions as to the luminiferous ether.

To think now of ponderable matter, imagine for a moment

that we make a rude mechanical model. Let
this be an infinitely rigid spherical shell ; let
there be another absolutely rigid shell inside
of it, and so on, as many as you please. Na-
turally, we might think of something more
continuous than that, but I only wish to call
your attention to a crude mechanical ex-
planation, possibly sufficient to account for dispersion. Suppose
we had luminiferous ether outside, and that this hollow space is
of very small diameter in comparison with the wave length. Let
zig-zag springs connect the outer rigid boundary with boundary

INTRODUCTORY. 13

number two. T use a zig-zag, not a spiral spring possessing the
helical properties for which we are not ready yet, but which must
be invoked to account for such properties as sugar and quartz
have in disturbing the luminiferous vibrations.
Suppose we have shells 2 and 3 also connected /^ \^ ^\\
by a sufficient number of zig-zag springs and
so on ; and let there be a solid nucleus in the
centre with spring connections between it and
the shell outside of it. If there is only one of
these interior shells, you will have one definite
period of vibration. Suppose you take away everything except
that one interior shell ; displace that shell and let it vibrate while
you hold the outer sheath fixed. The period of the vibration is
perfectly definite. If you have an immense number of such
sheaths, with movable molecules inside of them distributed through
some portion of the luminiferous ether, you will put it into a
condition in which the velocity of the propagation of the wave
will be different from what it is in the homogeneous luminiferous
ether. You have what is called for, viz., a definite period ; and
the relation between the period of vibration in the light con-
sidered, and the period of the free vibration of the molecule will
be fundamental in respect to the attempt of a mechanism of that
kind to represent the phenomena of dispersion.

If you take away everything except one not too massive
interior nucleus, connected by springs with the outer sheath, you
will have a crude model, as it were, of what Helmholtz makes the
subject of his paper on anomalous dispersion. Helmholtz, besides
that, supposes a certain degree or coefficient of viscous resistance
against the vibration of the nucleus, relatively to the sheath.

If we had only dispersion to deal with there would be no
difficulty in getting a full explanation by putting this not in a
rude mechanical model form, but in a form which would commend
itself to our judgment as presenting the actual mode of action of
the particles, of gross matter, whatever they may be upon the
luminiferous ether. It is difficult to imagine the conditions of the
luminiferous ether in dense fluids or liquids, and in solids; but
oxygen, hydrogen, and gases generally, must, in their detached
particles, somehow or other act on the luminiferous ether, have
some sort of elastic connection with it; and I cannot imagine
anything that commends itself to our ideas better than this

14 LECTURE I.

multiple molecule which I have put before you. By taking
enough of these interior shells, and by passing to the idea of
continuous variation from the density of the ether to the enor-
mously greater density of the molecule of grosser matter imbedded
in it, we may come as it were to the kind of mutual action that
exists between any particular atom and the luminiferous ether.
It seems to me that there must be something in this molecular
hypothesis, and that as a mechanical symbol, it is certainly not
a mere hypothesis, but a reality.

But alas for the difficulties of the undulatory theory of light ;
— refraction and reflection at plane surfaces worked out by Green
differ in the most irreduceable way from the facts. They cor-
respond in some degree to the facts, but there are differences that
we have no way of explaining. A great many hypotheses have
been presented, but none of them seems at all tenable.

First of all is the question, are the vibrations of light per-
pendicular tOy or are they m, the plane of polarization — defining
the plane of polarization as the plane through the incident and
refracted rays, for light polarized by reflection ? Think of light
polarized by reflection at a plane surface and the question is, are
the vibrations in the reflected ray perpendicular to the plane of
incidence and reflection, or are they in the plane of incidence and
reflection ? I merely speak of this subject in the way of index.
We shall consider very fully, Green's theory and Lord Rayleigh's
work upon it, and come to the conclusion with absolute certainty,
it seems to me, that the vibrations must be perpendicular to the
plane of incidence and reflection of light polarized by reflection.

Now there is this difficulty outstanding —the dynamical theory
of refraction and reflection which gives this result does not give it
rigorously, but only approximately. We have by no means so
good an approach in the theory to complete extinction of the
vibrations in the reflected ray (when we have the light in the
incident ray vibrating in the plane of incidence and reflection) as
observation gives. I shall say no more about that difficulty, be-
cause it will occupy us a good deal later on, except to say that the
theoretical explanation of reflection and refraction is not satisfac-
tory. It is not complete; and it is unsatisfactory in this, that we
do not see any way of amending it

But suppose for a moment that it might be mended and there
is a question connected with it which is this : Is the difference

INTRODUCTORY. 16

between two mediums a difference corresponding to difference of
rigidity, or does it correspond to difference of density? That is an
interesting question, and some of the work that had been done
upon it seemed most tempting in respect to the supposition that
the difference between two mediums is a difference of rigidity and
not a difference of density. When fully examined, however, the
seemingly plausible way of explaining the facts of refraction and
reflection by difference of rigidity and no difference of density I
found to be delusive, and we are forced to the view that there is
difference of density and very little difference of rigidity.

In working out this subject very carefully, and endeavouring to
understand Lord Bayleigh's work upon it, and to learn what had
been done by others, for a time I thought it too much of an
assumption that the rigidity was exactly the same and that the
whole effect was due to difference of density. Might it not be, it
seemed to me, that the luminiferous ether on the two sides of the
interface at which the refraction and reflection takes place, might
differ both in rigidity and in density. It seemed to me then by a
piece of work (which I must verify, however, before I can speak
quite confidently about it *) that by supposing the luminiferous
ether in the commonly called denser medium to be considerably
denser than it would be were the rigidities equal, and the rigidity
to be greater in it than in the other medium, we might get a
better explanation of the polarization by reflection than Green s
result gives. Green's work ends with the supposition of equal
rigidities and unequal densities. He puts the whole problem in
his formulae to begin with, but he ends with this supposition and
his result depends upon it.

Not to deal in generalities, let us take the case of glass and a
vacuum, say. It seemed to me that by supposing the effective
rigidity of luminiferous ether in glass to be greater than in vacuum
and the density to be greater, but greater in a greater proportion
than the rigidity, so that the velocity of propagation is less in glass
than in vacuum, we should get a better explanation of the details
of polarization by reflection than Green's result gives.

It is only since I have left the other side of the Atlantic that I
have worked at this thing, and going into it with keen interest,
I inquired of everybody I met whether there were any observa-
tions that would help me. At last I was told that Prof Rood had

* See Appendix A.

16 LECTURE I.

done what I desired to know, and on looking at his paper, I found
that it settled the matter.

My question was this: Has there been any measurement of the
intensity of light reflected at nearly normal incidence from glass or
water showing it to be considerably greater than FresneFs formula

gives? Fresnel gives I- — ^ j for the ratio of the intensity of the

reflected ray to the intensity of the incident ray in the case of
normal incidence, or incidence nearly normal. I wanted to find
out whether that had been verified by observation. It seems that
nobody had done it at all until Prof. Rood, of Columbia College,
New York, took it up. His experiments showed to a rather
minute degree of accuracy an agreement with Fresnel's formula,
so that the explanation T was inclined to make was disproved by
it. I myself had worked with the reflection of a candle from a
piece of window glass, during a pleasant visit to Dr Henry Field
in your beautiful Berkshire bills at the end of August ; and had
come to the same conclusion, even through such very crude and
roughly approximate measurements. At all events, I satisfied
myself that there was not so great a deviation from Fresnel's law
as would allow me to explain the diflScultiea of refraction and
reflection by assuming greater rigidity, for example, in glass than
in air. We are now forced very much to the conclusion from
several results, but directly from Prof Rood's photometrical ex-
periments, that the rigidity must be very nearly equal in the two.

There is quite another supposition that might be made that
would give us the same law, for the case of normal incidence ; — the
supposition that the reflection depends wholly upon diCFerence
of rigidity and that the densities are equal in the two. That gives
rise to the same intensity of perpendicularly reflected light, so that
the photometric measurement does not discriminate between these
two extremes, but it does prevent us from pushing in on the other
side of generally accepted result, a supposition of equal rigidities,
in the manner that I had thought of

We may look upon the explanation of polarization by reflection
and refraction as not altogether unsatisfactory, although not quite
satisfactory : —and you may see [pointing to diagrams which had
been chalked on the board : see pp. 12 and 13 above] that this
kind of modificatiou of the luminiferous ether is just what would
give us the virtually greater density. How this gives us precisely

INTRODUCTORY.

17

the same effect as a greater density I shall show when we work
the thing out mathematically. We shall see that this supposition
is equivalent to giving the luminiferous ether a greater density,
while making the addition to virtual density according to the period
of the vibration.

I am approaching an end : I had hoped to get to it sooner.
We have the subject of double refraction in crystals, and here
is the great hopeless difficulty.

If we look into the matter of the distortion of the elastic solid, Molar.
we may consider, possibly, that that is not wonderful ; but
Fresnel's supposition as to the direction of the vibration of light, is
that the conclusion that the plane of vibration is perpendicular to
the plane of polarization proves, if it is true, that the velocity
of propagation of light in uniaxial crystals depends simply on
the direction of vibration and not otherwise on the plane of
the distortion. In the vibrations of light, we have to consider
the medium as being distorted and tending to recover its shape.
Let this be a piece of uniaxial crystal, Iceland spar, for instance,
a round or square column, with its length in the direction of
the optic axis, which I will represent on the board by a
dotted line.

Now the relation between light polarized by passing
through Iceland spar on the one hand and light polarized
by reflection on the other hand, shows us that if the line
of vibration is perpendicular to the plane of polarization, the
velocity of propagation of light in different directions through
Iceland spar must depend solely on the line of vibration and not
on the plane of distortion.

There is no way in which that can be explained by the rigidity Case I.
of an elastic solid. Look upon it in this way. Take a cube of ^jJI^^J*"
Iceland spar, keeping the same direction of the axis
as before. Let the light be passing downwards, as
indicated by the arrow-head. What would be the
mode of vibration, with such a direction of propaga-
tion? Let us suppose, in the first place, the vibra-
tions to be in the plane of the diagram. Then the dis-
tortion of that portion of matter will be of the kind, and in the
direction indicated by Fig. 2. A portion which was rectangular
swings into the shape represented by the dotted lines. The force
tending to x»iuse a piece of matter which has been so displaced to

T.L. 2

Fig. 1.

direotio
of axis:
yibratio
in plane
duNgran

Fig. 2.

Aj :ar iini- r Tit r**3saL T^ St What

y*°*^ — iifw.i>n T^aiHT ^cll a. lie naokf flc "Hit £itcnm ?

rr- isuui^. i >iTrxriL "viiira ^:»^ ?^:!<5yiirxi^ic '•nl ■«jii£ iato this
«aac#r jmiiiar^i "ly "itr a.or^x iaea- Ji Fjc 3L Tit rsmra foice

'rZ, 'Jiks. it^mi -o».a a Qs&rr^«ia if tojS rnt£ Bat a

Tlz. 2 _ iai£ iif* rtsil- iircsc *>?. x -^ ^dSert Sfw»i» upon the

of pr>:fa^':L':c 3. '^ii :asi* AOii izL t^w; :at£if Fs£& $ acd •>
* III, O/Lsi'i^ sh-JatIt liir riscjroros w^:oi?^ed bx waTes con-

uwtiiig of Tf&cKiric:* p>ifT<egfi>.rg AT V -la* piiaAe vrf the diagimms,
the arrov'bea*!^ in Fi^. f as^i 5 sdTl rrcr^s^aiTing directkHis of
firopdgatioD. Tl^ dis^i^rtkn in ibe cfc« <i Fijr. 2 will stiU be
£M by fiheariog perpecHxicTilar to ihe axis., boi in the direction
p^^rpendicular to the |dane of th« diagrank instead of that repre-
Hcrit^;d by aid of the dotted lines of Ilg. 2. H^ioe on the homo-
geneous elastic solid theory, and in the cade of an axially isotropic
crystal, the velocity of propagation in this case would be the same
<|aMlV. OH in each of the first two cases. But with vibrations perpen-
Uoii iKif.' dicMjlar to the plane of the diagram, and juropagation perpendicular
jwiialoiilAr to ilnj axiH, the distortion is by shearing in the plane perpendicular
vIlimiltiiM to thn lino of the arrow in Fig. 3, and in the direction perpen-
Cur ill *"*''»'*^'' <<> <ho phinc of the diagram. In the supposed crystal, the
iiUitnor riKJility nuxluhiH for this shearing would be different from the
\ mf\%\\\, (.j^jjiiiy inodiilnM for th<3 shearings of cases I., II. and III. Hence
(MiNOH III. mill IV. would c(^rrespond to the extraordinary ray, and
s^mwH I. hhd II. ()i(t ordinary ray.

Now nliitprvp. (hiit light ixilarizcHl in a plane through the axis,
i^ohMlihilim (ho oiiliniirv my. and light })olarized in the plane per-
)iohdlr\ihn lo (ho |iliiho of \\\\\ ray and the axis constitutes the
ostH^ohlliMM) \\\s. Thoi'o Im thon^fon* an ontMnnding difficulty in

INTRODUCTORY. 19

the assumption that the vibration is perpendicular to the plane of Molar.
polarization, which is absolutely inexplicable on the bare theory
of an elastic solid.

The question now occurs, may we not explain it by loading the
elastic solid ? But the difficulty is, to load it unequally in different
directions. Lord Rayleigh thought that he had got an explanation
of it in his paper to which I have referred. He was not aware
that Rankine had had exactly the same idea. Lord Rayleigh at
the end of this paper puts forward the supposition that difference of
effective inertias in different directions may be adduced to explain
the difference of velocity of propagation in Iceland spar. But
if that were the case the wave propagation would not follow
Huyghens' law. It would follow the law according to which
the velocity of propagation would be inversely proportional to what
it is according to Huyghens* law. Huyghens' geometrical construc-
tion for the extraordinary ray in Iceland spar gives us an oblate
ellipsoid of revolution according to which the velocity of propagation
of light will be found by drawing from the centre of the ellipsoid
a perpendicular to the tangent plane. For
example (Fig. 4), CN will correspond to the
velocity of propagation of the light when the
front is in the direction of this (tangent) line.
If the velocity is different in different directions ^^' ^'

in virtue of an effective inertia, Lord Rayleigh's idea is that Molecular.
the vibrating molecules might be like oblate spheroids vibrating in
a frictionless fluid. The medium would thus have greater effective
inertia when vibrating in the direction of its axis (perpendicular to
its flat side), and less effective inertia when vibrating in its equa-
torial plana That is a very beautiful idea, and we shall badly want
it to explain the difficulty. We would be delighted and satisfied
with it if the pushing forward of the conclusions from it were veri-
fied by experiment. Stokes has made the experiment. Rankine
made the first suggestion in the matter, but did not push the ques-
tion frirther than to give it as a mode of getting over the difficulty
in double refrtu^tion. Stokes took away the possibility of it. He ex-
perimented on the refracting index of Iceland spar for a variety of
incidences, and found with minute accuracy indeed that Huyghens'
construction was verified and that therefore it was impossible to
account for the unequal velocity of propagation in different direc-
tions by the beautiful suggestions of Rankine and Lord Rayleigh.

20 LECTUBE L

T^^fiolMr, I have not been able to make a persistent suggestion of explan-
ation, but I have great hopes that these spring arrangements are
going to help us out of the diflSculty. I will, just in conclusion,
give you the idea of how it might conceivably do so.

We can easily suppose these spring arrangements to have
different strengths in different directions; and their directional
law will suit exactly. It will give the fundamental thing we want,
which is that the velocity of propagation of light shall depend on
the direction of vibration, and not merely on the plane of the
distortion. And it will obviously do this in such a manner as to
verify Huyghens' law — giving us exactly the same shape of wave-
surface as the aeolotropic elastic solid would give.

But alas, alas, we have one difficulty which seems still insuper-
able and prevents my putting this forward as the explanation. It
is that I cannot get the requisite difference of propagational
velocity for different directions of the vibration to suit different
periods. If we take this theory, we should have, instead of the very
nearly equal difference of refractive index for the different periods
in such a body as Iceland spar, with dispersion merely a small
thing in comparison with those differences : we should, I say, have
difference of refractive index in different directions comparable
with dispersion and modified by dispersion to a prodigious degree,
and in fact we should have anomalous dispersion coming in between
the velocity of propagation in one direction and the velocity of
propagation in another. The impossibility of getting a difference
of wave velocity in different directions sufficiently constant for the
different periods seems to me at present a stopper.

So now, I have given you one hour and seven minutes and
brought you face to face with a difficulty which I will not say
is insuperable, but something for which nothing ever has been
done from the beginning of the world to the present* time that
can give us the slightest promise of explanation.

I shall do to-morrow, what I had hoped to do to-day, give you
a little mathematics, knowing that it is not going to explain every-
thing. But I think that in relation to the wave-theory of light we
really have an interest in working out the motions of an elastic
solid and obtaining a few solutions that depend on the equations of
motion of an elastic solid. I shall first take the case of zero

♦ [I retract this stateinent. See an Appendix near the end of the present
volume. W. T. Oct. 28, 1889.]

INTRODUCTORY. 21

rigidity ; that will give us souud. We shall take the most elemen- Molecular,
tary sounds possible, namely a spherical body alternately expand-
ing and contracting. We shall pass from that to the case of a
single globe vibrating to and fro in air. We shall pass from that
to the case of a tuning-fork, and endeavour to explain the cone of
silence which you all know in the neighbourhood of a vibrating
tuning-fork. I hope we shall be able to get through that in a
short time and pass on our way to the corresponding solutions
of the motions of a wave proceeding from a centre in respect to the
wave-theory of light.

LECTURE II.

Oct 2, 5 P.M.

Part I. In the first place, I will take up the equations of
motion of an elastic solid. I assume that the fundamental prin-
ciples are familiar to you. At the same time, I should be very
gla^l if any person present would, without the slightest hesitation,
ask for explanations if anything is not understood. I want to be
on a conferent footing with you, so that the work shall be rather
something between you and me, than something in which I shall
be making a performance before you in a matter in which many
of you may be quite as competent as I am, if not more so.

I want, if we can get something done in half an hour, on these
problems of molar dynamics as we may call it, to distinguish from
Molecular dynamics, to come among you, and talk with you for
a few moments, and take a little rest; and then go on to a problem
of molecular dynamics to prepare the way for motions depending
on mutual interference among particles under varying circum-
stances that may perhaps have applications in physical science and
particularly to the theory of light.
Molar. The fundamental equations of equilibrium of elastic solids are,

of course, included in D'Alembert's form of the equations of
motion. I shall keep to the notation that is employed in Thomson
and Tait's Natural Philosophy, which is substantially the same
notation as is employed by other writers.

Let a, 6, c denote components of distortion : a, is a distortion in
the plane perpendicular to OX produced by slippings parallel to
either or to both of the two co-ordinate planes which intersect
in OZ.

Let us consider this state of strain, in which, without other
change, a portion of the solid in the plane yz which was a

DYNAMICS OF ELASTIC SOLID. 23

s(|uare becomes a rhombic figure. The measurement of that state MoUr.
of strain is given fully in Thomson and Tait's geometrical pre-
liminary for the theory of elastic solids, (Natural Philosophy,
§§ 169—177, and § 669, or Elements, §§ 148—156 and § 640). It
is called a simple shear. It may be measured .
either by the rate of shifting of parallel planes per
unit distance perpendicular to them, or, which
comes to exactly the same thing, the change of
the angle measured in radians. Thus I shall write
down inside this small angular space the letter a,
to denote the magnitude of the angle, measured
in radians; the particular case of strain considered
being a slipping parallel to the plane YOX.

I use the word "radian"; it is not, hitherto, a very common
word; but I suppose you know what I mean. In Cambridge
in the olden time we used to have a very illogical nomen-
clature,— "the unit angle" — ^a very absurd use of the definite
article "the". It is illogical to talk of an angle being measured in
terms of " the " unit angle ; there is no such thing as measuring
anything, except in terms of "a" unit. The degree is a unit
angle; so is the minute; so is the second; so is the quadrant;
so is the round ; all of these are units in frequent use for angular
measurement. The unit in which it is most generally convenient
to measure angles in Analytical Mechanics ia the angle whose arc
is radius. That used to be called at Cambridge " the unit angle."
My brother, James Thomson, proposed to call it " the radian \

There are three principal distortions, a, 6, c, relative to the
axes of OX, OY, 0Z\ and again, three principal dilatations — con-
densations of course if any one is negative, e, f, g, which are
the ratios of the augmentation of length to the length.

The general equation of energy will of course be an equation in
which we have a quadratic function of e, f, g, a, b, c, the ex-
pression for which will be

i (lle*+ 12e/+ 13eg + 146a + loeb + 16ec + 21^/+ 22/*+ 2S/g +. . .).

We do not here use 11, 12,... etc., as numbers but as symbols
representing the twenty-one coefficients of this quadratic subject
to the conditions 12 = 21, etc. If we denote this quadratic function
by E, then

^=116+ 12/+ ISg + 14a + 156 + 16c.

24

LELTTRE II. PAST L

r.

This is a component of the normal force required to produce this
compound strain e, j\ g, a. 6, c. According to the notation of
Thomson and Tait, let

** €lE ^ dE wv dE

We have then, the relation

the well-known dynamical interpretation of which you are of course
familiar with. A little later we shall consider these 21 coefficients,
first, in respect to the relations among them which must be
imposed to produce a certain kind of symmetry relative to the
three rectangular axes ; and then see what further conditions must
be imposed to fit the elastic solid for performing the functions
of the luminiferous ether in a crystal

Before going on to that we shall take the case of a perfectly
isotropic material. We can perhaps best put it down in tabular
form in this way :

1

1

«

0
0
0

0

0

0
0
0

»

0

0
0

0

0

0

n

0
n
0

0
0
n

0
0

0

0

0

0

In the first place in this lower right-hand comer-square which
has to do with the distortions a, 6, c, alone, if we let n represent the
rigidity-modulus the three main diagonal terms will each be n, and
those not in the diagonal will be zero. Six of the 21 coefficients
ure thus determined as follows:

44 = 55 = 66=n, and 45 = 46 = 56 = 0,

and other nine of them, by the zeros in the upper right and lower
left corner-squares, —

14 = 15 = 1 6 = 24 = 25 = 26 = 34 = 35 = 36 = 0.

DYNAMICS OF ELASTIC SOLID. 25

To verify these zeros of the upper right-hand and lower left-hand MoUr.
comer-squares, let us consider what possible relations there can be
for an isotropic body between longitudinal strains and distortions.
Clearly none. No one of the longitudinal strains can call into
play a tangential force in any of the faces ; and conversely, if the
medium be isotropic, no distortion produced by slipping in the
faces parallel to the principal planes can introduce a longitudinal
stress — a stress parallel to any of the lines OX, OY, OZ, There-
fore we have all zeros in these two squares. We know that
11 = 22 = 33; and each of these will be represented by Saxon A (^).
Now consider the effect of a longitudinal pull in the direction
of OX, If the body be only allowed to yield longitudinally, that
clearly will give rise to a negative pull in the directions parallel to
Oy, Oz. We have then a cross connection between pulls in the
directions OX, OF, OZ, Isotropy requires that the several
mutual relations be all equal, so that we have just one coefficient
to express these relationa That coefficient is denoted by Saxon B
(13). Thus we fill up our 36 squares, which represent but 21 co-
efficients in virtue of the relations 12 = 21, etc. We can now
write down our quadratic expression for the energy,

^= i[a(e' +/* + /) +223 (/5r+5^e + 6/) + n(a' + 6' + c';].

Instead of these Saxon letters ^, i3, which have very distinct and
obvious interpretations, we may introduce the resistance of the
solid to compression, the reciprocal of what is commonly called
the compressibility, or, what we may call the bulk modulus, k.
Then it is proved in Thomson and Tait, and in an article in the
EncyclopcBdia Britannica which perhaps some of you may have*,
that^ = i + Jw, 33 = A; — frt. The considerations which show
these relations with the bulk modulus also show us that we must
have n = j^(® — J3). This is most important. Take a solid cube
with its edges parallel to OX, 0 Y, OZ. Apply a pull along two
faces perpendicular to OX and an equal pressure on two faces
perpendicular to OF; that will give a distortion in the plane xy.
Find the value of that simple shear; it is done in a moment. Find
the shearing force required to produce it calculated from ^ and 23,
and equate that to the force calculated from the rigidity modulus
n, and then you find this relation. The relations for complete

* Beprint of Mathematical and Physical Papers^ VoL ni. Art. xcii. now in the
Press. [W. T. Aug. 7. 1886.]

26 LECTURE II. PART I.

mjiropy are exhibited here in this quadratic expression for the
energy, if in it we take H^ " ^) ^^ place of n.

We shall pass on to the formation of the equations of motion.
For e<{uilibrium, the component parallel to OX of the force applied
at any point x, y, z of the solid, reckoned per unit of bulk at that

point must be equal to

(dP^dU dT\

\dx dy dz)^
if the body be held distorted in any way, by bodily forces applied
all through the interior ; because the resultant of the elastic force
on an infinitesimal portion of matter at the point x,y^z\& obviously
dp dU dT

(

- +

dT\
+ ^ - ) dxdydz. To prove this remark that if the pull

dx dy dz )

augments as you go forward in the direction OX there will, in

virtue of that, be a resultant forward pull

dP

-j- dx.dydz upon the infinitesimal ele-
ment. The two tangential forces, IT per-
pendicular to OYf and U perpendicular
to OX on the one pair of forces and the
pair of forces equal and in opposite di-
rections on the other faces constitute two
balancing couples, as it were. If this
tangential force parallel to OX increases
as we proceed in the direction y positive, there will result a positive
force on the element, because it is pulled to left by the smaller
and to right by the larger, and thus the force in the direction

of OX receives a contribution^- dy.dzdx. Quite similarly we

find, '-T'dz.dxdy as a third and last contribution to the force

parallel to OX.

Now, let there be no bodily forces acting through the material,
but let the inertia of the moving part^ and the reaction against
acceleration in virtue of inertia, constitute the equilibrating re-
action against elasticity. The result is, that we have the equation

dP dU dT^ ^

dx dy dz '^ df*
if by p we denote the density and by f we denote the displace-
ment from equilibrium in the direction OX of that portion of
matter having x, y, z for co-ordinates of its mean position.

DYNAMICS OF ELASTIC SOLID. 27

I said I would use the notation of Thomson and Tait who MoUr.
employ a, /8, 7 to denote the displacements; but errors are too
common when a and a are mixed up, especially in print, so we
will take ^, 17, (f instead. I have had trouble in reading Helm-
holtz*s paper on anomalous dispersion, on this account, very fre-
quently not being able to distinguish with a magnifying glass
whether a certain letter was a or a.

The values of S, T, U, we had better write out in full, although
the others may be obtained from the value of any one of the
three by symmetry. The expenditure of chalk is often a saving
of brains. They are :

^-»(£-|). ^HMl ^-»(I-S)-

We have P=^e + 23 (/+5r). There are two or three other forms
which are convenient in some cases, and I will put them down
(writing m for A; 4- Jn)

P.(»..)g.(.-.,(^^4£).(»-.,(l^l4f)...g

\dx dy dz) \dx dy dz) '

We shall denote very frequently by S the expression

dx dy dz '
so that for example, the second of these expressions is

ax
If we want to write down the equations of a heterogeneous medium,
as will sometimes be the case, especially in following Lord Ray-
leigh's work on the blue sky, we must in taking dP/dx, dP/dy, &c.,
to find the accelerations, keep the symbols m, n inside of the
symbols of differentiation ; but for homogeneous solids, we treat m
and n as constant. I forgot to say that B is the cubic dilatation or
the augmentation of volume per unit volume in the neighbour-
hood of the point x, y, z, which is pretty well known, and helps us
to see the relations to compressibility. If we suppose zero rigidity,
P = mS is the relation between pressure and volume. In order
to verify this take the preceding expression for P and make n = 0
and we obtain P = 7nB, the equation for the compression of a
compressible fluid, in which m has become the bulk modulus.

28

LECTURE II. PART U.

lar. This sort of work I have called molar dynamics. It is the

dynamics of continuous matter; there are in it no molecules, no
heterogeneousnesses at all. We are preparing the way for dealing
with heterogeneousnesses in the most analytical manner by sup-
posing m and n to be functions of x, y, z. Lord Bayleigh
studied the blue sky in that way, and very beautifully ; his treat-
ment is quite perfect of its kind. He considers an imbedded
particle of water, or dust, or unknown material, whatever it is
that causes the blue sky. To discover the effect of such a particle,
on the waves of light, he supposes a change of rigidity and of
density from place to place in the luminiferous ether; not an
absolutely sudden change, but confined to a space which is small
in comparison with the wave-length.

Ten minuted interval.

lecular. Part II. I want to take up another subject which will pre-
pare the way to what we shall be doing afterward, which is the
particular dynamical problem of the movement of a system of con-
nected particles. I suppose most of you know the linear equations
of motion of a connected system — whose integral always leads to
the same formula as the cycloidal pendulum; this result being
come to through a determinant equated to zero, giving an alge-
braical equation whose roots are essentially real for the square
of the period of any one of the fundamental modes of simple
harmonic motion.

As an example, take three weights, one of 7 pounds, another of
14 pounds, and another of 28 pounds, say. The lowest weight is

hung upon the middle weight by a spiral spring;
the middle is hung upon the upper by a spiral
spring, and the upper is attached to a fixed point
by a spiral (or a zig-gag) spring. It is a pretty
illustration; and I find it very useful to myself.
I am speaking, so to say, to professors who sym-
pathize with me, and might like to know an
experiment which will be instructive to their
pupils.

Just apply your finger to any one of the

weights, the upper weight, for example. You

soon learn to find by trial the fundamental periods. Move it up

and down gently in the period which you find to be that of the

INTRODUCTION TO PART II. 29

three all moving in the same direction. You will get a very Moleeoli
pretty oscillation, the lowest weight moving through the greatest
amplitude, the second through a less, and the upper weight through
the smallest. That is No. 1 motion, corresponding to the greatest
root of the cubic equation which expresses the solution of the
mathematical problem. No. 2 motion will come after a little
practice. You soon learn to give an oscillation a good deal quicker
than before, the first; a second mode, in which the lowest weight
moves downward while the two upper move upwards, or the two
lower move downwards while the upper moves upwards, or it
might be that the middle weight does not move at all in this
second mode, in which case the excitation must be by putting the
finger on the upper or the lower weight. These periods depend
upon the magnitude of the weights, and the strength of the
springs that we use, and are soon learned in any particular set of
weights and springs. It might be a good problem for junior
laboratory students to find weights and springs which will in-
sure a case of the nodal point lying between the upper and
middle weights, or at the middle weight, or between the middle
and lower weights. The third mode of vibration, corresponding
to the smallest root of the cubic equation, is one in which you
always have one node in the spring between the upper and middle
weights, and another node in the spring between the middle and
lowest (the first and third weight vibrating in the same direction, and
the middle weight in an opposite direction to the first and third).
It is assumed that there is no mass in the springs. If you
want to vary your laboratory exercises, take smaller masses for the
weights, and more massive springs, and if you omit the attached
masses altogether, you pass on to a very beautiful illustration
of the velocity of sound. For that purpose a long spiral spring
of steel wire, the spiral 20 feet long, hung up, say, if you have
a lofty enough room, will answer, and you will readily get
two or three of the gi-aver fundamental modes without any
attached weights at all. In the special problem which we have
been considering we have three separate weights and not a con-
tinuous spring ; and we have three, and only three modes of vibra-
tion, when the springs are massless. We have an infinite number
of modes when the mass of the springs is taken into account. In
any convenient arrangement of heavy weights, the stiffness of the
springs is so great and their masses so small that the gravest

30 LECTURE n. PART IL

eoalar. period of vibration of one of the springs by itself will be very short ;
but take a long spring, a spiral of best pianoforte steel wire, if you
please, and hang it up, with a weight perhaps equal to its own,
on its lower end, and you will find it a nice illustration for getting
several of the graver of the infinite number of the fundamental
modes of the system.

I want to put down the dynamics of our problem for any
number of masses. Tou will see at once that that is just the case
that I spoke of yesterday, of extending Helmholtz's singly vibrat-
ing particle connected with the luminiferous ether to a multiple
vibrating heavy elastic atom imbedded in the luminiferous ether,
which I think must be the true state of the case. A solid mass
must act relatively to the luminiferous ether as an elastic body
imbedded in it, of enormous mass compared with the mass of the
luminiferous ether that it displaces. In order that the vibrations
of the ether may not be absolutely stopped by the mass, there
must be an elastic connection. It is easier to say what must
be than to say that we can understand how it comes to be. The
result is almost infinitely difficult to understand in the case of
ether in glass or water or carbon disulphide, but the luminiferous
ether in air is very easily imagined. Just think of the molecules
of oxygen and nitrogen as if each were a group of ponderable par-
ticles mutually connected by springs, and imbedded in homogene-
ous perfectly elastic jelly constituting the luminiferous ether.
Tou do not need to take into account the gaseous motions of the
particles of oxygen, nitrogen, and carbon dioxide in our atmosphere
when you are investigating the propagation of luminous waves
through the air. Think of it in this way : the period of vibration
in ultra-violet rays in luminous waves and in infra red heat
waves so far as known, is from the 1600 million-millionth of
a second to the 100 million-millionth of a second. Now think
how far a particle of oxygen or nitrogen moves, according to the
kinetic theory of gases, in the course of that exceedingly small
time. You will find that it moves through an exceedingly small
fraction of the wave-length. For example, think of a molecule
moving at the rate of 50000 centimetres per second. In the period
of orange light it crawls along 10"" of a centimetre which is only
1/600000 of the wave-length of orange light. I am fully confident
that the wave motion takes place independently of the translatory
motion of the particles of oxygen and nitrogen in performing

DYNAMICS OF A ROW OF CONNECTED PARTICLES.

31

their functions according to the kinetic theory of gases. You may Moleenla
therefore really look upon the motion of light waves through our
atmosphere as being solved by a dynamical problem such as this
before us, applied to a case in which there is so little of effective
inertia due to the imbedded molecules, that the velocity of light is
not diminished more than about one-thirty-third per cent by it
More difficulties surround the subject when you come to consider
the propagation of light through highly condensed gases, or trans-
parent liquids or solids.

In our dynamical problem, let the masses of the bodies be
represented by m,, m,,...wi,. I am going to suppose the sevei-al
particles to be acted upon by connecting springs. I do not want
to use spiral springs here. The helicalness of the spring in these
experiments has no sensible effect; but we want to introduce
a spiral for investigating the dynamics of the helical properties, as
shown by sugar. It is usually called the rotatory property, but this
is a misnomer. The magneto-optical property which was dis-
covered by Faraday is rotational, the property exhibited by quartz
and sugar and such things, has not the essential elements of
rotation in it, but has the characteristic of a spiral spring (a helical
spring, not a flat spiral), in the constitution of the matter that ex-
hibits it. We apply the word helical to the one and the word
rotational to the other.

I am going to suppose one other connect-
ed particle P, which is moved to and fro
with a given motion whose displacement
downwards from a fixed point 0, we shall call
f . Let c^ be the coefficient of elasticity of
the first spring, connecting the particle P
with the particle m^; c, the coefficient of
elasticity of the next spring connecting m^
and m,; c^^, the coefficient of elasticity of
the spring connecting m^ to a fixed point.
We are not taking gravity into account ; we
Jiave nothing to do with it. Although in the
experiment it is convenient to use gravity,
it would be still better if we could go to the
centre of the earth and there perform the
experiment. The only difference would be,
these springs would not be pulled out by the weights hung upon

32 LECTURE II. CONCLUSION.

>lfieiilar. them. In all other respects the problem would be the same, and
the same symbols would apply.

We are reckoning displacements downwards as positive, the
displacement of the particle m^ being a?,. The force acting upon
m, in virtue of the spring connection between it and P is c, (f — a?,) ;
and in virtue of the spring connection between it and m, is the
opposing pull - c, (x^ ~ a?,) ; so that the equation of motion of the
first particle is

For No. 2 particle we have

m, -^ = c, (a?,- a?,) -c^(x^-x^); and so on.

Now suppose P to be arbitrarily kept in simple harmonic mo-
tion in time or period r ; so that f = const, x cos — . We assume

that every part of the apparatus is moving with a simple harmonic
motion, as will be the case if there were infinitesimal viscous
resistance and the simple harmonic motion of P is kept up long

enough; so that we can write a?j = const, x cos — , etc. I am

going to alter the w's so as to do away with the 47r' which comes

in from diflferentiation. I will let 7-^ denote the mass of the first

47r

particle, and j-* the mass of the second particle, etc. The result

will be that the equations of motion become

-^^1 = ^1 (f-^i)-c«(^i-^»)' etc.

Our problem is reduced now to one of algebra. There are
some interesting considerations connected with the determinant
which we shall obtain by elimination from these equations. To
find the number of terms is easy enough ; and it will lead to some
remarkable expressions. But I wish particularly to treat it with
a view to obtaining by very short arithmetic the result which can
be obtained from the determinant in the regular way only by
enormous calculation. We shall obtain an approximation, to the
accuracy of which there is no limit if you push it far enough,

LECTURE II. CONCLUSION. 33

that will be exceedingly convenient in performing the calcu- Moleonla
latious.

In the next lecture, resuming the molar problem, we shall
begin with the solution for sound, of the equations that are now
before you on the board. We shall next try to go on a step further
with this molecular problem, of the vibrations of our compound
molecule.

T. L.

LECTURE III.

Oct. 3, 5 p.m.

KoUur. We will now go on with the problem of Molar dynamics, the

propagation of sound or of light, from a source. I advise you all
who are engaged in teaching, or in thinking of these things for
yourselves, to make little models. If you want to imagine the
strains that were spoken of yesterday, get such a box as this
covered with white paper and mark upon it the directions of the
forces 8, 7, U, I always take the directions of the axes in a
certain order so that the direction of positive rotation shall be
from y to a, from z to x^ from a? to y. What we call positive is the
same direction as the revolution of a planet seen from the northern
hemisphere, or opposite to the motion of the hands of a watch. I
have got this box for another purpose, as a mechanical model of an
elastic solid with 21 independent moduluses, the possibility of
which used to be disproved, and after having been proved, was still
disbelieved for a long time.

Let us take our equations,

d^i^dP dU dT ^^^^^
'^ df dx dy dz *

I dx dy dz)'

We shall not suppose that m and n are variables, but take them
constant. If we do not take them constant we shall be ready for
Lord Rayleigh's paper on the blue sky, already referred to. I will
do the work upon the board in full, as it is a case in which
the expenditure of chalk saves brain ; but it would be a waste to

WAVES OR VIBRATIONS OF ELASTIC SOLID. 35

print such calculations, for the reason that a reader of mathematics Molar,
should always have pencil and paper beside him to work the thing
out ♦ ♦ ♦ ♦ The result is that

^§=^S + ^V'f <1>-

We take the symbol

In the case of no rigidity, or w = 0, the last term goes out. We
shall take solutions of these equations, irrespectively of the
question of whether we are going to make n = 0 or not, and we
shall find that one standard solution for an elastic solid is in-
dependent of n and is therefore a proper solution for an elastic
fluid.

I have in my hand a printed report* of a Royal Institution
lecture of Feb. 1883, on the Size of Atoms, containing a note on
some mathematical problems which I set when I was examiner for
the Smith's Prizes at Cambridge, Jan. 30, 1883. One was to show
that the equations of motion of an isotropic elastic solid are what
we have here obtained, and another to show that so and so was a
solution. We will just take that, which is : Show that every pos-
sible solution of these three equations [(1) etc.] is included in the
following :

where <f>, % v, w, are some functions of x, y,z, t] with the condition
that w, V, w are such that

^^4.^+^ = 0 (3)

dx dy dz ^'

If we calculate the value of the cubic dilatation, we find

c. ^. du dv dw a , ...

Again, by using f = j^ + ^ i^ (1)» we find (bearing in mind

• Reprinted in Vol. I. of Sir W. Thomson's Popular Lectures and Addresses.
(MacmiUan, 1889).

3—2

36 LECTURE III. PART I.

Now we may take

4S=<-+«)^S («)•

The foil justification and explanation of this procedure is
reserved. [See commencement of Lecture IV. below.] Multiply
(6) by dx, and the corresponding y and z equations by rfy, ck, and
add. We thus get a complete differential; in other words, the
relation which <f> must satisfy is

And (2) shows that if <f> satisfies (7) we have u, v, w, satisfying
equations of the same form, but with n instead of (m-^n); viz,

^^ = '*^''' ^5? = ^^^' ^d? = ^V^ ^^>-

By solving these four similar equations, one involving (m + ti),
and three involving n, we can get solutions of (1), that is certain.
That we get every possible solution, I shall hope to prove
to-morrow. The velocity of the sound wave, or condensational

is a/ . The velocity of the wave of distortion in the

elastic solid is a/ - . I shall not take this up because I am very

anxious to get on with the molecular problem ; but you see
brought out perfectly well the two modes of waves in an isotropic
homogeneous solid, the condensational wave and the distortional
wave. The condensational wave follows the equations of motion
of sound, which is the same as if n were null ; and this gives the
solution of the propagation of sound in a homogeneous medium,
like air, etc. The solution is worked out ready to hand for the
distortional wave because the same forms of equations give us
separate components %v,w\ the same solution that gives us the
velocity potential for the condensational waves, gives us the
separate components of displacement for the distortional waves.

What I am going to give you to-morrow will include a solution
which is alluded to by Lord Rayleigh. There is nothing new in it.
I am going to pass over the parts of the solution which interpreted
by Stokes explain that beautiful and curious experiment of
Leslie's. Lord Rayleigh quotes from Stokes, ending his quotation
of eight pages with "The importance of the subject and the masterly
manner in which it has been treated by Prof. Stokes will probably

wave

P

WAVES OR VIBRATIONS OP ELASTIC FLUID. 37

be thought suflBcient to justify this long quotation." I would just
like to read two or three things in it Lord Rayleigh says {Theory
of Sound, Vol II. p. 207), "Prof. Stokes has applied this solution
to the explanation of a remarkable experiment by Leslie, according
to which it appeared that the sound of a bell vibrating in a
partially exhausted receiver is diminished by the introduction
of hydrogen. This paradoxical phenomenon has its origin in the
augmented wave length due to the addition of hydrogen in conse-
quence of which the bell loses its hold (so to speak) on the
surrounding gas." I do not like the words " paradoxical pheno-
menon;" "curious phenomenon," or "interesting phenomenon"
would be better. There are no paradoxes in science. Lord
Rayleigh goes on to say, "The general explanation cannot be
better given than in the words of Prof Stokes : ' Suppose a person
to move his hand to and fro through a small space. The motion
which is occasioned in the air is almost exactly the same as it
would have been if the air had been an incompressible fluid.
There is a mere local reciprocating motion in which the air
immediately in front is pushed forward and that immediately
behind impelled after the moving body, while in the anterior space
generally the air recedes from the encroachment of the moving
body, and in the posterior space generally flows in from all sides
to supply the vacuum that tends to be created ; so that in lateral
directions, the flow of the fluid is backwards, a portion of the
excess of the fluid in front going to supply the deficiency
behind.*'* It will take some careful thought to fBllow it. I
wish I had Green here to read a sentence of his. Green
says, "I have no faith in speculations of this kind unless they
can be reduced to regular analysis." Stokes speculates, but is
not satisfied without reducing his speculation to regular analysis.
He gives here some very elaborate calculations that are also
important and interesting in themselves, partly in connection
with spherical harmonics, and partly from their exceeding instruc-
tiveness in respect to many problems regarding sound. Passing
by all that five or six pages of mathematics — I will not tax your
brains with trying to understand the dynamics of it in the course
of a few minutes ; I am rather calling your attention to a thing
to be read than reading it — Stokes comes more particularly to
Leslie's experiments. Instead of a bell vibrating, Stokes con-
siders the vibrations of a sphere becoming alternately prolate and

38 LtXTTLBE ilL PABT IL

Molar. oblate ; and he shows that the principles are the same. Bead all
this for yourselves. I have intended merely to arouse an interest
in the subject.

Ten minvies interml.

Molecular. To return to the consideration of our molecules connected by
springs, we will suppose a good fixing at the top, so firm and stiff
that the changing pull of the spring does not give it any sensible
motion. For any one of the springs let there be a certain change
of pull, c per unit change of length. This coefficient c measures
what I call the longitudinal rigidity of the spring: its effective
stiffness in fact. It is not the slightest consequence whether the
spring is long or short, only, if it is long, let it be so much the
stiffer ; but long or short, thick or thin, it must be massless. I
mean that it shall have no inertia. The masses may be equal

or unequal, and are connected by springs. Let us
attach here something like the handle of the bell
pull of pre-electric ages ; — something that you can
pull by. Call it P. This, in our application to the.
luminiferous ether, will be the rigid shell lining be-
tween the luminiferous ether and thefirst moving mass.
The equation of motion for the first mass be-
comes, on bringing f to the left-hand side,

and similarly for the second mass ; I shall use %
to denote any integer. I find the letter % too useful
for that purpose to give it up, and when I want

to write the imaginary J — ly I use a. Let us call
the first coefficient on the right a^, the similar
coefficient in the next equation a,, and so on, so that

Tlic tth equation will thus be

Now write down all these j equations supposing the whole number
of the springs to hej ; form the determinant by which you find all
of the others in terms of f , and the problem is solved.

If we had a little more time I would like to determine the
number of terms in this determinant. We will come back to

VIBRATIONS OF SERIAL MOLECULE. dd

that because it is exceedingly interesting ; but I want at once to Moleoala
put the equations in an interesting form, taking a suggestion from
Laplace's treatment of his celebrated Tidal problem. What we
want is really the ratios of the displacements, and we shall there-
fore write

---- = u.

introducing the sign minus, so that when the displacements are
alternately positive and negative the successive ratios will be all
positive. We have then,

w, = a, ; (uj^, = 00 ).
We can now form a continued fraction which, for the case that
we want, is rapidly convergent. K this be differentiated with
respect to t"*, we find a very curious law, but I am afraid we
must leave it for the present. The solution is

c:

u. = a, 1-^

a. —

• 1

Thus if we are given the spring connections and the masses,
everything is known when the period is known. If you develop
this, you simply form the determinant ; but the fractional form
has the advantage that in the case when the masses are larger
and larger, and the spring connections are not larger in pro-
portion, we get an exceedingly rapid approximation to its value
by taking the successive convergenta The differential coeflBcient
of this continued fraction with respect to the period is essentially
negative, and thus we are led beautifully from root to root, and
see the following conditions : — First, suppose we move P to and
fro in simple harmonic motion of very short period; then when
the whole has got into periodic movement, it is necessary that P
and the first particle move in opposite directions. The vibrations
of the first particle needs to be " hurried up " (if you will allow
me an expressive American phrase) when the motion of P is
of a shorter period than the shortest of the possible inde-
pendent motions of the system with P held fixed. Now if you
want to hurry up a vibrating particle, you must at each end of
its range press it inwards or towards its middle position.

40 LECTUKE III. PART II.

joular. You meet this principle quite often ; it is well known in
the construction of clock escapements. To hurry up the vibra-
tory motion of our system we must add to the return force of
particle No. 1 by the action of the spring connected to the
handle P, by moving P always in the direction opposite to the
motion of m. From looking at the thing, and learning to under-
stand it hy feeling the experiment, if you do not understand it by
brains alone, you will see that everything that I am saying is
obvious. But it is not satisfactory to speak of these things in
general terms unless we can submit them to a rigorous analysis.

I now set the system in motion, managing, as you see, to get
it into a state of simple harmonic vibration by my hand applied
to P. That, which you now see, is a specimen of the configuration
in which the motion of P is of a shorter period than the shortest
of the independent motions with P fixed. Suppose now, the
vibration of P to be less rapid and less rapid ; a state of things
will come, in which, the period of P being longer and longer, the
motion of the first particle will be greater and greater. That is
to say, if I go on augmenting the period of P we shall find for
the same range of motion of P, that the ranges of motion of m^
and of the other particles generally will be greatly increased
relatively to the range which I give to P. In analytical words,
if we begin with a configuration of values corresponding to t
very small, and then, if we increase t to a certain critical

value, we shall find -^ will become infinite. In the first place, we

begin with u^, ii,,...i^^ all positive; and r small enough will make
them all positive as you see. Now take the differential coefficient
of u^ with respect to t and it will be found to be essentially nega-
tive. In other words, if we increase t, we shall diminish w,, w, ....
In every case u^ will first pass through zero and become negative.

CD

When u^ is zero we have the first infinity -^=00. If we diminish

T a little further u^ will pass through zero to negative while w^ is
still negative. Diminish r a little further and u^ will become
zero and pass to negative, while u^ is still negative; but in the
mean time u^ may have reached a negative maximum and passed
through zero to positive, or it may not yet have done so. We
shall go into this to-morrow ; but I should like to have you know
beforehand what is going to come from this kind of treatment of
the subject

LECTURE IV.

Oct 4, 3.45 p.m.

We found yesterday

Molar.

dS

(28

(1);

dz

and we saw that we get two solutions, which when fully inter-
preted, correspond to two different velocities of propagation, on
the assumptions that were put before you as to a condensational
or a distortional wave. We will approach the subject again from
the beginning, and you will see at once that the sum of these
solutions expresses every possible solution.

In one of our solutions of yesterday, we took, instead of f , fj, f,
other symbols u, v, w, which satisfied the condition,

du dv dw ^ ^
dx dy dz

In other words, the u, v, w of yesterday express the displacements
in a case in which the dilatation or condensation is zero. Now, just
try for the dilatation in any case whatever, without such restriction.
This we can do as follows: Differentiate (1) with respect to x
(taking account of the constancy of m and n) and the correspond-
ing equations with respect to y and z, and add. We thus find

p ^^ = (m + n) v'S = (A; + Jn) v'8

,(2).

42 LECTURE IV.

'riiif« <H|niition, you will remember, is the same as we had yesterday
for 0. Wo slmll consider solutions of this equation presently;
lint how rouuirk, that whatever be the displacements, we have
u (lilut^Uioh i\)rn\<i>onding to some solution of this equation. It
iim,v Ih« tvtw hut it must satisfy (2). Now in any actual case, 8 is
14 tlott^nuiiuito funotivui of jt, y, ^, ^; and whether we know it or
uol. wo lUHV t«iko ^ to donote a function such that

VV =• ^ through all space (3).

TW\H t\inotiitu, ^. is dotonuinate. It is in fact given explicitly
(uu iH woU kui»wu in tho theory of attraction) by the equation

whuru B' douotoH tho vuluo of B at (x\ y\ z). This formula (4) is
iiu|Mirtaht IU4 giving ^ oxplicitly; and exceedingly interesting on
woiumnt t)f tlu> rolutions to tho theory of attraction; but in the
wavt^-prohlom. whou h is given, (3) gives 0 determinately and in
tlio ouHioMt piMMiblo way. l\itting now

f-:L*-"' '-^+«. t-g+» w.

s-v-*.:i^.|.g (6„

...a . w„„, b, p), J;+|+^-o (7).

Now, romeuiberiug that (1) are satisfied by d<f>/dx, d<f>/dy, d<f>/dz
in place of f, rj, f, wo see, by multiplying the first by da?, or the
second by dy, or the thin! by dz, and iutegratiug, that

P^? = (i- + |H)vV (8),

and we find the three equations for

^ dx' "^ dy' ^"dz

reduced, in virtue of (7), to the following :

d'u , rf'v . d^w
p -^^ = HV u, p ^^ = nyV, p ^^ = HV"^ (9).

For any possible solutions of equations (1), we have a value
: i which is a function of iv,y,z\ take the above volume integral

WAVES OR VIBRATIONS IN ELASTIC SOLID. 43

correspondiDg to this value of 8 through all points of space x'yz\ MoUr.
and we obtain the corresponding <^ function which fulfils the

condition v*<^ = S. Now, let us compound displacements — -^ , etc.,

¥rith the actual displacements and denote the resultant as follows:

and remarking that therefore

du dv dw
— -|. _ ^_ *iz: = 0
dx dy dz '

we see the proposition that we had before us yesterday established.
Hence to solve the three equations (1) we have simply to find
h by solution of the one equation

and to deduce ^ from h, by (3); or to find ^ direct by solution
of the equation

and w, v, w from the three separate similar equations with n in the
place of (m + w), subject to the conditions

^ .dv dw _ ^
dx dy dz

We shall take our <^ equation and see how we can from it
obtain different forms of <t> solutions. We can do that for the
purpose of illustrating different problems in sound, and in order
to familiarize you with the wave that may exist along with the
wave of distortion in any true elastic solid which is not incom-
pressible. We ignore this condensational wave in the theory of
light. We are sure that its energy at all events, if it is not null,
is very small in comparison with the energy of the luminiferous
vibrations we are dealing with. But to say that it is absolutely
null would be an assumption that we have no right to make.
When we look through the little universe that we know, and think
of the transmission of electrical force and of the transmission of
magnetic force and of the transmission of light, we have no right
to assume that there may not be something else that our philo-

44 LKCTUEE IV.

Hophy does not dream of. We have no right to assume that there
may not be condensational waves in the luminiferous ether. We
only do know that any vibrations of this kind which are excited
by the reflection and refraction of light are certainly of very small
energy compared with the energy of the light from which they
proceed. The tajct of the case as regards reflection and refraction
ia this, that unless the luminiferous ether is absolutely incom-
pressible, the reflection and refraction of light must generally
give rise to waves of condensation. Waves of distortion may
exist without waves of condensation, but waves of distortion cannot
be reflected at the bounding surface between two mediums without
exciting in each medium a wave of condensation. When we come
to the subject of reflection and refraction, we shall see how to
deal with these condensational waves and find how easy it is to
get quit of them by supposing the medium to be incompressible.
But it is always to be kept in mind as to be examined into, are
there or are there not very small amounts of condensational waves
generated in reflection and refraction, and may after all, the pro-
pagation of electric force be by these waves of condensation ?

Suppose that we have at any place in air, or in luminiferous
ether (I cannot distinguish now between the two ideas) a body
that through some action we need not describe, but which is con-
ceivable, is alternately positively and negatively electrified; may
it not be that this will give rise to condensational waves ? Suppose,
for example, that we have two spherical conductors united by a
fine wire, and that an alternating electromotive force is pro-
duced in that fine wire, for instance by an ''alternate current"
dynamo-electric machine ; and suppose that sort of thing goes on
away from all other disturbance — at a great distance up in the
air, for example. The result of the action of the dynamo-electric
machine will be that one conductor will be alternately positively
and negatively electrified, and the other conductor negatively and
positively electrified. It is perfectly certain, if we turn the
machine slowly, that in the air in the neighbourhood of the con-
ductors we shall have alternately positively and negatively directed
electric force with reversals of, for example, two or three hundred
per second of time with a gradual transition from negative
through zero to positive, and so on; and the same thing all
through space; and we can tell exactly what the potential and
what the electric force is at each instant at any point. Now, does

ELECTRIC WAVES. 45

any one believe that if that revolution was made fast enough
the electro-static law of force, pure and simple, would apply to
the air at different distances from each globe? Every one believes
that if that process be conducted fast enough, several million times,
or millions of million times per second, we should have large devia-
tion from the electrostatic law in the distribution of electric force
through the air in the neighbourhood. It seems absolutely certain
that such an action as that going on would give rise to electrical
wavea Now it does seem to me probable that those electrical
waves are condensational waves in luminiferous ether; and pro-
bably it would be that the propagation of these waves would be
enormously faster than the propagation of ordinary light waves.

I am quite conscious, when speaking of this, of what has
been done in the so-called Electro-Magnetic theory of light
I know the propagation of electric impulse along an insulated
wire surrounded by gutta percha, which I worked out myself
about the year 1854, and in which I found a velocity comparable
with the velocity of light*. We then did not know the relation
between electro-static and electro-magnetic units. If we work
that out for the case of air instead of gutta percha, we get simply
"v," (that is, the number of electrostatic units in the electro-
magnetic unit of quantity,) for the velocity of propagation of the
impulse. That is a very different case from this very rapidly
varying electrification I have ideally put before you : and I have
waited in vain to see how we can get any justification of the way
of putting the idea of electric and magnetic waves in the so-called
electro-magnetic theory of light.

I may refer to a little article of mine in which I gave a sort
of mechanical representation of electric, magnetic, and galvanic
forces — ^galvanic force I called it then, a very badly chosen name.
It is published in the first volume of the reprint of my papers.
It is shown in that paper that the static displacement of an elastic
solid follows exactly the laws of the electro-static force, and that
rotatory displacement of the medium follows exactly the laws of
magnetic force. It seems to me that an incorporation of the
theory of the propagation of electric and magnetic disturbances
with the wave theory of light is most probably to be arrived at
by trying to see clearly the view that I am now indicating. In
the wave theory of light, however, we shall simply suppose the

* (See an Appendix near the end of the present volome.)

46 LECTURE IV.

Molar, resistance to compression of the luminiferous ether and the
velocity of propagation of the condensational wave in it to be
infinite. We shall sometimes use the words ''practically infinite"
to guard against supposing these quantities to be absolutely
infinite.

I will now take two or three illustrations of this solution for
condensational waves. Fart of the problem that I referred to
yesterday says : — prove that the following is a solution of (7), the
equation of motion,

*-J*x('-V'7^)^

P

or, if we put for brevity,

*=^-^' (11).

The question might be put into more analytical form : — to find
a solution of (7) isotropic in respect to the origin of co-ordinates ;
or to solve (7) on the assumption that ^ is a function of r and t.
Taking this then as our problem, remark that we now have (be-
cause ^ is a function of r)

v>=-^(-S)='^* (-)•

Hence (7), with both sides multiplied by r*, becomes

P^ = (* + |n)4/-^ (13).

of which the general solution is

r<^ = i' (r - v*) +/(r + vt)\

where ^"/"T^" 1 ^^'^'

and F and /denote two arbitrary functions.

This result simply expresses wave disturbance of r^, with
velocity of propagation V[(^ + Jw)/p]« ^i^d it proves (9), being
merely the case of a simple harmonic wave disturbance propagated
in the direction of r increasing, that is to say outwards from the
origin.

CONDENSATIONAL WAVES IN SOLID OR FLUID. 47

Here then is the determination of a mode of motion which Molar,
is possible for an elastic solid. We shall consider the nature of

this motion presently. The factor - in (9) prevents it from being

a pure wave motion. Passing over that consideration for the
present, we note that it is less and less effective, relatively to the
motion considered the farther we go from the centre.

In the meantime, we remark that the velocity of propagation
in an elastic solid is but little greater than in a fluid with the
same resistance to compression, k is the bulk modulus and
measures resistance to compression, n is the rigidity modulus.
I may hereafter consider relations between k and n for real solids.
h is generally several times n, so that ^n is small in comparison
with A, and therefore in ordinary solids the velocity of propagation
of the condensational wave is not greatly greater than if the solid
were deprived of rigidity and we had an elastic fluid of the same
bulk modulus.

I shall want to look at the motion in the neighbourhood of
the source. That beautiful investigation of Stokes, quoted by
Lord Rayleigh, has to do entirely with the region in which the

change of value of the factor (-1 from point to point is consider-
able. Without looking at that now, let us find the components of
the displacement and their resultant, and study carefully all the
circumstances of the motion.

ddi did> diw . •,

^ , -T- , ^ , are the three components of the displacement.

Clearly, therefore, the displacement will be in the direction of the
radius because everything is symmetrical ; and its magnitude will

be ^: and from (11) and (10) we find
dr

dS -1 . . 27rl

^ = _8m3+ --COS? (15).

Having obtained this solution of our equations, let us see what
we can make of interpreting it. When r is great in comparison

with ^ , the first term becomes very small in comparison with

48 LECTURE IV.

[olar. the second and we have

t-Xf-' w-

Therefore, when the distance from the origin is a great many

2ir 1
wave-lengths, the displacement is sensibly equal to -;r — cos },

and is therefore approximately in the inverse proportion to the
distance ; and the intensity of the sound if the solution were
Uy be applied to sound, would be inversely as the square of the
distance from the source.

I want now to get a second and a third solution. Take

as the velocity potential for a fresh solution. I take it that you
all know that if we have one solution <f>, for the velocity potential,
wo can gi»t another solution by yjr any linear function of

d^ d<l> d<f>
dx * dy ' dz'

Now lot UB find the displacements

dy^ d'^ d'^
dx' dy ' dz'

]{oro I want to prove that though this solution is no longer S3rm-
niotri(Mil with refi]>cct to r, so that there will be motions other
than rmlial in the neighbourhood of the source, yet still the
motion is approximately radial at great distances from the source.
Work it out, and you will find that

d^ 27r8in<7rir* / \ Vr*- ac*! cos^ r*- ar* ,-q.

dx^^\T\7^''\2^)—r^ J"^~p-^^ ^^^^'

The principal term here is — -5- -j sin 5. We might go on to the

third and fourth and higher differential coeflScients of ^, with
their larger and larger numbers of terms. The interpretation of
this multiplicity of terms, of the terms other than those which
I am now calling the " principal terms," is all-important in respect
to the motion of the air in the neighbourhood of the source. It

* I ase = to denote approximate equality.

t X/2ir is introdaced merely for convenienoe. The solation differs from
^=d0/dx, only by a constant factor.

AERIAL WAVES OF SOUND. 49

is dealt with in that splendid work of Stokes, one of the finest Molar,
things ever written in physical mathematics, of which I read to
you this afternoon, with reference to the effect of an atmosphere
of hydrogen round a bell killing its sound. But we will drop
those terms and think only of the terms which express the
efficiency of the vibrator at distances great in comparison with
the wave-length.

Thus, for the a;-component displacement of the motion now
considered, we have

f=-T?-^'S-') <")•

This approximate equality is true for distances from the centre
great in comparison with the wave-length. Let me remark, it is
the differentiation of cos q that gives the distantly effective terms
of the displacement ; and in differentiating -^ with respect to y,
you have simply to differentiate cos q and to take the differential
coefficient of r now with respect to y, instead of a; as formerly.
So that we may write down the principal terms of the y and z
displacements by taking yjx and zjx of the second member of (19)
as follows : —

17 = - ~^sm}, ? = ---psmg.

The three component displacements being proportional to x, y, z,
shows that the resultant displacement is in the direction of the

radius ; and its magnitude is — r— -3 sin g. If we write a? = r cos i,

this becomes

27r cos t . .Q^.

-^-7 sing (20),

or the displacement is inversely proportional to the distance. If

t = 0 we have a maximum ; if t = 5- we have zero. The upshot of

it is that the displacement is a maximum in the axis OX, zero
everywhere in the plane of OY, OZ; and symmetrical all round
the axis OX.

A third solution is got by taking -r? as our velocity potential.

At a distance from the origin, great in comparison with the wave-
T. L. 4

48 LECTURE IV.

M^dftr. the second and we have

^r-Tl^" <")••

Therefore, when the distance from the origin is a great many

wave-lengths, the displacement is sensibly equal to — — cos },

and is therefore approximately in the inverse proportion to the
distance; and the intensity of the sound if the solution were
to be applied to sound, would be inversely as the square of the
distance from the source.

I want now to get a second and a third solution. Take

, \fd6 x/ X . \ .^^.

^ = 2^e5 = r^(««?-&^«^?j <17).

as the velocity potential for a fresh solution. I take it that you
all know that if we have one solution ^, for the velocity potential,
we can get another solution by yjr any linear function of

cUf} d<f> d<f>
dx ' dy * dz'

Now let us find the displacements

dy^ dy^ dyp"
dx* dy* dz'

Here I want to prove that though this solution is no longer sym-
metrical with respect to r, so that there will be motions other
than radial in the neighbourhood of the source, yet still the
motion is approximately radial at great distances from the source.
Work it out, and you will find that

d^_ 27r sin q \a? / \ V r^ - Za?"] cosq r^-Sa? .

dx"\ r lr*'^\2irr)~t^' y 7^ ^ ^^^^'

The principal term here is — — -, sin q. We might go on to the

third and fourth and higher differential coefficients of ^, with
their larger and larger numbers of terms. The interpretation of
this multiplicity of terms, of the terms other than those which
I am now calling the " principal terms," is all-important in respect
to the motion of the air in the neighbourhood of the source. It

* I UBe = to denote approximate equality.

t X/2ir is introdaced merely for conyenience. The solation differs from
^=d^ldXf only by a constant factor.

AERIAL WAVES OF SOUND. 49

is dealt with in that splendid work of Stokes, one of the finest Molar,
things ever written in physical mathematics, of which I read to
you this afternoon, with reference to the effect of an atmosphere
of hydrogen round a bell killing its sound. But we will drop
those terms and think only of the terms which express the
efficiency of the vibrator at distances great in comparison with
the wave-length.

Thus, for the a;-component displacement of the motion now
considered, we have

f-T?-^S-;) »

This approximate equality is true for distances from the centre
great in comparison with the wave-length. Let me remark, it is
the differentiation of cos q that gives the distantly effective terms
of the displacement ; and in differentiating '^ with respect to y,
you have simply to differentiate cos q and to take the differential
coefficient of r now with respect to y, instead of a? as formerly.
So that we may write down the principal terms of the y and z
displacements by taking yjx and zjx of the second member of (19)
as follows : —

^irxy , ^ 2'7rxz .

The three component displacements being proportional to x, y, z,
shows that the resultant displacement is in the direction of the

radius ; and its magnitude is — — -^ sin q. If we write a; = r cos i,

this becomes

27r cos % . .. _ .

- X" r ^^^^ ^^^^'

or the displacement is inversely proportional to the distance. If

1 = 0 we have a maximum ; if i = ^ we have zero. The upshot of

it is that the displacement is a maximum in the axis OX, zero
everywhere in the plane of OY, 0Z\ and symmetrical all round
the axis OX,

A third solution is got by taking -r? as our velocity potential.

At a distance from the origin, great in comparison with the wave-
T. L. 4

50 LBCTTRE IT.

krtigtb, tbe dispbcemenl is in the directioD of the radios, and its
magDitode is -^ -^^ .

Now the interpretation of these cases is as fidlows : — ^The first
vJation, (Tekcitv potential ^) a globe alteraalelj becoming burger
an'l sioaller; the second solution, (relocitj potmtial d^'dx) a
globe vibrating to and fro in a straight line ; the third solution,
^velocity potential cP^dlx^ a chaiacieristic constituent of the
motion of the air produced bj two globes vibrating to and fro
in tbe line of their centres, or bv the prongs of a vibrating fork.

This last requires a little nice consideration^ and we shall take

it up in a subsequent lecture. The third mode does not quite

represent the motion in the neighbourhood of the pair of vibrating

globes, or of the prongs of a vibrating fork ; there must be an

unknown amount of the first mode compounded with the third

mfftle for this purpose. The expression for the vibration in the

neighbourhood of a tuning fork, going so far from the ends of it

that we will be undisturbed or but little disturbed, by the general

shape of the whole thing, will be given by a velocity potential

(PS
A^+ -j^ . That will be the velocity potential for the chief terms,

tbe terms which alone have effect at great distances. The dif-
ferentiation will be performed simply with reference to the r in
the term sin q or cos q ; and will be the same as if the coeflScient
of sin 9 or cos q were constant. A differentiation of this velocity
potential will show that the displacement is in the direction of
the radius from the centre of the system, and the magnitude of

the displacement will be ^ IA4> + ^] .

^ is an unknown quantity depending upon the tuning fork.
I want to suggest this as a junior laboratory exercise, to try tuning
forks with different breadths of prongs. When you take tuning
forks with prongs a considerable distance asunder you have much
less of the 0 in the solution : try a tuning fork with flat prongs,
pretty close together, and you will find much moro of the ^
The ^ part of the velocity potential corresponds to the alternate
swelling and shrinking of the air between the two prongs of the
tuning fork. The larger and flatter the prongs are the greater is
the proportion of tbe ^ solution, that is to say, the larger is the

AERIAL VIBRATIONS ROUND A TUNING FORK. 51

value of A in that formula; and the smaller the angle of the cone Molar,
of silence.

The experiment that I suggest is this : take a vibrating tuning
fork and turn it round until you find the cone of silence, or find
the angle between the line joining the prongs and the line going to
the place where your ear must be to hear no sound. The sudden-
ness of transition from sound to no sound is startling. Having
the tuning fork in the hand, turn it slowly round near one ear
until you find its position of silence. Close the other ear with your
band. A very small angle of turning round the vertical axis from
that position gives you a startlingly loud sound. I think it is very
likely that the place of no sound will, with one and the same fork,
depend on the range of vibration. If you excite it very powerfully,
you may find less inclination between the line of vibration of the
prongs and the line to the place of silence ; less powerfully, greater
inclination. It will certainly be different with different tuning forks.

4-2

LECTURE V.

Saturday, Oct. 4, 5 pjl

Molar: I STATED in the last lecture that the second solution, corre-

Becapitn- » .

lation. spending to the velocity potential -j^ , would represent the effect,

at a great distance from the mean position, of a single body vibrat-
ing to and fro in a straight line. I said a sphere, but we may take
a body of any shape vibrating to and fro in a straight line; and at
a very great distance from the vibrator, the motion produced will

be represented by the velocity potential ^, provided the period

of the vibration is great in comparison with the time taken by
sound to travel a distance equal to the greatest diameter of the

body. Then the velocity potential -r^ , in the third solution,

would, I believe, represent (without an additional term A<f>) the
motion at great distances, when the origin of the sound consists
in two globes, let us say, for fixing the ideas, placed at a distance
from one another very great in comparison with their diameters
and set to vibrate to and fro through a range small in comparison
with the distance between them, but not necessarily small in
comparison with their diameters : provided always that the period
of the vibration is great in comparison with the time taken by
sound to travel the distance between the vibrators. Suppose
this is a globe in one hand, and this is one in the other. I now
move my hands towards and from each other — the motion of the
air produced by that sort of motion of the exciting bodies would, at
a very great distance, be expressed exactly by the velocity potential

da?'

But when you have two globes, or two flat bodies, very near
one another, you need an unknown amount of the ^ vibration to

AERIAL VIBRATIONS ROUND A TUNING FORK. 53

represent the actual state of the case. That unknown amount Molar:
might be determioed theoretically for the case of two spherea ution^
The problem is analogous to Poisson's problem of the distribution
of electricity upon two spheres, and it has been solved by Stokes
for the case of fluid motion (see Mem. de Flnst., Paris, 1811, pp. 1,
163; and Stokes' Papers, Vol. I., p. 230 — " On the resistance of a
fluid to two oscillating spheres"). Tou can thus tell the motion
exactly in the neighbourhood of two spheres vibrating to and fro
provided the amplitudes of their vibrations are small in comparison
with the distance between them; and you can find the value of
A for two spheres of any given radii and any given distance
between them. For such a thing as a tuning fork, you could
not, of course, work it out theoretically ; but I think it would be
an interesting subject for junior laboratory work, to find it by
experiment.

I suppose you are all now familiar with the zero of sound in
the neighbourhood of a tuning fork; but I have never seen it
described correctly anywhere. We have no easy enough theo-
retical means of determining the inclination of the line going to
the position of the ear for silence to the line joining the prongs ;
but we readily see that it is dependent upon the proportions
of the body. In turning the tuning fork round its axis, you
can get with great nicety the position for silence; and a sur-
prisingly small turning of the tuning fork from the position of
silence causes the motion to be heard. It would be very curious
tu find whether the position of zero sound varies perceptibly with
the amplitude of the vibrations. I doubt whether any perceptible
difference will be found in any ordinary case however we vary the
amplitude of the vibrations. But I am quite sure you will find
considerable difference, according as you take tuning forks with
cylindrical prongs, or with rectangular prongs of such proportions
as old Marlowe used to make, or tuning forks like the more
modern ones that Koenig makes, with very broad flat prongs.

Now for our molecular problem. Moleculi

I want to see how the variable quantities vary, when we vary
the period. Remember that

«,= 3-c,-c,^j (1);

54 LECTURE V.

Molecular, so that ,, ' = m, (2).

a(T ) *

Write for the moment 3 for j- ^ , and differentiate the equation
for u^ ; we find

Substitute successively, and we find,

»»<«

+ f9m£m]\i +_(9mj::iSL]\ (3).

This is our expression, and remark the exceedingly important
property of it that it is essentially positive, i.e. the variation of u^
with increase of t"* is essentially positive. Now

Cj+l _ '«'hj

^i+^ __ ^<+S

"w «.

^*« ^<M

also

The result (3) therefore is equivalent to the following expression
(4) for the differential coefficient of u^ with respect to the period.

This is certainly a very remarkable theorem, and one of great
importance with reference to the interpretation of the solution of
our problem. Remember that x^ is the displacement of m^ at any
time of the motion. You may habitually think of the maximum
values of the displacements, but it is not necessary to confine your-
selves to the maximum values. Instead of ajj, a?,, ... Xj we may
take constants equal to the maximum values of the xs, each

multiplied into sin , because the particles vibrate according to

T

the simple harmonic law, all in the same period and the same
phase, that is all passing through zero simultaneously, and reach-
ing maximums simultaneously, every vibration. The masses are
positive, and we have squares of the displacements in the several
terms of (4) ; so that the second member of (4) is essentially

VIBRATIONS OF SERIAL MOLECULE. 55

negative. Hence, as we augment the period, each one of the ratios MolecoU
u^ decreases.

Let us now consider the configurations of motion in our spring
arrangement, for different given periods of the exciting vibrator, P.
I am going to suppose, in the first place, that the period of
vibration is very small, and is then gradually increased. As you
increase the period, we have seen that the value of each one of the
quantities Uj,u,, ... decreases. It is interesting to remark that
this is so continuously throughout each variety of the configura-
tions found successively by increasing t from 0 to oo . But we
shall find that there are critical values of r, at which one or other
of the u's, having become negative decreases to — oo ; then suddenly
jumps to + 00 as r is augmented through a critical value; and
again decreases, possibly again coming to + oo , possibly not, while
T is augmented farther and farther, to infinity. In the first place,
T may be taken so small that the t^'s are all very large positive

ft% c*

quantities; for u^ being equal to -y — c, — c^j, — *— may be cer-

tainly made as very large positive as we please by taking r small
enough, if, at the same time the succeeding quantity, w^j, is large,
a condition which we see is essentially fulfilled where t is very
small, because we have Uj = mj/T^ — Cj — c^^, which makes Uj very
great.

Observe that the us all positive implies that ^y x^, x^, x^, ,,, x^
are alternately positive and negative. In other words the handle
P and the successive particles m,, m^, ... m^, are each moving in a
direction opposite to its neighbour on either side. Since the
magnitudes of the ratios u^^ u^, , . , u^ o{ the successive amplitudes
decrease with the increase of the period, the amplitude of particle
m^ is becoming smaller in proportion to the amplitude of the suc-
ceeding particle m^j, as long as the vibrations of the successive
particles are mutually contrary-wards. I am going to show you
that as every one of these quantities u^ decreases, the first that
passes through zero is necessarily u^ ; corresponding to a motion of
each particle of the system infinitely great in comparison with
the motion of the handle P ; that is to say, finite simple harmonic
motion of the system with P held fixed. This is our first critical
case. It is the only one of the j fundamental modes of vibration of
the system with P fixed, in which the directions of vibration of the
successive particles are all mutually contrariwise, and it is the one

56 LECTURE V.

[olecular. of thetn of which the period is shortest. After that, as we in-
crease T, Wj becomes negative, and the motion of P comes to be in
the direction of the motion of the first particle. As we go on
increasing the period we shall find that the next critical case that
comes is one in which particle m^ has zero motion, or

M, = '— = — 00 .

* —a?

To prove this, and to investigate the further progress, let us look

at the state of things when a positive decreasing u^ has approached

c*
very near to zero. We shall have u^j, being equal to a^^ — ^ , a

very large negative quantity. This alone shows that u^_^ must
have preceded u^ in becoming zero, since it must have passed
through zero before becoming negative. Therefore, as we augment

T, the first of the w*s to become zero is m = -^ ; or, as I said

— X

before, the motion of particle m^ and also of each of the other
particles is infinite in comparison with the motion of P. Just
before this state of things all the particles P, m,, ... m^ are, as we
saw, moving each contrary-wards to its neighbour; just after it, P
has reversed its motion with reference to the first particle, and is
moving in the same direction with it.

This continues to be the configuration, till just before the
second critical case, in which we have t^^ large negative, tx, small
positive, t^j, ... Mj, all positive. At this critical case, we have

u. = — ^ = + 00 : or a;, = ± 0 X f .

The period of motion of P that will produce this state of things is
equal to the period of the free vibration of the system of particles,
with mass m^ held at rest, and each of the other masses moving
contrary-wards to its neighbour on each side. When the period of
the simple harmonic motion of P is equal to a period of motion
of the system with the first particle held at rest, then the only
simple harmonic motion which the system with all the particles
unconstrained can have is in that period, and with the amplitude
of vibration of the second particle in one direction just so great as
to produce by spring No. 2, a pull in that direction, on m^, equal
to the pull exercised on it through spring No. 1, by P in the
opposite direction ; so as to let the first particle be at rest Sup-

VIBRATIONS OF SERIAL MOLECULE. 57

pose now t to be continuously increased through this critical value. Molecular
Immediately after the critical case, u, has changed from large
negative, through + oo , to large positive, and u, from small positive,
through ± 0, to small negative ; or the first particle has reversed
the direction of its motion and come to move same-wards with P
and contrary-wards to the second particle.

The third critical case might be that of the second particle
coming to rest, (m, = T » , w, = ± 0); or it might be m^ = 0 a second
time : it must be either one or other of these two cases. But we
must not stop longer on the line of critical cases at present*. I
will just jump over the remaining critical cases to the final con-
dition.

It would be curious to find the solution when the period is

infinitely great out of our equations. When t is infinite, -3

[Note added; Jan. 11, 1886, Netherhall, Largs.]

* As we go on increasing r from the first critical valae (that which made %=0),
the essential decreasings of Uj (negative) and u^ (still positive) bring Uj to - ao and
ti, to 0 simultaneously. With farther increase of r, the decreasings of u, (now
negative) and of u, (still positive) bring u, to - ao and u, to 0, simultaneously; and
so on, in succession from u^ to u^ which passes through zero to negative, but cannot
become - oo and remains negative for all greater values of r, diminishing to the
value - {cj + c^i) as r is augmented to infinity.

But %, after decreasing to - ao , must pass to + oo and again become decreasing
positive. It must again pass through zero; and thus there is started another
procession of zeros along the line from P, through nij, m^,... successively, but ending
in ">>->' ^^^ ^^ ^^J-\ ^^ose amplitude {u/ijlcj) is made zero and negative by the
conclusion of the first procession, before the second procession can possibly reach
it. Thus Uj^i passes a second and last time through zero and diminishes to the

limiting values - Cj_. I — +- + — |/( — + — i.asris augmented to 00 .

A third procession of zeros, similarly commencing with P, passing along the

line, m^, }iL2,...and ending with m/.,, makes Uj_^ zero for a third and last time, and

leaves it to diminish to its limit (shown by the formula reported in the text below,

from the lecture,) as r augments to oo . Similarly procession after procession, in

all J processions, commence with in^. The last begins and ends in m^, and leaves u^

to go from zero to its negative limiting value

/ 1 1 1 1\// 11 1\

-cA — + +...+- + -)/( — + - + ...+-),

as r augments to 00 . There is no general rule of precedence in respect to magni-
tude of r, of the dififerent transitional zeros of the different processions. For
example, there is no general rule as to order of commencement of one procession,
and termination of its predecessor. The one essential limitation is that no
collision can take place between the front of one procession and the rear of its
predecessor.

58 LECTURE V.

Moleoolar. vanishes, and ct^ = — c^ — c^^^, That applied to the equations for the
u*8 ought to find the solution quite readily.

You know, when you think of the dynamics of this case, that
when T is infinitely great, P is moving infinitely slowly, so that
the inertia of each particle has no sensible effect ; and all the
particles are in equilibrium. Let F be the force, then, on the
spring ; that is to say, pull P slowly down with a force F and hold
it at rest. What will be the displacements of the different

particles ? Answer, Xj = — , Xj_^ = 1 — , and so on. Particle

number j is displaced to a distance equal to the force, divided by
the coefficient of elongation of the spring. To obtain the displace-
ment of particle j — I, we have to add the displacement resulting
from the elongation of the next spring Cj, and so on. The general
equation then is

It is a curious but of course a very simple and easy problem to
substitute the value of a^ = •^'C^'- c^^ in the continued fraction
which gives u^, and verify this solution. No more of this now
however.

It is fiddling while Rome is burning ; to be playing with
trivialities of a little dynamical problem, when phosphorescence
is in view, and when explanation of the refraction of light in
crystals is waiting. The difficulty is, not to explain phosphor-
escence and fluorescence, but to explain why there is so little of
sensible fluorescence and phosphorescence. This molecular theory
brings everything of light, to fluorescence and phosphorescence.
The state of things as regards our complex model-molecule would
be this: Suppose we have this handle P moved backwards and
forwards until everything is in a perfectly periodic state. Then
suddenly stop moving P. The system will continue vibrating for
ever with a complex vibration which will really partake something
of all the modes. That I believe is fluorescence.

But now comes Mr Michelson's question, and Mr Newcomb's
question, and Lord Rayleigh's question ; — the velocity of groups
of waves of light in gross matter. Suppose a succession of luminous

FINITE SEQUENCES OF WAVES IN A DISPERSIVE MEDIUM. 59

vibrations commencea In the commencement of the luminous vi- Moleoula
brations the attached molecules imbedded in the luminiferous ether,
do not immediately get into the state of simple harmonic vibration
which constitutes a regular light. It seems quite certain that
there must be an initial fluorescence. Let light begin shining on
uranium glass. For the thousandth of a second, perhaps, after
the light has begun shining on it, you should find an initial state
of things, which differs from the permanent state of things some-
what as fluorescence differs from no light at all.

There is still another question, which is of profound interest,
and seems to present many difficulties, and that is, the actual
condition of the light which is a succession of groups. Lord
Rayleigh has told us in his printed paper in respect to the
agitated question of the velocity of light, and then again, at the
meeting of the British Association at Montreal, he repeated very
peremptorily and clearly, that the velocity of a group of waves
must not be confounded with the wave velocity of an infinite
succession of waves, and is of necessity largely different from the
velocity of an infinite succession of waves, in every dispersively
refracting medium, that is to say, medium in which the velocity
is different for lights of different period. It seems to be quite
certain that what he said is true. But here is a difficulty which
has only occurred to me since I began speaking to you on the
subject; and I hope, before we separate, we shall see our way
through it. All light consists in a succession of groups. We are
all, already, familiar with the question; — Why is all light not
polarized ? and we are all familiar with its answer. We are now
going to work our way slowly on until we get expressions for
sequences of vibrations of existing light. Take any conceivable
supposition as to the origin of light, in a flame, or a wire made
incandescent by an electric current, or any other source of li^rht ;
we shall work our way up from these equations which we have
used for sound, to the corresponding expression for light from
any conceivable source. Now, if we conceive a source con-
sisting of a motion kept going on with perfectly uniform period-
icity; the light from that source would be plane polarized, or
circularly polarized, or elliptically polarized, and would be abso-
lutely constant. In reality, there is a multiplicity of successions
of groups of waves, and no constant periodicity. One molecule,
of enormous mass in comparison with the luminiferous ether that

60 LECTURE V.

MoleMilar. it displaces, gets a shock and it performs a set of vibrations until
it comes to rest or gets a shock in some other direction ; and it
is sending forth vibrations with the same want of r^ularity that
is exhibited in a group of sounding bodies consisting of bells,
tuning forks, organ pipes, or all the instruments of an orchestra
played independently in wildest confusion, every one of which is
sending forth its sound which, at large enough distances from the
source, is propagated as if there were no others. We thus see
that light is essentially composed of groups of waves ; and if the
velocity of the front or rear of a group of waves, or of the centre
of gravity of a group, differs from the wave-velocity of absolutely
continuous sequences of waves, in water, or glass, or other dispersively
refracting mediums, we have some of the ground cut from under us
in respect to the velocity of waves of light in all such mediums.

I mean to say, that all light consists of groups following one
another, irregularly, and that there is a difficulty to see what to
make of the beginning and end of the vibrations of a group : and
that then there is the question which was talked over a little in
Section A at Montreal, — will the mean of the effects of the
groups be the same as that of an infinite sequence of uniform
waves, and will the deviation from regular periodicity at the
beginning and end of each group have but a small influence on
the whole ? It seems almost certain that it must have but a
small influence from the known facts regarding the velocity of
light proved by the known, well-observed, and accurately measured,
phenomena of refraction and interference. But I am leading
you into a muddle, not however for you, I hope, a slough of
despond; though I lead you into it and do not show you the
way out. You will all think a good deal along with me about the
connections of this subject.

LECTURE VI.

Monday, Oct 6, 5 p.m.

I WANT to ask you to note that when I spoke of A? + Jn not Molar,
differing very much from k for most solids, I was rather under the
impression for the moment that the ratio of n to A; was smaller
than it is; and also you will remember that we had k-k-^n on
the board by mistake for k + |n. The square of the velocity of a
condensational wave in an elastic solid is (k + f n)//!>. For solids
fulfilling the supposed relation of Navier and Poisson between
compressibility and rigidity we have n = |A: ; and for such cases the
numerator becomes ^k. It would be A; if there were no rigidity ;
it is ^k if the rigidity is that of a solid for which Poisson's ratio
has its supposed value.

Metals are not enormously far from fulfilling this condition,
but it seems that for elastic solids generally n bears a less pro-
portion to k than this. It is by no means certain that it fulfils it
even approximately for metals ; and for india rubber, on the other
hand, and for jellies, 7i is an exceedingly small fraction of k, so
that in these cases the velocity of the condensational wave is but

yk
- . The velocity of propagation of a

distortional wave is / - ; so that for jellies, the velocity of propa-
gation of condensational wave is enormously greater than that of
distortional waves.

I am asked by one of you to define velocity potential. Those
who have read German writers on Hydrodynamics already know
the meaning of it perfectly well. It is a purely technical expres-
sion which has nothing to do with potential or force. " Velocity
potential " is a function of the co-ordinates such that its rate of
variation per unit distance in any direction is equal to the

62 LECTURE VI. PART 1.

)lar. component of velocity in that direction. A velocity potential
exists when the distribution of velocity is expressible in this way ;
in other words when the motion is irrotationaL The most con-
venient analytical definition of irrotational motion is, motion such
that the velocity components are expressed by the differential co-
efficients of a function. That function is the velocity potential.
When the motion is rotational there is no velocity potential.

This is the strict application of the words " velocity potential "
which I have used. A corresponding language may be used for
displacement potential. It is not good language, but it is con-
venient, it is rough and ready. And when we are speaking of
component displacements in any case, whether of static displace-
ment in an elastic solid or of vibrations, in which the components
of displacement are expressible as the differential coefficients of a
function, we may say that it is an irrotational displacement If from
the differentiation of a function we obtain components of velocity,
we have velocity potential ; whereas, if we so get components of
displacement, we have displacement potential. The functions ^,
that we used, are not then, strictly speaking, velocity potentials
hut displacement potentials.

I want you in the first place to remark what is perfectly well
known to all who are familiar with Differential equations, that

taking the solution <^ = - . sin qr as a primary, where

we may derive other solutions by differentiations with respect to
the rectangular co-ordinates. The first thing I am going to call
attention to is that at a distance from the origin, whatever be
the solution derived from this primary by differentiation, the cor-
responding displacement is nearly in the direction through the
origin of co-ordinates.

Take any differential coefficient whatever, j-^^-j » i; the term

of this which alone is sensible at an infinitely great distance is
that which is obtained by successive differentiation of sin q. That
distance term in every case is as follows :

l+i+t

/M-- /^y /^y /dry 1 siB

\ \ / \cUc/ \dyj \dzj r cos ^

CONDENSATIONAL WAVE. 63

It will be sin 9 or COS 9, according as i + j + ii; is even or odd. We Molar.

do not need to trouble ourselves about the algebraic sign, because

we shall make it positive, whether the differential coefficient is

. ^T dr X dr y dr z nr%
positive or ne&:ative. Now -=-=-, -^="^, -=- = -. Thus our
^ ° ax r ay r dz r

type solution becomes, -j^^i 9. This expresses the most

general type of displacement potential for a condensational wave
proceeding from a centre, and having reached to a distance in
any direction from the centre, great in comparisoD with the wave-
length. I have not formally proved that this is the most general
type, but it is very easy to do so. I am rather going into the
thing synthetically. It is so thoroughly treated analytically by
many writers that it would be a waste of your time to go into
anything more, at present, than a sketch of the manner of treat-
ment, and to give some illustrations.

But now to prove that the displacement at a distance from the
origin of the disturbance is always in the direction of the radius
vector. Once more, the differential coefficient of this displace-
ment potential, which has several terms depending upon the
differentiation of the r's, ic's, etc. has one term of paramouut

importance, and that is the one in which you get as a factor.

The smallness of \ in proportion to the other quantities makes

27r
the factor — - give importance to the term in which it is found.

A

The distance terras then for the components of the displacement
are

^ a;V^ 27r a; cos r,x cos

* ^H^+*+i ' \ ' r sin ^ r sin ^'

D y cos ^ j^z cos
r sin ^ r sm ^

These are then the components of a displacement which is radial ;
and the expression for the amplitude of the radial displacement is

The sum of any number of such expressions will express the
distance effect of sound proceeding from a source. It is interest-

64 LECTURE VL PART I.

[)lar. ing to see how, simply by making up an algebraic function in the
numerator out of the a?'8, y's and /s, we can get a formula that
will express any amount of nodal subdivision where silence is felt.
The most general result for the amplitude of the radial displace-
ment is 12 = S T^j^i- Remark that - , ^, - are merely angular

functions and may be expressed at once as sin 0 cos -^^ sin 0 sin y^,
cos 5; and therefore R is an integral algebraic function of sin^cos-^,
sin 0 sin y^^ cos 0, It is thus easy to see that you can vary inde-
finitely the expressions for sound proceeding firom a source with
cones of silence and corresponding nodes or lines in which those
cones cut the spherical wave surfiEice. It is interesting to see that
even in the neighbourhood of the nodes the vibration is still
perpendicular to the wave surface; so that we have realized in
any case a gradual falling off of the intensity of the wave to zero
and a passing through zero, which would be equivalent to a change
of phase, without any motion perpendicular to the radius vector.

The more complicated terms that I have passed over are those
that are only sensible in the neighbourhood of the source. Sup-
pose, for instance, that you have a bell vibrating. The air slipping
out and in over the sides of the bell and round the opening gives
rise to a very complicated state of motion close to the bell ; and
similarly with respect to a tuning fork. If you take a spherical
body, you can very easily express the motion in terms of spherical
harmonics. You see that ib the neighbourhood of the sounding
body there will be a great deal of vibration in directions perpen-
dicular to the radius vector, compounded with motions out and in ;
but it is interesting to notice that all except the radial component
motions become insensible at distances from the centre large in com-
parison with the wave-length. It is the consideration of the motion
at distances that are moderate in comparison with the wave-length
that Stokes has made the basis of that very interesting investiga-
tion with reference to Leslie's experiment of a bell vibrating in a
vacuum, to which I have already referred. (Lecture in., pp. 36, 37
above.)

We may just notice, before I pass away from the subject, two
or three points of the case, with reference to a tuning fork, a bell,
and so on. Suppose the sounding body to be a circular bell. In
that case clearly, if the bell be held with its lip horizontal, and if
it be kept vibrating steadily in its gravest ordinary mode, the

VIBRATIONS OF A BELL. 65

kind of vibration will be this : a vibration from a circular figure Molar.

into an elliptic figure \.y\)) along one diameter, and

a swinging back through the circular figure \^T"^ ^^^ ^^

elliptic figure n ^ I) along the diameter at right angles to the

first. Clearly there would be practically a plane of silence
here and another at right angles to it here (represented on the
diagrams by dotted lines). Hence the solution for the radial
component corresponding to this case, at a considerable distance

fi"om the bell, is jB = (J — cos*-4) — ^ , in order that the component

may vanish when cos"il = J, or ^ = ± 45"*; A being an azimuthal
angle if the axis of the bell is vertical.

On the other hand, consider a tuning-fork vibrating to-and-fro
or an elongated (elliptic) bell, which I got from that fine old
Frenchman, Koenig's predecessor, Marloye. It makes an ex-
ceedingly loud sound and has an advantage in acoustic experi-
ments over a circular bell. If you set a circular bell vibrating
and leave it to itself you always hear a beating sound, because
the bell is approximately but not accurately symmetrical. Excite
it with a bow, and take your finger off, and leave it to itself: and
if you do not choose a proper place to touch it, for a fundamental
mode, when you take your finger off it will execute the re-
sultant of two fundamental modes.

I do not know whether the corresponding experiment with
circular plates is familiar to any of you. I would be glad to know
whether it is. I make it always before my own classes, in illus-
trating the subject. Take a circular plate — just one of the ordinary
circular plates that are prepared for showing vibration in acoustic
illustrations. Excite it in the usual way with a violoncello bow, and
putting a finger, or two fingers, to the edge to make the quadrantal
vibration. K sand is sprinkled on the plate, the vibrations toss it
into sand-hills with ridges lying along two diameters of the disc,
perpendicular to one another, one of them through the point or
T. L. 0

66 LECTUBE VL PART L

Molar. two of them through the points of the edge touched by finger. Now
cease bowing and take your finger off the edge of the disa The
sand-hills are tossed up in the air and the sand is scattered to
considerable distances on each side, and is continually tossed up
and not allowed to rest anywhere. At the same time you hear a
beating sound. But by a little trial, I find one place where if I
touch with the finger, and ply the bow so as to make a quadrantal
vibration, and if I then cease bowing and take off my finger, the
sand remains undisturbed on two diameters at right angles, and no
beat is beard. Then having found one pair of nodal diameters,
I know there will be another pair got by touching the plate here,
45° from the first place. Now touch therefore with two fingers, at
two points 90° from one another midway between the first and
second pairs of nodal diameters : cause the plate to vibrate and
then take off your fingers, and stop bo¥dng : you will hear very
marked beats; a sound gradually waxing from absolute silence
to loudest sound, then gradually waning to silence again, and so
on, alternating between loudest sound and absolute silence with
perfect gradualness and regularity of waxing and waning.

Take a division of the circumference into six equal parts by
three diameters, and you find the same thing over again. Go on
by trial touching the plate at two points 60° or 120° asunder, and
bowing it 30° from either; and you will see the sand resting
on the three diameters determined by your fingers. Take off
your fingers and you will in general see the sand scattered and
hear a beat Follow your way around, little by little — it is very
pretty when you come near a place of no beat. The moment you
take off your fingers you see the lines of nodes swaying to-and-
fro on each side of a mean position, with a slow oscillation;
and you hear a very distinct beat, though of a soft but perfectly
regular character from loudest to least loud sound. Qet exactly
the mean position, steadying the nodal sand-hills while stiU plying
the bow : then cease bowing and suddenly remove your finger or
fingers from the plate ; you will see the nodal lines remaining
absolutely still and you will hear a pure note without beats. If
you touch at exactly 30° from the nodal lines first found you wUl
have the strongest beat possible, which is a beat from loud sound
to silence. Advance your fingers another 30° and you will again find
the sand-hills remain absolutely still when you remove your fingers.
You may go on in this way with eight and ten subdivisions, and so

YIBBATIONS OF APPBOXIMATELT CIRCULAR PLATES. 67

on ; but you must not expect that the places for the sextan tal, Molar,
octantal, and higher, subdivisions correspond to the places for the
quadrantal subdivisions. The places for quadrantal subdivision
will not in general be places for octantal subdivision. You must
experiment separately for the octantal places, and you will find
generally that their diameters are oblique to the quadrantal

The reason for all this is quite obvious. In each case, the
plate being only approximately circular and symmetrical, the
general equation for the motion has two approximately equal
roots corresponding to the nodes or divisions by one, two, three, or
four diameters, and so on. These two roots always correspond to
sounds differing a little from one another. The effect of putting
the finger down at random is to cause the plate, as long as your
finger is on it, to vibrate forcedly in a simple harmonic vibration
of period greater than the one root and less than the other. But
as soon as you take your finger off, the motion of the plate follows
the law of superposition of fundamental modes ; each fundamental
mode being a simple harmonic vibration. I have often, in showing
this experiment, tried musicians with two notes which were veiy
nearly equal, and said to them, '' Now, which of the two notes is
the graver ?" Rarely can they tell. The difference is generally too
small for a merely musical ear, and the verdict is that the notes
are " the same ;" musicians are not accustomed to listen to sounds
with scientific ears and do not always say rightly which is the
graver note, even when the difference is perceptible. Any person
can tell, after having made a few experiments of the kind, that
this is the graver and that the less grave note, even though he
may have what, for musicians, is an uncultivated ear, or truly a
very bad ear for music, not good enough in fact to guide him in
sounding a note with his voice, or to make him sing in tune if he
tries to sing. It is very curious, when you have two notes which
you thoroughly know are different, that if you sound first one and
then the other, most people will say they are about the same.
But sound them both together, and then you hear the discord of
the two notes in approximate unison.

In every case of a circular plate vibrating between diametral
lines of nodes, there is an even number of planes of silence in the
surrounding air; being the planes perpendicular to the plate,
through the nodal diameters. If you take a square plate or bell
vibrating in a quadrantal mode, for instance, then you have two

5—2

68 LECTURE VI. PART I.

Molar. vertical planes of silence at right angles to one another. If you
make it vibrate with six or more subdivisions, you will have a
corresponding number of planes of silence.

With reference to the motions in the neighbourhood of the
tuning-fork, you get this beautiful idea, that we have essentially
harmonic functions to express them. Essentially algebraic func-
tions of the co-ordinates appear in these distant terms, but in the
other terms which Prof. Stokes has worked out, and which have
been worked out in Prof. Rowland's paper on Electro-Magnetic
Disturbances*^ quite that kind of analysis appears, and it is most
important I have not given you a detailed examination of that
part of our general solution, but only called your attention specially
to the "distance terms,* partly because of their interest for sound
and partly because the consideration of them prepares us for our
special subject, waves of light.

To-morrow we shall begin and try to think of sources of waves
of light. I want to lead you up to the idea of what the simplest
element of light is. It must be polarized, and it must consist of
a single sequence of vibrations. A body gets a shock so as to
vibrate; that body of itself then constitutes the very simplest
source of light that we can have ; it produces an element of light.
An element of light consists essentially in a sequence of vibrations.
It is very easy to show that the velocity of propagation of
sequences in the pure luminiferous ether is constant. The sequence
goes on, only varying with the variation of the source. As the
source gradually subsides in giving out its energy the amplitude
evidently decreases ; but there will be no throwing off of wavelets
forward, no lagging in the rear, no ambiguity as to the velocity of
propagation. But when light, consisting as it does of sequences
of vibrations, is propagated through air or water, or glass, or
crystal, what is the result ? According to the discussion to which
I have referred, the velocity should be quite uncertain, depending
upon the number of waves in the sequence, and all this seems
to present a complicated problem.

But I am anticipating a little. We shall speak of this here-
after. One of you has asked me if I was going to get rid of the
subject of groups of waves. I do not see how we can ever get rid
of it in the wave theory of light. We must try to make the best
of it, however.

* Phil. Mag. xvn., 1884, p. 418. Am. Jour. Math, yi., 1884, p. 359.

NUMBER OF TERMS IN DETERMINANT. 69

Ten minutes* interval.

This question of the vibration of connected particles is aMolecolai
peculiarly interesting and important problem. I hope you are
not tired of it yet. You see that it is going to have many ap-
plications. In the first place remark that it might be made the
base of the theory of the propagation of waves. When we take
our particles uniformly distributed and connected by constant
springs we may pass from the solution of the problems for the
mutual influence of a group of particles to the theory, say, of the
longitudinal vibrations of an elastic rod, or, by the same analysis,
to the theory of the transverse vibrations of a cord.

I am going to refer you to Lagrange's Micanique Analyttque^
[Part II. p. 339]. The problem that I put before you here is given
in that work under the title of vibrations of a linear system of
bodies. Lagrange applies what he calls the algorithm of finite
differences to the solution. The problem which I put before you
is of a much more comprehensive kind ; but it is of some little
interest to know that cases of it may be found, ramifying into each
other.

I wish to put before you some properties of the solution which
are of very great importance. I want you to note first the number
of terms.

We have :

All the x's being expressed in this way successively in terms of Xj,
Let Nt be the number of terms in Xj.f, These terms are obtained
by substituting the values of iCy.^^.i, Xj.^^^ i^ t,he formula

None of the terms can destroy one another except for special
values, and the conclusion is that we have the following formula
for obtaining the number of terms :

This is an equation of finite differences. Apply the algorithm

70 LECTURE VL PART I.

Molecular, of finite differences, as Lagrange says ; or, which is essentially the
same, we may try for solutions of this equation by the. following
formula : JV, = zN^^y We thus find

z = z-\-\,ot z^ — 5 — •

We can satisfy our equation by taking either the upper or the
lower sign. The general solution is, of course,

^,.c(i±^7.c(i^)'

where C, (J are to be determined by the equation -A/^ = 1, N^=^\,
It is rather curious to see an expression of this kind for the
number of terms in a determinant. You will find that, of the more
general equation

the following is a solution:

where r, 8 are the two roots of the equation a:* = cw? + 6. Remark
that the coefficients of N^, N^, being symmetrical functions of the
two roots, are, as they must of course be, integral functions of a
and b.

If one of the roots, 5, for example, be less than unity, we may
omit the large powers of 8, and therefore for large values of t we
may be sure of obtaining iV, to within a unit, and therefore the
absolutely correct value, by calculating the integral part of

h •
r

It is interesting to remark that the numerical value of this
formula differs less and less from an integer the greater is i, and
differs infinitely little from an infinitely large integer when t is
infinitely great.

The values of N^ up to t = 12 for the case of our problem
(a = 6=iV; = i^^ = l)are,

1 = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
iV, = 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.

LECTURE VIL

Tuesday, Oct. 7, 3.30 p.m.

Lagrange, in the second section of the second part of his Me- Moieonli
canique Analytiqus on the Oscillation of a Linear System of Bodies,
has worked out very fully the motion in the first place for bodies
connected in series, and secondly for a continuous cord. The case
that we are working upon is not restricted to equal masses and
equal connecting springs, but includes the particular linear system
of Lagrange, in which the masses and springs are equal. I hope to
take up that particular case, as it is of great interest. We shall
take up this subject first to-day, and the propagation of disturbances
in an elastic solid second.

It was pointed out by Dr Franklin that, for the particular case
iVj = aN^ which is the case of our particular question as to the
number of terms in our determinant, the formula becomes

r* - s' ~'-i -'-1^

[a +6

N.

and may be thus simplified.

We have r" = ar + 6, or multiplying by r'~*, 7^^ = 01^ + ftr*"*. So
that the expression simplifies down to

N,^K

^i+l _ s^i

r — s

This may be obtained directly, by determining C, C\ in terms of
JV,, with iVl, = 0, to make i\r, = Or* + CV.

We have, in our case, a = 6 = 1 : whence

r-« = 75,r = ^-±^ = l-618, « = --618.

72 LECTURE VIL PART I.

Molecular. If we work this out by very moderate logarithms for the case

" r — «

dropping «*', we find

13 log 1-618 - log VS = 13 X -209 - -3496 = 2-3675 = log 233,

which comes out exact; and this working with only 4-place-
logarithma

I want to call your attention to something far more important
than this. The dynamical problem, quite of itself, is very interest-
ing and important, connected as it is with the whole theory of
modes and sequences of vibration ; but the application to the
theory of light, for which we have taken this subject up, gives to
it more interest than it could have as a mere dynamical problem.
I want to justify a fundamental form into which we can put our
solution, which is of importance in connection with the application
we wish to make.

""•1/
Algebra shows that we must be able to throw -^* into the form

where q^, q^t»»*qj are determinate constants, and k^, k^.».Kj are the

values of the period t for which —^ becomes infinite. We can

put it into this form certainly, for if a\ and f be expressed in terms of
Xj, they will be functions of the (j — 1)"* and/** degrees, respectively,

in 3 . This is easily seen if we notice that aj^.j = — ^ Xj is of the
t" Cj

first degree in -^ , and that the degree of each x is raised a unit
above that of the succeeding x by the factor cLj^--^ — c^ — c^j in
the equation — c^^.i = a^,+ c^i^<+i. Therefore, writing z for — ,,

T

we have

c,f Bz'^B'z^-'-t-... '

SOLUTION IN TERMS OF BOOTS. 73

This, on being expanded into partial fractions, becomes Molecular.

Z ^ m Z ^ n Z ^ u

which takes the previously written form if we put iT/c * = j,.

We know that the roots of the equation of the t*** degree in z

which makes —^ become infinite are all real; they are the

periods of vibration of a stable system of connected bodies. We
have formal proof of it in the work which we have gone through
in connection with such a system. I am putting our solution in
this form, because it is convenient to look upon the characteristic
feature of the ratio of r to one or other of the fundamental
periods. In the first place it is obvious that if we know the roots
iCj, #e,,..., the determination of g^, g,,... is algebraic. Another form
which I shall give you is an answer to that algebraic question,
what are the values of q^, ?«•••? It is an answer in a form that is
particularly appropriate for our consideration because it introduces
the energy of the vibrations of the several fundamental modes in a
remarkable manner. We will just get that form down distinctly.

Take the differential coefficients of -*^ with respect to

1 K K

-3, and denoting J — 1> -I - 1> for brevity, by D,, D, we

find

K.'

For the case t = «,, our diflFerential coefficient becomes - , which
determines

1

Now you will remember that we had

For the moment, take the expression for the simple harmonic
motion, and you see at once that that comes out in terms of the
energy. Adopt the temporary notation of representing the

74 LECTURE VII. PART I.

Molecular, maximum value by an accented letter. Then we have at any time

of the motion x^ = x. sin — , if we reckon our time from an era of

each particle passing through its middle position, remembering
that all the particles pass the middle position at the same instant.
We have therefore for the velocity of particle No. 1,

2ir , 27r€
a?, = — a? - cos •

The energy which at any time is partly kinetic and partly

potential, will be all kinetic at the instant of passing through the

middle position. Take then the energy at that instant. For t = 0

27r
we have x^=0,x^ = — x\. Denoting the whole energy by E

T

(and remembering that the mass = j-^ J we have

Thus, the ratio of the whole energy to the energy of the first
particle / ,,\ being denoted by JTS we have

* CtT — X^

This is true for any value of t whatever. From this equation find

then the ratios of the whole energy to the energy of the first

particle when t = #Cj, #c,,.... Denoting these several ratios by 22^"*,

ic*R ic*R

ii,"*..., we find ?i = -^ — - , ?« = ~* — ^*> •••• Our solution becomes

then

m^ *• m^

This is a very convenient form, as it shows us everything in terms
of quantities whose determinations are suitable, and intrinsically
important and interesting, viz. the periods and the energy-ratios.

It remains, lastly, to show how, from our process without cal-
culating the determinants, we can get everything that is here
concerned. Our process of calculating gives us the n's in order,
beginning with w^.^. That gives us the «'s in order, and thus
we have all that is embraced in the differential coefficient with

DETERMINATION OF BOOTS. 75

respect to -^ . Everything is done, if we can find the roots. I Molecular

will show how you can find the roots from the continued fraction,
without working out the determinant at all. The calculation in
the neighbourhood of a root gives us tlie train of x's corresponding
to that root, and then by multiplying the squares of the ratios of
the of 8 to d?p by the masses, and adding, we have the corresponding
energy.

The case that will interest us most will be the successive
masses greater and greater; and the successive springs stronger
and stronger, but not in proportion to the masses so that the
periods of vibration of limited portions of the higher numbered
particles of the linear system shall be very large : so that if we
hold at rest particles 4 and 6, the natural time of vibration of
particle 5 will be longer than No. 2^8 would be if we held Nos. 1
and 3 at rest and set No. 2 to vibrate, and so on.

We will just put down once more two or three of our equations :

Without considering whether w^, is absolutely large or small, let
us suppose that it is large in comparison with c^^,;
u^ will then be of the order a^\ u^^^ of the order a^,; and so on.
We are to suppose that a^ a,, ...a, are in ascending order of

magnitude. Now, u^ u^.,,u^ = (— )' c, . . . c, — . We thus have this

^i

important proposition, that the magnitudes of the vibrations of

the successive particles decrease from particle No. 1 towards No. j;

and x^ is exceedingly small in comparison with ^, even though

there is only a moderate proportion of smallness with respect to

c c c

the ratios -*, -^, ... — . Thus see how small is the motion at a

considerable distance from the point at which the excitation is
applied, under the suppositions that we have been making.

Now, as to the calculations. I do not suppose anybody is
going to make these calculations*, but I always feel in respect to
arithmetic somewhat as Green has expressed in reference to

* Happily this negatiye prognostication was not fulfilled. See (Appendix G)
Nomerioal Solution with Illustrative Curves, by Prof. £. W. Morley, of the case of
seren connected masses, proposed in Lecture IX.

76 LECTURE VIL PART I.

Molecular, analysis. I have no satisfaction in formulas unless I feel their
arithmetical magnitude — at all events when formulas are intended
for definite dynamical or physical problems. So that if I do not
exactly calculate the formulas, I would like to know how I could
calculate them and express the order of the magnitudes concerned
in them. We are not going to make the calculations, but you will
remark that we have every facility for doing so. In the first
place, is the exceeding rapidity of convergence of the formulas.

The question is to find —^; everything, you will find, depends

upon that. The exceeding rapidity of the convergence is mani-
fest. Since u^ is large, u^ is equal to a^ with a small correction ;
similarly u^ = a, with a small correction, and so on ; so that two or
three terms of the continued fraction will be suflficient for cal-
culating the ratio denoted by u^. The continued fractions con-
verge with enormous rapidity upon the suppositions we have been
making. We thus know the value of the diflferential coefficient

du
Tj^^ . We can in this way obtain several values of u^ and begin

to find it coming near to zero. Then take the usual process.
Knowing the value of the differential coefficient allows you to
diminish very much the number of trials that you must make for
calculating a root The process of finding the roots of this con-
tinued fraction will be quite analogous to Newton's process for
finding the roots of an algebraic equation ; and I tell any of you
who may intend to work at it, that if you choose any particular
case you will find that you will get at the roots very quickly.

I should think something like an arithmetical laboratory would
be good in connection with class work, in which students might be
set at work upon problems of this kind, both for results, and in
order to obtain facility in calculation.

I hinted to you in the beginning about the kind of view that
I wanted to take of molecules connected with the luminiferous
ether, and affecting by their inertia its motions. I find since then
that Lord Rayleigh really gave in a very distinct way the first
indication of the explanation of anomalous dispersion. I will just
read a little of his paper on the Reflection and Refraction of Light
by intensely Opaque Matter (Phil. Mag., May, 1872). He com-
mences, '* It is, I believe, the common opinion, that a satis£a.ctory
mechanical theory of the reflection of light from metallic surfaces

BATLEIQH ON CAUCHY'S METALLIC REFLECTION. 77

has been given by Cauchy, and that his formubs agree very well MoleoaUr
with observation. The result, however, of a recent examination of
the subject has been to convince me that, at least in the case of
vibrations performed in the plane of incidence, his theory is
erroneous, and that the correspondence with fact claimed for it is
illusory, and rests on the assumption of inadmissible values for the
arbitrary constants. Cauchy, after his manner, never published
any investigation of his formulae, but contented himself with a
statement of the results and of the principles from which he
started. The intermediate steps, however, have been given very
concisely and with a command of analysis by Eisenlohr (Pogg.
AfiTL Vol. CIV. p. 368), who has also endeavoured to determine
the constants by a comparison with measurements made by Jamin.
I propose in the present communication to examine the theory of
reflection from thick metallic plates, and then to make some
remarks on the action on light of a tiiin metallic layer, a subject
which has been treated experimentally by Quincke.

" The peculiarity in the behaviour of metals towards light is
supposed by Cauchy to lie in their opdcity, which has the eflfect of
stopping a train of waves before they can proceed for more than a
few lengths within the medium. There can be little doubt that
in this Cauchy was perfectly right; for it has been found that
bodies which, like many of the dyes, exercise a very intense
selective absorption on light, reflect from their surfaces in ex-
cessive proportion just those rays to which they are most opaque.
Permanganate of potash is a beautiful example of this given by
Prof. Stokes. He found {Phil Mag., Vol. vl p. 293) that when
the light reflected from a crystal at the polarizing angle is ex-
amined through a Nicol held so as to extinguish the rays polarized
in the plane of incidence, the residual light is green ; and that,
when analyzed by the prism, it shows bright bands just where
the absorption-spectrum shows dark ones. This very instructive
experiment can be repeated with ease by using sunlight, and
instead of a crystal a piece of ground glass sprinkled with a little
of the powdered salt, which is then well rubbed in and buraished
with a glass stopper or otherwise. It can without difficulty be so
arranged that the two spectra are seen from the same slit one
over the other, and compared with accuracy.

"With regard to the chromatic variations it would have seemed
most natural to suppose that the opacity may vary in an arbitrary

78 LECTURE VII. PART L

Molecular, manner with the wave-length, while the optical density (on which
alone in ordinary cases the refraction depends) remains constant,
or is subject only to the same sort of variations as occur in trans-
parent media. But the aspect of the question has been materially
changed by the observations of Christiansen and Kundt (Pogg.
Ann. vols. CXLL, cxuiL, cxltv.) on anomalous dispersion in
Fuchsin and other colouring-matters, which show that on either
side of an absorption-band there is an abnormal change in the
refrangibility (as determined by prismatic deviation) of such a
kind that the refraction is increased below (that is, on the red
side of) the band and diminished above it. An analogy may be
traced here with the repulsion between two periods which frequently
occurs in vibrating systems. The effect of a pendulum suspended
from a body subject to horizontal vibration is to increase or diminish
the virtual inertia of the mass according as the natural period of
the pendulum is shorter or longer than that of its point of suspen-
sion. This may be expressed by saying that if the point of support
tends to vibrate more rapidly than the pendulum, it is made to go
faster still, and vice versd" — I cannot understand the meaning of the
next sentence at alL There is a terrible difficulty with writers in
abstruse subjects to make sentences that are intelligible. It is
impossible to find out from the words what they mean ; it is only
from knowing the thing* that you can do so — "Below the absorp-
tion-band the material vibration is naturally the higher, and hence
the effect of the associat'Cd matter is to increase (abnormally) the
virtual inertia of the aether, and therefore the refi-angibility. On
the other side the effect is the reverse." Then follows a note, "See
Sellmeier, Pogg. Ann. vol. cxuil. p. 272." Thus Lord Rayleigh
goes back to Sellmeier, and I suppose he is the originator of all
this. " It would be difficult to exaggerate the importance of these
facts fix)m the point of view of theoretical optics, but it lies
beside the object of the present paper to go further into the ques-
tion here."

There is the first clear statement that I have seen. Prof.
Rowland has been kind enough to get these papers of Lord
Rayleigh for me. I am most grateful to him and others among
you, by whom, with great trouble kindly taken for me, an

* In the next sentence, for "the refrangibility," sabstitnte its refractivity.
W. T. Feb. 9, 1892.

8ELLMEIER, HELMHOLTZ, LOMMEL. 79

immense number of books have been brought to rae, in every one MolecnUi
of which I have foimd something very important

Sellmeier, Lord Rayleigh, Heimholtz, and Lommei seem to be
about the order. Lommei does not quote Helmhoitz. I am rather
surprised at this, because Lommei comes three or four years after
Heimholtz: 1874 and 1878 are the respective dates. Lommel's
paper is published in Helmholtz's Journal {Ann, der Physik und
Chemie, 1878, voL in. p. 339), Helmholtz's paper is excellent.
Lommei goes into the subject still further, an4 has worked out
the vibrations of associated matter to explain ordinary dispersion.

I only found this forenoon that Lommei (Ann. der Ph. und
Chem, 1878, vol. iv. p. 55) also goes on to double refraction of light
in crystals — the very problem I am breaking my head against.
He is satisfied with his solution, but I do not think it at all satis-
&ctory. It is the kind of thing that I have seen for a long time,
but could not see that it was satisfactory ; and I do see reason for its
not being satisfactory. He goes on from that and obtains an
equation which would approximately give Huyghens' surface. I
have not had time to determine how far it may be correct, but I
believe it must essentially differ from Huyghens' wave-surface to
an extent comparable with that experimentally disproved by
Stokes in his experimental disproof of Rankine's theory explaining
optical aeolotropy by difference of inertia in different directions.
The exceedingly close agreement of Huyghens* surface with the
facts of the case which Stokes has found, absolutely cuts the
ground from under a large number of very tempting modes of ex-
plaining double refraction.

LECTURE VIII.

Tuesday, Oct 7, 5 p.m.

Molar. We shall take some fundamental solutions for wave motion

such as we have already had before us, only we shall consider them
as now applicable to non-condensational distortional waves, instead
of condensational waves. We can take our primary solution in the

form ^ = - sin — (r — c^), where c = a/ — if the wave is con-
densational, and a/- if the wave is distortional. But for a dis-

V p

tortional wave we must also have S = 0.

In the first place, if our value of c is a/ - , we know that ^

satisfies p -^ = ny*^. [I want very much a name for that symbol

y* (delta turned upside down). I do not know whether, Prof.
Ball, you have any name for it or not; your predecessor. Sir
William Hamilton, used it a great deal, and I think perhaps you
may know of a name for it] The conditions to be fulfilled by the
three components of displacement, ^, 17, ^, of a distortional wave
are, in the first pkce,

and we must have besides

dx dy dz

Thus ^, 17, ^ must be three functions, each fulfilling the same
equation. There is a fulfilment of this equation by functions ^ ;
and as we have one solution, we can derive other solutions from

ROTATIONAL OSCILLATION IN ORIGIN. 81

that by differentiation. Let us see then, if we can derive three Molar,
solutions from this value of <^ which shall fulfil the remaining con-
dition. It is not my purpose here to go into an analytical investi-
gation of solutions ; it is rather to show you solutions which are
of fundamental interest. Without further preface then, I will
put before you one, and another, and then I will interpret them
botL

Take for example the following, which obviously fulfils the

df dm dt ^
equation j^ + -j- + j^ = 0 :

^ ax ay dz

In each case the distance-terms only of our solution are what we
wish. Thus

dd> , 27r z 1^ dd> 2ir y

Remark that in this solution the displacement at a distance from
the source is perpendicular to the radius vector ; i.e. we have

Before going further, it will be convenient to get the rotation. It
is an exceedingly convenient way of finding the direction of vibra-
tion in distortional displacements. The rotations about the axes of
X, y, z will be :

, /df dv\ 27r»y' + ^ .

4y

1 fdfi df\ . 27r' xz .

dy

Su 1 X'U xz

These rotations are proportional to -g — , f , -3- ; that is

to say, besides an x component equal to — , we have an r com-

X

ponent equal to -j . We have a rotation around the radius vector
T.L. 6

82 LBCTUEE VIII. PART L

Molar. r, and a rotation around the axis of x, whose magnitudes are

X 1

proportional to -^ and - .

If you think out the nature of the thing, you will see that
it is this : a globe, or a small body at the origin, set to oscillating
rotationally about Ox as axis. You will have turning vibrations
everywhere ; and the light will be everywhere polarized in planes
through Ox. The vibrations will be everywhere perpendicular to
the radial plane through Ox.

omitting the constant factor — )
f = 0, i; = -pCos5r, C=^cos9.

Hence for (y = 0, z = 0,) the displacements are zero, or we have
zero vibration in the axis of x. Everywhere else the displacements
are not null and are perpendicular to Ox (since we always have
f = 0) ; and being perpendicular also to the radius vector, they are
perpendicular to the radial plane through the axis of x.

Let us consider the state of things in the plane yz. Suppose

we have a small body here at the origin or centre of disturbance,

and that it is made to turn forwards and backwards

^^ in this way (indicating a turning motion about an axis

^^J perpendicular to the plane of the paper) in any given

period. What is the result ? Waves will proceed out

in all directions from the source, and the intersections of the wave

fronts with the plane (yz) of the paper will be circles. We shall

have vibrations perpendicular to the radius vector ; of magnitude

2 , which is the same in all directions. The rotation (molecu-

A, T

lar rotation about the axis of x^ or in the plane yz^ ^^ \% — •

There is therefore zero displacement where the rotation or the
distortion is a maximum (positive or negative) : and vice versa^ at
a place of maximum displacement (positive or negative) there is
zero rotation and zero distortion. The rotation is clearly equal
to half the distortion in the plane yz by shearing (or differential
motion) perpendicular to rx, of infinitesimal planes perpendicular

BEALIZED ILLUSTRATION IN .TELLT. 83

to r. The result is polarized light consisting of vibrations in the MoUr.
plane yz and perpendicular to the radius vector, and therefore
the plane of polarization is the radial plane through OX.

Here we have a simple source of polarized light; it is the
simplest form of polarization and the simplest source that we can
have. Eveiy possible light consists of sequences of light from
simple sources. Is it probable that the shocks to which the par-
ticles are subjected in the electric light, or in fire, or in any
ordinaiy source of light, would give rise to a sequence of this kind ?
No, at all events not much ; because there cannot be much ten-
dency in these shocks or collisions, to produce rotatory oscillations
of the gross molecules. We can arbitrarily do it, for we can do
what we will with the particle. That privilege occurred to me
in Philadelphia last week, and I showed the vibrations by having
a large bowl of jelly made with a ball placed in the middle of it.
I really think you will find it interesting enough to try it for
yourselves. It allows you to see the vibrations we are speaking
o£ I wish I had it to show you just now, so that you might see
the idea realized. It saves brain very much.

I had a large glass bowl quite filled with yellowish transparent
jelly, and a red-painted wooden ball floating in the middle of it.

Try it, and you will find it a very pretty il-
lustration. Apply your hand to the ball, and
give it a turning motion round its vertical
diameter, and you have exactly the kind
of motion expressed by our equations. The
motion in any oblique direction, such as at
this point P (a;, y, z) represents that of polarized light vibrating
perpendicularly to the radial plane (or plane through the vertical
central axis). The amplitude of the vibration here (in the vertical

axis) is zero ; here at the surface (in the plane yz) it is - cos q ; and

if you use polar coordinates, calling this angle 0 (indicating on the

diagram) then the amplitude here (at P) is - cos q sin 6, giving

when tf is a right angle the previous expression.

I say that this is the simplest source and the simplest system of
polarized light that we can imagine. But it can scarcely be induced
naturally. The next simplest is a globe or small body vibrating to-

6—2

i

H^ LECTVUE niL PABT L

aod-fro in Que line, Vfe will take the soiutioD ffw that frestmlT.
fkill we bare not got up to the e«eodal oomplexm' <c>f the
vibration prodiu^ed naturaJlj bj the smfdesl Dstnnl Tibrator
with tnunovibd centre of inertia. I may take mj hand and.
instead of the torsonal oscillations which we hare be^i con-
mdering, I may gpive vertical oadllations to the globe in the
jelly (and that makes a very pretty modification of the expm-
ment), and people all call oat, ''O, there is the natural time of
the vibration, if you only leave the globe to itad^" ornlUting
np and down in the jelly. Bat the case is not pioper for an
iUwdraiim of andulatory vibrations spreading out from a centre.
We are troubled here also by reflection back, as it were, from the
containing bowl, just as in experiments on a stretched rope to
show waves running along it, we are troubled by the rope not
being infinitely long. You can always see sets of vibrations
running along the rtipe, and reflected back from the ends. But in
this experiment with the jelly in the bowl, you do not see the
waves travelling out at all because the distance to the boundary
is not large enough in comparison with the wave-length ; and what
you really see is a certain set of standing vibrations, depending on
the finiteness of the bowl. But just imagine the bowl to be
infinitely large, and that you commence making torsional oscilla-
tions ; what will take place ? A spreading outwards of this kind
of vibrations, the beginning being, as we shall see, abrupt. We
shall scarcely reach that to-day, but we shall perhaps another day
consider if the motion in the source begins and ends abruptly, the
consequent abruptness* of the beginnings and endings of the vibra-
tions throughout an elastic solid ; in every case in which the
velocity of propagation is independent of the wave-length.

When you apply your hands and force the ball to perform
those torsional vibrations, you have waves proceeding from it;
but if you then leave it to itself, there is no vibrating energy in
it at all, except the slight angular velocity that you leave it with.
A vibrator which can send out a succession of impulses independ-
ently of being forced to vibrate from without, must be a vibrator
with the means of conversion of potential into kinetic energy in
itself. A tuning-fork, and a bell, are sample vibrators for sound.
The simplest sample vibrator that we can imagine to represent the

* This ifi in faot proved by the solution of Leotare IV. expressed by (14) in terms
of an arbitrary function.

DOUBLE SOURCE WITH CONTRARY ROTATIONS. 85

origin of an independent sequence of light may be like a tuning-fork. Molar.
Two bodies, joined by a spring would be more symmetrical than a
tuning-fork. Two rigid globes joined by a spring — that will give
you the idea ; or (which will be a vibration of the same type still)
one elastic spherical body vibrating from having been drawn into
an oval shape, and let go.

I will look, immediately, at a set of vibrations produced in an
elastic solid by a sample vibrator. But suppose you produce
vibrations in your jelly elastic solid by taking hold of this ball
and moving it to-and-fro horizontally, or again moving it up and
down vertically and think of the kinds of vibrations it will make
all around. Think of that, in connection with the formulas, and
it will help us to interpret them. This is in fact the kind of
vibration produced in the ether by the rigid spherical containing
shell of our complex molecule (Lecture II.). Or think of the
vibrations due to the higher though simpler order of vibrator, of
which we have taken as an example a very dense elastic globe
vibrating from prolate to oblate and back periodically. We might
also have those toi'sional vibrations; but among all the possible
vibrations of atoms in the clang and clash of atoms that there is
in a flame, or other source of light, a not very rare case I think
would be that which I am going to speak of now. It consists of
opposite torsional vibrations at the two ends of an elongated mass.
To simplify our conception for a moment, imagine two globes con-
nected by a columnar spring; twist them in opposite directions, and
let them go. There you have an imaginable source of vibrations.
If in any one of our cases the potential energy of the spring is very
large in comparison with the energy that is carried off in a
thousand, or a hundred thousand vibrations, you will have a
nearly uniform sequence of vibrations such as those we have been
considering, but gradually dying down.

Before passing on to the to-and-fro vibrator we will think of
this motion for a moment, but we will not work it out, because it
is not so interesting. To suit our drawing we shall suppose one
globe here, and another upon the opposite side on a level with the
first, so that the line of the two is perpendicular to the board.
Give these globes opposite torsional vibrations about their common
axis, and what will the result be? A single one produces zero
light in the axis and maximum light in the equatorial plane. The
two going in opposite directions will produce zero light in the

86 LECTURE VIII. PAKT I.

Molar, equatorial plane and zero light in the axis; so that you will
proceed from zero in the equatorial plane to a miaimum between
the equatorial plane and the poles, and zero at the poles ; and you
will have opposite vibrations in each hemisphere. That constitutes
a possible case of vibrations of polarized light, proceeding from a
possible independent vibrator.

One of the most simple and natural suppositions in respect to
an independent vibrator is afforded by the illustration of a bell, or
a tuning-fork, or an elastic body defoimed from its natural shape
and left to vibrate. In all these cases, as also in the case of our
supposed complex molecule (Lecture I.), remark that the centre
of gravity of the vibrator is at rest ; except for the comparatively
very small reaction of the ether upon it ; and this is essential to an
independently acting vibrator. The vibrator must have potential
energy in itself, for many thousand vibrations generating waves
travelling outwards through ether ; and its centre of gravity must
be at rest, except in so far as the reaction of the medium upon
it causes a slight motion of the centre of gravity.

I will put down the solution which corresponds to a to-and-fro
vibration in the axis of ^, viz. :

^" A.* '^"^'da^' ^'dydx' ^ dzdx'
4> is our old friend,

but with n now in place of the n + f i' which we had formerly when
we were dealing with condensational waves. First remark that

we know that

are satisfied, because ^ and all its differential coefficients satisfy
this relation. We have therefore only to verify that the dilatation is
zero. Instead of merely going through the verification, I wish to help
you to make the solution your own by showing you how I obtained
it. I will not say that there is anything novel in it, but it is simply
the way it occurred to me. I obtained it to illustrate Stokes'
explanation of the blue sky. I afterwards found that Lord Rayleigh
had gone into the subject even more searchingly than Stokes, and
I read his work upon it.

WAVES GENERATED BY TO-AND-FRO VIBRATOR. 87

The way I found the solution was this : -^ is clearly the dis- Molar,

placement-potential for an elastic fluid, corresponding to a source
of the kind, constituted by an immersed solid moving to-and-fro
along the axis of x. The displacement function of which the
displacements are the differential coefiicients would take simply that
form if the question were of sound in air or other compressible
fluid, and not of light, or of waves in an incompressible solid, or of
waves of distortion in a compressible solid. It was a question of
condensational vibrations with us several days ago. I did not go
into the matter in detail then, but we saw that for condensational
vibrations proceeding from a vibrator vibrating to-and-fro along

the axis of x that y^ was the displacement potential ; and it is

obvious, if we start from the very root of the matter that it must be so.

Hence we may judge that the differential coefficients j- > j" » j-

of ^ , with n in place of n + f A;, must therefore be at all events

constituents of the components of displacement in the case of
light, or of distortional waves, from such a source : but neither they
nor the differential coefficients of any function can be simply
equal to the displacement-components of our present problem, in
which the motion is essentially rotational. The irrotational dis-
placements in the condensational wave problem are displacements
which fulfil certain of the conditions of our present problem : but
they do not fulfil the condition of giving us a purely distortional
wave, unless we add a term or terms in order to make the dilata-
tion zero. This is done in fact, as I found, by the addition of the

spherically symmetric term -, <^ to -v- -7- , for the a;-component of

displacement. Just try for the dilatation. We have

in which we may substitute ~ for <f>. Thus

dx^ \* ' dx'

88 LECTURE VIII. PART 1.

We verify therefore in a moment that the displacements given by
the formulas satisfy

dx dy dz '

and thus we have made up a solution which satisfies the con-
ditions of being rigorously non-condensational, (no condensation
or rarefaction anjrwhere,) and of being symmetrical round the axis
of a?.

In the first plaoe, taking the distant terms only, we have

It is easy to verify that these displacements are perpendicular to
the radius vector, i.e. that we have x^ + yrj + z^^ 0. Just look at
the case along the axis of x, and again in the plane yz. It is
written down here in mathematical words painting as clearly and
completely as any non-mathematical words can give it. Take
y = 0, 2r = 0, and that makes f = 0, i; = 0, ? = 0. Therefore, in the
direction of the axis of x there is no motion. That is a little
startling at first, but is quite obviously a necessity of the funda-
mental supposition. Cause a globe in an elastic solid to vibrate
to-and-fro. At the very surface of the globe the points in which
it is cut by Ox have the maximum motion ; and throughout the
whole circumference of the globe, the medium is pulled, by hypo-
thesis, along with the globe. But this is not a solution for that
comparatively complex, though not difficult, problem. I am only
asking you to think of this as the solution for the motion at a great
distance. It may not be a globe, but a body of any shape moved to-
and-fro. To think of a globe will be more symmetrical. In the im-
mediate neighbourhood of the vibrator there is a motion produced in
the line of vibration ; the motion of the elastic solid in that neigh-
bourhood consists in a somewhat complex, but very easily imagined
state of things, in which we have particles in the axis of x, moving
out and in directly along the radius vector; in all other places
except the plane of yz, slipping around with motions oblique to the
radius vector; and in the plane of y-^ moving exactly perpendicular

WAVES GENERATED BY TO-AND-FRO VIBRATOR. 89

to the radius vector. All, however, except motions perpendicular MoUr.

to the radius vector, become insensible at distances very great in

comparison with the wave-length. We have taken, simply, the

leading terms of the solution. These represent the motion at

great distances, quite irrespectively of the shape of the body,

and of the comparatively complicated motion in the neighbourhood

of the vibrating body.

Take now ^ = 0, and think of the motions in the plane yz.

The vibrator is supposed to be vibrating perpendicular to this

plane. We have

4^ 1
f^^-sinj, iy-0, C=0.

What does that mean ? Clearly, that the vibrations are perpen-
dicular to the plane yz. We have the wave spreading out uni-
formly in all directions in that plane, and "polarized in'' that
plane, the vibrations being perpendicular to it. That is exactly
what Stokes supposed was of necessity the dynamical theory of
the blue light of the sky. Lord Rayleigh showed that it was not
so obvious as Stokes had supposed. He elaborately investigated
the question, '* Whether is the blue light of the sky, (which we
assume to be owing to particles in the air,) due to the particles
being of density different from the surrounding luminiferous ether,
or being of rigidity different from the surrounding luminiferous
ether?" The question would really be, If the particles are water,
what is the theory of waves of light in water ; does it differ from
air in being, as it were, a denser medium with the same effective
rigidity, or is it a medium of the same density and less eft'ective
rigidity, or does it differ from ether both as to density and as to
rigidity ?

Lord Rayleigh examined that question very thoroughly, and
finds, if the fact that the cause were, for instance, little spherules
of water, and if in the passage of light through water the propaga-
tion is slower than in air were truly explained by less rigidity and
the same density we should have something quite different in the
polarization of the sky from what we would have on the other
supposition. On the other hand, the observed polarization of the
sky supports the other supposition (as far as the incertitude
of the experimental data allows us to judge) that the particles,
whether they be particles of water, or motes of dust, or whatever

90 LECTURE VIII. PART I.

Molar. they may be, act as if they were little portions of the luminiferous
ether of greater density than, and not of diflFerent rigidity from,
the surrounding ether. Hence our present solution, which has for
us such special interest as being the expression of the disturbance
produced in the ether by our imbedded spring-molecule, acquires
farther and deeper interest as being the solution for dynamical
action which according to Stokes and Rayleigh is the origin cause
of the blue light coming from the sky. I will call attention a
little more to Lord Rayleigh's dynamics of the blue sky in a
subsequent lecture. Meantime, returning to our solution, we
may differentiate once more with respect to x, in order to get
a proper form of function to express the motion from a double
vibrator vibrating to-and-fro like this — vibrating my hands to-
wards and from each other. Then we shall have a solution which
will express another important species of single sequence of vibra-
tions, of which multitudes may constitute the whole, or a large
part, of the light of any ordinary source.

A question is now forced upon us, — what is the velocity of a
group of waves in the luminiferous ether disturbed by ordinary
matter ? With a constant velocity of propagation, as in pure ether,
each group remains unchanged. But how about the propagation
of light-sequences in a transparent medium like glass ? It is a
question that is more easily put than answered. We are bound
to consider it most carefully. I do not despair of seeing the
answer. I think, if we have a little more patience with our
dynamical problem we shall see something towards the answer.

Moleonlar. Here is a perfectly parallel problem. Commence suddenly to
give a simple harmonic motion through the handle P to our
system of particles m^, mj,...tny, which play the part of a molecule.
If you commence suddenly imparting to the handle a motion of
any period whatever, only avoiding every one of the fundamental
periods, if there be a little viscosity it will settle into a state of
things in which you have perfectly regular simple harmonic vibra-
tion. But if there be no viscosity whatever, what will the result
be? It will be composed of simple harmonic motions in the
period of our applied motion at the bell-handle P ; with the ampli-
tude of each calculated from our continued fraction ; and super-
imposed upon it, a jangle as it were, consisting of coexistent simple
harmonic vibrations of all the fundamental periods. If there is no
viscosity, that state of things will go on for ever. I cannot satisfy

SUDDEN AND GRADUAL COMMENCEMENTS OF VIBRATION. 91

myself with viscous terms in these theories not only because the Molecular
assumption of viscosity, in molecular dynamics, is a theoretic
violation of the conservation of energy ; but because the smallest
degree of viscosity of ether sufficing to practically rid us of any
of this jangling, or to have any sensible influence in any of the
motions we have to do with in sources or waves of light, would not
allow a light-sequence through ether to last, as we know it lasts,
through millions of millions of millions of millions of vibrations.
But if we have no viscosity at all, whatever energy of any vibra-
tions, regular or irregular, we have at any time in our complex
molecule must show in the vibrations of something else, and
that is what ? In studying that sort of vibration with which we
have been occupied in the molecular part of our course, we must
account for these irregular vibrations somehow or other. The
viscous terms which Helmholtz and others have introduced re-
present merely an integral effect, as it were, of actions not followed
in detail, not even explained, in the theory. By viscous terms, I
mean terms that assume a resistance in simple proportion to
velocity.

But the state of things with us is that that jangling will go on
for ever, if there is no loss of energy ; and we want to coax our
system of vibrators into a state of vibration with an arbitrarily
chosen period without viscous consumption of energy. Begin
thus: commence suddenly acting on P just as we have already
supposed, but with only a very small range of motion of P. The
result will be just as I have said, only with very small ranges
of all the constituent motions. After waiting a little time increase
the range of the motion of P\ after waiting a little longer, in-
crease the range farther, and so go on, increasing the range by
successive steps. Each of those will superimpose another state of
vibration. There would be, I believe, virtually an addition of the
energies, not of the amplitudes, of the several janglings if you make
these steps quite independent of one another.

For example, suppose you proceed thus: In the first place,
start right off into vibrations of your handle P through a space,
say of 30 inches. You will have a certain amount of energy, J,
in the irregular vibrations (the "jangling"). In the second place,
commence with a range of three inches. After you have kept
P vibrating three inches through many periods, suddenly increase
its range by three inches more, making it six inches. Then, some-

92 LECTURE VIII. PART I.

Molecular, time after, suddenly increase the range to nine inches ; and so on
in that way by ten steps. The energy of the jangle produced by
suddenly commencing through the range of three inches, which
is one-tenth of 30 inches, will be exactly one-hundredth of J, the
energy of jangle which you would have if you commenced right
away with the vibration through 30 inches. Each successive
addition of three inches to the range of P will add an amount
of energy of jangling of which the most probable value is the one-
hundredth of J\ and the result is that if you advance by these
steps to the range of 30 inches, you will have in the final jangle
ten-hundredths, that is to say one-tenth, of the energy of jangle
which you would get if you began at that range right away.
Thus, by very gradually increasing the range, the result will be
that, without any viscosity at all there will be infinitely little of
the irregular vibrations.

But there are cases in which we have that tremendous jangling
of the molecules concerned in luminous vibrations; for instance,
the fluorescence of such a thing as uranium glass or sulphate
of quinine which lasts for several thousandths of a second after
the exciting light is taken away, and then again in phosphorescence
that lasts for hours and days. There have been exceedingly
interesting beginnings, in the way of experiments already made,
in these subjects, but nobody has found whether initial refraction
is exactly the same as permanent refraction. For this purpose
we might use Becquerel's phosphoroscope or we might use methods
such as those of Fizeau or Foucault, or take such an appliance as
Prof. Michelson has been recently using, for finding the velocity of
light, and so get something enormously more searching than even
BecquereFs phosphoroscope, and try whether in the first hundredth
of a second, or the first millionth of a second, there is any
indication of a diflFerent wave velocity from that which we find
from the law of refiraction. when light passes continuously through
a transparent liquid or solid. If, with the methods employed for
ascertaining the velocity of light in a transparent body (to take
account of the criticisms that they have received at the British
Association meeting, to which I have referred several times), we
combine a test for instantaneous refraction, it seems likely that
we should not get negative results, but rather find phenomena
and properties of ultimate importance. We might take not only
ordinary transparent solids and liquids, but also bodies in which.

FLUORESCENCE AND PH08PH0BESCENCE. 93

like uraDium glass, the phosphorescence lasts only a few Molecular
thousandths of a second ; and then again bodies in which phos-
phorescence lasts for minutes and hours. With some of those
we should have anomalous dispersion, gradually fading away after
a time. I cannot but think that by experimenting, in some such
way, we should find some very interesting and instructive results
in the way of initial fluorescence.

LECTURE IX.

Wednesday, Oct, 8, 5p3I.

We shall go on for the present with the subject of the
propagation of waves from a centre. Let us pass to the case of
two bodies vibrating in opposite directions, by superposition of
solutions such as that which we have already found for a single
to-and-fro vibrator, which was expressed by

^ __ 4f7T^ d d4> __ d d(l> y__ ^ ^^

We verified that

dx dy dz

so that this expresses rigorously a distortional wave. It is obvious
that this expresses the result of a to-and-fro motion through the
origin in the line OX. Remark, for one thing, that in the
neighbourhood of the origin, at such moderate distance from it
that the component motion in the direction OX is not insensible,
we have on the two sides of the origin simultaneously positive
values, f is the same for a positive value of a; as for the negative
of that value. At distances from the origin in the line OX which
are considerable in comparison with the wave-length the motion
vanishes as we have seen.

Pass on, now, to this case : a positive to-and-fro motion on the
one side of the origin, and a simultaneous negative to-and-fro
motion on the other side of the origin ; that is to say, two simul-
taneous co-periodic vibrations of portions of matter on the two
sides of the origin moving simultaneously in opposite directions.
I will indicate these motions by arrow-heads, continuous arrow-
heads to indicate directions of motion at one instant of the period.

CONTBABT VIBRATORS IN ONE LINE.

95

and dotted at the instant half a period earlier or later. The first Molar,
case already considered

Fig.l.

y

^^

X;

the second case

Fig. 2.
Y

•<^

^y^

X.

The effect in the first case being expressed by the displacements
(f V* ^> already given, the effect in the second case will be
expressed by displacement-coefficients respectively equal to

d^ drj d^
dx^ dx* dx'

This configuration of displacement clearly implies a motion of
which the component parallel to OX has opposite signs, and the
components perpendicular to OX are equal with the same sign, for
equal positive and negative values of ^ ; it is a simultaneous out
and in vibration on the two sides of the origin in the line OX, and
a simultaneous in and out vibration perpendicular to OX, every-
where in the plane yz. A motion of the matter at distances from
the origin moderate in comparison with the wave-length will be
accurately expressed by these functions. Passing now from Fig. 2
which shows the germ from which we have developed the idea of
this configuration of motion, and the functions expressing it ; look

96 LECTURE IX. PART I.

Molar. to Fig. 3 illustrating the in and out vibration perpendicular to OX

which accompanies the out and in vibration along OX from which

we started. The configuration of arrow-heads on the circle in

Fig. 3 shows the component motions perpendicular to the radius

vector at any distance, small or great, from the origin ; which

constitute sensibly the whole motion at the great distances. To

express this motion, take only the *' distance- terms '* (as in previous

— Stt*
cases,) and drop the factor -—^ , from the differential coefficients

indicated above. We thus find, for the three components of the
displacement at great distances from the origin,

i^.x^ ^^ ^cosg, i; == ;jj^ cos g, ?-^cosj.

To satisfy ourselves that the radial component of the displace-
ment is zero verify that we have a^ + yrj -{- z^ = 0.

To think of the kind of " polarization " that will be found, when
the case is realized in a sequence of waves of light, remark that the
motion is everywhere symmetrical around the axis of x, and is in
the radial plane through OX, Therefore, we have light polarized
in the plane through the radius to the point considered and
perpendicular to the plane through OX.

This is, next to the eflFect of a single to-and-fro rigid vibrator,
the simplest set of vibrations that we can consider as proceeding
from any natural source of light As I said, we might conceive of
a pair of equal and opposite torsional motions at the two ends of
a vibrating molecule. That is one of the possibilities, and it
would be rash to say that any one possible kind of motion does not
exist in so remarkably complex a thing as the motion of the
particles from which light originates.

The motion we have just now investigated is perhaps the most
interesting, as it is obviously the simplest kind of motion that can
proceed from a single independent non-rotating vibrator with
unmoved centre of inertia. If you consider the two ends of a
tuning-fork, neglecting the prongs, so that everything may be
symmetrical around the two moving bodies, you have a way by
which the motion may be produced. Or our source might be
two balls connected by a spring and pulled asunder and set to
vibrating in and out ; or it might be an elastic sphere which has
experienced a shock. An infinite number of modes of vibration
are generated when an elastic ball is struck a blow, but the gravest

SUDDEN BEGINNINGS AND GRADUAL SUBSIDENCES. 97

mode, which is also no doubt the one in which the energy is Molar.
greatest, if the impinging body be not too hard, consists of the
globe vibrating from an oblate to a prolate figure of revolution ;
and this originates in the ether the motion with which we have
just now been occupied.

The kind of thing that an elemental source of light in nature
consists of, seems to me to be a sudden initiation of a set of
vibrations and a sequence of vibrations from that initiation which
will naturally become of smaller and smaller amplitude. So that
the graphic representation of what we should see if we could see
what proceeds from one element of the source, the very simplest
conceivable element of the source, would consist of polarized waves
of light spreading out in all directions according to some such law
as we have here. In any one direction, what will it be ? Suppose
that the wave advances from left to right ; you will then see what
is here represented on a magnified scale.

I have tried to represent a sudden start, and a gradual falling
oflf of intensity. Why a sudden start ? Because I believe that
the light of the natural flame or of the arc-light, or of any other
kuown source of light, must be the result of sudden shocks upon
a number of vibrators. Take the light obtained by striking two
quartz pebbles together. You have all seen that. There is one
of the very simplest sources of light. Some sort of a chemical or
ozoniferous effect connected with it which makes a smell, there must
be. As to what the cause of this peculiar smell may be, I suppose
we are almost assured, now, that it, proceeds from the generation
of ozone. What sort of a thing can the light be that proceeds
from striking two quartz pebbles together? Under what cir-
cumstances can we conceive a group of waves of light to begin
gradually and to end gradually ? You know what takes place in
the excitation of a violin string or a tuning fork by a bow. The
vibrations gradually get up from zero to a maximum and then,
when you take the bow oflf, gradually subside. I cannot
see anything like that in the source of light. On the contrary,
it seems to me to be all shocks, sudden beginnings and gradual
subsidences; rather like the excitation of a harp string plucked
in the usual manner, or of a pianoforte string struck by the

T. L. 7

98 LECTURE IX. PART I.

Molar. hammer ; and left to itself to give away all its energy gradually
in waves of sound.

I say this, because I have just been reading very interesting
papers by Lommel and Sellmeier*, both touching upon this subject.
Helmholtz remarks that Sellmeier gets into a diflSculty in his
dynamics and does not show clearly what becomes of the energy
in a certain case ; but it seems to me that Sellmeier really takes
hold of the thing with great power. He goes into this case
very fully, and in a way with which we are all now more or
less familiar. He remarks that Fizeau obtained a suite of 50,000
vibrations interfering with one another, and judges from that that,
though ordinary light consists of polarized light, circularly-, or ellipti-
cally-, or plane-polarized as I said to you myself, one or two days ago,
with (what I did not say) the plane of polarization, or one or both
axes of the ellipse if it be elliptically polarized, gradually varying,
and the amplitude gradually changing, the changing must be
so gradual that the whole amount of the change, whether of
amplitude or of mode of polarization or of phase, in the course
of 50,000 or 100,000, or perhaps several million vibrations cannot
be so great as to prevent interference. In fact, I suppose there is
no perceptible difference between the perfectness of the annul-
ments in Fizeau's experiment, with 50,000 vibrations and with
1,000 ; although I speak here not with confidence and I may be
corrected. You have seen that with your grating, have you not.
Prof Rowland ?

Prof. Rowland. Yes; but it is very difficult to get the
interferences.

Sir Wm. Thomson. But when you do get them, the black
lines are very black, are they not ?

Prof. Rowland. I do not know. They are so very faint
that you can hardly see them.

Sir Wm. Thomson. What do you infer from that ?

Prof. Rowland. That there is a large number. The narrow-
ness of the lines of the spectrum indicates how perfectly the light
interferes ; and with a grating of very fine lines I jir\d exceedingly
'perfect interference for at least 100,000 periods I should think.

Sir Wm. Thomson. That goes further than Fizeau. Sellmeier
says that probably a great many times 50,000 waves must pass
before there can be any great change. He goes at the thing very

* Sellmeier ; Ann. der Phy, u, Chem, 1872, Vols, oxly., oxLvn.

AMOUNT OF ENERGY IN THE VIBRATION OF AN ATOM. 99

admirably for the foundation of his dynamical explanation of Molar.
absorption and anomalous refraction. The only thing that I do
not fully agree with him in his fundamentals is the gradualness of
the initiation of light at the source. I believe, in the majority of
cases at all events, in sudden beginnings and gradual endings.
Prof Rowland has just told us how gradual the endings are.
Fizeau could infer that the amplitude does not fall off greatly in
50,000 vibrations. It is quite possible from all we know, that the
amplitude may fall off considerably in 100,000 vibrations, is
it not ?

Prof. Rowland. The lines are then very sharp.

Sir Wm. Thomson. It would not depend on the sharpness of
the lines, would it ?

Prof. Rowland. 0, yes. It would draw them out of
fineness.

Sm Wm. Thomson. Would it broaden them out, or would
it leave them fine, but throw a little light over a place that should
be dark?

Prof. Rowi^and. It would broaden them out.

Sir Wm. Thomson. It is a very interesting subject; and
from the things that have been done by Prof Rowland and others,
we may hope to see, if we live, a conquering of the difficulties
quite incomparably superior to what we have now. I have no
doubt, however, but that some now present will live to see
knowledge that we can have hardly any conception of now, of the
way of the extinction of vibrations in connection with the origin
and the propagation of light. We are perfectly certain that the
diminution of amplitude in the majority of sequences in any
ordinary source, must be exceedingly small — practically nil — in
1,000 vibrations; we can say that probably it is practically nil
in 50,000 vibrations; we know that it is nearly nil in 100,000
vibrations. Is it practically nil in two or three hundred thousand
vibrations, or in several million vibrations? Possibly not. Dy-
namical considerations come into play here. We shall be able
to get a little insight into these things by forming some sort of an
idea of the total amount of energy there can possibly be in one
elemental vibrator, in a source of light, and what sequences of
waves it can supply. That the whole energy of vibration of
a single freshly excited vibrator in a source of light is many
times greater than what it parts with in the course of 100,000

7-2

100 LECTUBE IX. PART L

Molar. molar-vibratioDR, is a most interesting experimental conclusion,
drawn from Fizeau's and Rowland's grand observations of inter-
ference.

In speaking of Sellmeier's work, and Helmholtz's beautiful
paper which is really quite a mathematical gem, I must still say
that I think Helmholtz's modification is rather a retrograde step.
It is not so perhaps in the mathematical treatment; and at the
same time Helmholtz is perfectly aware of the kind of thing that
is meant by viscous consumption of energy. He knows perfectly
well that that means, conversion of energy into heat ; and in
introducing viscosity he is throwing up the sponge, as it were,
so far as the fight with the dynamical problem is concerned.

Mr Mansfield brought me another quarter hundred weight of
books on the subject last night. I have not read them all
through. I opened one of them this forenoon, and exercised
myself over a long mathematical paper. I do not think it will
help us very much in the mathematics of the subject. What we
want is to try and see if we cannot understand more fully what
Sellmeier has done, and what Lommel has done. I see that both
stick firmly to the idea that we must in the particles themselves
account for the loss of energy from the transmitted wave. That is
what I am doing; and we shall never have done with it until we
have explained every line in Prof. Rowland's splendid spectrum.
If we are tired of it, we can rest, and go at it again.

Lommell and Sellmeier do not go very fully into these
multiple modes of vibrations, although they take notice of them.
But they do indicate that we must find some way of distributing
the energy without supposing annulment of it. That is the
reason why I do not like the introducing of viscous terms in our
equations. It is very dangerous, in an ideal sense, to introduce
them at alL This little bit of viscosity in one part of the system
might run away with all our energies long before 50,000 vibrations
could be completed. If there were any sensibly efiFective viscosity
in any of the material connected with the moving particle it might
be impossible to get a sequence of one-hundred thousand or a million
vibrations proceeding from one initial vibration of one vibrator.

What the dynamical problem has to do for us is to show how
we can have a system capable of vibrations in itself and acted upon
by the luminiferous ether, that under ordinary circumstances does
not absorb the light in millions of vibrations, as for transparent

DYNAMICS OF ABSORPTION FIRST TAUGHT BY STOKES. 101

liquids or solids, or in himdieds of thousands of millions of vibra- Molar,
tions as in our terrestrial atmosphere. That is the case with
transparent bodies ; bodies that allow waves to pass through them
one-hundred feet or fifty miles, or greater distances ; transparent
bodies with exceedingly little absorption. If we take vibrators,
then, that will perform their functions in such a way as to give a
proper velocity of propagation for light in a highly transparent
body, and yet which, with a proper modification of the magnitudes
of the masses or of the connecting springs, will, in certain complex
molecules, such as the molecules of some of those compounds that
give rise to fluorescence and phosphorescence, take up a large
quantity of the energy, so that perhaps the whole suite of
vibrations from a single initiation may be absolutely absorbed
and converted into vibrations of a much lower period, which will
have, lastly, the effect of heating the body, I think we shall see
a perfectly clear explanation of absorption without introducing
viscous terms at all; and that idea we owe to SelUneier.

I would like, in connection with the idea of explaining
absorption and refraction, and lastly, anomalous refraction and
dispersion, to just point out as a matter of history, the two
names to which this is owing, — Stokes and Sellmeier. I would
be glad to be corrected with reference to either, if there is any
evidence to the contrary; but so far as I am aware, the very first
idea of accounting for absorption by vibrating particles taking up,
in their own modes of natural vibration, all the energy of those
constituents of mixed light trying to pass through, which have
the same periods as those modes, was from Stokes. He taught
it to me at a time that I can fix in one way indisputably.
I never was at Cambridge once from about June 1852 to May
1865; and it was at Cambridge walking about in the grounds
of the colleges that I learned it from Stokes. Something was
published of it from a letter of mine to Helmholtz, which he
communicated to Kirchhoff and which was appended by Kirchhoff
in his postscript to the English translation (published in Phil,
Mag., July 1860) of his paper on the subject which appeared in
Poggendorff's Annalen, Vol. cix. p. 275.

In the postscript you will find the following statement taken
from my letter : —

" Prof. Stokes mentioned to me at Cambridge some time ago,
probably about ten years, that Prof Miller had made an experiment

104 LBCTURE IX. PART U.

If oleoolar. values of z, making { = 0, which implies, and is secured by, u^ = 0.
We have

aj = -? — 3, a, = 4^ -5, a, = 16-^ — 7, a^ = 64^-9,

a, = 256^ - 11, a. = 1024^ - 13, a, = 4096-? - 15.

You will have to take values of z by trial until you get near a
root. The convergence of the continued fraction (p. 39) will be
so rapid that you will have very little trouble in getting the
largest roots. Begin then with the largest i?-root, corresponding
to the shortest of the critical periods r, and proceed downwards,
according to the indications of pp. 55 — 58. In the course of
the process, you will have the whole series of the u's for each
root ; by multiplying these in order, you have the a's for each par-
ticular root, and then you can calculate the energy ratios for each
root. We shall then be able to put our formula into numbers ; and
I feel that I understand it much better when I have an example
of it in numbers than when it is merely in a symbolic form.

I want to show you now the explanation of ordinary refraction.
Let us go back to our supposition of spherical shells, or, if you like,
our rude mechanical model. Suppose an enormous number of
spherical cavities distributed equally through the space we are con-
cerned with. Let the quantity of ether thus displaced be so
exceedingly small in proportion to the whole volume that the
elastic action of the residue will not be essentially altered by that.
These suppositions are perfectly natural. Now, what is unnatural
mechanically, is, that we suppose a massless rigid spherical lining to
this spherical cavity in the luminiferous ether connected with an
interior rigid massive shell, 7/ij, by springs — ^in the first place
symmetrical We shall try afterwards to see if we cannot do some-
thing in the way of aeolotropy ; but as I have said before I do not
see the way out of the difficulties yet. In the meantime, let us

^^"-^^g^-v. suppose this first shell m^ to

to^heriaScavity In thel / /^iC^^"^ \ ^ isotropically conuectod by

luminiferoua ether. ) / /^ur^^^^ \ • -.i.i ••iin

L I f ^kui \ini ^P^^S^ ^^^*^ *^® ngid shell

sbeu No. 1, mi xA W^^^ J J 1 lining of the spherical cavity

^^'''^<-^^^ \^^>^^y in the ether. When I say

\^^^]^^x^ isotropically connected I
mean distinctly this: that if you draw this first shell m^ aside
through a certain distance in any direction, and hold it so, the

DYNAMICAL EXPLANATION OF ORDINARY REFRACTION. 105

required force will be independent of the direction of the displace- Molecular,
ment. Certain springs in the drawing — the smallest number
would be three — placed around in proper positions will rudely
represent the proper connections for us. Similarly, let there
be another shell here, m,, in the interior of m^, isotropically
connected with it by springs ; and so on.

This is the simplest mechanical representation we can give of a
molecule or an atom, imbedded in the luminiferous ether, unless
we suppose the atom to be absolutely hard, which is out of the
question. If we pass from this problem to a problem in which we
shall have continuous elastic denser matter instead of a series of
connections of associated particles, we shall be, of course, much
nearer the reality. But the consideration of a group of particles
has great advantage, for we are more familiar with common algebra
than with the treatment of partial differential equations of the
second order with coefficients not constant, but functions of the
independent variable, — which are the equations we have to deal
with if we take a continuous elastic molecule, instead of one made
up of masses connected by springs as we have been supposing.

Let us suppose the diameters of these spherical cavities to be
exceedingly small in comparison with the wave length. Practically
speaking, we suppose our structure to be infinitely fine-grained.
That will not in the least degree prevent its doing what we want.
The distance also from one such cavity, containing within it a
series of shells, to another such cavity, in the luminiferous ether, is
to be exceedingly small in comparison with the wave length, so
that the distribution of these molecules through the ether leaves
us with a body which is homogeneous when viewed on so coarse a
scale as the wave length ; but it is, if you like, heterogeneous
when viewed with a microscope that will show us the millionth or
million-millionth of a wave length. This idea has a great advan-
tage over Cauchy^s old method, in allowing an infinitely fine-
grainedness of the structure, instead of being forced to suppose
that there are only several molecules, ten or twelve, to the wave
length, as we are obliged to do in getting the explanation of
refraction by Cauchy's method.

I wish to show you the effect of molecules of the kind now
assumed upon the velocity of light passing through the medium.

Let 7-^ denote the sum of all the masses of shells No. 1 in any

106 LECTURE IX. PART 11.

Bioleoolar. volume divided by the volume; let j-p d6^<>te the sum of the masses

of No. 2 interior shell in any volume divided by the volume ; and
so on. We will not put down the equations of motion for all
directions, but simply take the equations corresponding to a set of
plane waves in which the direction of the vibration is parallel to
OX, and the direction of the propagation is parallel to OT,

If we denote by ^j£^ the density of the vibrating medium, (I am

taking -^-^ instead of the usual p for the reason you know, viz. : to

get rid of the factor 4w^ resulting from differentiation). Let

-7-3, (instead of n as formerly,) denote the rigidity of the

luminiferous ether. The d3mamical equation of motion of the ether
and embedded cavity-linings will clearly be

For waves of period T, we have f = const. X sin 27r (r — m) •
The second differential coefficients of this with respect to t and x
will be — mi f I - -TT f respectively. Therefore our equation be-
comes ^ = 5r« + ^i ( 1 "" "fc ) • ^^ ^s fi^^ Ti ' which is the reciprocal

of the square of the velocity of propagation. You may vrrite it -^
if you like, or /li", the square of the refractive index. We have,

S-H'-'-^M)}-

Substitute our value (Lecture VII.) for — a?,/f ;
and this becomes

This is the expression for the square of the refractive index as
it is affected by the presence of molecules arranged in the way we

DYNAMICAL EXPLANATION OF ORDINARY REFRACTION. 107

have supposed. It is too late to go into this for interpretation Moleoalar.
just now, but, I will tell you that if you take T considerably less
than /tfp and very much greater than /c,, you will get a formula
with enough of disposable constants to represent the index of
refraction by an empirical formula, as it were; which, from what we
know, and what Sellmeier and Ketteler have shown, we can accept
as ample for representing the refractive index of ordinary trans-
parent substances.

We shall look into this farther, a little later, and I will point
out the applications to anomalous dispersion. We must think a
good deal of what can become of vibrations in a system of that
kind when the period of the vibration of the luminiferous ether
is approximately equal to any one of the fundamental periods that
the internal complex molecule could have were the shell lining in
the ether held absolutely at rest.

106 LECTURE IX. PART 11.

Moleoolar. volume divided by the volume; let j3 denote the sum of the masses

of No. 2 interior shell in any volume divided by the volume ; and
so on. We will not put down the equations of motion for all
directions, but simply take the equations corresponding to a set of
plane waves in which the direction of the vibration is parallel to
OX, and the direction of the propagation is parallel to OY.

If we denote by -j—^ the density of the vibrating medium, (I am

taking ^p. instead of the usual p for the reason you know, viz. : to

get rid of the factor 4w^ resulting from differentiation). Let

-7-3, (instead of n as formerly,) denote the rigidity of the

luminiferous ether. The d3mamical equation of motion of the ether
and embedded cavity-linings will clearly be

p (Pf_ I cff ^.

For waves of period T, we have f = const. X sin 27r (r — 77) •
The second differential coefficients of this with respect to t and x
will be — jsfi f , — -^ f respectively. Therefore our equation be-
comes ^ = ^Ts + ^i ( 1 "" "fc ) • ^^ ^8 find r-j , which is the reciprocal

of the square of the velocity of propagation. You may write it —^
if you like, or /it*, the square of the refiractive index. We have,

?.J{,-c,r.(i-|)}.

Substitute our value (Lecture VII.) for — xj^\

c.T*

and this becomes

c.r / M*B,

^ */^.

This is the expression for the square of the refractive index as
it is affected by the presence of molecules arranged in the way we

DYNAMICAL EXPLANATION OF ORDINABY KEFRACTION. 107

have supposed. It is too late to go into this for interpretatiou Molecmlar.
just now, but, I will tell you that if you take T considerably less
than K^y and very much greater than /c„ you will get a formula
with enough of disposable constants to represent the index of
refraction by an empirical formula, as it were; which, from what we
know, and what Sellmeier and Ketteler have shown, we can accept
as ample for representing the refractive index of ordinary trans-
parent substances.

We shall look into this farther, a little later, and I will point
out the applications to anomalous dispersion. We must think a
good deal of what can become of vibrations in a system of that
kind when the period of the vibration of the luminiferous ether
is approximately equal to any one of the fundamental periods that
the internal complex molecule could have were the shell lining in
the ether held absolutely at rest.

LECTURE X.

Thursday, October 9, 6 p.m.

We shall now think a little about the propagation of waves
with a view to the question, what is the result as regards waves
at a distance from the source, the source itself being discontinuous
in its action. In the first place, we will take our expression for a
plane wave. The factor in our formulas showing diminution of
amplitude at a distance from a source does not have effect
when we come to consider plane waves. So we just take the
simple expression for plane harmonic waves propagated along the
axis of y with velocity v ;

f = a cos "Y" (y — v<)-

Let us consider this question: — what is the work done per
period by the elastic force in any plane perpendicular to the line
of propagation of the wave. We shall think of the answer to
that question with the view to the consideration of the possibility
of a series of waves advancing through space previously quiescent.
Suppose I draw a straight line here for the line of propagation
and let this curve represent a succession of waves travelling from
left to right and penetrating into an elastic solid previously

It

quiescent Take a plane perpendicular to the line of propa-
gation of the waves, and think of the work done by the elastic
solid upon one side of this plane upon the elastic solid on the

ENERGY OF WAVES. 109

other side, in the course of a period in the vibration. We shall Molar,
take an expression for the tangential force in the plane XOZ,
and in the direction OX, which we denote by T (according
to our old notation of 8, T, U, P, Q, R). We shall virtually in-
vestigate here the formula for the propagation of the wave in-
dependently of our general formula in three dimensions. Taking
T to denote the tangential force of the elastic medium on the one
side of the plane XOZ, the downward direction of the arrow-head
which I draw being that direction in which the medium on the
left pulls the medium on the right, I put infinitely near that in
the medium on the right another arrow-head. Imagine for the
moment a split in the medium to indicate the reaction which the
medium on the right exerts on the medium on the left by this
plane ; and imagine the medium on the left taken away, and that
you act upon the plane boundary of the medium on the right,
with the same force as in the continuous propagation of waves.
The medium upon the left acts in this way upon the plane inter-
face : — that is an easy enough conception. I correctly represent
it in my diagram by an arrow-head pointing down infinitely near
to the plane on the left-hand side. The displacement of the
medium is determined by a distortion from a square figure to an
oblique figure, and there is no inconsistency in putting into this

little diagram an exaggeration of the obliquity, so as to

L-r-''' .[ show the direction of it. The force required to do that

,." is clearly as our diagram lies, upward on the right and

downward on the left.
Let us consider now the work done by that force. Calling
f the displacement of a particle from its mean position, T. ^ is

the work done by that tangential force per unit of time, -p is

the shearing strain experienced in the medium so that

dy

In this particular position which we have taken, ^ increases with y
so that the sign minus is correct according to the arrow heads.

Let there be simple harmonic waves propagated from left to
right with velocity v. This is the expression for it

indicating ^— Acob— {y — vt)\ .

110 LECTUBE X. PART I.

Molar. Hence,

f=-m8iny; ^ = -_asinj;

and the rate of doing work is

47r* , . ,
—5- avn sm* q.

That is the rate at which this plane, working on the elastic solid
on the right-hand side of it, does work ("per unit area of the
plane" understood). Multiply this by dt and integrate through
a period t = X/v. Now

I Bm^qdt=j 5 (1 — cos2y)</f = I ^dt — ^r.

The rate of doing work then, per period, is

27r* , 27r*a"n

— j- a WIT = — ^ — .

If it is possible for a set of waves to advance uniformly into
space previously undisturbed, then it is certain that the work done
per period must be equal to the energy in the medium per wave
length. Let us then work out the energy per wave length.

It is easily proved that, in waves in a homogeneous elastic
solid, the energy is half potential of elastic stress, and half kinetic
energy; and it will shorten the matter, simply to calculate the
kinetic energy and double it, taking that as the energy in the
medium per wave length. In our notation of yesterday, we took

^j as the density. Multiply this by dy, to get the mass of an

infinitesimal portion (per unit of area in the plane of the wave).
The kinetic eneigy of this mass is

oA-«^yr = o^ sin'g.dy.

2 47r«^* 2 X'

Integrating this through a wave length, and doubling it so as to

1 ^9
get the whole energy, we have ^ ~— . Compare that with the

^ At

work done per period, viz. ^-- I, it j—^ denote as yesterday the
rigidity instead of n. We see that they are equal, because (velocity

ENERGY OP WAVES. Ill

of propagation) v = a/ - as we find from the elementary equation Molar,
of the wave motion,

df'pdy''

Thus the work done per period is equal to the energy per wave
length.

This agrees with what we know from the ordinary general
solution of the equation of motion by arbitrary functions that it
is possible for a discontinuous series of waves to be propagated
into the elastic medium, previously quiescent: — and is coex-
tensive with the case of velocity of propagation independent of
wave length, for a regular simple harmonic endless succession of
waves. But if our present energy equation did not verify, it
would be impossible to have a discontinuous series of waves
propagated forward without change of form into a medium
pi-eviously quiescent I wanted to verify the energy equation for
the case of the homogeneous elastic solid, because we are con-
cerned with a case in which this is not verified ; that is to say,
when we put in our molecules. In this case, the work done per
period is less than the energy in the medium per wave length,
and therefore it is impossible for the waves to advance without
change of form.

Before we go on to that, let us stay a little longer in a
homogeneous elastic solid, and look at the well-known solution by
discontinuous functions. The equation of motion is

Although I said I would not formally prove this now, it is in
reality proved by our old equation

I took the liberty of asking Professor Ball two days ago whether
he had a name for this symbol v' > ^^^ ^^^ ^^ mentioned to me
nahla, a humorous suggestion of Maxwell's. It is the name of an
Egyptian harp, which was of that shape. I do not know that
it is a bad name for it Laplacian I do not like for several reasons
both historical and phonetic. [Jan. 22, 1892. Since 1884 I have
found nothing better, and I now call it Laplacian.]

112 LECTURE X, PART I.

I should have told you that this is the case of a plane wave
propagated in the direction of OY, with the plane of the wave
parallel to XZ\ for which case, nahla of f (that is to say v'f)

becomes simply ,^ . The time-honored solution of this equa-
tion is

where / and F are arbitrary functions. You can verify that by
diflFerentiatioa This solution in arbitrary functions proves that a
discontinuous series of waves is possible ; and knowing that a discon-
tinuous series is possible, you could tell without working it out, that
the work done per period by the medium on the one side of the
plane which you take perpendicular to the line of propagation
must be equal to the energy of the medium per wave length.

Before passing on to the energy solution for the case in which
we have attached molecules, in which this equality of energy and
work does not hold, with the result that you cannot get the
discontinuous single pulse or sequence of pulses, I want to suggest
another elementary exercise for the anticipated arithmetical labo-
ratory. It is to illustrate the propagation of waves in a medium
in which the velocity is not independent of the wave length, and
to contrast that with the propagation of waves when the velocity
is independent of the wave length in order that you may feel
for yourselves what these two or three symbols show us, but which
we need to look at from a good many points of view before we
can make it our own, and understand it thoroughly. To realize

that this equation -^ = const, x -t4 gives us constant velocities

for all wave lengths, and that constant velocities for all wave
lengths implies this equation, and to see that that goes along with
the propagation of a discontinuous pulsation without change of
figure, or a discontinuous succession of pulsations without change
of character, I want an illustration of it, by the consideration of a
case in which the condition of constancy of velocity for different
wave lengths is not fulfilled.

I ask you first to notice the formula

S = ^ -\} .-• =i + ccosflr-f e'cos 2o+ ...

I — 2e cos q-\-fr * ^ ^

ENERGY OF WAVEa

113

which is familiar to all mathematical readers as leading up to Molar.

Fourier's harmonic series of sines and cosines. It is proved by

taking

2 cos y = €^ + €~*«,

and resolving S into two partial fractions. Poisson and others*
make this series the foundation of a demonstration of Fourier's
theorem. If e < 1 the series is convergent ; when e = 1 it ceases

to converge. If we take q = — - and draw the curve whose de-
pendent coordinate is a? = /S, what have we ?

Take t = 0 and measure off lengths from the origin

y = a, 2a, ...
The curve represented will be this (heavy curve).

The heavy curve is

28

^ = ^ — o o — (a = 1, e = i).

6 — 3 cos 27ry ^ ^'

It is here drawn by the points

(y, X) = (0, 1), (i, A), (i, ^), (i. i),

and symmetrical continuation.

The dotted curve is

3/2

a? =

(a = l, e = J).

6 — 4 cos 27ry
It is here drawn by the points

(y. -c) = (0, 1), (i. -^), (i, A). (J, i),

and symmetrical continuation.

I want the arithmetical laboratory to work this out and give

* See Thomson and Tait's Natural Philoiophy, § 77.
T.L. 8

114

LECTURE X. PART h

tfolar.

Kfolar: Di-
ipression,
Deep-sea
Waves.

graphic representations of the periodic curves for several different

values of e. The particular numerical case
that I am going to suggest is one in which
the curve will be more like this second
curve which I draw ; it is much steeper
and comes down more nearly to zero. Take
the extreme case of 6 = 1, and what happens ?
8 is infinitely great for q infinitely small, and
is infinitely small for all other values of q less
than a. For any value of e, the maximum
and minimum ordinates of the curve (corre-
sponding respectively to q = 0°, and q = 180°)
are

h-

l+e

and ^.

1-e

1-e' — "l+e

and therefore the minimum is

(l-ey/il+ey

of the maximum. Thus, if for example we take e =■ '9, we find the
minimum ordinate to be 1/361 of the maximum. I suggest, as a
mathematical exercise, to draw the curve for this case by the
finite formula: and, as an arithmetical exercise, to calculate as
many as you please of the ordinates by the series. You will find
its convergence tediously slow.

[(•9)43 = -0108, (-9)** = -0097]:

you must take more than 43 or 44 terms to reach an accuracy of
one per cent, in the result. So I do not think you will be inclined
to calculate very many of the ordinates by the series.

I would also advise those who have time to read Poisson's and
Cauchy's great papers on deep-sea waves. (Poisson's M^moire sur
la th^orie des ondes. Paris, Mdm. Acad, Sci. i., 1816, pp. 71 — 18G ;
Annal. de Chimie, v., 1817, pp. 122 — 142. Cauchy, M^moiro sur la
th^orie de la propagation des ondes k la surface d'un fluide pesant
d*une profondeur inddfinie [1815], Paris, M^m, Sav, Strang, i.,
1827, pp. 3 — 312.) Those papers are exceedingly fine pieces of
true mathematics; and they are very strong. But you might
have the hydrodynamical beginnings presented much more fasci-
natingly. If you know the elementary theory of deep-sea waves,
well and good: then take Poisson and Cauchy for the higher
analytical treatment. Those who do not know the theory of

ENERGY OF WAVES. 116

deep-sea waves may read it up in elementary books. The best Molar: Di-
text-books I know for Hydrokinetics are Besant's and Lamb's. Deep-sea

The great struggle of 1815, not that fought out on the plains ^*^®*'
of Belgium, was, who was to rule the waves, Cauchy or Poisson.
Their memoirs seem to me of very nearly equal merit. I have no
doubt the judges had good reason for giving the award to Cauchy,
but Poisson's paper also is splendid. I can see that the two writers
respected each other very much, and I suppose each thought the
other's work as good as his own (? and sometimes better !).

The problem which they solve is this, in their high analytical
style : Every portion of an infinite area of water is started initially
with an arbitrarily stated infinitesimal displacement from the
level and an arbitrarily stated velocity up or down from the level,
and the inquiry is, what will be the result ? It is obvious that
you have the solution of that problem firom the more elementary
problem, what is the result of an infinitesimal displacement at
a single point, such as may be produced by throwing a stone
into water ? Let a solid, say, cause a depression in any place, the
velocity of the solid performing the part of giving velocity and
displacement to the surface of the water : then consider the solid
suddenly annulled. The same thing in two dimensions is exceed-
ingly simple. Take, for example, waves in an infinitely deep
canal with vertical sides. Take a sudden disturbance in the canal,
equal over all its breadth, and inquire what will the result be ?

I wish now to help you toward an understanding of Cauchy's
and Poisson's solutions. They only give symbols and occasionally
numerical results : they do not give any diagrams or graphical repre-
sentations ; and I think it would repay any one who is inclined to go
into the subject to work it out with graphic representations thus : —
first I must tell you that the elementary Hydrokinetic solution for
deep-sea waves is simply a set of waves, or a set of standing vibra-
tions (take which you please): the propagational velocity of the waves

being >y/f-> and therefore for different waves directly propor-
tional to the square root of the wave-length ; and the vibrational

/27r\
period of waves or of standing vibrations being ^ , also

directly proportional to the square root of the wave-length.
Thus, so far as concerns only the varying shape of the disturbed
water surface, the whole result of the elementary Hydrokinetics of

8—2

116 LECTURE X. PART I.

Molar: Di- deep-sea wave-motion is expressed by one or other of these

gression, *" i • i. t -4. j

Deep-sea equations which I wnte down ; —

, Stt / ^ ^ /g\\ , sin Sttv sin . /27ror

\ V V iirj cos X cos V \

Superposition of two motions represented by either formula
gives a specimen of single motion represented by the other, as you
all know well by your elementary trigonometry.

And now, in respect to Poisson's and Cauchy's great mathe-
matical work on deep-sea waves, it will be satisfactory to you
to know that it consists merely in the additions of samples
represented by whichever of these formulas you please to take.
The simple formula, or summation of it with different values of h
and \, represents waves with straight ridges, or generally straight
lines of equal displacement : or, as we may call it, two-dimensional
wave-motion. Every possible case of three-dimensional wave-
motion (including the circular waves produced by throwing a
stone into water) is represented by summation of the samples of
the formula, as it stands, and with z substituted for y ; y and z
representing Cartesian coordinates in the horizontal plane of the
undisturbed water-surface, and x representing elevation of the
disturbed water-surface above this plane.

And now, confining ourselves to the two-dimensional wave-
motion, I suggest to our arithmetical laboratory, to calculate and
draw the curve represented by

« = i + e cos ji -h 6' cos 2^2 + e* cos 3y, + . . . e* cos iq^i -h . . .

where o* = — (y — Vit) ; with v^ = -y. .

^ a ^ ^ Ji

Calculate the curve corresponding to any values you please of
e and of t Also calculate and draw curves representing the sum
of values of « with equal positive and negative values of Vi,

I suggest particularly that you should perform this last
described calculation for the cases, c = J and e = ^. You have
already the curves for ^ = 0 in these cases shown on the diagram
of page 113, and you will find it interesting to work out, in con-
siderable detail, curves for other values of t which you can do
without inordinately great labour, as the series are very rapidly
convergent. You will thus have graphic representations of the
two-dimensional case of Cauchy and Poisson's problem for infinitely
deep water between two fixed pamllel vertical planes.

ENERGY OF WAVES. 117

Lastly, I may say that if there are some among you who Molar: Di-
will not shrink from the labour of calculating and adding forty or ^^!^
fifty terms of the series, I advise you to do the same for c = '9 ; Waves,
and you will have splendid graphical illustrations of the two-
dimensional problem of deep-sea waves initiated by a single
disturbance along an endless straight line of water. If you do so,
or if you spend a quarter-of-an-hour in planning to begin doing so,
you will learn to thank Cauchy and Poisson for their magnificent
mathematical treatment of their problem by definite integrals,
and for their results from which, with very moderate labour, you
may calculate the answer to any particular question that may be
reasonably put with reference to the subject ; and may work out
very thorough graphical illustrations of all varieties of the problem
of deep-sea waves*.

We are going to take our molecules again, and put them in the Moleealar.
ether; and look at the question, what is the velocity of propagation
of waves through it under some suppositions which we shall make
as to the masses of these embedded molecules, and how much they
will modify the velocity of propagation from what it is in pure
ether. Then we shall look at the matter, with respect to the
question of the work done upon a plane perpendicular to the line
of propagation, and we shall see that the energy per wave-length
is greater than the work done per period, and that therefore it is
impossible under these conditions for waves to advance uniformly
into space previously occupied by quiescent matter.

You will find, in Lord Rayleigh's book on sound, the question
of the work done per period, and the energy per wave length, gone
into: and the application of this principle, with respect to the
possibility of independent suites of waves travelling without change
of form, thoroughly explained.

To-morrow we shall consider investigations respecting the
difference of velocity of propagation in different directions in an
aeolotropic elastic solid, for the foundation of the explanation of
double refraction on mere elastic solid idea. The thing is quite

• [Note of May, 1898. For »ome of these see my papers: — *'0n Stationary
Waves in Flowing Water," Phil. Mag. 1886, Vol. 22. pp. 353, 446, 617; 1887,
Vol. 23, p. 62. **0n the Front and Bear of a Free Procession of Waves in Deep
Water," Phil. Mag, 1887, Vol. 23, p. 113; and, "On the Waves produced by a
Single Impulse in Water of any depth or in a Dispersive Medium," Phil, Mag,
1887, Vol. 23, p. 262.]

118 LECTURE X. PART II.

Molecular, familiar to many of you, no doubt, and you also know that it is a
failure in regard to the explanation of the propagation of light in
biaxal crystals. It is, however, an important piece of physical
dynamics, and I shall touch upon it a little, and try to show it in
as clear a light as I can.

Ten minutes interval.

Now for our proper molecular question. The distance from
cavity to cavity in the ether is to be exceedingly small, in
comparison with the wave-length, and the diameter of each
cavity is to be exceedingly small, in comparison with the distance
from cavity to cavity. Let the lining of the cavity be an
ideal rigid massless shell. Let the next shell within be a rigid

fn,,

shell of mass ^— ^. I represent the thing in this diagram as

if we had just two of these massive shells
and a solid nucleus. The enormous mass of
the matter of the grosser kind which exists
in the luminiferous ether when permeated
by even such a comparatively non-dense
body as air, would bring us at once to very
great numbers in respect to the masses which
we will suppose inside this cavity, in comparison with the masses
of comparable bulks of the luminiferous ether. If there is time
to-morrow, we shall look a little to the possible suppositions as to
the density of the luminiferous ether, and what limits of greatness
or smallness are conceivable in respect to it. At present we have
enough to go upon to let us see that, even in air of ordinary
density, the mass of air per cubic centimetre must be enormously
great, in comparison with the mass of the luminiferous ether per
cubic centimetre. We must therefore have something enormously
massive in the interior of these cavities. We shall think a good
deal of this yet, and try to find how it is we can have the large
quantity of energy that is necessary to account for the heating of
a body such as water by the passage of light through it, or for
the phosphorescence of a body which is luminous for several days
after it has been excited by light. I do not think we shall have
the slightest difficulty in explaining these things. These are not

DIFFICULTIES IN WAVE THEORY OF LIGHT. 119

the difficulties. The difficulties of the wave theory of light are Moleculi
difficulties which do not strike the popular imagination at all.
They are the difficulties of accounting for polarization by reflection
with the right amount of light reflected ; and of accounting for
double refraction with the form of wave-surface guessed by
Huyghens and proved experimentally by Stokes. With the
general character of the phenomena we have no difficulty what-
ever ; the great difficulty, in respect to the wave theory of light,
is to bring out the proper quantities in the dynamical calculation
of these effects.

There is no difficulty in explaining the energy required for
heating a body by radiant heat passing through it, nor how it is
that it sometimes comes out as visible light and, it may be, so
slowly that it may continue appearing as light for two or three
days. All these properties, wonderful as they are, seem to come
as a matter of course from the dynamical consideration. So much
so that any one not knowing these phenomena would have dis-
covered them on working out the subject dynamically. He would
discover anomalous dispersion, fluorescence, phosphorescence, and
the well-known visible and invisible radiant heat of longer periods
emitted by a body which has been heated and left to cool. All
these phenomena might have been discovered by dynamics ; and
a dynamical theory that discovers what is afterwards verified by
experiment is a very estimable piece of physical dynamics.

I speak with confidence in this subject because I am ashamed
to say that I never heard of anomalous dispersion until after I
found it lurking in the formulas. And, when I looked into the
matter, I found to my shame that a thing which had been known
by others for fifteen or twenty years* I had not known until I
found it in the dynamics.

Take our concluding formula of yesterday (p. 106 above), with
some changes of notation

^-n^ ,r 1^ + m, (^T* _ «^ + T« - *; + '^■)\ '
where f denotes the propagational velocity of waves of period t ;

* Leroux, *' Dispersion anomale de la vapeur d'iode," Comptes Rendus, lv., 1862,
pp. 126—128: Pogg, Ann. cxvii., 1862, pp. 659, 660. Christiansen, "Ueber die
Brechungsverhaltnisse einer weingeistigen Losung des Fuchsins," Ann. Phys. Chem.
CXLI, 1870, pp. 479, 480: Phil. Mag., xli., 1871, p. 244; Annates dc Chimie, xxv.,
1872, pp. 213, 214.

120 LECTURE X. PART II.

Bfoleoular. n, p, and mi denote now respectively the rigidity of the ether,

the mass of the ether in unit volume of space, and the sum
of the masses of the first interior shells of the embedded
molecules in unit volume of space ;

Ci the force of the first spring, per unit elongation, multiplied by
the number of molecules embedded in the ether per imit
volume of space ;

/e, /c^, /c^,, &c., in order of ascending magnitude, the fundamental
periods of the molecule when the outer shell is held fixed ;

R, R,, R,„ &c., denote for the separate fundamental vibrations the
ratio of the energy of the first interior shell to the whole
energy of the complex vibrator.

Let us consider what the wave-period t may be relatively to
the fundamental periods k, tc^, /e^^, ... of the vibrator on the sup-
position of the bounding shell held fixed, to give us a good
reasonable explanation of dispersion, in accordance with the facts
of observation with respect to the diflference of velocity for
different periods. To help us with this consideration, take our
previous auxiliary formulas

f 471^ \,T«- /t«^ T«- Ky J '

where | and Xi denote respectively the simultaneous maximum
displacements of the outermost massless shell and the first of the
massive shells within it.

If T were less than the smallest of the fundamental periods,

■^ would be negative, the wave- velocity would be greater than in

firee ether, and the refi:tictive index would be less than unity.
But in all known cases the refiractive index is greater than unity ;

and when this is so, -7 — I must be positive. I want to see if

we can get our formula to cover a range, including all light from
the highest ultra-violet photographic light of about half the
wave-length of sodium light down to the lowest we know of,
which is the radiant heat from a Leslie cube with a wave-length

REFRACnVITY THROUGH WIDE RANGE. 121

that I hear fix)m Prof. Langley since I spoke to you on the MoleonU
subject a week ago is about jjfuu ^^ ^ centimetre or 17 times the
wave-length of sodium light. That will be a range of about
forty to one. The highest invisible ultra-violet light hitherto
determined, by its photographic action, has a period about 1/40
of the period of the lowest invisible radiation of radiant heat thsCt
has yet been experimented upon.

It is probable that all or many colourless transparent liquids
and solids are mediums for which throughout every part of that
range there are no anomalous dispersions. I think it is almost
certain that for rock-salt, in the lower part of the range, there
are no anomalous dispersions at all In fact Langley's experiments
on radiant heat are made with rock-salt ; and in all experiments
made with rock-salt, it seems as if little or no radiant heat is
absorbed by it. At all events, we could not be satisfied unless we
can show that this kind of supposition will account for dispersion
through a range of period from one to forty. It is obvious that
if we are to have continuous refraction without anomalous dis-
persion through that wide range of periods, there cannot be any
of the periods tc, tc,, k,^, ... within it.

To-morrow we shall consider the case in which the wave-period
is longer than the longest of the molecular periods ; and we shall
find that on this supposition our formula serves well to represent
all we have hitherto known by experiment regarding ordinary
dispersion. [Added July 7, 1898. This was true in October 1884 ;
but measurements by Langley of the refractivity of rock-salt for
radiant heat of wave-lengths (in air or ether) from 43 of a mikron
to 6*3 mikrons, (the " mikron " being 10*^ of a metre or 10"* of a
centimetre), published in 1886 {Phil, Mag. 1886, 2nd half-year),
showed that there must be a molecular period longer than that
corresponding to wave-length 5*8. In an addition to Lecture
XII, Part II, we shall see that subsequent measurements of
refractivity by Rubens, Paschen, and others, extend the range
of ordinary dispersion by rock-salt to wave-lengths of 23 mikrons ;
and give results in splendid agreement with our formula, which
is identical with Sellmeier's expression of his own original theory,
through a range of from '4 of a mikron to 23 mikrons, and
indicate 56 mikrons as being the probable wave-length of radiant
heat in ether of which the period is a critical period for rock-salt.]

LECTURE XI.

Friday, October 10, 3.30 p.m.

ifolar. We shall now take up the subject of an elastic solid which is

not isotropic. As I said yesterday, we do not find the mere
consideration of elastic solid satisfactory or successful for explain-
ing the properties of crystals with reference to light. It is, how-
ever, to my mind quite essential that we should understand all
that is to be known about homogeneous elastic solids and waves
in them, in order that we may contrast waves of light in a crystal
with waves in a homogeneous elastic solid.

Aeolotropy is in analogy with Cauchy's word isotropy which
means equal properties in all directions. The formation of a word
to represent that which is not isotropic was a question of some
interest to those who had to speak of these subjecta I see the
Germans have adopted the term anisotropy. If we used this in
English we should have to say : '* An anisotropic solid is not an
isotropic solid " ; and this jangle between the prefix an (privative)
and the article an, if nothing else, would prevent us from adopting
that method of distinguishing a non-isotropic solid from an
isotropic solid. I consulted my Glasgow University colleague
Prof. Lushington and we had a good deal of talk over the subject.
He gave me several charming Greek illustrations and wound up
with the word aeolotropy. He pointed out that al6\o<; means
variegated; and that the Greeks used the same word for variegated
in respect to shape, colour and motion ; example of this last, our
old friend '' KopvdaLoKos '^EKTtopJ' There is no doubt of the
classical propriety of the word and it has turned out very con-
venient in science. That which is different in different directions,
or is variegated according to direction, is called aeolotropic.

The consequences of aeolotropy upon the motion of waves, or
the equilibrium of particles, in an elastic solid is an exceedingly

AEOLOTROPY IN AN ELASTIC SOLID. 123

interesting fundamental subject in physical science ; so that there Molar,
is no need for apology in bringing our thoughts to it here except,
perhaps, that it is too well known. On that account I shall be
very brief and merely call attention to two or three fundamental
points. I am going to take up presently, as a branch of molar
dynamics, the actual propagation of a wave ; and in the mathe-
matical investigation, I intend to give you nothing but what is
true of the propagation of a plane wave in an elastic solid, not
limited to any particular condition of aeolotropy ; in an elastic solid,
that is to say, which has aeolotropy of the most general kind.

Before doing that, which is strictly a problem of continuous or
molar dynamics, I want to touch upon the somewhat cloud-land
molecular beginning of the subject, and refer you back to the old'
papers of Navier and Poisson, in which the laws of equilibrium or
motion of an elastic solid were worked out from the consideration
of points mutually influencing one another with forces which are
functions of the distance. There can be no doubt of the mathe-
matical validity of investigations of that kind and of their interest
in connection with molecular views of matter ; but we have long
passed away from the stage in which Father Boscovich is accepted
as being the originator of a correct representation of the ultimate
nature of matter and force. Still, there is a never-ending interest
in the definite mathematical problem of the equilibrium and motion
of a set of points endowed with inertia and mutually acting upon
one another with any given forces. We cannot but be conscious
of the one splendid application of that problem to what used to be
called physical astronomy but which is now more properly called
dynamical astronomy, or the motions of the heavenly bodies. But
it is not of these grand motions of mutually attracting particles
that we must dow think. It is equilibriums and infinitesimal
motions which form the subject of the special molecular dynamics
now before us.

Many writers [Navier (1827), Poisson (1828), Cauchy, F. Neu-
mann, Saint-Venant, and others] who have worked upon this
subject have come upon a certain definite relation or set of
relations between moduluses of elasticity which seemed to them
essential to the hypothesis that matter consists of particles acting
upon one another with mutual forces, and that the elasticity of a
solid is the manifestation of the forcive required to hold the
particles displaced infiuitesimally from the position in which the

124 LECl'URE XI.

oUr. mutual forces will balance. This, which is sometimes called
Navier 8 relation, sometimes Poisson s relation, and in connection
with which we have the well-known "Poisson's ratio," I want to
show you is not an essential of the hypothesis in question. Their
supposed result for the case of an isotropic body is interesting,
though now thoroughly disproved theoretically and experimentally.
Doubtless most of you know it ; it is in Thomson and Tait, and
I suppose in every elementary book upon the subject. I will just
repeat it.

An isotropic solid, according to Navier s or Poisson's theory,
would fulfil the following condition : if a column of it were pulled
lengthwise, the lateral dimensions would be shortened by a
quarter of the proportion that is added to the length; and the
proportionate reduction of the cross-sectional area would therefore
be half the proportion of the elongation. Stokes called attention
to the viciousness of this conclusion as a practical matter in
respect to the realities of elastic solids. He pointed out that jelly
and india-rubber and the like, instead of exhibiting lateral
shrinkage only to the extent of one quarter of the elongation, give
really enough of shrinkage to cause no reduction in volume at all.
That is to say, india-rubber and such bodies vary the area of the
cross-section in inverse proportion to the elongation so that the
product of the length into the area of the cross-section remains
constant. Thus the proportionate linear contraction across the
line of pull is half the elongation instead of only quarter as
according to Navier and Poisson.

Stokes* also referred to a promise that I made, I think it
was in the year 1856, to the efiect that out of matter fulfilling
Poisson's condition a model may be made of an elastic solid,
which, when the scale of parts is sufficiently reduced, will be a
homogeneous elastic solid not fulfilling Poisson's condition. That
promise of mine which was made 30 years ago, I propose this
moment to fulfil, never having done so before.

Let this box help us to think of 8 atoms placed at its 8
corners, with the box annulled. The kind of elastic model I am
going to suppose is this : particles or atoms arranged equidistantly
in equidistant parallel rows and connected by springs in a certain

* [Report to Brit, Association, Cambridge, 1862. *'0d Doable Kefraction/'
p. 262, at bottom.]

green's theory realised in mechanical model. 125

definite way. I am going to show you that we can connect MoUr.
neighbouring particles of a Boscovich elastic solid with a special
appliance of cord, and a sufficient number of springs, to fulfil the
condition of giving 18 independent moduluses ; then by trans-
forming the coordinates to an orientation in the solid taken at
random, we get the celebrated 21 coefficients, or moduluses, of
Green's theory. I suppose you all know that Green took a short-
cut to the truth ; he did not go into the physics of the thing at
all, but simply took the general quadratic expression for energy in
terms of the 6 strain components, with its 21 independent coeffi-
cients, as the most general supposition that can be made with
regard to an elastic solid.

To make a model of a solid having the 21 independent coeffi-
cients of Green's theory, think of how many disposable springs we
have with which to connect 8 particles at the comers of a parallel-
epiped. Let them be connected by springs first along the 12
edges of the parallelepiped. That clearly ydll not be sufficient to
give any rigidity of figure whatever, so far as distortions in the
principal planes are concerned. These 12 springs connecting in
this way the 8 particles would give resistances to elongations
in the directions of the edges; but no resistance whatever to
obliquity ; you could change the configuration from rectangular,
if given so as in the box before you, into an oblique parallelepiped,
and alter the obliquity indefinitely, bringing if you please all
the 8 atoms into one plane or into one line, without calling any
resisting forces into play. What then must we have, in order
to give resistance against obliquity? We can connect particles
diagonally. We have in the first place, the two diagonals in each
face although we shall see that the two will virtually count as but
one ; and then we have the four body diagonals.

Now let me see how many disposables we have got. Remark
that each edge is common to four parallelepipeds of the Boscovich
assemblage. Hence we have only a quarter of the number of
the twelve edge springs independently available. Thus we have
virtually three disposables from the edge springs. Each face is
common to two parallelepipeds ; therefore from the two diagonals
in each face we have only one disposable, making in the six faces,
six disposables. We have the four body diagonals not common to
any other parallelepipeds and therefore four disposables fi om them.
We have thus now 13 disposables in the stiffnesses of 13 springs.

126

LECTURE XI.

Moltf. And we have two more dispoeables in the ratios of edges of
the parallelepiped. Lastly we have three angles of three of the
oblique parallelograms constituting the £EM;es of our oblique parallel-
epiped. Thus we have in all 18 disposables. But these 18
disposables cannot give us 18 independent moduluses because it is
obvious that they cannot give us infinite resistance to compression
with finite isotropic rigidity, a case which is essentially included
in 18 independent moduluses. Hence I must now find some
other disposable or disposables that will enable me to give any
compressibility I please in the case of an isotropic solid, and to
give Green's 21 independent coeflScients, for an aeolotropic solid.
For this purpose we must add something to our mechanism that
can make the assemblage incompressible or give it any compres-
sibility we please ; so that, for example, we can make it represent
either cork or india-rubber, the extremes in respect to elasticity of
known natural solids.

I must confess that since 1856 when I promised this result I
have never seen any simple definite way of realising it until a few
months ago when in making preparations for these lectures I
found I could do it by running a cord twice round the edges of
our parallelepiped of atoms as you see me now doing on the model
before you. It is easier to do this thus with au actual cord and
with rings fixed at the eight comers of a cubic box, than to
imagine it done. There is a vast number of ways of doing it ;
I cannot tell you how many, I wish I could. It is a not unin-
teresting labyrinthine puzzle to find them all and to systematise
the finding. You see now we are finding one way.

Here it is expressed in terms of the coordinates of the comers
as we have taken them in succession.

(000)(001)(011)(010)(000)(001)(011)(010)(000)(100)(110)(010)
(110)(111)(011)(111)(101)(001)(101)(111)(110)(100)(101)(100)

[April 14, 1898. For the accompanying very clear diagram

(fig, 1) representing another of the vast number of ways of laying

a cord round the edges of a parallelepiped, I am indebted to

IC Brillouin who has added abstracts of some of the present

to a translation by M. Lugol of Vol. i. of my Popular

and Addresses. M. Brillouin describes his diagram as

" J'ai modifi^ Tordre indique par Thomson en permutant

iilto 8^ sommet, pour que la corde suive ehaque c6t^ en sens

grben's theory realised in mechanical model. 127

oppos^ k ses deux trajets. Dans I'ordre primitif, la face (000) Molar.
(001) (Oil) (010) ^tait parcounie deux fois dans le mfime sens.**]

You see I have drawn the cord just three times through each
corner ring and I now tighten it over its whole length and tie the
ends together. Remark now that if the cord is inextensible it
secures that the sum of the lengths of the 12 edges of the parallel-
epiped remains constant, whatever change be given to the relative
positions of the 8 comers. This condition would be fulfilled for
any change whatever of the configuration ; but it is understood
that it always remains a parallelepiped, because our application of
the arrangement is to a homogeneous strain of an elastic solid
according to Boscovich. A cord must similarly be carried twice
round the 4 edges of every one of the contiguous parallelepipeds ;
and eight of these have a common comer, at which we suppose
placed a single ring. Hence every ring is traversed three times

(0117

(010)

Fig. L

by each one of eight endless cords. Each edge is common to four
contiguous parallelepipeds and therefore it has two portions of
each of four endless cords passing along it.

Suppose now for example the parallelepiped to be a cube.
Inextensible cords applied in the manner described between
neighbouring atoms, keep constant the sum of the lengths of the
12 edges of each cube, and therefore secure that its volume is
constant for every infinitesimal displacement. Consider for a
moment the assemblage of ring atoms thus connected by endless
(jords. In itself and without further application to molecular
theory we see a very interesting structure which, provided the
cords are kept stretched, occupies a constant volume of space and
yet is perfectly without rigidity for any kind of distortion. You

128

LECTURE XI.

see if I eloDgate it in one of the three directions of the pcurallel
edges of the cubes, it necessarily shrinks in the perpendicular
directions so as to keep constant volume. This kind of deforma-
tion gives us two of the five components of distortion without
change of bulk ; and the other three are given by the shearings
which we have called a, 6, c ; of which, for instance, a is a distor-
tion such that the two square parallel faces of the cube perpen-
dicular to OX become rhombuses while the other four remain
square.

[April 14, 1898. The cube (fig. 1) with an endless cord twice
along each edge is, at least mechanically, somewhat interesting in
realising an isolated mechanism for securing constancy of volume
of a hexahedron, without other restriction of complete liberty of
its eight comers than constancy of volume requires: provided only
that the hexahedron is infinitely nearly cubic, and that each face
is the (plane or curved) surface of minimum area bounded by four
straight lines. Two years ago in preparing this lecture for the
press from Mr Hathaway 's papyrograph, a much simpler mechan-
ism of cords for securing constancy of volume, occupied by an
assemblage of a vast number of points given in cubic order, than
is provided by the linking together of cubes separately fulfilling
this condition, occurred to me. It was not till some time later

Fig. 2.

that I found myself anticipated by M. Brillouin. I am much
pleased to find that he has interested himself in the subject and
has introduced a new idea, by which the condition of constant
volume is realised in the exceedingly simple and beautiful mechan-
of endless cords represented in the annexed diagram (fig. 2),
fix>m one of his articles in the volume already referred to.
diagram represents endless cords of which just one is shown

green's theory realised in mechanical model. 129

complete. These cords pass through rings not shown in fig. 2. Molar.
Fig. 3 shows one of the rings viewed in the direction of a diagonal
of the cube and seen to be traversed twelve times by four endless
cords of which one is shown complete. In an assemblage of a vast
number of rings thus connected by endless cords and arranged in
cubic, or approximately cubic, order, one set of three conterminous
edges of each cube is kept constant, and therefore for the exactly

Fig. 3.

cubic order the volume is a maximum. This method, inasmuch
as it implies only two lines of cord along each edge common to
four cubes, is vastly simpler than the original method described
above as in my lecture at Baltimore, which required eight lines of
cord along each common edge. The kinematic result with its
dynamic consequences, is the same in the two methods.

To avoid the assumption of an inextensible elastic cord, place
at each comer, common to eight cubes, six bell-cranks properly
pivoted to produce the eflfect of cords running, as it were, round
pulleys, which we first realised by the six cords running through
the ring of fig. 3. And instead of each straight portion of cord,
substitute an inextensible bar of rigid matter, hooked at its two
ends to the arm-ends of the proper bell-cranks ; just as are the
two ends of each copper bell-wire, so well known in the nineteenth
and preceding centuries, but perhaps to be forgotten early in the

t.l. 9

130 LBCTUBE XI.

Kcdtf. twentieth century, when children grow np who have never seen
bell-cranks and bell-wires and only know the electric belL Bemark
now that onr connecting-rods, being rigid, can transmit posh as
well as poll ; instead of merely the pull of the flexible cofd
with which we commenced. Our model now may be constructed
wholly of matter fulfilling the Poisson-Navier condition, and it
gives us a molecular structure fur matter violating that c(xxiiti<XL
Suppose now we ideally introduce repulsions in the lines <^
the body diagonals between the four pairs of comers of each cube.
This will keep stretched, all the cords between rings, or con-
necting-rods between arm-ends of bell-cranks; and will give a
cubically isotropic elastic solid with a certain definite aeolotropic
quality. If besides we introduce mutual forces in the edges of
the cubes between each atom and its nearest neighbour, we can
give complete isotropy, or any prescribed aeolotropy ccHisistent
with cubic isotropy. All this is for an incompressible solid. But
lastly, by substituting india-rubber elastics for the cords, or ideal
attractions or repulsions (Boscovichian) instead of the connecting-
rods between bell-crank-arm-ends, we allow for any degree of
compressibility, and produce if we please a completely isotropic
elastic srJid with any prescribed values for the moduluses of
rigidity and resistance to compression, fulfilling or not fulfilling
Vffwi¥^\ r%tUf. It is mechanically interesting to work out details
U^ thi* pf^^Wern for the case suggested by cork, that is, a perfectly
ivytf'yjn^j ^U«tic wJid, having rigidity much greater in proportion
U» f^\nfAi!nfj>i Up r/>mpression than in the case of Poisson's ratio,
ari/l jri*t Anffident to produce constancy of cross-section in a
f/Aurnn c/fmprf^m^], or elongated, by forces applied to its enda
It is als/; int^e^.ing to go further and produce a solid of which a
column *hall shrink transversely when compressed merely by
longitudinal frirce.

In my lecture at Baltimore I indicated, without going into
details, how by taking a parallelepiped of unequal edges instead
of a cube and introducing different degrees of elasticity in the
portions of the cords lying along the different edges of each
parallelepiped, and by introducing also forces of attraction or
repulsion among neighbouring atoms, we can produce a model
elastic solid with the twenty-one independent moduluses of
Green's theory.

But the method by cords and pulleys or bell-cranks which we

green's theory realised in mechanical model. 131

have been considering, though highly interesting in mechanics, Molar,
and dignified by its relationship to Lagrange's original method of
proving his theorem of " virtual velocities " (law of work done) in
mathematical dynamics, lost much of its interest for molecular
physics when I found* that the restriction to Poisson's ratio in
an elastic solid held only for the case of a homogeneous assemblage
of single Boscovich point-atoms : and that, in a homogeneous as-
semblage of pairs of dissimilar atoms, laws of force between the
similar and between the dissimilar atoms can readily be assigned,
so as to give any prescribed rigidity and any prescribed modulus of
resistance to compression for an isotropic elastic solid: and for
an aeolotropic homogeneous solid. Green's 21 independent modu-
luses of elasticity, or 18 when axes of coordinates are so chosen
as to reduce the number by three.]

I want now to go through a piece of mathematical work with
you which, though indicated by Green+, has not hitherto, so far
as I know, been given an)rwhere, except partially in my Article
on "Elasticity" in the Encyclopcedia Britannica, It is to find the
most general possible plane wave in a homogeneous elastic solid of
the most general aeolotropy possible, expressed in terms of Green's
21 independent modulusesj. Taking Green's general formula

* [Proc. R, S. E, July 1 and 16, 1889 : Math, and Phyt. Papers, Vol. ra. Art.
xcTii., p. 395 : also " On the Elasticity of a Crystal according to Boscovich,** Proc,
R, S., Jane 15, 1893, and republished as an Appendix to present volume.]

t Green's Mathematical Papers (Macmillan, 1871), pp. 307, 308.

% [June 16, 1898. — Through references in Todhunter and Pearson's EUuticity
I have recently found three very important and suggestive memoirs by Blanohet in
LiouviUe*s Journal, Vols. v. and vii. (1840 and 1842), in which this problem is
treated on the foundation of 36 independent coefficients in the six linear equations
expressing each of the six stress-components in terms of the six strain-components.
In respect to the history of the doctrine of energy in abstract dynamics, it is curious
to find in a Beport to the French Academy of Sciences by Poisson, Coriolis, and
Sturm {Comptes Rendus, Vol. vii., p. 1143) on the first of these memoirs (which had
been presented to the Academy on August 8, 1838), the following sentence: — "Les
^nations diff^rentielles auxqueUes sont assuj^tis les d^placements d'un point
quelconque du milieu 6cart6 de sa position d*6quilibre renferment 36 coefficients
constants, qui dependent de la nature du milieu, et qu*on ne pourrait riduire h un
moindre nonibre sans faire des hypotheses sur la disposition des moUcules et sur les
lois de leurs actions mutuelles" (The italicising is mine.) Blanohet's second
memoir, also involving essentiaUy 36 independent coefficients, was presented to the
Academy of Sciences on June 14, 1841, and was reported on by Cauohy, LiouviUe,
and Duhamel without any protest against the 36 coefficients. In Green's memoir
**0n the Propagation of Light in crystaUized Media," read May 20, 1839 to the
Cambridge Philosophical Society, the expression for the energy of a strain as a

9—2

132

Ifokr.

LeesaielL

jllV

with the Dotation vha v^ick I vas h heSxi jov m
Part L (pp. 22. 23 24.. »* fai *

P = lis - IV*^ 13^ - 14s - ISi > l«rt
Q = 12* - 2^- 23? - 24a - Si - afe'
£ = 13« -r 23r - 3»r -^ Ma - S3& ^ aSe

r = 16« + 26r + 36* ->- 4«a • 3«& -^ «««:

Now, coDsiderxng an uiemi mfinhesEBoal pumlLelepipod of tbe
solid ixiyiz. remark that:, in virtoe *^ ibse scresE cccaponeiilB
P, Q, R, S. T, r. rt eiperieiKes pair? of .jppcan^ fbrees panfld to
OX on its three pairs of £bces a^ fi^&^wx

and tbe total resoltant component parallel to OX is th^^^ore

(

dP dU rfTN , - -

.(2).

Ffftnce if we denote by (jr + f, y + iy. r + f) the coordinates at
time f, of a p^>int of the solid of which (x. _y, z) is the equilibrium,
wft hav*:, dm we fonn.l in Lecture II. (p. 26 aboreK

^i^dp du dry

dp dx dy dz

.(3V

dp dx dy dz

#f dT dS dR
^ df" dx'^ dy'^ dz.

M6W a p\firt4i wAvft, or a ^accession of plane waves, or the
firi/yt.i/ifi fi-moltiri^ from th^j superposition of sets of plane waves

^plMrMlA fiMMti/in /^if tfM ftif Alritifiw^omponentA had been fandftmentally used ; and
tfHfl^ AftMrn A#|i)*liiiMi am/>nK th* 3^ coefficients in the linear equations for stress
iff fSTMM 6l slratA, ft-4nt:ittfi the nnmber of independent coefficients to 21, bad been

QREEK^S THEORY REALISED IN MECHANICAL MODEL. 133

travelling in the same or in contrary directions, may be defined Molar,
generally as any motion of the solid in which every infinitely thin
lamina parallel to some fixed plane experiences a motion which is
purely translational ; or in other words, a motion in which f , 17, f
are functions of (p, t), where p denotes the perpendicular from 0
to the plane through (x, y, z) parallel to the wave-front. If /, m, n
denote the direction-cosines of this perpendicular, we have

p — Ix-k-my -\' nz (4),

and therefore

^ _ 7 ^ .
dx dp*

d d

dy dp*

£^ I.
dz dp

(5),

when these symbols are applied to any function of (p, t).

Hence, according to the definitions of e, /, g, a, 6, c which I
gave you in Lecture II. (pp. 22, 23), we have

,df /. dvi dt

"-'dp' f-^dp> ^=»^ ^g^

dp dp dp dp dp dpi

Hence by (3), (5), (1) and (6) we find

r

in

where

A
B
C

A'
B'
C

111' + 66to» + o5n« + 2 X I6lm + 2 x 56mn + 2 x 15nl

66i» + 22»i» + 44n» + 2 x 26lm + 2 x 24mn f 2 x 4iQnl

■■ 55P + 44to« + 3.3n» + 2 x 4,51m + 2 x 34mn + 2 x 35nl

Now, for our plane waves travelling in either direction, f, i}, ^

118 LECTURE X. PART 11.

Molecular, familiar to many of you, no doubt, and you also know that it is a
failure in regard to the explanation of the propagation of light in
biaxal crystals. It is, however, an important piece of physical
dynamics, and I shall touch upon it a little, and try to show it in
as clear a light as I can.

Ten minutes interval.

Now for our proper molecular question. The distance from
cavity to cavity in the ether is to be exceedingly small, in
comparison with the wave-length, and the diameter of each
cavity is to be exceedingly small, in comparison with the distance
from cavity to cavity. Let the lining of the cavity be an
ideal rigid massless shell. Let the next shell within be a rigid

shell of mass 7—5. I represent the thing in this diagram as

if we had just two of these massive shells
and a solid nucleus. The enormous mass of
the matter of the grosser kind which exists
in the luminiferous ether when permeated
by even such a comparatively non-dense
body as air, would bring us at once to very
great numbers in respect to the masses which
we will suppose inside this cavity, in comparison with the masses
of comparable bulks of the luminiferous ether. If there is time
to-morrow, we shall look a little to the possible suppositions as to
the density of the luminiferous ether, and what limits of greatness
or smallness are conceivable in respect to it. At present we have
enough to go upon to let us see that, even in air of ordinary
density, the mass of air per cubic centimetre must be enormously
great, in comparison with the mass of the luminiferous ether per
cubic centimetre. We must therefore have something enormously
massive in the interior of these cavities. We shall think a good
deal of this yet, and try to find how it is we can have the large
quantity of energy that is necessary to account for the heating of
a body such as water by the passage of light through it, or for
the phosphorescence of a body which is luminous for several days
after it has been excited by light. I do not think we shall have
the slightest difficulty in explaining these things. These are not

DIFFICULTIES IN WAVE THEORY OF LIGHT. 119

the difficulties. The difficulties of the wave theory of light are Moleculi
difficulties which do not strike the popular imagination at all.
They are the difficulties of accounting for polarization by reflection
with the right amount of light reflected ; and of accounting for
double refraction with the form of wave-surface guessed by
Huyghens and proved experimentally by Stokes. With the
general character of the phenomena we have no difficulty what-
ever ; the great difficulty, in respect to the wave theory of light,
is to bring out the proper quantities in the dynamical calculation
of these effects.

There is no difficulty in explaining the energy required for
heating a body by radiant heat passing through it, nor how it is
that it sometimes comes out as visible light and, it may be, so
slowly that it may continue appearing as light for two or three
days. All these properties, wonderful as they are, seem to come
as a matter of course from the dynamical consideration. So much
so that any one not knowing these phenomena would have dis-
covered them on working out the subject dynamically. He would
discover anomalous dispersion, fluorescence, phosphorescence, and
the well-known visible and invisible radiant heat of longer periods
emitted by a body which has been heated and left to cool. All
these phenomena might have been discovered by dynamics ; and
a dynamical theory that discovers what is afterwards verified by
experiment is a very estimable piece of physical dynamics.

I speak with confidence in this subject because I am ashamed
to say that I never heard of anomalous dispersion until after I
found it lurking in the formulas. And, when I looked into the
matter, I found to my shame that a thing which had been known
by others for fifteen or twenty years* I had not known until I
found it in the dynamics.

Take our concluding formula of yesterday (p. 106 above), with
some changes of notation

where f denotes the propagational velocity of waves of period t ;

* Leroux, ** Dispersion anomale de la vapeur d'iode," Comptes Rendus, lv., 1862,
pp. 126—128: Pogg, Ann. cxvii., 1862, pp. 659, 660. Christiansen, "Ueber die
Brechungsverhaltnisse einer weingeiHtigen Losang des Fuchsins," Ann. Phys. Chem.
CXLI, 1870, pp. 479, 480: Phil. Mag., xli., 1871, p. 244; Annalea de Chimie, xxv.,
1872, pp. 213, 214.

120 LECTURE X. PART II.

Moleoalar. n, p, and mj denote now respectively the rigidity of the ether,

the mass of the ether in unit volume of space, and the sum
of the masses of the first interior shells of the embedded
molecules in unit volume of space ;

Ci the force of the first spring, per unit elongation, multiplied by
the number of molecules embedded in the ether per imit
volume of space ;

f^9 >^,f i^,n ^^'> ^^ order of ascending magnitude, the fundamental
periods of the molecule when the outer shell is held fixed ;

R, R,t R^^y &c., denote for the separate fundamental vibrations the
ratio of the energy of the first interior shell to the whole
energy of the complex vibrator.

Let us consider what the wave-period t may be relatively to
the fundamental periods /c, #c^, #c^^, ... of the vibrator on the sup-
position of the bounding shell held fixed, to give us a good
reasonable explanation of dispersion, in accordance with the facts
of observation ydth respect to the difference of velocity for
different periods. To help us with this consideration, take our
previous auxiliary formulas

f« w"^47r«nVf

Xi Ci / i^Rt^ k

where f and oo^ denote respectively the simultaneous maximum
displacements of the outermost massless shell and the first of the
massive shells within it.

If r were less than the smallest of the fundamental periods,

J would be negative, the wave-velocity would be greater than in

firee ether, and the reflective index would be less than unity.
But in all known cases the reflective index is greater than unity ;

and when this is so, -^ — 1 must be positive. I want to see if

we can get our formula to cover a range, including all light from
the highest ultra-violet photographic light of about half the
wave-length of sodium light down to the lowest we know of,
which is the radiant heat from a Leslie cube with a wave-length

REFRACTIVITY THROUGH WIDE RANGE. 121

that I hear from Prof. Langley since I spoke to you on the Moleeak
subject a week ago is about j^j^ of a centimetre or 17 times the
wave-length of sodium light. That will be a range of about
forty to one. The highest invisible ultra-violet light hitherto
determined, by its photographic action, has a period about 1/40
of the period of the lowest invisible radiation of radiant heat thflCt
has yet been experimented upon.

It is probable that all or many colourless transparent liquids
and solids are mediums for which throughout every part of that
range there are no anomalous dispersions. I think it is almost
certain that for rock-salt, in the lower part of the range, there
are no anomalous dispersions at all. In fact Langley's experiments
on radiant heat are made with rock-salt ; and in all experiments
made with rock-salt, it seems as if little or no radiant heat is
absorbed by it. At all events, we could not be satisfied unless we
can show that this kind of supposition will account for dispersion
through a range of period from one to forty. It is obvious that
if we are to have continuous refraction without anomalous dis-
persion through that wide range of periods, there cannot be any
of the periods #c, /e^, #c,^, ... within it.

To-morrow we shall consider the case in which the wave-period
is longer than the longest of the molecular periods ; and we shall
find that on this supposition our formula serves well to represent
all we have hitherto known by experiment regarding ordinary
dispersion. [Added July 7, 1898. This was true in October 1884 ;
but measurements by Langley of the refractivity of rock-salt for
radiant heat of wave-lengths (in air or ether) from 43 of a mikron
to 5'3 mikrons, (the " mikron " being 10~* of a metre or 10~* of a
centimetre), published in 1886 {Phil, Mag, 1886, 2nd half-year),
showed that there must be a molecular period longer than that
corresponding to wave-length 5*3. In an addition to Lecture
XII, Part II, we shall see that subsequent measurements of
refractivity by Rubens, Paschen, and others, extend the range
of ordinary dispersion by rock-salt to wave-lengths of 23 mikrons ;
and give results in splendid agreement with our formula, which
is identical with Sellmeier s expression of his own original theory,
through a range of from '4 of a mikron to 23 mikrons, and
indicate 66 mikrons as being the probable wave-length of radiant
heat in ether of which the period is a critical period for rock-salt.]

LECTURE XI.

Friday, October 10, 3.30 p.m.

Molar. We shall now take up the subject of an elastic solid which is

not isotropic. As I said yesterday, we do not find the mere
consideration of elastic solid satisfactory or successful for explain-
ing the properties of crystals with reference to light. It is, how-
ever, to my mind quite essential that we should understand all
that is to be known about homogeneous elastic solids and waves
in them, in order that we may contrast waves of light in a crystal
with waves in a homogeneous elastic solid.

Aeolotropy is in analogy with Cauchy's word isotropy which
means equal properties in all directions. The formation of a word
to represent that which is not isotropic was a question of some
interest to those who had to speak of these subjects. I see the
Germans have adopted the term anisotropy. If we used this in
English we should have to say : '* An anisotropic solid is not an
isotropic solid " ; and this jangle between the prefix an (privative)
and the article an, if nothing else, would prevent us from adopting
that method of distinguishing a non-isotropic solid from an
isotropic solid. I consulted my Glasgow University colleague
Prof. Lushington and we had a good deal of talk over the subject.
He gave me several charming Greek illustrations and wound up
with the word aeolotropy. He pointed out that ato\o9 means
variegated; and that the Greeks used the same word for variegated
in respect to shape, colour and motion ; example of this last, our
old fiiend "KopvOaioXo^ "E/ctoi/o." There is no doubt of the
classical propriety of the word and it has turned out very con-
venient in science. That which is different in diflFerent directions,
or is variegated according to direction, is called aeolotropic.

The consequences of aeolotropy upon the motion of waves, or
the equilibrium of particles, in an elastic solid is an exceedingly

AEOLOTROPY IN AN ELASTIC SOLID. 123

interesting fundamental subject in physical science ; so that there Molar,
is no need for apology in bringing our thoughts to it here except,
perhaps, that it is too well known. On that account I shall be
very brief and merely call attention to two or three fundamental
points. I am going to take up presently, as a branch of molar
dynamics, the actual propagation of a wave ; and in the mathe-
matical investigation, I intend to give you nothing but what is
true of the propagation of a plane wave in an elastic solid, not
limited to any particular condition of aeolotropy ; in an elastic solid,
that is to say, which has aeolotropy of the most general kind.

Before doing that, which is strictly a problem of continuous or
molar dynamics, I want to touch upon the somewhat cloud-land
molecular beginning of the subject, and refer you back to the old*
papers of Navier and Poisson, in which the laws of equilibrium or
motion of an elastic solid were worked out from the consideration
of points mutually influencing one another with forces which are
functions of the distance. There can be no doubt of the mathe-
matical validity of investigations of that kind and of their interest
in connection with molecular views of matter ; but we have long
passed away from the stage in which Father Boscovich is accepted
as being the originator of a correct representation of the ultimate
nature of matter and force. Still, there is a never-ending interest
in the definite mathematical problem of the equilibrium and motion
of a set of points endowed with inertia and mutually acting upon
one another with any given forces. We cannot but be conscious
of the one splendid application of that problem to what used to be
called physical astronomy but which is now more properly called
dynamical astronomy, or the motions of the heavenly bodies. But
it is not of these grand motions of mutually attracting particles
that we must now think. It is equilibriums and infinitesimal
motions which form the subject of the special molecular dynamics
now before us.

Many writers [Navier (1827), Poisson (1828), Cauchy, F. Neu-
mann, Saint-Venant, and others] who have worked upon this
subject have come upon a certain definite relation or set of
relations between moduluses of elasticity which seemed to them
essential to the hypothesis that matter consists of particles acting
upon one another with mutual forces, and that the elasticity of a
solid is the manifestation of the forcive required to hold the
particles displaced infiuitesimally from the position in which the

124 LECl'URE XI.

olwr. mutual forces will balance. This, which is sometimes called
Navier s relation, sometimes Poisson's relation, and in connection
with which we have the well-known "Poisson's ratio," I want to
show you is not an essential of the hypothesis in question. Their
supposed result for the case of an isotropic body is interesting,
though now thoroughly disproved theoretically and experimentally.
Doubtless most of you know it ; it is in Thomson and Tait, and
I suppose in every elementary book upon the subject. I will just
repeat it.

An isotropic solid, according to Navier's or Poisson's theory,
would fulfil the following condition : if a column of it were pulled
lengthwise, the lateral dimensions would be shortened by a
quarter of the proportion that is added to the length; and the
proportionate reduction of the cross-sectional area would therefore
be half the proportion of the elongation. Stokes called attention
to the viciousness of this conclusion as a practical matter in
respect to the realities of elastic solids. He pointed out that jelly
and india-rubber and the like, instead of exhibiting lateral
shrinkage only to the extent of one quarter of the elongation, give
really enough of shrinkage to cause no reduction in volume at all.
That is to say, india-rubber and such bodies vary the area of the
cross-section in inverse proportion to the elongation so that the
product of the length into the area of the cross-section remains
constant. Thus the proportionate linear contraction across the
line of pull is half the elongation instead of only quarter as
according to Navier and Poisson.

Stokes* also referred to a promise that I made, I think it
was in the year 1856, to the effect that out of matter fulfilling
PoLBSon's condition a model may be made of an elastic solid,
which, when the scale of parts is suflBciently reduced, will be a
homogeneous elastic solid not fulfilling Pdisson's condition. That
promise of mine which was made 30 years ago, I propose this
moment to fulfil, never having done so before.

Let this box help us to think of 8 atoms placed at its 8
comers, with the box annulled. The kind of elastic model I am
going to suppose is this : particles or atoms arranged equidistantly
in equidistant parallel rows and connected by springs in a certain

* [Report to Brit, AMociation, Cambridge, 1862. **0n Doable Refraction,"
p. 262, at bottom.]

grekn's theory realised in mechanical model. 125

definite way. I am going to show you that we can connect MoUr.
neighbouring particles of a Boscovich elastic solid with a special
appliance of cord, and a sufiScient number of springs, to iiilfil the
condition of giving 18 independent moduluses ; then by trans-
forming the coordinates to an orientation in the solid taken at
random, we get the celebrated 21 coefficients, or moduluses, of
Green's theory. I suppose you all know that Green took a short-
cut to the truth ; he did not go into the physics of the thing at
all, but simply took the general quadratic expression for energy in
terms of the 6 strain components, with its 21 independent coeffi-
cients, as the most general supposition that can be made with
regard to an elastic solid.

To make a model of a solid having the 21 independent coeffi-
cients of Green's theory, think of how many disposable springs we
have with which to connect 8 particles at the comers of a parallel-
epiped. Let them be connected by springs first along the 12
edges of the parallelepiped. That clecurly will not be sufficient to
give any rigidity of figure whatever, so far as distortions in the
principal planes are concerned. These 12 springs connecting in
this way the 8 particles would give resistances to elongations
in the directions of the edges; but no resistance whatever to
obliquity; you could change the configuration from rectangular,
if given so as in the box before you, into an oblique parallelepiped,
and alter the obliquity indefinitely, bringing if you please all
the 8 atoms into one plane or into one line, without calling any
resisting forces into play. What then must we have, in order
to give resistance against obliquity ? We can connect particles
diagonally. We have in the first place, the two diagonals in each
face although we shall see that the two will virtually count as but
one ; and then we have the four body diagonals.

Now let me see how many disposables we have got. Remark
that each edge is common to four parallelepipeds of the Boscovich
assemblage. Hence we have only a quarter of the number of
the twelve edge springs independently available. Thus we have
virtually three disposables from the edge springs. Each face is
common to two parallelepipeds ; therefore from the two diagonals
in each face we have only one disposable, making in the six faces,
six disposables. We have the four body diagonals not common to
any other parallelepipeds and therefore four disposables fi om them.
We have thus now 13 disposables in the stiflFnesses of 13 springs.

142 LCCTURE XIL PABT L

HtuM'uffi of tu4f(hihu¥m of elasticity. We shall work ap firom an
imfifffjnc M/;lid Uf the mrist gr^eral iiolid ; and we shall w(^k down
from the uitmi gen#.Tal sr^lid to an isotropic solid. We shall take
fimt the Uitmi general vahie for the compressibility ; we shall then
(UfUt*3 Up thin mibj^sct again of assuming incompressibility. We
Mhall then li<;gin with the most general solid possible, and see what
«;^iiiditions we must impose to make it as symmetrical as is necessary
tor th«) Frefinel wave-surface. The molecular problem will prepare
your way a g^Kxl deal for this.

f had int<5nd(id to prepare something about the mass of the
lufnifiifnrouH ether. I have not had time to take it up, but I
I'^frtainly shall do ho before we have done with the subject. We
shall go into the question of the density of the luminiferous ether,
giving superior and inferior limits. We shall also consider what
fraction of a granuno may be in one of these molecules and show
whjit an enonnously smaller fraction of a gramme we may suppose
it to diH[)laco in the luminiferous ether. We shall try to get into
the notion of thiH, that the molecule must be elastic and that there
must ho an (enormous mass in its interior. Its outer part feels
and tourhoH the huniuiforous ether. It is a very curious sup-
{Mmition to nuikO) of a molecular cavity lined with a rigid spherical
sholl ; but that something exists in the luminiferous ether and
acits u|H)n it in tho mivnnor that is faultily illustrated by our
miH'hanionl moilol, I absolutely believe.

Just t'hink of tho effect of a shock consisting say of a collision
WtwiH'n that, and anothci* molecule. Instead of its being broken
inU> bitu, lot us suppose an imbroken spherical sheath around it.
It will bound away, vibrating. Just imagine the central nucleus
vibniting in ono dinH>tion while the shell vibrates in the other, and
you Imvo a moUvulo with two parts going in opposite directions ;
but difforing from what I thought of the other day (Lecture IX.
|V \>i\ alMn*o> in that ono part is inside the other. The ether gets
it* n\otion fr^^m tho o\it«ido jvart Therefore I say that the most
(Vuu)amont4^l 8\ipjHvation wo can make with reference to the origi-
nating JHumv of a jHHjuonco of wavo« of light is that illustrated
by a gioW vibmtiug to and fi\> in a straight line.

Wo haw alroadv invt^stigated ^Lecture VIIL pp, 86 — S9> the
3^^\it ion o^^mv^jv^mling to thats Consider the spherical waves : no
vibniti^^^!^ ft^r {vuntc^ in ono cort^in diameter of the sphere ; maxi-
mum vibr«it.i\^^s in all )XMUt« of the equatorial plane of that

POLARISED SOURCE OF LIGHT. 143

diameter and perpeodicular to that plane ; for all points in the Moleonlai
quadrant of an arc of the spherical surface extending from axis to
equator, vibrations in the plane of and tangential to the arc ; and
of magnitude proportional to the cosine of the latitude or angular
distance from the equator and of intensity proportional to the
square of the cosine of the latitude. Then in a wave travelling
outwards, let the amplitude vary inversely as the distance from
the centre, and therefore the intensity inversely as the square of
the distance from the centre ; and you have a correct word-painting
of the very simplest and most frequent sequence of vibrations
constituting light.

Ten minutes interval.

Let us return to the consideration of the dynamics of refittction,
ordinary dispersion, anomalous dispersion, and absorption. Begin-
ning with our formulas as we left them yesterday (p. 120 above),
let us consider what they become for the case t = oo ; that is to
say, for static displacement of the containing shell. The inertia
of the molecules will now not be called into play, and the case
becomes simply that of the equilibrium of the set of springs whose
stiffnesses we have denoted by Ci, c„ ... , Cj, Cj+i, when S, the end
of Ci remote from mj, is displaced through a space f, and F, the
end of Cj+i remote from m^, is fixed.

mj iny_j my., wi, m^

Denoting by X the force with which S pulls and F resists,
which, as inertia does not come into play, is therefore the force
with which each spring is stretched, we have

X = Cj+iiCj = Cj (x^i - iTj) = ... = Ca (^1 - arj) = Ci (f - a?i)...(l) ;

or, as we may write it,

whence

11 .11

- + -+... +T + 7—
Xi Ca C3 Cj Cf^i

?""1 11 11

C] C2 C) Cj Cj^i

(3).

144 isgcrrtoL xn, f-Mta u.

^

H^fhf'A, nttUtm *t Uam^ 'W^ f4 ^,, ^^ .,, , ^j^ it jhi»^ -^ — 1 h

^mfi^ Up f /, . U^ditM % large iOifm^ finite valoe of r

^fl»/>, mik) All bkrger ralnen fA 7^ make p, Begatire Tins is a

ymMmtXM, ninth Mi$ exireiyipe, eaue r>f a rerj iin|wriaot result with
irir^fv w^ ^ail hare mneh to do hUer, in fioihywii^ the course of
fffif ffffffffiin whmi the peri/jd r/f the light h increased from anj
m^fiMfuifif pmiffi Uf the next greater mokcakr period ; and we
ffmf K^ ^uh fff it UfT the present bj assnmii^ e^, to be zero. We
umy f^Uff^ it again, anrl perceive its tme physical significance,
hy mifffffmiun m^ Up be infinitely great; and therefore we lose
UffiMun iff girfiifrality by taking c^, » 0. This, in oar formulas of

UfiUft timktm X mO and J - 1. The latter, by patting t = 00 in

//lir liMit fomiula of Lecture X. (p. 120 above), gives

1 - J;^ (^i« + ^;i2, + ^,;ii. + &c.) (4).

NutiiriM)iltig IIiIm from our last formula of Lecture X. with

r- « -. "; ( r'S + J^\+5^-.+&c.) (6);

wIlMlioP, liy Uii< |ii'o(<t<iiiiif( f<»rmula of Lecture X., (p. 120)

A iMtitVt>iii(i|tl> iittnliiU'niioti of thin formula is got by putting in it

.(7X

k

•^««i <s-*;fr' (8x

Tlwin ^ ^l^^uoitvn ihi^ ivft^oti\*o index of the medium; and c^
^li^W^^^^ tho |H^vuMi whioh m^ xwuld have as a vibrator, if the
nh^^U Uui^ViJ: tho t^thov \%^i>i> held fixixi and if the elastic connection
\\^^i\\^v^^ my ^\\\\ uxttv^n^MT ^\\ajt»e9^ >jrx^ri^ t<MnfK\mnly annulled With
|.h%^^ W\nUI\\^\* V^^ be(s%i^\e*

DYNAMICS OF DISPERSION. 145

When the period of the light is very long in comparison with MoleonUr.
the longest of the molecular periods of the embedded molecules, it
is obvious that each material particle will be carried to and fro
with almost exactly the same motion as the shell, in fact almost
as if it were rigidly connected with the shell ; and therefore the
velocity of light will be sensibly the same as if the masses of the
particles were distributed homogeneously through the ether without
any disturbance of its rigidity.

Let us consider now how, when the period of the light is not
infinitely long in comparison with any one of the molecular periods,
the internal vibrations of the molecule modify the transmission
of light through the medium. For simplicity at present let us
suppose our molecule to have only one vibrating particle, the mi of
our formulas, which we will now call simply m,
being the sum of the masses of the vibrators
per unit volume of the ether. Imagine it con-
nected with the shell or sheath, Sy surrounding
it, by massless springs (as in the accompanying
diagram), through which it acts on the ether
surrounding the sheath. Its influence on the
transmission of light through the medium can
be readily understood and calculated from the diagrams of p. 147,
representing the well-known elementary dynamics of a pendulum
vibrating in simple harmonic motion, when freely hung from a
point 8, (corresponding to the sheath lining our ideal cavity in
ether), which is itself compelled by applied forces to move horizon-
tally with simple harmonic motion. Figs. 1 and 2 illustrate the
cases in which the period of the point of suspension, 8, is longer
than the period of the suspended pendulum with 8 fixed ; and
figs. 3 and 4 cases in which it is shorter. In each diagram OM
represents the length of an undisturbed simple pendulum whose
period is equal to that of the motion of 8, the point of support of
our disturbed pendulum 8M, Thus if we denote the periods by
T and K respectively, we have 0M/8M= t^/k^. Hence if f and x
denote respectively the simultaneous maximum displacements of
8 and M, we have

f 08 T>-/c« ^^"^•

If now for a moment we denote by w the mass of the single
vibrating particle of our diagrams, the horizontal component of the
T. L. 10

^44^ lbcttbc xn. PAKT n.

MolMmliir. pall exerted by the thread MS oa S xt the instant of mazimam
duplacement in w(2fwlrfx ; and with m in place of v, we have
the Sam of the forceti exerted by all the nM>ieciiIes par anit Yolnme
of the ether at the inatant of maxiinnni displacement of ether and
molecalea at any point. This foree is sabtracted firom the re-
stitational fioroe of the etber^a elasticity, when the period of the
h^i is greater than the nK^ecolar period (figs. 1 and 2) ; and »
added to it, when the period of the light is leas than the molecular
period (figs. 3 and 4X

Now if the rigidity of the ether be taken as unity, its dastie
force per unit of Tolume at any point at the instant of its nuudmam
displacement is {inrjtf^ where I is the wafe-length of light in
the ether with its embedded molecules. Hence the total actual
restitutional force on the ether is

(Tl'f-^'^'-

which must be equal to [ — ] ( if the density of the ether be

taken as unity. Hence dividing both members of our equation by
{iwlrfJ^, and denoting by ^ the velocity of light through the
medium consisting of ether and embedded molecules, and by m its
refractive index, we have

/»'=^=(9'=i+"»| (11);

whence by (10),

By quite analogous elementary djmamical considerations we find

if, instead of one set of equal and similar molecules, there are
several such sets, the molecules of one set differing firom those
of the others. The period /r, or /c„ or k„, &c., of any one of the
wots in Himply the period of vibration of the interior mass,
whnn th(» rigid spherical lining of the ether around it, which for
brevity wn shall hcmcoforth call the sheath of the molecule, is
hohi fixed. Ttmtoiui of calling this sheath massless as .hitherto
wn Mhnll MUppnso it to have mass equal t^ that of the displaced

DYNAMICS OP DISPERSION.

147

Holeenlar.

10—2

148 LECTURE XII. PART II.

Ifoleeolar. ether ; so that if there were no interior masses the propagation
of light would be sensibly the same as in homogeneous un-
disturbed ether. The condition which we originally made, that
the distance from molecule to molecule must be very great in
comparison with the diameter of each molecule and very small
in comparison with the wave-length, must be fulfilled in respect
to the distances between molecules of different kinds. The
number of molecules of each kind in a cube of edge equal to
the wave-length must be very great; and the numbers in any
two such cubes must be equal, this last being the definition of
homogeneity. These are substantially the conditions assumed by
Sellmeier*, and our equation (13) is virtually identical with his
original expression for the square of the refractive index.

It is interesting to remark that our formula (9) for the effect
on the velocity of regular periodic waves of light, of a multitude
of equal and similar complex vibrating molecules embedded in
the ether, shows that it is the same as the effect of as many
sets of different simple vibrators as there are Amdamental periods
in our one complex vibrator; and that the masses of the equi-
valent simple vibrators are given by the equations

^, = !?h^^; ,,, = ^': ,;,, = '^';&C....(14).
/>/C,* pK^* pK,*

But although the formula for the velocity of regular periodic
waves is the same, the distribution of energy, kinetic and potential,
is essentially different on the two suppositions. This difference
is of great importance in respect to absorption and fluorescence,
as we shall see in considering these subjects later.

[Added Sept 23f , 1898. The dynamical theory of dispersion,
as originally given by Sellmeier*, consisted in finding the velocity
of light as affected by vibratory molecules embedded in ether,

• SeUmeier, Pogg. Ann., Vol. 146, 1872, pp. 899, 620; Vol. 147, 1872, pp. 387,
626.

t [The sabstance of this was oommunioaied to Seo. A of the British Association
ftl Btifitol on September 9, 1898, m two papers nnder the titles " The Dynamical
Theory of Refraction, Dispersion, and Anomalous Dispersion," and ** Continuity in
Undolatory Theory of Condensational-rarefactional Waves in Gases, Liquids, and
Solids, of Distortional Waves in Solids, of Electric and Magnetic Waves in all
Sabttancea capable of transmitting them, and of Radiant Heat, Visible Light,
.Ulim-^olet Light and Rdntgen Rays."]

sellmbier's theory of dispersion. 149

such as those which had been suggested by Stokes* to account Moleonlar.
for the dark lines of the solar spectrum. Sellmeier's mathematical
work was founded on the simplest ideal of a molecular vibrator,
which may be taken as a single material particle connected by a
massless spring or springs with a rigid sheath lining a small vesicle
in ether. He investigated the propagation of distortional waves,
and found the following expression (which I give with slightly
altered notation) for the square of the refractive index of light
passing through ether studded with a very large number of
vibratory molecules in every volume equal to the cube of the
wave-length : —

'H T^ T*

where t denotes the period of the light ; /r, /c„ /r„, &c., the vibratory
periods of the embedded molecules on the supposition of their
sheaths held fixed ; and m, m„ m,,, &c., their masses. He showed
that this formula agreed with all that was known in 1872 regard-
ing ordinary dispersion, and that it contained what we cannot
doubt is substantially the true dynamical explanation of anomalous
dispersions, which had been discovered by Fox-Talbotf for the
extraordinary ray in crystals of a chromium salt, by LerouxJ for
iodine vapour, and by Christiansen§ for liquid solution of fuchsin,
and had been experimentally investigated with great power by
Kundt**.

Sellmeier himself somewhat marred fi" the physical value of his
mathematical work by suggesting a distinction between refractive
and absorptive molecules (" refractive und absorptive Theilchen "),
and by seeming to confine the application of his formula to cases
in which the longest of the molecular periods is small in com-
parison with the period of the light. But the splendid value
to physical science of his non-absorptive formula has been quite
wonderfully proved by Rubens (who, however, inadvertently

* See Kirchhoff-Stokes-Thomson, Phil, Mag.^ Maroh and July 1S60.
t Fox-Talbot, Proc, Roy. 8oc. Edin,, 1870—71.
t LeroQX, Comptes rendu$, 55, 1862, pp. 12d— 128.

§ Christiansen, Ann, Phys. Chem,, 141, 1870, pp. 479, 480; Phil Mag,, 41,
1871, p. 244 ; AnnaU$ de Chimie, 25, 1872, pp. 218, 214.
** Kundt, Pogg, Ann,, Vols. 142, 143, 144, 145, 1871—72.
t+ Pogg, Ann,, Vol. 147, 1872, p. 625.

\r>() LECTURE XII. PABT IL

^/ylMnUr. (-|fK>t^^^* it aa if due U> Ketteler> Fomteen years ago Langlejf
ha/l rn^^AAurerl the refractivity of rock-salt for light and radiaot
h#;at /^ wave-lf.TigthH (in air or ether) from '43 of a mikrom^ to
5*8 m'lkrfmtH (the mikrom being 10^ of a metre, or l(r* of a
centimetre), and without measuring refractivities farther, had
metmm(A wave-lengths as great as 15 mikroms in radiant heat
Within the last six years measurements of refractivity by Rubens,
Faw^hen, and others, agreeing in a practically perfect way with
liangley s through his range, have given us very accurate know-
Ifxlge of the refractivity of rock-salt and of sylvin (chloride of
|:K;tassiurfi) through the enormous range of from '4 of a mikrom to
29 mikroms.

Hiih<;ris Ixjgan by using empirical and partly theoretical
formiihiM which hod been suggested by various theoretical and
ex|Hiriffjontal writcrSi and obtained fairly accurate representations
of tho refractivities of flint-glass, quartz, fluorspar, sylvin, and
rfK3k-HHlt through ranges of wave-length from *4 to nearly 12
inikrofiiH§. Two years later, further experiments extending the
trioasurn of n;fractivities of sylvin and rock-salt for light of wave-
Inii^MiH u[) to 28 mikroms, showed deviations from the best of the
prnviouH eiii[)irical formulas increasing largely with increasing

* Whd, AHti,^ Vol. 68, 1894, p. 367. In the formula quoted by Bnbens from
Knitntprt MtiliMiliuto for Mod^^^* vhIuo of fi found by patting r=aom Sellmeier's
forttiuiii, M\i\ Ki>tt«lor'M formula beoomos identical with Sellmeier's. Bemark that
Kntti*li*r> *' M '* in H(«lltnoier*8 ** mic* " aooordiug to my notation in the text

f liatiKl«\V, PhiL Mao,, 1H80, *iiid half-year.

X \Vm a miiait unit of length Langley, fourteen years ago, used with great
aiWauiagi* and (^onvmlttnoo the word **mikron" to denote the millionth of a
tni>tn*. Thi> littler n ban no plaoo in tlio metrical system, and I venture to suggest
a Dhaii|t« of HiHtUing to «* mikrom *' for the millionth of a metre, after the analogy
of ihi» Kitgliith UKagi^ for millionihs (mikrohm, mikro-ampere, mikrovolt). For a
t«i>itviMtloial>' sniall oorrospondiiig unit of time I fUrther venture to suggest
** iniohin^ii '' to donotA ih« |>ori(Kl of vibration of light whose wave-length in ether
in I lutkrttiti. Thus, tho voU)ci(y of light in ether being S x 10" metres per second,
ibi» miohn^u (m | \ tO *^ of a ncoimd, and the velocity of light is 1 mikrom of space
)Hvv ntloht^m of \.\\\\p. ThuN thf> fr<»quency o( the highest ultra-violet light investi-
9!k\p\\ bv H^'bumMitn r\ of a nukr\>m wavc»-)<>ngth, Sittmn^$her. d. h, GfseU*ch, d,
M'«iiirf-nj»f*A. t^ H'fVH, \*n. |>p. 41 A and G3A, 181UI) is 10 i^oriods per michron of time.
Tb<« \H^\^^>^\ of mvhuui liithl (m^n of Unen />) is *58.t^ of a michron ; the periods
of ih«' ** UfiiUht^hbMr' of i\vk wiUl and ^vlvin found by Rubens and Aftchkinaas
^Ht*y*. Aitn. \.\\. 0^^)» P« *41) ai>» M'S and 6M mkhrons respectively. No
|>raHi«H^l mot\n\^niMi<H» <Hin ^^vm* arine (h>m any possible confasion, or momentary
fiMV0i(\Oni^H, in )>w)>ivt to \\w stmilarily of iKvund between michn^njt of time and

i Uiib^nn. H ff<W« Ann., VoK ^H, .\4« linU 4^V

RUBENS' RESULTS FOR ROCK-SALT AND SYLVIN. 151

wave-leDgths. Rubens then fell back* on the simple unmodified Molecular.
Sellmeier formula, and found by it a practically perfect expression
of the refractivities of those substances from '434 to 22*3 mikroma

And now for |he splendid and really wonderful confirmation of
the dynamical theory. One year later a paper by Rubens and
Aschkinassf describes experiments proving that radiant heat after
five successive reflections from approximately parallel surfaces of
rock-salt ; and again of sylvin ; is of mean wave-length 51*2 and
61*1 mikroms respectively. The two formulas which Rubens had
given in February 1897, as deduced solely from refractivities
measured for wave-lengths of less than 23 mikroms, made /i«
negative for radiant heat of wave-lengths from 37 to 55 mikroms
in the case of reflection from rock-salt, and of wave-lengths
from 45 to 67 mikroms in the case of reflection from sylvin!
(ji* negative means that waves incident on the substance cannot
enter it, but are totally reflected).

These formulas, written with a somewhat important algebraic
modification serving to identify them with Sellmeier s original
expression, (13) above, and thus making their dynamical meaning
clearer, are as follows : —

Rock-salt ^. = 11875 + 11410^,-^ + 2-8504 ^-4^.

Sylvin M' = 1-5329 + -6410 ^,-^ + 2-3792 ^-^-jy^j .

The accompanying diagrams (pp. 152, 153) represent the squares of
the refractive index of rock-salt and sylvin, calculated from these
formulas through a range of periods from '434 of a michron to
100 michrons. In each diagram the scale both of ordinate and
abscissa from 0 to 10 michrons is ten times the scale of the
continuation from 10 to 100 michrons. Additions and sub-
tractions to keep the ordinates within bounds are indicated on
the diagrams. The circle and cross, on portion 6 of each curve,
represent respectively the points where ft' = 1 and /i* = 0. The
critical periods exhibited in the formula are

for rock-salt,

•1273 and 561 16, being the square roots of 01621 and 3149*3;

for sylvin,

•1529 and 67-209, „ „ „ „ '0234 ,,45171.

* Rubens and NichoU, Wied, Ann,, Vol. 60, 1896—97, p. 454.
t Rubens and Aschkinass, Wied. Ann., Vol. 65, 1898, p. 241,

LBCTUHB XII. PART IL

J

3
S
S

A

^

—

]

■^

1 —

■

' ■

1

i

.

^

^

K^

S
^

/

/

/

/

f\

t. «

Vi

r + 51 T I 1

|-- ■

—

Qi ^1 ^ "q, »,,

1 . . . . 1

]

■^

y

% % %

RCTBENS' RESULTS FOR ROCK-8ALT AND EITLVIN. 153

1

/

i

>^

— ■

r

-^

-i

—

^

/

^

^

/

<■

/

1

s s

R 1

n •«

1-

s

J _

J —

\

H

%

s

\ ' ' ' ' '

1

11,

J

' L 1 J

i

i

E

%

a

%

■ s

~

? '

o

154 LECTURE Xn. PART 11.

Molecular. The value of /x' passes from — » to + » as t rises through these
values, and we have accordingly asymptotes at these points,
represented by dotted ordinates in the diagrams.

The agreement with observation is absolutely perfect through
the whole range of Langley, Rubens, and Paschen from '434 of a
michron to 22-3 michrons. The observed refractivities are given
by them to five significant figures, or four places of decimals ; and
in a table of comparison given by Rubens and Nichols* the
calculated numbers agree with the observed to the last place of
decimals, both for rock-salt and sylvin.

While in respect to refi:iictivity there is this perfect agreement
with Sellmeier's formula through the range of periods from *434
of a michron to fifty-one times this period (corresponding to
nearly six octaves in music), it is to be remarked that with
radiant heat of 22*3 michrons period, Rubens found, both for
rock-salt and sylvin, so much absorption, increasing with increasing
periods, as to prevent him from carrying on his measurements
of refractivity to longer periods than 22 or 23 michrons, and to
extinguish a large proportion of the radiant heat of 23 michrons
period in its course through his prisms. Hence although Sellmeier's
formula makes no allowance for the large absorptivity of the trans-
parent medium, such as that thus proved in rock-salt and sylvin with
radiant heat of periods less than half the critical period k in each
case, it is satisfactory to know that large as it is the absorptivity
produces very little effect on the velocity of propagation of radiant
heat through the medium. This indeed is just what is to be
expected from dynamical theory, which shows that the velocity
of propagation is necessarily affected but very little by the forces
which produce absorption, unless the absorptivity is so great that
the intensity of a ray is almost annulled in travelling three or
four times its wave-length through the medium.

In an addition to a later lecture I intend to refer again to
Helmholtz*s introduction of resisting forces in simple proportion
to velocities, by which he extended Sellmeier's formula to include
true bodily absorption, and to recent modifications of the extended
formula by himself and by Ketteler. Meantime it is interesting,
and it may correct some misapprehension, to remark that so far

* Wied. Ann,, Vol. 60, 1896-97, p. 464.

THREE KINDS OF CRITICAL PERIOD. 155

as the dynamics of Sellmeier's single vibrating masses, or of my Molecular,
complex molecule, goes, there is no necessity to expect any
absorption at all, even for light or radiant heat of one of the chief
critical periods /r, as we shall see by the following general view of
the circumstances of this and other critical periods. We shall see
in £Etct that there are three kinds of critical period,

(1) a period for which /a' = 1, or velocity in the medium
equal to velocity in pure undisturbed ether ;

(2) a period for which /a' = 0, or wave-velocity infinite in
the medium ;

(3) a period (any one of the /e-periods) such that if we
imagine passiog through it with augmenting period, fj^ changes
from — 00 to + 00 .

The dynamical explanation of points where ^' = 1 (marked
by circles on the curves), is simpler with my complex molecule
than with Sellmeier's several sets of single vibrators. With my
complex molecule it means a case in which No. 1 spring, that
between m^ and the sheath, experiences no change of lengtL In
order that this may be the case, the period of the light must be
equal to some one of the free periods of the complex molecule,
detached from the sheath (ci temporarily annulled). With Sell-
meier's arrangement, it is a case in which tendency to quicken
the vibrations of the ether by one set of vibrators, is counter-
balanced by tendency to slow them by another set, or other seta
It is a case which could not occur with only one set of equal and
similar simple vibrators.

The case we have next to consider, /a' = 0 (or ?' = oo ), marked
with crosses on the curve, occurs essentially just once for a single
set of simple vibrators ; and it occurs as many times as there are
sets of vibrators, if there are two or more sets with different
periods; or, just once for each of the critical periods /e, /e,, /e„, &c.
of my complex molecule. The dynamical explanation is particu-
larly simple for a single set of Sellmeier vibrators ; it is the whole
ether remaining undistorted and vibrating in one direction, while
the masses m all vibrate in the opposite direction; the whole
system being as it were two masses E and M connected by
a single massless spring, and vibrating to and fro in opposite
directions in a straight line with their common centre of inertia

156 LECTURE XIL PART II.

Moleonlar. at rest. Thus we see exactly the meaning of wave-velocity
becoming infinite for the critical cases marked with a cross on
the curves.

Increase of the period beyond this critical value makes fi-
negative until we reach the next one of our critical values ic> K,y
K„, &c., or the one of these, if there is only one. The meaning
clearly is that light cannot penetrate the medium and is totally
reflected from it wherever it &lls on it. Thus for light or radiant
heat of all periods corresponding to the interval between a point
on one of our curves marked with a cross and the next asymptotic
ordinate, the medium acts as silver does to visible light; that
is to say, it is impervious and gives total reflection. When we
increase the period through one of our chief critical values k, tc,,
K,„ &c., fi^ passes from — oo to + oo . With exactly the critical
period we have infinitely ^mall velocity of propagation of light
through the medium, and still total reflection of incident light.
There would be infinitely great amplitude of the molecular
vibrators, if the light could get into the medium ; but it cannot
get in. With period just a little greater than one of these critical
values, the reflection of the incident light is very nearly total ;
the velocity of propagation of the light which enters is very
small; and the energy (kinetic and potential) of the molecular
vibrators is very great in comparison with the energy (kinetic
and potential) of the ether.

Lastly, if there were no greater critical period than this which
we have just passed, we should now have ordinary refraction with
ordinary dispersion, at first large, becoming less and less with
increased period, and fi^ diminishing asymptotically to the value

^Tf^ — , where p denotes the density of the ether, and M the

sum of all the masses within the sheaths and connected to
them by springs. But when we consider that the whole mass
of the ponderable matter is embedded in the ether, and we
cannot conceive of merely an infinitesimal portion of it clogging
the ether in its luminiferous or electric vibrations, and that
therefore for glass, or water, or even for rarefied air, M must be
millions of millions of times p, we see how utterly our d3mamical
theory fails to carry us with any tolerable comfort, in trying to
follow and understand the nature of waves and vibrations of
periods longer than the 60 michrons (or 2 x 10~^' of a second)

CONTINUITY IN UNDULATORY THEORY. 167

touched by Rubens and Aschkinass*, up to 10~* of a second, or Molar,
further up to one thousandth of a second, or up to still longer
periods.

Tet we must try somehow to find and thoroughly understand
continuity in the undulatory theory of condensational-rarefactional
waves in gases, liquids, and solids, of distortional waves in solids,
of electric and magnetic waves in all substances capable of trans-
mitting them, and of radiant heat, visible light, ultra-violet light
and RSntgen rays.

Consider the following three analogous cases : — I. mechanical ;
II. electrical ; III. electromagnetic.

I. Imagine an ideally rigid globe of solid platinum of 12
centim. diameter, hung inside an ideal rigid massless spherical
shell of 13 centim. internal diameter, and of any convenient
thickness. Let this shell be hung in air or under water by a very
long cord, or let it be embedded in a great block of glass, or rock,
or other elastic solid, electrically conductive or non-conductive,
transparent or non-transparent for light.

I. (1) By proper application of force between the shell and
the nucleus cause the shell and nucleus to vibrate in opposite
directions with simple harmonic motion through a relative total
range of 10""' of a centimetre. We shall first suppose the shell
to be in air. In this case, because of the small density of air
compared with that of platinum, the relative total range will be
practically that of the shell, and the nucleus may be considered
as almost absolutely fixed. If the period is ^ of a second,
(firequency 32 according to Lord Rayleigh's designation), a humming
sound will be heard, certainly not excessively loud, but probably
amply audible to an ear within a metre or half a metre of the
shell. Increase the frequency to 256, and a very loud sound of
the well-known musical character (Caw) will be heard f.

Increase the frequency now to 32 times this, that is to 8192
periods per second, and an exceedingly loud note 5 octaves higher
will be heard. It may be too loud a shriek to be tolerable ; if so,

• Wied. Ann,, Vol. 64, 1898.

t Lord Bayleigh has found that with frequency 256, periodic condensation and
rarefaction of the marvellously small amount 6 x 10~' of an atmosphere, or
"addition and subtraction of densities far less than those to be found in our highest
vacua," gives a perfectly audible sound. The amplitude of the aerial vibration, on
each side of zero, eorrespondinc; to this, is 1*27 x 10~^ of a centimetre, — Sound, Vol. ii.
p. 439 (2nd edition).

loS LBCTURE XTT. PART II.

diminish the range till the sound is not too loud. Increase the
frequency now successively according to the ratios of the diatonic
scale, and the well-known musical notes will be each clearly and
perfectly perceived through the whole of this octave. To some
or all eara the musical notes will still be clear up to the G (24756
periods per second) of the octave above, but we do not know
from experience what kind of sound the ear would perceive
for higher frequencies than 26000. We can scarcely believe
that it would hear nothing, if the amplitude of the motion is
suitable.

To produce such relative motions of shell and nucleus as we
have been considering, whether the shell is embedded in air, or
water, or glass, or rock, or metal, a certain amount of work, not
extravagantly great, must be done to supply the energy for the
waves (both condensational and rarefactional), which are caused
to proceed outwards in all directions. Suppose now, for example,
we find how much work per second is required to maintain vibra-
tion with a frequency of 1000 periods per second, through total
relative motion of 10~' of a centimetre. Keeping to the same
rate of doing work, raise the frequency to 10*, 10*, 10*, 10*, 10",
600 X lO**. We now hear nothing ; and we see nothing from any
point of view in the line of the vibration of the centre of the
shell which I shall call the axial line. But from all points of
view, not in this line, we see a luminous point of homogeneous
polarized yellow light, as it were in the centre of the shell, with
increasing brilliance as we pass from any point of the axial line
to the equatorial plane, keeping at equal distances from the centre.
The line of vibration is everywhere in the meridional plane, and
perpendicular to the line drawn to the centre.

When the vibrating shell is surrounded by air, or water, or
other fluid, and when the vibraticms are of moderate frequency, or
of anything less than a few hundred thousand periods per second,
the waves proceeding outwards are condensational-rarefactional,
with zero of alternate condensation and rarefaction at every point
of the equatorial plane and maximum in the axial line. When
the vibrating shell is embedded in an elastic solid extending to
vast distances in all directions from it, two sets of waves, distor-
tional and condensational-rarefactional, according respectively to
the two descriptions which have been before us, proceed outwards
with different velocities, that of the former essentially less than

CONTINUITY IN UNDFLATORY THEORY. 159

that of the latter in all known elastic solids*. Each of these Molar,
propagational velocities is certainly independent of the frequency
up to 10*, 10*, or 10*, and probably up to any frequency not so
high but that the wave-length is a large multiple of the distance
from molecule to molecule of the solid. When we rise to fire-
quencies of 4 x W\ 400 x 10", 800 x 10", and 3000 x W\ cor-
responding to the already known range of long-period invisible,
radiant heat, of visible light, and of ultra-violet light, what
becomes of the condensational-rarefactional waves which we have
been considering ? How and about what range do we pass from
the propagational velocities of 3 kilometres per second for distor-
tional waves in glass, or 5 kilometres per second for the condensa^
tional waves in glass, to the 200,000 kilometres per second for
light in glass, and, perhaps, no condensational wave? Of one
thing we may be quite sure ; the transition is continuous. Is it
probable (if ether is absolutely incompressible, it is certainly
possible) that the condensational-rarefactional wave becomes less
and less with frequencies of from 10* to 4 x 10", and that there is
absolutely none of it for periodic disturbances of frequencies of
from 4 X 10" to 3000 x 10" ? There is nothing unnatural or
fruitlessly ideal in our ideal shell, and in giving it so high a
frequency as the 500 x 10" of yellow light. It is absolutely
certain that there is a definite dynamical theory for waves of
light, to be enriched, not abolished, by electromagnetic theory;
and it is interesting to find one certain line of transition from our
distortional waves in glass, or metal, or rock, to our still better
known waves of light.

I. (2) Here is another still simpler transition from the dis-
tortional waves in an elastic solid to waves of light. Still think
of our massless rigid spherical shell, 13 centim. internal diameter,
with our solid globe of platinum, 12 centim. diameter, bung in its
interior. Instead of as formerly applying simple forces to produce
contrary rectilinear vibrations of shell and nucleus, apply now
a proper mutual forcive between shell and nucleus to give them
oscillatory rotations in contrary directions. If tb^ shell is hung
in air or water, we should have a propagation outwards of dis-
turbance due to viscosity, very interesting in itself; but we should
have no motion that we know of appropriate to our present
subject until we rise to frequencies of 10*, 10 x 10", 400 x 10",

• Math, and Phys, Papers, Vol. iii. Art. civ. p. 622.

160 LECTURE XII. PART II.

Molar. 800 X 10", or 3000 x 10^«, when we should have i-adiant heat, or
visible light, or ultra-violet light proceeding from the outer surface
of the shell, as it were from a point-source of light at the centre,
with a character of polarization which we shall thoroughly con-
sider a little later. But now let our massless shell be embedded
far in the interior of a vast mass of glass, or metal, or rock, or of
any homogeneous elastic solid, firmly attached to it all round, so
that neither splitting away nor tangential slip shall be possible.
Purely distortional waves will spread out in all directions except
the axial. Suppose, to fix our ideas, we begin with vibrations of
one-second period, and let the elastic solid be either glass or iron.
At distances of hundreds of kilometres (that is to say, distances
great in comparison with the wave-length and great in comparison
with the radius of the shell), the wave-length will be approxi-
mately 3 kilometres*. Increase the frequency now to 1000
periods per second : at distances of hundreds of metres the wave-
length will be about 3 metre& Increase now the frequency to
10* periods per second; the wave-length will be 3 millim., and
this not only at distances of several times the radius of the shell,
but throughout the elastic medium from close to the outer surface
of the shell; because the wave-length now is a small fraction
of the radius of the shell. Increase the frequency further to
1000 X 10* periods per second ; the wave-length will be 3 x 10"' of
a millim., or 3 mikroms, if, as in all probability is the case, the
distance between the centres of contiguous molecules in glass and
in iron is less than a five-hundredth of a mikrom. But it is
probable that the distance between centres of contiguous molecules
in glass and in iron is greater than 10'' of a mikrom, and there-
fore it is probable that neither of these solids can transmit waves
of distortional motion of their own ponderable matter, of so short
a wave-length as 10~' of a mikrom. Hence it is probable that if
we increase the frequency of the rotational vibrations of our shell
to one hundred thousand times 1000 x 10*, that is to say, to
100 X 10", no distortional wave of motion of the ponderable
matter can be transmitted outwards ; but it seems quite certain
that distortional waves of radiant heat in ether will be produced
close to the boundary of the vibrating shell, although it is also
probable that if the surrounding solid is either glass or iron,
these waves will not be transmitted far outwards, but will be

* Math, and Phys, Papert, Vol. ni. Art. civ. p. 522.

CONTINUITY OF UNDULATORF THEORY. 161

absorbed, that is to say converted into non-undulatory thermal Molar,
motions, within a few mikroms of their origin.

Lastly, suppose the elastic solid around our oscillating shell
to be a concentric spherical shell of homogeneous glass of a few
centimetres, or a few metres, thickness and of refractive index
1'5 for D light. Let the frequency of the oscillations be increased
to 5*092 X 10" periods per second, or its period reduced to '58932
of a michron : homogeneous yellow light of period equal to the
mean of the periods of the two sodium lines will be propagated
outwards through the glass with wave-length of about | x '58982
of a mikrom, and out from the glass into air with wave-length
of '58932 of a mikrom. The light will be of maximum intensity
in the equatorial plane and zero in either direction along the axis,
and its plane of polarization will be everywhere the meridional
plane. It is interesting to remark that the axis of rotation of the
ether for this case coincides everywhere with the line of vibration
of the ether in the case first considered ; that is to say, in the
case in which the shell vibrated to and fro in a straight line,
instead of, as in the second case, rotating through an infinitesimal
angle round the same line.

A full mathematical investigation of the motion of the elastic
medium at all distances from the originating shell, for each of the
cases of I. (1) and L (2), will be found later (p. 190) in these
lecturea

II. An electrical analogy for 1.(1) is presented by sub-
stituting for our massless shell an ideally rigid, infinitely massive
shell of glass or other non-conductor of electricity, and for our
massive platinum nucleus a massless non-conducting globe elec-
trified with a given quantity of electricity. For simplicity we
shall suppose our apparatus to be surrounded by air or ether.
Vibrations to and fro in a straight line are to be maintained by
force between shell and nucleus as in I. (1). Or, consider simply
a fixed solid non-conducting globe coated with two circular caps of
metal, leaving an equatorial non-conducting zone between them,
and let thin wires from a distant alternate-current dynamo, or
electrostatic inductor, give periodically varying opposite electri-
fications to the two caps. For moderate frequencies we have
a periodic variation of electrostatic force in the air or ether sur-
rounding the apparatus, which we can readily follow in imagination,
and can measure by proper electrostatic measuring apparatus.

T. L. 11

162 LECTURE Xn. PART II.

Molar. Its phase,, with moderate frequencies, is very exactly the same
as that of the electric vibrator. Now suppose the frequency of
the vibrator to be raised to several hundred million million
periods per second. We shall have polarized light proceeding as
if from an ideal point-source at the centre of the vibrator and
answering fully to the description of I. (1). Does the phase of
variation of the electrostatic force in the axial line outside the
apparatus remain exactly the same as that of the vibrator ? An
affirmative answer to this question would mean that the velocity
of propagation of electrostatic force is infinite. A negative
answer would mean that there is a finite velocity of propagation
for electrostatic force.

III. The shell and interior electrified non-conducting massless
globe being the same as in II., let now a forcive be applied
between shell and nucleus to produce rotational oscillations as
in I. (2). When the frequency of the oscillations is moderate,,
there will be no alteration of the electrostatic force and no
perceptible magnetic force in the air or ether around our ap-
paratus. Let now the frequency be raised to several hundred
million million periods per second ; we shall have visible polarized
light proceeding as if from an ideal point-source at the centre
and answering fully to the description of the light of I. (2).
The same result would be obtained by taking simply a fixed
solid non-conducting globe and laying on wire on its surface
approximately along the circumferences of equidistant circles of
latitude, and, by the use of a distant source (as in II.), sending
an alternate current through this wire. In this case, while there
is no manifestation of electrostatic force, there is strong alter-
nating magnetic force, which in the space outside the globe is
as if from an ideal infinitesimal magnet with alternating magneti-
zation, placed at the centre of the globe and with its magnetic
axis in our axial line.]

LECTURE XIII.

Saturday, October 11, 8 p.m.

Prof. Morlet has already partially solved the definite dyna- Moleoala
mical problem that I proposed to you last Wednesday (p. 103
above) so &r as determining four of the fundamental periods;
and you may be interested in knowing the result. He finds roots,
KT*, ie^-«, &c., = 3-46, 1005, -298, -087; each root not being very
different from three times the next after it. I will not go into the
affair any further just now. I j ust wish to call your attention to what
Prof. Morley has already done upon the example that I suggested
for our arithmetical laboratory. I think it will be worth while
also to work out the energy ratios* (p. 74). In selecting this
example, I designed a case for which the arithmetic would of
necessity be highly convergent. But I chose it primarily because
it is something like the kind of thing that presents itself in
the true molecule : — An elastic complex molecule consisting of a
finite number of discontinuous masses elastically connected (with
enormous masses in the central parts, that seems certain): the
whole embedded in the ether and acted on by the ether in virtue
of elastic connections which, unless the molecule were rigid and
embedded in the ether simply like a rigid mass embedded in jelly,
must consist of elastic bonds analogous to springs.

I think you will be interested in looking at this model which,
by the kindness of Prof. Rowland, I am now able to show you.
It is made on the same plan as a wave machine which I made
many years ago for use in my Qlasgow University classes, and
finally modified in preparations for a lecture given to the Royal

^ This was done by Mr Morley who kindly gave his results to the *< Coeffi-
cients '' on the 17th Oct. See Lecture XEL. below.

11—2

104

LECTUIUC XIII.

H'.W-'iiw. IrjNtitiitiob aly^iit twi. y»rarfe ago on "The Siae of AtomB*." I
thifjk th'>><r wh'/ an: ixitenrt^ted in the illuBtntion of dynamical

mm^:

L

pmblf ins will find this a very nice and convenient method. If
Y<Mi will look at it, you will see how the thing is done : PianofiMte

wire, bent around three pins in the way yon
see here, supports each bar. These pins are
slanted in such a wav as to cause the wire
to press in close to the bar so as to hold it
quit'C firm. The wood is slightly cut away
to prcvt'iit the wire from touching it above and below the pins,
so that thtTi' may bo no impairment of elastic action due to slip of
HttM'l on wimkI. The wire used is fine steel pianoforte wire; that
is \hv most I'liistic substance available, and it seems to me, indeed,
by far thr most I'lastie of all the materials known to us [except
crystals: Jan If), \S\)U]. A heavy weight is hung on the lower
end of thr win- to kt'tp it tightly stretched.

Pnil. K<iwIaiHl is going to havr ani»thor machine made, which
J think you will In* iilrast^l with — a continuous wave machine.
I nis <fi tnnii- IN hot II wiwr machine, but a machine for illus-
tiatin;^ ihf vibr/iiiniiN of m finite' gnmp of several elastically
'v/htit ri4t\ jmitiiliN 'ri,,, riintiirting springs are represented by
^ i« t'/r>ii,i,ii| •*|ifiii^Mi„nn ,.( i|,r ihiit' poftious of conuccting wire

yi:,\iiy loiiint.nf.c, ttniUtu^ i., i }„. HVict except to stretch
^r, t-ii..i,.|, ||„, „i,,, |«.iw,'„i, twt) sides of a portable

^A /^ ; \.,, , j; ;,;/ '*«"•' *"-* *•■*• ^"^ » l» \^ x Popular l^U^^

^OLOTROPy OF ELASTIC QUALITY. 165

frame, we might take our model to an ideal lecture-room at the Moleonlar.
centre of the earth, and it would work exactly as you see it working
now. You will understand that these upper masses correspond
to mi, TTi^, nh* In all we have four masses here, of which the
lowest represents the spherical shell lining the ether around
our ideal cavity. I will just apply a moving force to this lower
mass, P. To realize the circumstances of Our case more fully, we
should have a spring connected with a vibrator to pull P with,
and perhaps we may get that up before the next lecture. [Done
by Professor Rowland; see Lecture XIV.] I shall attempt no
more at present than to cause this first particle to move to and
fro in a period perceptibly shorter than the shortest of the three
fundamental periods which we have when the lowest bar is held
fixed. The result is scarcely sensible motion of the others.
I do not know that there would be any sensible motion at all if
I had observed to keep the greatest range of this lowest particle
to its original extent on the two sides of its mean position.

The first part of our lecture this evening I propose to be a
continuation of our conference regarding seolotropy. The second
part will be molecular dynamics. I propose to look at this question
a little, but I want to look very particularly to some of the points
connected with the conceivable circumstances by which we can
account for not merely regular refraction, but anomalous dispersion,
and both the absorption that we have in liquids and very opaque
bodies and such absorption as is demonstrated by the very fine
dark lines of the solar spectrum which are now shown more
splendidly than ever by Prof. Rowland's gratings.

I shall speak now of aeolotropy. The equations by which Molar.
Green realized the condition that two of the three waves having
fronts parallel to one plane shall be distortional, are in this respect
equivalent to a very easily understood condition that I may
illustrate first by considering the more general problem. That
problem is similar to another of the very greatest simplicity,
which is the well-known problem of the displacement of a particle
subject to forces acting upon it in diflerent directions from fixed
centres. An infinitesimal displacement in any direction being
considered, the question is, when is the return force in the
direction of the displacement ? As we know, there are three
directions at right angles to one another, in which the return
force is in the direction of the displacement. The. sole difference

166 LECTURE XIII.

between that very trite problem and our problem of yesterday
(Lecture XII. p. 135), is that in yesterday's the question is
put with reference to a whole infinite plane in an infinite
homogeneous solid, which is displaced in any direction between
two fixed parallel planes on its two sides, to which it is always
kept pfiu:^llel. Considering force per unit of area, we have the
same question, when is the return force in the direction of the
displacement ? And the answer is, there are three directions
at right angles to one another in which the return force is in
the direction of the displacement. Those three directions are
generally oblique to the plane; but Green found the conditions
under which one will be perpendicular to the plane, and the
other two in the plane.

I shall now enter upon the subject more practically in
respect to the application to the wave theory of light, and begin
by preparing to introduce the condition of incompressibility.
Take first the well-known equations of motion for an isotropic
solid and express in them the condition that the body is incom-
pressible. The equations are :

Suppose now the resistance to compression is infinite, which
means, make A; = x at the same time that we have S = 0. What
then is to become of the first term of the second members of
these equations? We simply take (A;+ ^n)S=jo, and write the

second members ^ + nV'f, &c., accordingly. This requires no

hypothesis whatever. We may now take A; = oo , S = 0, without
interfering with the form of our equations. These equations,
without any condition whatever as to f, 17, f, with the condition
p = {k-\-^n)S, are the equations necessary and sufficient for the
problem. On the other hand, if A; = 00 , the condition that that
involves is

dx dy dz *

FRENCH CLASSICAL NOMENCLATURE. 167

which gives four equations in all for the four unknown quantities Molar.

Precisely the same thing may be done in respect to equations
(1), (2), (3) of p. 132 above for a solid with 21 independent
coefficients. We will have this equation, S = 0, again for an
»olotropic body, and a corresponding equality to infinity. I am
not going to introduce any of these formulas at present. In
the meantime, I tell you a principle that is obvious. In

order to introduce the condition j^ + j^ + ^ = 0 into our

ax ay dz

general equation of energy with its 21 coefficients, which involves

a quadratic expression in terms of the six quantities that we

have denoted by e, /, g, a, 6, c, we must modify the quadratic

into a form in which we have (e +/+ gY with a coefficient That

coefficient equated to infinity, and c+/ + 5r = 0, leave us the

general equations of equilibrium of an incompressible seolotropic

elastic solid.

I want to call your attention to the kind of deviation from
isotropy which is annulled by Qreen's equations among the
coefficients expressing that two out of the three waves shall be
purely distortional, and the third shall be condensational-rare-
factional.

The next thing to an isotropic body is one possessing what
Bankine calls cyboid asymmetry. Rankine marks an era in
philology and scientific nomenclature. In England, and I
believe in America also, there has been a classical reaction,
or reform, according to which, instead of taking all our
Greek words through the French mill changing k (kappa) into
c, and V (upsilon) into y, we spell in English, and we pronounce
Greek words, and even some Latin words, more nearly according
to what we may imagine to be the actual usage of the ancients.
We cannot however in the present generation get over Kuros
instead of Cyrus, Kikero instead of Cicero. Rankine is a curious
specimen of the very last of the French classical style. Rankine
was the last writer to speak of cinematics instead of kinerruUica.
Cyboid, which he uses, is a very good word, but I do not know
that there is any need of introducing it instead of Ctibic, Cubic
is an exception to the older classical derivation in that u is not
changed into y; it should be cybic, and cube should be cybe
(I suppose /cuffo^ to be the Greek word). Cyboid obviously means

168 LECTURE XIIL

foUr. cube-like, or cubic, and it is taken firom the Greek in Rankine's
manner, now old-fashioned.

Rankine gives the equations that will leave cubic asymmetry.
He afterwards makes the very apposite remark that Sir David
Brewster discovered that kind of variation from isotropy in
analcime. I only came to this in Rankine two or three days ago.
But I remember going through the same thing myself not long
ago, and I said to Stokes — (I always consult my great authority
Stokes whenever I get a chance) — " Surely there may be some-
thing found in nature to exemplify this kind of asymmetry;
would it not be likely to be found in crystals of the cubic class ?"
Stokes — he knows almost everything — instantly said, " Sir David
Brewster thought he had found it in cubic crystals, but there
is another explanation ; it may be owing to the effect of the
cleavage planes, or the separation of the crystal into several
crystalline lamiuas'' — I do not remember all that Stokes said,
but he distinctly denied that Brewster's experiment showed a true
instance of cubic optical asymmetry. He pointed out that an
exceedingly slight deviation from cubic isotropy would show
very markedly on elementary phenomena of light, and might be
very readily tested by means of ordinary optical instruments.
The fact that nothing of the kind has been discovered is
absolute evidence that the deviation, if there is any, from optic
isotropy in a crystal of the cubic class, is exceedingly small in
comparison with the deviation from isotropy presented by ordinary
doubly refracting crystals.

ifoleoalar. As a matter of fact, square aeolotropy is found in a pocket
handkerchief or piece of square-woven cloth, supposing the warp
and woof to be accurately similar, a supposition that does not
hold of ordinary cloth. Take wire- cloth carefully made in
squares; that is symmetrical and equal in its moduluses with
reference to two axes at right angles to one another. There
will be a vast diflFerence according as you pull out one side
and compress the other, or pull out one diagonal and compress the
other. Take the extreme case of a cloth woven up with inex-
tensible frictionless threads, and there is an absolute resistance to
distortion in two directions at right angles to one another, and
no resistance at all to distortion of the kind that is presented
in changing it from square to rhombic shape. That is to say, a
framework of this kind has no resistance to shearing parallel

RANKINE S NOMENCLATURE.

169

to the sides ; in other words, to the distortion produced Mol/Doola
by lengthening one diagonal and shortening the
other. Now imagine cut out of this pocket hand-
kerchief, a square with sides parallel to the dia-
gonals, making a pattern of this kind. There is
a square that has infinite resistance to shearing parallel to its
sides, and zero resistance to pulling out in the
direction perpendicular to either pair of sides. This
is not altogether a trivial illustration. Surgeons
make use of it in their bandages. A person not
familiar with the theory of elastic solids might cut a strip of
cloth parallel to warp or woof; but cut it obliquely and you have
a conveniently pliable and longitudinally semielastic character
that allows it to serve for some kinds of bandage.

Imagine an elastic solid made up in that kind of way, with
that kind of deviation from isotropy ; and you have clearly two
different rigidities for different distortions in the same plane.
I remember that Rankine in one of his early papers proved this
to be impossible! He proved a proposition to the effect that
the rigidity was the same for all distortions in the same plane.
That was no doubt founded on some special supposition as to
arrangement of molecules and may be true for the particular
arrangement assumed ; but it is clearly fallacious in respect to
true elastic solids. Rankine in his first paper made too short work
of the elastic solid in respect to possibilities of «olotropy. He
soon after took it up very much on the same foundation as Green
with his 21 coefficients, but still under the bondage of his old
proposition that rigidity is the same for all distortions in the
same plane. Yet a little later he escaped from the yoke, and
took his revenge splendidly by giving a fine Greek name " cyboid
asymmetry'* to designate the special crystalline quality of which
he had proved the impossibility !

I must read to you some of the fine words that Rankine has Molar,
introduced into science in his work on the elasticity of solids.
That is really the first place I know of, except in Green, in
which this thing has been gone into thoroughly. It is not
really satisfactory in Rankine except in the way in which he
carries out the algebra of the subject, and the determinants
and matrices that he goes into so very finely. But what I want
to call attention to now is his grand names. I do not know

170 LKCTU&E XIIL

whether Prof. Sylvester ever looked at these names ; I think he
would be rather pleased with them. " Thlipsinomic transforma-
tions/' " Umbra! snrbces" and so on. Any one who will learn the
meaning of all these words will obtain a large mass of knowledge
with respect to an elastic solid. The simple, good words, '' strain
and stress/' are due to Rankine; "potential" energy also. Hear
also the grand words "Thlipsinomic, Tasinomic, Platythliptic,
Euthytatic, Metatatic, Heterotatic, Plagiotatic, Orthotatic, Pan-
tatic, Cybotatic, Goniothliptic, Euthythliptic," &c.

You may now understand what cyboid asymmetry is, or as I
prefer to call it, cubic seolotropy. Rankine had not the word
aeolotropy; that came in from myself* later. Cyboid or cubic
flBolotropy is the kind of aeolotropy exhibited by a cubic grating ;
as it were a structure built up of uniform cubic frames. There
[Feb, 14, 1899 ; a skeleton cube, of twelve equal wooden rods with
their ends fixed in eight india-rubber balls forming its corners]
is a thing that would be isotropic, except for its smaller rigidity
for one than for the other of the two principal distortions in each
one of the planes of symmetry.

I will go no further into that just now ; but I hope that in
the next lecture, or somehow before we have ended, we may be
able to face the problem of introducing the relations among the
21 moduluses which are sufficient to do away with all obliquities
with reference to three rectangular axes. But you can do this in
a moment — equate to zero enough of the 21 coefficients to fulfil
two conditions, (1) that if you compress a cube of the body by
three balancing pairs of pressures, equal or unequal, perpendicular
to its three pairs of faces, it will remain rectangular, and (2) that
if you apply, in four planes meeting in four parallel edges, balanc-
ing tangential forces perpendicular to these edges, the angles at
these edges will be made alternately acute and obtuse, and the
angles at the other eight edges will remain right angles. [Feb. 14,
1899 ; here are the required annulments of coefficients to fulfil
those two conditions, with OX, OY, OZ taken parallel to edges of
the undisturbed cube, and with the notation of Lecture II. p. 23 : —

42 = 0; 43 = 0; 51=0; 53 = 0; 61 = 0; 62 = 01 . .
41=0; 52 = 0; 63 = 0; 56 = 0; 64 = 0; 45 = 0] ••^^'

* See p. lis above.

JiOLOTROPlC ELASTICITY WITHOUT SKEWNESS. lYl

Thus the formula for potential energy becomes reduced to

+ 44a« + 556« -h 66c» (2).

Molar.

With these nine coefficients, 11, 22, 33; 23, 31, 12; 44, 55,
66 ; all independent ; the elastic solid would present two different
rigidities for the two distortions of each of the planes (yz), (zx), (xy),
one by shearing motion parallel to either of the two other planes,
the other by shearing motion parallel to either of the planes
bisecting the right angles between those planes. The values of
these rigidities are as follows for the cases of sheanDg motions
parallel, and at 45°, to our principal planes : —

Distortion

Plane of Distortion

Line of Motion

Rigiditj

a

f-g

(y')

y or z
45'' to y and z

44
i{i(22+33)-23}

h

9-^

{zx)

z or X
45" to z and x

55

i{i(33+ll)-31}

c

(^)

X or y
45** to X and y

66

i{i(ll+22)-12}

From the fact that squares only of a, b, c appear in the equa-
tion of energy, we see that in equilibrium* these distortions are
separately balanced by the tangential stresses S, T, U, And we
conclude that there can be plane waves of purely longitudinal
motion (condensational-rarefactional) and waves of pure distortion
travelling in the directions x, y, z with their fronts perpendicular
to these directions ; and that their velocities f are as shown in the
following table : —

* Leotore n. p. 24.

t Equations (1) above, with equations (4), (7), (8) of Lecture XI. p. 133, applied
respectively to the oases

m=0, n=0, {=0; n=0, Z=0, ij=0; f=0, m=0, ^-=0.

172

LVCTDBK XnL

W»Te-front

Line of Vibrmtkm

Quftlity

Ydoeity

X

conde&satioiud-rarefK^tioiial

n/t

(yz)

y

INirdj distortioiud

v^

z

n n

v^

y

oondeDsatioDal-rarefoctioiud

V?

(«:)

z

purely distortioDal

^/?

X

M W

n/?

z

oondeosatioDal-rarefactioiial

n/?

(^)

X

purely difitortional

/65

y

w »

/44

V7

Selecting from these the purely distortional waves, and taking
them in pairs having equal velocities, we have the following
convenient table : —

Wave-front

{xz)

Line of Vibration

Plane of Distortion

Velocity

y

z

(y^)

-J-,

z

X

{zx)

^-:

{zx)

X

y

(^)

n/?

EQUALISATION OF VELOCITIES FOR ONE PLANE OF DISTORTION. 178

In this table we find one and the same velocity, V44/p, for two Molar,
different waves with fronts parallel to x, and having their lines of
vibration parallel to y and z respectively, and therefore each having
yz for its plane of distortion. If now we apply the formulas of
Lecture XI., p. 133, to investigate the velocities of other waves
having the same plane of distortion — for instance, waves with
fronts parallel to x and lines of vibration at 45"* to y and z, — we
find different propagational velocities unless

44 = i{i(22 + 33)-23}*,

which makes them all equal. Thus, and by similar considerations

relatively to y and z, we see that each of the three propagational

velocities given in the preceding table is the same for all waves

having the same plane of distortion, if the following conditions

are fulfilled: —

44 = i{i(22 + 33)-23}\

55=i{i(33 + ll)-31} (3).

66 = i{i(n + 22)-12}^

These three equations simply express the condition that in
each of the three coordinate planes the rigidity due to a shear
parallel to either of the two other coordinate planes is equal to
the rigidity due to a shear parallel to either plane bisecting the
right angle between them.

Green f found fourteen equations among his 21 coefficients to
express that there can be a purely distortional wave with wave-
front in any plane whatever. Three more equationsj express
further that the planes chosen for the coordinates are planes
of symmetry. The conditions which we have considered have
given us, in (1) and (3) above, fifteen equations which are identical
with fifteen of Green's. His other two are

11 = 22 = 33 (4).

With all these seventeen equations among the coefficients, the
equation of energy becomes reduced to

2£=ll(e+/+5r)H44(a«-4/gf)+55(6»-45re)-f-66(c«-4«/)...(5).

* Compare with formnla n = |(^ - IS) in Lecture n. p. 25, which iR the condition
that the rigidity is the same for all distortions in any one of the three coordinate
planes for the case of 11 = 22 = 33, there considered.

t CollecUd Papers, p. 309.

X Ibid,, pp. 303, 309.

174 LECTURE XIII.

Molar. It is easy now to verify Green's result. Remember that if we

denote by ^, 17, ^ infinitesimal displacements of a point of the
solid from its undisturbed position (a?, y, z)y we have

,(6).

,(7).

*~di' ■'~dy'' ^~dz

dy dz^ dz dx^ dx dyj

Denoting now by 1 1 1 dxdydz integration throughout a volume

F, with the condition that f , 1;, f are each zero at every point of
the boundary, we find by a well-known method of double integra-
tion by parts*,

And by aid of this transformation we find

.///«w.{**(|-S)%-(i-2)'..6(S-D-}

(8).

Let now V be the space between two infinite planes, paralle^
to the fronts of any series of purely distortional waves traversing
the space between them, these planes being taken at places at
which for an instant the displacement is zero. They may, for
instance, be the planes, half a wave-length asunder, of two
consecutive zeros of displacement in a series of simple harmonic
waves. Let P be any plane of the vibrating solid parallel to
them ; let p be its distance from the origin of coordinates ; and

q, its displacement at any instant. Then ^ ^ is the molecular

* Compare Math, and Phy$, Papen, Art. sen. Vol. m. p. 448 ; and '* On the
Beflexion and Befraction of Light,'' Phil Mag. 1888, 2nd half-year.

green's dynamics for fresnel's wave-surface. 175

rotation of the solid infinitely near to this plane on each side ; and Molw.
by the rectangular resolution of rotations, we have

dp dy dz^ dp dz dx^ dp dx dy"

where i, m, n denote the direction cosines of the line in P perpen-
dicular to the direction of q, this being the axis of the molecular
rotation. Using these new values in (8), and remarking that the
first chief term vanishes because the displacement is purely
distortional, we have

2 jjjdxdydz E = (44P + 55m« + 66n») jjjdxdydz (^\. . .(10).

Hence we see that do condensation or rarefaction accompanies
our plane distortional waves, and that their velocity of propagation
is

'44P + 55m« + 667i>

/

(11).

This is Fresnel's formula for the propagational velocity of a
plane wave in a crystal with (I, m, n) denoting the direction of
vibration ; while in Green's theory (I, m, n) is the line in the
wave-front perpendicular to the direction of vibration. The
wave-surface is identical with Fresnel's.]

I will read to you Green's own statement of the relative
tactics of the motion in his and in Fresnel's waves. Here it
is at the bottom of page 304 of Green's Collected Papers : " We
"thus see that if we conceive a section made in the ellipsoid
" to which the equation (10) belongs, by a plane passing through
"its centre and parallel to the wave's front, this section, when
" turned 90 degrees in its own plane, will coincide with a similar
"section of the ellipsoid to which the equation (8) belongs, and
"which gives the directions of the disturbance that will cause
"a plane wave to propagate itself without subdivision, and the
"velocity of propagation parallel to its own front. The change
" of position here made in the elliptical section is evidently
" equivalent to supposing the actual disturbances of the ethereal
" particles to be parallel to the plane usually denominated as the
'* plane of polarization.'*

Before we separate this evening, return for a few minutes
to our problem of vibratory molecules embedded in an elastic

176 LECTURE XIII.

Molecular, solid ; and let us consider particularly the application of this
dynamical theory to the Fraunhofer double dark line D of sodium-
vapour.

[March 1, 1899.* For a perfectly definite mechanical repre-
sentation of Sellmeier's theory, imagine for each molecule of
sodium-vapour a spherical hollow in ether, lined with a thin rigid
spherical shell, of mass equal to the mass of homogeneous ether
which would fill the hollow. This rigid lining of the hollow we
shall call the sheath of the molecule, or briefly the sheath.
Within this put two rigid spherical shells, one inside the other,
each moveable and each repelled from the sheath with forces, or
distribution of force, such that the centre of each is attracted
towards the centre of the hollow with a force varjring directly as
the distance. These suppositions merely put two of Sellmeier's
single-atom vibrators into one sheath.

Imagine now a vast number of these diatomic molecules, equal
and similar in every respect, to be distributed homogeneously
through all the ether which we have to consider as containing
sodium-vapour. In the first place, let the density of the vapour
be so small that the distance between nearest centres is great in
comparison With the diameter of each molecule. And in the
first place also, let us consider light whose wave-length is very
large in comparison with the distance from centre to centre of
nearest molecules. Subject to these conditions we have (Sellmeier's
formula)

Ci)'-

i + ^rr^+,.^. • (1);

where m, m, denote the ratios of the sums of the masses of one
and the other of the moveable shells of the diatomic molecules in
any large volume of ether, to the mass of undisturbed ether filling
the same volume ; Ky k^ the periods of vibration of one and the
other of the two moveable shells of one molecule, on the supposition
that the sheath is held fixed; v^ the velocity of light in pure
undisturbed ether; r, the velocity of light of period t in the
sodium-vapour.

* This replaces the concluding portion of the Lecture as originally delivered. It
was read before the Koyal Society of Edinburgh on Feb. 6, 1899, and reprinted in
Phil. Mag. for March, 1899, under the title " Application of Sellmeier's Dynamical
Theory to the Dark Lines Dj, D, produced by Sodium-Vapour.'*

DTKAMICS OF FRAUNHOFER'S DARK LINES DiD,. 177

For sodium-vapour, according to the measurements of Rowland Molecnlar.
and Bell*, published in 1887 and 1888 (probably the most
accurate hitherto made), the periods of light corresponding to the
exceedingly fine dark lines Di, D^ of the solar spectrum are
-589618 and '589022 of a michronf. The mean of these is so
nearly one thousand times their difference that we may take

* = H* + *,) (l - 20oo) ' *' = *<* + *'>(^ + 200o) ^^^'

Hence if we put

T = i(^ + 0(l + i^) (3);

and if a? be any numeric not exceeding 4 or 5 or 10, we have

whence

T* ^ 1000 , T* 1000
T«-.^« ' 2xJ^V T»-/t;""2d?-l ^^'

Using this in (1), and denoting by fi the refractive index from
ether to an ideal sodium-vapour with only the two disturbing
atoms rw, w,, we find

©■-"■=

, lOOOm . lOOOro, ,„.

1 + 2^T-1+2^1 ^^>-

When the period, and the corresponding value of x according
to (3), is such as to make fi* negative, the light cannot enter the
sodium-vapour. When the period is such as to make /i" positive,
the proportion, according to Fresnel and according to the most
probable dynamics, of normally incident light which enters the
vapour would be

1

-^^ <'^

if the transition from space, where the propagational velocity is
Ve, to medium in which it is v,, were infinitely sudden.

Judging from the approximate equality in intensity of the

* Rowland, Phil. Mag. 1887, first hftlf-year; Bell, Phil Mag. 1888, first half-
year.

t See footnote, p. 150.

T. L. 12

178 LECTURE XIII.

Moleonlar. bright lines Di, D, of incandescent sodium-vapour; and from the
approximately equal strengths of the very fine dark lines Di, Dj
of the solar spectrum; and from the approximately equal strengths,
or equal breadths, of the dark lines Di, D, observed in the
analysis of the light of an incandescent metal, or of the electric
arc, seen through sodium- vapour of insufficient density to give
much broadening of either line ; we see that m and m, cannot be
very different, and we have as yet no experimental knowledge to
show that either is greater than the other. I have therefore
assumed them equal in the calculations and numerical illustrations
described below.

At the beginning of the present year I had the great pleasure
to receive from Professor Henri Becquerel, enclosed with a letter
of date Dec. 31, 1898, two photographs of anomalous dispersion
by prisms of sodium-vapour*, by which I was astonished and
delighted to see not merely a beautiful and perfect demonstration
of the " anomalous dispersion" towards infinity on each side of the
zero of refractivity, but also an illustration of the characteristic
nullity of absorption and finite breadth of dark lines, originally
shown in Sellmeier's formulaf of 1872 and now, after 27 years,
first actually seen. Each photograph showed dark spaces on
the high sides of the Dj, D, lines, very narrow on one of the
photographs; on the other much broader, and the one beside
the i>2 line decidedly broader than the one beside the A line;
just as it should be according to Sellmeier's formula, according
to which also the density of the vapour in the prism must have
been greater in the latter case than in the former. Guessing
from the ratio of the breadths of the dark bands to the space
between their D^, D, borders, and from a slightly greater breadth
of the one beside D,, I judged that m must in this case have been
not very different from '0002 ; and I calculated accordingly from
(6) the accompanying graphical representation showing the value

of 1 , represented by y in fig. 1. Fig. 2 represents similarly

1

the value of 1 — for m = '001, or density of vapour five times

* A description of Professor Beoquerel's experiments and resalts will be found
in CompUi Rendtu, Dec. 5, IS98, and Jan. 16, 1899.

t Sellmeier, Pogg. Ann, Vol. cxlv. (1872) pp. 899, 620; Vol. cxlvii. (1872)
pp. 887, 625.

DTNAHICS OF ¥RA.VSR0¥SB.'8 DARK LIKES D,Dt. 179

Fig. 1*. m=-0l)09. HolMDlar.

1

1

~

tt

0.

B

6

?

^

f

n

9

^

"v

/f

w^

l'■^ ws

-X

^

^

EJ

^

^

1

'le-

9.

n

= ■001

a

D,

-

S

,

,

•<\

--

_

JL

XHO

f-

Dl

35

r-

--

M4

\

\

-

-

* In Bgi. 1 uid 2 th« Dj, D, Uoei ue touched by oottm ot finite oi
y = 4- 1 ; vdA in figt. S, 4, uid 6 at y=0. The left-hand side of each &az\ baod ii
Ml asTinptota to the enrrei of figs. 1 and 2, and a tangent at y = 0 to the onrves of
figs. 3, 1, and 6. The diignmi ooald not ahow these ohaneteristiM olearly onkM
on a maeh larger Male.

12—2

• 'Z. . Ts?. •: inri -t represent
':ia£tiu:n:4i 7 lunnailv' incident lisfat

i=r<eai.

=1^

= -^1.

^ M

»=-a5.

■■s-

« X

••i.-a *

•H^>

>•

■M/'.^

■Viii

*>»j

>o

*■

•K«J ..

■ Li

■ ••

•»tas

•

■

v;^

-||IS1W

'^•y

s,*s

-|li£K.

-A»>.

'M»U

^'«

DYNAMICS OF PRAtTNHOPEB's DARK LINES D,t>,. 181

According to Sellraeier's formula the light transmitted through MolMolar,
k layer of sodium -vapour (or any transparent substance to which

yu

m

'

1

^m.

-

-1

■

^

1

-

i

P

1

■nil*

■4

9

y

J

the formula is applicable) is the same whatever be the thickness
of the layer (provided of course that the thickness is many times
the wave-length). Thus the A, D, lines of the spectrum of solar
light, which has traversed from the source a hundred kilometres
of sodium-vapour in the sun's atmosphere, must be identical
in breadth with those seen in a laboratory experiment in the
spectrum of light transmitted through half a centimetre or a
few centimetres of sodium -vapour, of the same density as the
densest part of the sodium-vapour in the portion of the solar
atmosphere traversed by the light analysed in any particular
observation. The question of temperature cannot occur except
in so tar as the density of the vapour, and the clustering in groups
of atoms or non -clustering (mist or vapour of sodium), are con-
cerned.

A grand inference &om the experimental foundation of Stokes'
and KircbhofiTs original idea is that the periods of molecular
vibration are the same to an exceedingly minute degree of
accuracy through the great differences of range of vibration pre-
sented in the radiant molecules of an electric spark, electric arc,
or 6ame, and in the molecules of a comparatively cool vapour
or gas giving dark lines in the spectrum of light transmitted
through it.

It is much to be desired that laboratory experiments be made,
notwithstanding their extreme difficulty, to determine the density
and pressure of sodium -vapour through a wide range of temperature,
and the relation between density, pressure, and teniperatui-e of
gaseous sodium.

it^

vidi c k1+*-

Ar/3 f«rx-^*rr:.''>rr.Li' :i:-a'. lie dark iiirt or band enends tkniwh
If*'; rw.;?*: '/ vi -i*A f'^ wLki • ?, »• '^ i* i«iCTt and di« l^;»i^
i* z^o at tL*: h:^-ffr ir^rfer, w^ see &*:-& • 1 > d>« *« ** ^"^

—'— (8X

TSKC to T

^1+M

Au an *:XMU]At: KQitJ^ble to illustrate the broadening of the
<lark lin«; by iacrfr^hed deiwity of the gas, I take »asaxl(r^,
abd tak'2 a ti^jiuh m<AHmUi nurueric not greater than 10 « 20-
Thii» giv<i8 for th'; rang'; of the dark band from

t^kU, T = iicn-iaxia-^) W'

and for largi* valuc-H of t it makes the refractive mdex
1 +4tt X 10 ♦, and therefore the refractivity, Ja x lO"^. If for
example we Uke a « 6, the refractivity would be -0003, which is
nearly the Harne jih the refractivity of common air at oidinaiy
atmoHpheric denHity.

KiK. 7.

/»

\

AitymptoU, y = 1*0002

_i

AtymptoU ysl

li

\)^

la

U\l.l

(Otni

iwi

j_jj

IOCS

DYNAMICS OF A SINGLE DARK LINB. 183

Taking k = 1000, we have, for values of t not differing from Moleonla
1000 by more than 10 or 20,

T^^IOOO ^j^^^ ar = T-1000 (10).

T* — /c" 2a? .

Thus we have

V

i+K (>')•

In fig. 7 the curve marked /it represents the values of the refractive
index corresponding to values of r through a small range above
and below k, taking a = 4. The other curve represents the
proportionate intensity of the light entering the vapour, calculated
from these values of /it by (7) above.

The table on next page shows calculated values for the ordi-
nates of the two curves ; also values (essentially negative) for the
formula of intensity calculated from the negative values of /i
algebraically admissible from (11).

The negative values of fi have no physical interpretation for
either curve ; but the consideration of the algebraic prolongations
of the curves through the zero of ordinates on the left-hand side
of the dark band illustrates the character of their contacts. The
physically interpreted part of each curve ends abruptly at this
zero ; which for each curve corresponds to a maximum value of x.
The algebraic prolongation of the fi curve on the negative side is
equal and similar to the curve shown on the positive side. But
the algebraic prolongation of the intensity curve through its zero,
as shown in the table, differs enormously from the curve shown on
the positive side. To the degree of approximation to which we
have gone, the portions of the intensity curve on the left and
right hand sides of the dark band are essentially equal and
similar. This proves that so far as Sellmeier s theory represents
the facts, the penumbras are equal and similar on the two sides
of a single dark line of the spectrum uninfluenced by others. It
is also interesting to remark that according to Sellmeier as now
interpreted, the broadening of a single undisturbed dark line,
produced by increased density of the gas or vapour, is essentially
on the high side of the finest dark line shown with the least
density, and is in simple proportion to the density of the gas.]

— V

1 ' -».

4

0 0

-A*

'/^

-r^^

-l-s^-*

LECTURE XIV.

Monday, October 13, 8 p.m.

At this lecture were seen, immediately behind the model Moleoiilar.
heretofore presented, two wires extending from the ceiling and

bearing a long heavy bai- about three feet above the floor by
means of closely-fitting rings. By slipping these rings along the
bar, the period of vibration of this bifilar suspension could be
altered at will. Two parallel pieces of wood, jointed at each end,

166 LECTURE XlV.

Moleoolar. served to transmit the azimuthal motion of this vibrator to the
lower bar P of the model

Let us look at this and see what it does. I have not seen it
before, and it is quite new to me. Oh, see, you can vary the
period; that is very nice, that is beautiful. We are going to
study these vibrations a little, just as illustrations. Prof. Rowland
has kindly made this arrangement for us, and I think we will all
be interested in experimenting with it. We have this bar P,
moved by this bifilar pendulum H, which is so massive that its
period is but little aflFected, I suppose, by being connected with P.
It takes some time before the initial firee vibrations in the model
are got quit of and the thing settles into simple harmonic motion
corresponding to the period of the bifilar pendulum. If we keep
this pendulum going long enough through nearly a constant range,
the masses P, m^, fn,, m^ will settle into a definite simple har-
monic motion, through the subsidence of any firee vibrations which
may have been given to them in the start. We see the whole
apparatus now performing very nearly a' simple harmonic motion.
We will now superimpose another vibration on this by altering
the period of the pendulum very slightly. That, you see, seems
to have diminished very much the vibrations of the system. They
are now increasing again. That will go on for a long time. I
shall give this pendulum a slight impulse when I see it flagging,
to keep its range constant. When it is in its middle position,
I apply a working couple. We will give no more attention to it
than just to keep it vibrating, while we look at these notes which
you have in your hands, and which I have prepared for you so
as to shorten our work on the black-board.

[These notes related to the tasinomic treatment of seolotropy.
The discussion of them was interrupted at intervals to continue
experiments and observations on Professor Rowland's model. That
part of the Lecture has been omitted bom the print as the subject
has been treated in the addition of date February 14, 1899 to
Lecture XIIL, pp. 170—175.]

Let us stop and look at our vibrating apparatus. It has been
going a considerable time with the exciter kept vibrating through
a constant range, and you see but small motion transmitted to
the system. That is an illustration of the most general solution
of our old problem*. Our *' handle " P is in firm connection with

* Lecture III. Pi. 2, p. 3S.

ROWLAND^S MODEL VIBRATOR. 187

the bifilar pendulum (the exciter) and is forced to agree with it. Moleooli
Let us bring the system to rest. Now start the exciter and keep
it going. In time the viscosity will annul the system of vibrations,
representing the difference between zero and the permanent state
of vibration which the three particles will acquire. If there were
no loss of energy whatever, the result would be that this initial
jangle, which you now see, would last for ever; consisting of
a simple harmonic motion in the exciter and a compound of the
three fundamental modes of these three particles viewed as a
vibrating system with the bar P held fixed. Let the system with
the bar P held fixed be set to vibrate in any way whatever; then its
motion will be merely a compound of those three fundamental
modes. But now set the exciter going, and the state of the case
may be looked on as thus constituted ; — the exciter and the whole
system in simple harmonic motion of the same period, and, superim-
posed upon that, a compound of the three modes of simple harmonic
vibration that the system can perform with the exciter fixed. We
cannot improve on the mathematical treatment by observation;
and really a model of this kind is rather a help or corrective to brain
sluggishness than a means of observation or discovery. In point
of fact, we can discover a great deal better by algebra. But
brains are very poor after all, and this model is of some slight
use in the way of making plain the meaning of the mathematics
we have been working out.

The system seems to have come once more into its permanent
state. Let us stop the exciter and see how long the system will
hold its vibration. The reaction on the exciter is very slight, it is
very nearly the same as if that massive bar H were absolutely
fixed. But the motion actually communicated to it, since it is
not absolutely fixed, will correspond to a considerable loss of
energy. A very slight motion of H with its great length and
mass has considerable energy compared even with the energy of
our particle of greatest mass ; so that our system will come to
rest far sooner than if H were absolutely fixed. The model is at
present illustrating phosphorescence. You see the particles (t^^,
ma, mj) have gone on vibrating for a whole minute, and rrii must
have performed a couple of dozen vibrations at least. A true
phosphorescence of a hundred seconds' duration is quite analogous
to the giving back of vibrations which you see in our model,
only instead of our two or three dozen vibrations, we have in

188 LKCrURE XIV.

[oleenlar. phosphorescence 40,000 million-million vibrations during a hundred
seconds. Now, we cannot get 1000 residual vibrations in our
model because of the dissipation of energy arising from imperfect
elasticity in the wire, friction between parts of the model, and the
resistance of the air. That dissipation of energy is simply the
conversion of energy from one state of motion, (the visible motions
which we have been watching), into another (heat in the wire,
heat of frictions, and heat in the air). In molecular dynamics, we
have no underground way of getting quit of energy or carrying it
off. We must know exactly what has been done with it when the
vibration of an embedded molecule ends, even though this be
not before a thousand million-million periods have been performed.
[March 6, 1899. Imagine a homogeneous mass of rock — ^granite or
basalt, for example — as large as the earth, or as many times larger as
you please, but with no mutual gravitation between its parts to
disturb it Let there be, anywhere in it very far from a boundary,
a spherical hollow of 5 cms. radius, and let a violin-string be
stretched between two hooks fixed at opposite ends of a diameter
of this hollow, and tuned to vibrations at the rate of 1000 per
second. Let this string be set in vibration (for the present, no
matter how) according to its gravest fundamental mode, through a
range of one millimetre. Let the elasticity of the string and of the
granite be absolutely perfect, and let there be no air in the hollow
to resist the vibrations. They will not last for ever. Why not ?
Because two trains of waves, respectively condensational-rarefac-
tional and purely distortional, will be caused to travel outwards,
carrying away with them the energy given first to the vibrating
string (see below § 28 of addition to Lecture XIV.).] We must
suppose the elasticity of our matter and molecules to be perfect, and
we cannot in any part of our molecular dynamics admit unaccounted
for loss of energy ; that is to say, we cannot admit viscous terms
unless as an integral result of vibrations connected with a part of
the system that is not convenient for us to look at.

In three minutes our system has come very nearly to rest.
We infer therefore that in three minutes from a commencement of
vibration of the exciter we shall have nearly reached the permanent
state of things.

Now we vary the period of the exciter, making it as nearly as
we can midway between two fundamental periods of our complex
molecule. We will keep this going in an approximately constant

MOTION OF ETHER WITH EMBEDDED MOLECULE& 189

range for a while and look at the vibrations which it produces in Molecular,
the system.

Now you see very markedly the difference in the vibrations
of our system after it has been going for several minutes with
the exciter in a somewhat shorter period of vibration than that
with which we commenced. Here is another still shorter. In
the course of two or three minutes the superimposed vibrations
will die out. See now the tremendous difference of this case in
which the period of the exciter is approximately equal to one
of the fundamental periods of the system, or the periods for the
case in which the lowest bar is held absolutely fixed.

I had almost hoped that I would see some way of explaining
double refraction by this system of molecules, but it seems more
and more diflBcult. I will take you into my confidence to-morrow,
if you like, and show you the difficulties that weigh so much upon
me. I am not altogether disheartened by this, because of the
fact that such grand and complicated and highly interesting
subjects as I have named so often, absorption, dispersion and
anomalous refiraction, are all not merely explained by their
means but are the inevitable results of this idea of attached
molecules.

There is one thing I want to say before we separate, and that
is, when I was speaking last of the subject, I saw what seemed
to me to be a difficulty, but on further consideration, I find it
no difficulty at all. Not very many hours after I told you it
was a difficulty, I saw that I was Avrong in making it appear
to be a difficulty at all. I do not want to paint the thing any
blacker than it really is and I want to tell you that that question
I put as to the ether keeping straight with the molecules is
easily answered when there is a large number. Our assumption
was a large number of spherical cavities, lined with rigid spherical
shells and masses inside joined by springs or what not : and the
distance from cavity to cavity small in comparison with the wave-
length. It then happens that the motion of the medium rela-
tively to the rigid shells will be exceedingly small and a portion
of the medium that will contain a large number of these shells will
all move together (see below, addition to Lee. XIV.). If the distance
from molecule to molecule is very small in comparison with the
wave-length, then you may look upon the thing as if the structure
were infinitely fine, and you may take it that the ether moves

190 LECTURE XIV.

liolecular. quite straight with them all, and not in and out among them,
as I said. It is evident on the other hand, when the wave-
length of the light traversing the medium is moderate in
comparison with the distance between the molecules, that it
must move out and in among them. But if the stifihess of the
medium is such as to make the wave-length large in comparison
with the distance from molecule to molecule, this stiffness is suf-
ficient to keep them all together, and you may regard these rigid
shells as bonds of attachment by which the molecule is pulled
this way and that way, so that we may suppose our reactionary
forces, of which Ci{^ — Xi) is a sample, to be absolutely the same
in their effect upon the medium as if they were uniformly
distributed through it.

That takes away one part of our discontent. The only diflB-
culty that I see just now is that of explaining double refraction.
The subject grows upon us terribly, and so does our want of
time. If it is not too much for you I must have one of our
double lectures to-morrow.

[Twice* in this Lecture, and indeed many times in preceding
and subsequent Lectures, I felt the want of a full mathematical
investigation of spherical waves originating in the application of
force to an elastic solid within a limited space. I have therefore
recently undertaken this work "I", and I give the following statement
of it as an addition to Lecture XIV.

§ 1. The complete mathematical theory of the propagation of
motion through an infinite elastic solid, including the analysis of
the motion into two species, equivoluminal and irrotational, was
first given by Stokes in his splendid paper ''On the Dynamical
Theory of Diflfraction J." The object of the present communication
is to investigate fiiUy the forcive which must be applied to the
boundary, S, of a hollow of any shape in the solid, in order to
originate and to maintain any known motion of the surrounding
solid; and to solve the inverse problem of finding the motion

* p. 18S, and pp. 189, 190.

t Comnmnioated to B. S. E. on May 1, 1899, and pablished in Phil. Mag. May,
Aug. and Oct., 1899 under the title ** On the Application of Force within a Limited
Space, required to produce Spherical Solitary Waves, or Trains of Periodic Waves,
of both species, Equivoluminal and Irrotational, in an Elastic Solid."

X Stokes, Mathematieal Papen, Vol. u. p. 248.

SPBEmCAL WAVES IN ELAJ3TIC SOLID.

191

when the forcive on, or the motion of, S is given, for the particular Molar,
case in which 5 is a spherical surface kept rigid.

§ 2. Let f, 17, t denote the infinitesimal displacement at any
point of the solid, of which (x, y, 2) is the equilibrium position.
The well-known equations of motion* are

(1).

where 8 denotes -r +^ + ^- Using the notation of Thomson

ax ay dz

and Taitf for strain-components (elongations; and distortions),
^i f* g\ fit, 6> c ; we have

e

dz dy dx dz'

dy dx

(2);

and with the corresponding notation P, Q, R; S, T, U, for stress-
components (normal and tangential forces on the six sides of an
infinitely small rectangular parallelepiped), we have

iJ = (* + in)5r+(&-|n)(6+/) [ (3).

Let now a be an infinitesimal area at any point of the surface

S ; X, /A, 1/ the direction-cosines of the normal ; and Xa, Ta, Zc

the components of the force which must be applied from within to

produce or maintain the specified motion of the matter outside.

We have

"X^PX-^ UfjL-hTp

- Z = jBi/ + TX + S/A .

(4);

>i

* See my paper ** On the Beflexion and Refraction of Solitary Plane Waves, &o,
Proe. R. S, E, Deo. 189S, and PJnl, Mag, Feb. 1899 ; reprinted below in the present
volnme.

t Thomson and Tait*s Natural Philoiophy, § 669, or ElemenU, § 640.

in

192 LECTURE XIV.

Molar. whence by (3)

- Z = ( Jfc - |n) \8 + n ( 2\c + |AC + I*)

-F=(&-}n)/iS + n(2/i/+i;a + Xc)V (5).

-Z =(*- Jn)i/S + n (2i^ + X6 + Ata)!

These equations give an explicit answer to the question, What
is the forcive ? when the strain of the matter in contact with S is
given. We shall consider in detail their application to the case in
which S is spherical, and the motions and forces are in meridional
planes through OX and symmetrical round this line. Without
loss of generality we may take

z = 0; giving i/ = 0. a = 0, 6 = 0, Z = 0 (6).

Equations (5) therefore become

- Z = (i - |n) \8 + n (2\e + /ic)

- F= (A: - |fi) /iS + n (2/I/+ Xc) ' "

§ 3. In §§ 5 — 26 of his paper already referred to, Stokes gives
a complete solution of the problem of finding the displacement
and velocity at any point of an infinite solid, which must follow
from any arbitrarily given displacement and velocity at any
previous time, if after that, the solid is left to itself with no
force applied to any part of it. In a future communication I
hope to apply this solution to the diffraction of solitary waves,
plane or spherical. Meantime I confine myself to the subject
stated in the title of the present communication, regarding which
Stokes gives some important indications in §§ 27 — 29 of his paper.

§ 4. Poisson in 1819 gave a complete solution of the equation

^=«^^'«' (8)

in terms of arbitrary functions of x, y, z, representing the initial
values of w and -jj ; and showed that for every case in which w
depends only on distance (r) from a fixed point, it takes the form

where F and / denote arbitrary functions. In my Baltimore
Lectures of 1884 (pp. 46, 86, 87 above) I pointed out that solutions

SPHERICAL WAVES IN ELASTIC SOLID. 193

expressing spherical waves, whether equivoluminal (in which there Molar,
is essentially different range of displacement in different parts of
the spherical surface) or irrotational (for which the displacement
may or .may not be different in different parts of the spherical
surface), can be very conveniently derived from (9) by differen-
tiations with respect to x, y, z. It may indeed be proved, although
I do not know that a formal proof has been anywhere published, that
an absolutely general solution of (8) is expressed by the formula

K0(0(£)'{-i['('-3^/(-3]}-'

r = V[(^ - xj + (y - yj + (z - zj] (10),

where 2 denotes sums for different integral values of A, i, j, and
for any different values of x\ y\ z\

§ 5. I propose at present to consider only the simplest of all
the cases in which motion at every point {x, 0, 0) and (0, y, z) is
parallel to X'X ; and for all values of y and z, f is the same for
equal positive and negative values of x. For this purpose we of
course take a?' = y' = / = 0 ; and we shall find that no values of A,
*, j greater than 2 can appear in our expressions for f , 17, f, because
we confine ourselves to the simplest case fulfilling the specified
conditions. Our special subject, under the title of this paper,
excludes waves travelling inwards from distant sources, and there-
fore annuls f{t + r/v).

§ 6. In §§ 5 — 8 of his paper Stokes showed that any motion
whatever of a homogeneous elastic solid may, throughout every
part of it experiencing no applied force, be analysed into two
constituents, each capable of existing without the other, in one of
which the displacement is equivoluminal, and in the other it is
irrotational. Hence if we denote by (^1, 771, fi) the equivoluminal
constituent, and by (^21 Vn* Sa) ^^^ irrotational constituent, the
complete solution of (1) may be written as follows : —

f=fiH-|2; ^-^i + ^a; r=?i + ?, (11),

where fi, 171, ?i and ^2, ^a, S2 fulfil the following conditions, (12)
and (13), respectively : —

dx dy dz '

«8V«!J =r^' M«V«« -^' u^V^t -^. U^--

"" ^' dt'' ^^^1- dt' ' ^'''W ~p)

T. L. 13

. ...(12);

(13).

194 LECTURE XIV.

Molar. t —— — ^^ y —^

^^"d^' "^'^ dy' ^^~ dz'
w being any solution of

dt* ' p

The first equation of (12) shows that in the (f,, rji, 5i) con-
stituent of the solution there is essentially no dilatation or con-
densation in any part of the solid ; that is to say, the displacement
is equivoluminal. The first three equations of (13) prove that in
the (f,, i7j, fa) constituent the displacement is essentially irrota-
tional.

§ 7. We can now see that the most general irrotational
solution fulfilling the conditions of § 5 is

giving

d^i d7f2 , dfa _ 1 d F^ .- .,.

Tv^Ty^Tz-v^dx^ ^'*^'

and the most general equivoluminal solution fulfilling the same
conditions is

^'"da^r u^r' '^''' dxdy r ' ^'^ dxdzr ^ ^'

giving

i+l'+f-" "^').

where J?\ and F^ are put for brevity to denote arbitrary functions
of it — 1 and it j respectively. Hence the most general solu-
tion fulfilling the conditions of § 5 is

where for brevity ^ denotes a function of r and t, specified as
follows : —

</.(r,0 = ^.(«-3 + ^,(«-^) (17).

Denoting now by accents differential coefficients with respect to r.

SPHERICAL WAVES IN ELASTIC SOLID.

195

and retaining the Newtonian notation of dots to signify differential
coefficients with reference to t, we have

Molar

,(18).

Working out now the differentiations in (16), we find

§ 8. For the determination of the force-components by (7), we
shall want values of S, e,/, and c. Using therefore (2) and going
back to (16) we see that

.(19).

(20).

Hence, and by (19), we find

+4^+-.-ii (21).

By (16), (14'), and (15') we find

»' da; r

1£\ 1^1

1 j»*, 1 i\

and by (19) directly used in (2) we find

(22);

e — X '

•^ ^' (^ V r* r» ■*■ r« r" J i"" r* r* J

(23).

Remark here how by the summation of these three formulas we
find for 6 +/+5r the value given for h in (22).

13—2

196

LECTURE XIV.

§ 9. These formulas (22) and (23), used in (5), give the force-
components per unit area at any point of the boundary (/S of § 1)
of a hollow of any shape in the solid, in order that the motion
throughout the solid around it may be that expressed by (19).
Supposing the hollow to be spherical, as proposed in § 2, let its

radius be q. We must in (23) and (5) put

*
x=q\\ y^qfi^'y z=^qv (24);

and putting i/ = 0, we have, as in (7), the two force-components for
any point of the surface in the meridian z = 0, expressed as
follows : —

Z = (Jk-§n)a,V -n[2V^+2(2\«+l)J5 + (X=+l)
T = (ifc -fn) a,X/A- n\fi(2A + 45 + C.,)

where

4>'" 6*" 15*' 154>
q q^ (t q"

^^|...(25),

<^'_3;^ 3^

• • • • •

C = — 4- ^*

* qu^ q^u*

C ^—^ 4- ^

^ qv' 5 V
<E> and S denoting </> and F with q for r.

y

(26),

§ 10. Returning now to (19), consider the character of the
motion represented by the formulas. For brevity we shall call
XX' simply the dans, and the plane of YY\ ZZ' the equatorial
plane. First take y = 0, <^ = 0, and therefore a? = r. We find by
aid of (18)

(axial) | = i5^ + ^(5 + f)+^(^i + ^.); ^ = 0; r=0...(27).

Next take x = 0 and we find

iP, 1 //, ^;

(equatorial) f=-lg- ^ (J^ ^'j-^Ci-. +i.,); ,=

0;?=0
(28).

SPHERICAL WAVES IN ELASTIC SOLID. 197

Hence for very small values of r we have Molar.

(axial)|=^(^, + ^3)

(equatorial) ^ = --^(^1 + ^2)
and for very large values of r,

.(29);

r r

.(30).

(equatorial) f %-;-^

Thus we see that for very distant places, the motion in the
axis is approximately that due to the irrotational wave alone ; and
the motion at the equatorial plane is that due to the equivoluminal
wave alone : also that with equal values of F^ and F^ the equi-
voluminal and the irrotational constituents contribute to these
displacements inversely as the squares of the propagational
velocities of the two waves. On the other hand, for places very
near the centre, (29) shows that both in the axis and in the
equatorial plane the irrotational and the equivoluminal con-
stituents contribute equally to the displacements.

§ 11. Equations (25) and (26) give us full specification of
the forcive which must be applied to the boundary of our spherical
hollow to cause the motion to be precisely through all time that
specified by (19), with F^ and F^ any arbitrary functions. Thus
we may suppose Fi {t — qju) and F^ (t — q/v) to be each zero for all
negative values of t, and to be zero again for all values of t exceed-
ing a certain limit t. At any distance r from the centre, the
disturbance will last during the time

from « = ^^^ to « = -— ? + T,|

\ (31).

and from t = to t= + t

u u

Supposing t; > w, we see that these two durations overlap by

an interval equal to

r-q r-q

h T

V u

(32).
if r — q<T

198

LECTURE XIV.

On the other hand, at every point of space outside the radius
5'4-t/(1/iz— 1/t;) the wave of the greater propagational velocity
passes away outwards before the wave of the smaller velocity
reaches it, and the transit-time of each wave across it is t. The
solid is rigorously undisplaced and at rest throughout all the spaces
outside the more rapid wave, between the two waves, and inside
the less rapid of the two.

§ 12. The expressions (25) and (26) for the components of the
sur&tce-forcive on the boundary of the hollow required to produce

• • • • • •

the supposed motion, involve ^^ and <v^a. Hence we should have
infinite values for ^ = 0 or i = t, unless F^ and F^ vanish for ^ = 0
and ^ = T, when r = 5^. Subject to this condition the simplest
possible expression for each arbitrary function to represent the two
solitary waves of § 11, is of the form

2^
-^=(1 -"%')'» where x = 1

.(33).

Hence, by successive differentiations, with reference to t,

^=-^x(i-xr '

48

y

(34).

The annexed diagram of four curves represents these four
functions (33) and (34).

§ 13. Take now definitively

where

fi{t-l)=o.<f(i-xi'y

(35),

(36).

SPHERICAL WAVES IN ELASTIC SOLID.

SMleo

J:.

i

Ecale of F.

a.

i.

M F.

„

Jf=

tIt

P-

200

LECTURE XIV.

Consider now separately the equivoluminal and the irrotational
motions. Using (19), (18), (35), (34), and taking the equi-
voluminal constituents, we have as follows: —

'Equatorial, x = 0; ^i = 0,

/Cone of latitude 45°, a^ = xy = ^r\ \

Axial, a^ = r*, 171 = 0,

.(37),

(38),

(39);

where

«^.=(i-xi')'; ^/=-(i-xi')'x.; ^."=-i+6xi'-5xi^..(40).

.(«);

,(42);

§ 14. Similarly for the irrotational constituents ;

I Equatorial, a? = 0; 172 = 0,

Cone of latitude 45°, a^='Xy = \i^,
f> = C2 (^i^<^9 + ^^V+^:^^2j»

1^72 - ica ^— ^2 + ^:^ ^2 +^^,^ ^^2 J ;

Axial, a? = r*, 172 = 0,

where <^a &c. are given by (40) with x-^ for x, ; Xi ^^^ X^ being
given by (36).

§ 15. The character of the motion throughout the solid, which
is fully specified by (19), will be perfectly understood after a
careful study of the details for the equatorial, conal, and axial

,(43);

SPHERICAL WAVES IN ELASTIC SOLID. 201

places, shown clearly by (37)... (43) for each constituent, the equi- Molar,
voluminal and the irrotational, separately. The curve S in the
diagram of § 12 shows the history of the motion that must bte given
to any point of the surface S, for either constituent alone, and
therefore for the two together, in any case in which g is exceed-
ingly small in comparison with the smaller of the two quantities
t4T, ^^^, which for brevity we shall call the wave-lengths. The curve
J^ shows the history of the motion produced by either wave when
it is passing any point at a distance from the centre very great in
comparison with its own wave-length. But the three algebraic
functions ^, S^ S all enter into the expression of the motion
due to either wave when the faster has advanced so far that its
rear is clear of the front of the slower, but not so far as to make
its wave-length (which is the constant thickness of the spherical
shell containing it) great in comparison with its inner radius.
Look at the diagram, and notice that in the origin at £f, a mere
motion of each point in one direction and back, represented by ^,
causes in very distant places a motion {S) to a certain displace-
ment (2, back through the zero to a displacement 1*36 x (2 in the
opposite direction, thence back through zero to d in the first
direction and thence back to rest at zero. Remark that the
direction of d is radial in the irrotational wave and perpendicular
to the radius in the equivoluminal wave. Remark also that the
d for every radial line varies inversely as distance from the
centre.

§ 16. Draw any line OPK in any fixed direction through
0, the centre of the spherical surface 8 at which the forcive
originating the whole motion is applied. In the particular

case of ^ 12... 15, and in any case in which ^lU — -J and

-F, u — -j are each assumed to be, from ^=0 to ^= r, of the

form ^{r — tyAiV, where i denotes an integer, the time-history
of the motion of P is 5o-h-Bi<+ ... -{-B^^il^^^ and its space-history
(t constant and r variable) is G^^r'^-^-C^r"* + ... + (75+fr*+*; the
complete formula in terms of t and r being given explicitly by (19).
The elementary algebraic character of the formula : and the exact
nullity of the displacement for every point of the solid for which
r>q-{'Vt\ and between r = g -f- v (^ — r) and r = g + w^, when

202 LECTURE XIV.

v(t — T)>ut; and between r = q and r = g + w (^ — t), when t>T;
these interesting characteristics of the solution of a somewhat
intricate dynamical problem are secured by the particular character
of the originating forcive at S, which we find according to §§ 8, 9
to be that which will produce them. But all these characteristics
are lost except the first (nullity of motion through all space out-
side the spherical surface r = q -\- vt), if we apply an arbitrary
forcive to 8*, or such a forcive as to produce an arbitrary deforma-
tion or motion of 8. Let for example jS be an ideal rigid spherical
lining of our cavity ; and let any infinitesimal arbitrary motion
be given to it. We need not at present consider infinitesimal
rotation of 8: the spherical waves which this would produce,
particularly simple in their character, were investigated in my
Baltimore Lecturesf , and described in recent communications to
the British Association and Philosophical MagazineX. Neither
need we consider curvilinear motion of the centre of 8, because
the motion being infinitesimal, independent superposition of ar-, y-,
;?-motions produces any curvilinear motion whatever.

§ 17. Take then definitively S{t\ or simply S, an arbitrary
function of the time, to denote excursion in the direction OX, of
the centre of 8 from its equilibrium-position. Let 3~*^, 3~*^

denote I dtS and I dt I dtS, Our problem is, supposing the solid

to be everywhere at rest and unstrained when ^ = 0, to find (f , 17, f )
for every point of the solid (r > q) at all subsequent time {t posi-
tive); with

at r=q, ^=^S(t), 17 = 0, f=0 (44).

These, used in (19), give

0 =

U' V* I? L ^ V J (jT* *\ /JJ

(45).

* If the space inside S is filled with solid of the same quality as outside, the
solution remains algebraic, if the forcive formula is algebraic, though discontinuous.
The displacement of S ends, not at time C=r when the forcive is stopped, but at
time t = r + 2g/u when the last of the inward travelling wave produced by it has
traveUed in to the centre, and out again to r=9.

t Pp. 81—83 and 169, 160 above.

X B,A. Report, 1898, p. 783; Phil Mag, Nov. 1898, p. 494.

SPHERICAL WAVES IN ELASTIC SOLID.

203

and

Molar.

^<')=4.[^^+"¥]-^t^'<'>+^'<'>4# <*'>•

Adding Sq x (46) to (45), we find

39^(e) 2 -^ + -j^

.(47);

whence

£llO ^ 2 "?i-^ + 3g9-'^(0 (48);

and by this eliminating ^a from (46),

[a* + - (u + 2t;) a + ^ (u» + 2t;«)1 ^, (0 = c^(0 (49),

where d denotes d/dt, and

^(0 = -?»'(l + y9-' + ^3-*)^(0

.(50).

§ 18. I hope later to work out this problem for the case of
motion commencing from rest at ^ = 0, and S{t) an arbitrary
function ; but confining ourselves meantime to the case of 8
having been, and being, kept perpetually vibrating to and fro in
simple harmonic motion, assume

^(^) = Asinft>^ (51).

With this, (50) gives

m = hqu^ |^(|J, - 1) sin a>« +^ cos a>t^ (52).

To solve (49) in the manner most convenient for this form of <^{t),
we now have

^i(0 =

3' + - (u + 2t;)3 + -, (u« + 2v^)
q o*

^{t)

3' + ^ (u' + 2v') --(u + 2v)d

2. 1 ^/(O

[9'+^,(«'+2t^)]'-[^(« + 2f)aJ

204

LECTURE XIV,

[olar. = hqtt* X

{^.<"*+^^) + ^.<"' + ^""-"') + "i"°"^+V(^'-^"')

COS a)t

If

(m» + 2«') - «»

« a)»

+ 7i (« + 2v)»

(53).

With ^I'thus determined, (48) gives J^, as follows,

^=?^-3^'sin«e (54).

tr w' o)"

For f , 17, f by (19) we now have

V = B{r,t)xy
^ = B{r, t) xz

where, with notation corresponding to (26) above,

1 J^.(«.)
r u*

(55).

r»L u»

+ ^ [^1 («.) + Mm ;

= ^-^--2:

U V

,(56).

§ 19. The wave-lengths of the equivoluminal and rotational

waves are respectively — and . For values of r very great

in comparison with the greater of these, the second members of
(55) become reduced approximately to the terms involving l^i and
^,. These terms represent respectively a train of equivoluminal
waves, or waves of transverse vibration, and a train of irrotational
waves, or waves of longitudinal vibration ; and the amplitude of
each wave as it travels outwards varies inversely as r.

§ 20. For the case of an incompressible solid we have v = oo ,
which by (53) gives

4?^ = - i*? sin a,< (57);

SPHERICAL WAVES IN ELASTIC SOLID.

205

and by (55) we have, for r very great,

f=-fAj sin ««(---)

Molar.

.(58),

V = -^hq sin mt ^ V

f = - fAg sin ft>e —

which fully specify, for great distances from the origin, the wave-
motion produced by a rigid globe of radius g, kept moving to and
fro according to the formula h sin tat

§ 21. The strictly equivoluminal motion thus represented
consists of outward-travelling waves, having direction of vibration
in meridional planes, and very approximately* perpendicular to

the radial direction, and amplitude of vibration equal to f A - sin 0,

where 0 denotes the angle between r and the axis. The gradual
change from the simple motion |= Asina>^ at the surface of the
rigid globe, through the elastic solid at distances moderate in
comparison with g, out to the greater distances where the motion
is very approximately the pure wave-motion represented by (58),
is a very interesting subject for detailed investigation and illustra-
tion. The formulas expressing it are found by putting t;= oo in
(45), and using this equation to determine ^2(0 ^^ terms o( Fi {t)',
then using (47) to determine F^ (t) in terms of ^(^) given by (51) ;
and then using (55) and (56), for which t; = oo makes t, = ^, to
determine f , 17, f. They are as follows : —

f = £(r, e)a?'

(qy 3h u ^ a 3A . .

— - IS cos (oti ■}- - -jr sin cjti ;

\rj 2 qo) r z

Su 1

— COSft)^

qo) J

V = B{r,t)xy', ^^B{r,t)xz

.(59),

* Bigoroasly so, if the wave-length and q are each infinitely smaU in comparison
with r.

[olar. where
fB(r,

')=©"¥[(

LECrUHE XIV.

1 — --. sin tot + ~r— .sin oa, coa tot

g'o)'/ qhoi' qto

+ ) s — COS o)i, —- -TT s\n mt,,
\r/ 2 qw ' r 2 ^

..(60).

+ 11

v^

■■

,.■

.■

/:" a*"

-- ^ c^A

J.-"^ ^>^'^

^ *?^^^

% - , .,^

^"-^ <^''<'

^^ -^i _

^ r-'ft-r

/ \ ^>t X-^ '■

'^ ^ 4^

/ _„

^ 3 jI ■

/ j_

ii'HM iiimmi MiiiMiij

Fig, 1.

For the particular case of the wave-length equal to the radius we
have

^-^ m.

SPHERICAL WAVES IN ELASTIC SOLID.

207

which enables us to write simply l/2Tr for u/qw in equations (59) Holar.
and (60). For graphic representation of this case we take z = 0,
which makes all the dittplacements lie in the plane (xy).

§ 22. The accompanying drawings help us to understand
thoroughly the character of the motion of the solid throughout
the whole infinite apace around the vibrating rigid globe. They
show displacements and motions of points of which the equilibrium
positions are in the equatorial plane, in the cone of 45° latitude.

1

1

"■ 2

T

7

Z^

l

:t "

J>

^.k

■>>

d5'^3

^v ^

^\ y

^

\.-\ ■

"* "■ - ^ ^ -. i 'h

■■>^£s

'

i ^i;

~~~-^

S-?

"^-^

■-

\ s

,-

\ IE i' i" '

,-

>

. t

» — -

4 ^

-■-\.\ \-\A..} ^\, 1 i LL-^ t

Fig. 2.

and in the axial line. Fig. 1 represents displacements at an
instant when the globe is moving rightwards through its middle
position. Fig. 2 shows displacemeuls a quarter- period later, when
the globe is at the end of its rightward motion. Each figure shows

208 LECTURE XIV.

Molar. also the orbit of a single particle of which the eqailibrium position
is in the 45° cone, at a distance f g from the centre of the globe.
The orbital motion is in the direction of the hands of a watch. It
is interesting to see illustrated in fig. 2 how the axial motion is
gradually reduced frt>m + A at the surface of the globe to a very
small range at distance q from the surface, or %q from the centre,
and we are helped to understand its gradual approximation to
zero at greater and greater distances by the little auxiliary
diagrams annexed, in which are shown by ordinates the magnitudes
of the axial displacements at the two chosen times.

§ 23. The gradual transition from motion h sin tot parallel to
the axis at the surface of the globe, to motion

— s - A sin 0 sin tat
2 r

at great distances from the globe in any direction, is interestingly
illustrated by the conal representations in the two diagrams for
the case 0 = 4o^ It should be remarked that in reality h ought
to be a small fraction of 5. the radius of the globe, practically not
more than y^, in order that the strains may be within the limits
of elasticity of the most elastic solid, and that the law of simple
proportionality of stresses to strains (Hooke s Ut tensio sic vis) may
be approximately true. In the diagram we have taken h = ^q]
but if we imagine every displacement reduced to ^ of the amount
shown, and in the direction actually shown, we have a true, highly
approximate representation of the actual motions, which would be
so small as to be barely perceptible to the eye, for a globe of 6 cms.
diameter.

§ 24. Return now to our solution (53), (54), (55), (56) for
arbitrary or periodic motion of a rigid globe embedded in an
isotropic elastic solid of finite resistance to compression and finite
rigidity. For distances from the globe very great in compaiison
with q, its radius, that is to say for q/r very small, (55) and (56)
become

f)'=^.B(r,t)xy\ mB(r,t)xz.

.(62);

SPHERICAL WAVKS IN ELASTIC SOLID.

209

r |_ u' "J

where

V

Molar.

(63).

Putting now in the equations (62) the value of B (r, t) from
(63), and eliminating ^2(^) hy (47), we find

fN-

r» - iF» J^, (e,)

t£=

^=5 ^

[2^l(^> + 3?^(«.)J

)-...(64).

?^

^"^■<^> + ^[2 4.^V3g^(«.)l

W^

14'

The terms of these formulas, having ^ and ^ respectively for
their arguments, represent two distinct systems of wave-motion,
the first equivoluminal, the second irrotational, travelliDg outwards
from the centre of disturbance with velocities u and v,

§ 25. I reserve for some future occasion the treatment of the
case in which S(t) is discontinuous, beginning with zero when
^ = 0 and ending with zero when ^= t. I only remark at present
in anticipation that ^j (t\ determined by the differential equation
(49), though commencing with zero at ^ = 0, does not come to zero
at t = T, but subsides to zero according to the logarithmic law
(e"**) as t goes on to infinity; and that therefore, as the same
statement is proved for ^j (t) by (48), neither the equivoluminal
nor the irrotational wave-motion is a limited solitary wave of
duration r, but on the contrary each has an infinitely long
subsidential rear.

§ 26. For the general problem of the globe kept in simple
harmonic motion, Asinw^, parallel to OX, we may write (53)
for brevity as follows : —

T. L.

14

^

^ r —

I

- e€t*i.

Si 1

*-^ ":..3

ti^ m J t *i

1 ' 1

f* VruM 'vf t^ irt<je;duiL ^M^ ^793^. fa- r^'SK nwfiarwvf in^ like

>

(661

T(m;«m; €^\%^\MHm rtfj0nitieui two isetB of simple harmoaic wares,
it4^mvh\Hmu$H\ anrl imAaXvmaX^ U/r which the ware-lengths are
rfmjf*'/^iv(:\y 2^rw/«, i/irvjw. The maximom displacements in the
iw// m'M Hi fffftuiM of the ama of semi-vertical angle ^ and axis OX,
tiftt ri*M\Hu:i\ytAy,

(<;/)iiivoliifriirml) Kin 0 ~2 ^(K^ + IJ) ;

r (67).

(irroUtion/iI) cam 0 ^^ ^jm - 2Kf + 4L»]

r

jK 5^7, TImi mil! of tmnmniHHioii of energy outwards by a
nIii^Im m<I, of wiivcm of (;ith<T HpocieH is equal, per period, to the
«MIM of iJin kinotic; (itid poUiiitiul energies, or, which is the same,
Ivvlnn iJio wlioln kiiiotic (Jncjrgy, (»f the medium between two
I'oni'niil.i'l^ Nplinric/il MurfiieoM, of radii differing by a wave-length.
Now Mm iivnmgo kiiintin oncirgy throughout the wave-length in
Hiiy |Miil. of l.ho HphiM'iciil nIk^II is half the kinetic energy at the
(iiMl.aiit of iiMuliiMiiu volonity. lieuco the total energy transmitted

SPHERICAL WAVES IN ELASTIC SOLID. 211

per period is equal to the wave-length multiplied into the surface- Molar.

integral, over the whole spherical surface, of the maximum kinetic

energy at any point per unit of volume : and therefore the energy

transmitted per unit of time is equal to the product of the propa-

gational velocity into this surface-integral. Thus we find that the

rates per unit of time of the transmission of energy by the two

sets of waves, of amplitudes represented by (67), are respectively

as follows: —

47r
(equivoluminal) -^ pA'gW {K* + X') u

2^ y (68)-

(irrotational) -^ pAYft>«[(3 - 2Ky + 4Z«]t;

§ 28. The sum of these two formulas is the whole rate of
transmission of energy per unit of time, and must be equal to
the average rate of doing work by the vibrating rigid globe upon
the surrounding elastic solid. Hence if w denote this rate, we
must have

w=^ AV«' {2tt (K^ + X«) + 1; [(3 - 2K)^ + 4Z»]) . . .(69).

§ 29. To verify this proposition, let us first find the resultant
force, P, with which the globe presses and drags the elastic
solid, and then the integral work which P does per period, and
thence the average work per unit time. Going back to § 9, we
see that P is the surface-integral of X over the spherical surface
of radius q. Hence by the first of equations (25), which, in virtue
of the equations

k'\'^n = pv^; n = pu* (70),

we may write as follows : —

Z = p{aaW-[2X»(il + (7,) + 2(2\» + l)fi-h(A.'+l)ai]ti^j ...(71),

we find

p = l!^ {Cf,t;«-2(il -h 55 + 2(7^ + a,)i^»}
S \qv q^ \qu q^/^

••• • •

= 9\2(v-u)^^-^2(^-^u^)^\ + 3vi-^ — s\ (72),

q [ qu^ ^u* q J

14—2

212

LECTURE XIV.

Molecular, where Q denotes ^^rq^p, being the mass which our rigid globe
would have if its density were equal to that of the elastic solid.
Hence by (65), for simple harmonic motion in period 27r/»,

P = Qa)«A \(2K''' "^

IV

J*ft>'

.21"^ + ^^^
qoi (for J

sin o>t

+ f2ir"^ + 2Z;'^ + ?^)oos«4 (72');

or, substituting the values of K and L from (65), and denoting by
D the common denominator,

D

X \ — -r (t/«+ 2o» + gW) - -r-Au^ - «»)« + -,- ,(4ut;- «») sin ©e

r2u/9t;* 3t;« -\ v /9w* 3u» ,\1 J

+ — (ir-4 + Ii 5+ 1 +- L* i + -T 8+ 1 C08G>4
L<jra> \g*ft)* g^o)' / ga> \g*ft>* g'©* /J J

(72").

This may be written for brevity

P = A(asinft>f + 6cosft)0 (72").

Finally for w we have, denoting the period by r,

w = - rdtPS{t) = h^bw - l^dt cos» (ot = A A^to (73)

^(2a,«A«r2i./9^ 31^ N .
2/) L? W®* g^o)^ / g

/ 9it* 3?^^

g*ft)* / g \g^ci>* g'o)*

...(73').

§ 30. To verify the agreement of this direct formula for the
work done, with (69) which expresses the effect produced in
waves travelling outwards at great distances from the centre,
is a very long algebraical process, with K and L in (69) given
by (65). But it becomes very simple by the aid of the following
modified formulas for ^^ {t) and J^j {t), which are also useful
for other purposes. From (48), (49), and (50), by eliminating
^1, we get an equation for ^^ similar to (49), viz..

where

ry+-(w+2t;)a + -'5(w« + 2t;«)1j^, = jr(0,

M(t) = qv'(

- 3m^ - . 2u
g g

? 9-^)^(0

y ...(74).

SPHERICAL WAVES IN ELASTIC SOLID.

213

We may now write (50) in the form

c^(0 = hqu^O sin (o>t + a), ]

and similarly • (75),

^(^) = hqv^H sin (ayt -/3)]

where the values of 0, H, a, )8, are given by the following
equations : —

G cos a = -^— , — 1 ; G sm a = —

Moleoulai

5'ft)'

qto

ircos)8 = l-?^,; irsin)8 = -l

gW q(Of

From the second of equations (53) we therefore have

,(76).

and

CY /^\ L ,>n, Jlfsin(ft>^ + a)-iVcos(a>e+a)\
^, (0 = hqu'G ^ WTN' ^ ■

J^,(t) = hqv'H-

Msiu (cot - )8) - JVTcos (tat - 13)

on

where M^ and iV' denote the terms of the denominator of the
third of equations (53). By the same method of investigation
as that which gave us (69), we now find for the sum of the rates
of transmission of energy

w^~ h^q
o

W

(78).

Substituting the values of G, H, M, N, and introducing the
notation Q, we obtain

w = ^Q(o'

2w/9v* 3t;» l^.!i/'^^ ^ A

— ,(i/«+2v«)-l

(79).

This agrees with the value of w given by (73') ; and thus the
verification is complete.

§ 31. In (78) the numerator of the last factor shows the
parts due to the equivoluminal and irrotational waves respectively.
Denoting by J the ratio of the energy of the equivoluminal wave
to that of the irrotational, we have

J =

vH

/9u* 3a« ,\

_ 2mG» _ \q*

.(80).

214

Lecture xiv.

§ 32. Consider the following four cases : —

(a) qc9 very large in comparison with the larger %

of u and v.

2u

J«

(6) qa> = V.

J =

26

- + — + -T-
u V tr

(c) qto = u.

of u and i;.

^_ 2 fu Sv 9tA
5© very small in comparison with the smaller

...(81)l

J=?^

it'

If V =00 , cases (a) and (6) cannot occur; and in cases (c), (ci)
we see by (81) that •/= oo ; that is to say, the whole energy is
carried away by the equivoluminal waves. If v is very small in
comparison with u, we find that although J is infinite in cases (a)
and (c), it is zero in cases (6) and (d). This to my mind utterly
disproves my old hypothesis* of a very small velocity for irrotational
wave-motion in the undulatory theory of light

§ 33. Let us now work out some examples such as that
suggested in an addition of March 6, 1899, to this Lecture
(p. 188), but with the simplification of assuming a rigid massless
spherical lining for the cavity, which for brevity I shall call the
sheath. But firnt let us work out in general the problem of
finding what force in simple proportion to velocity must be
applied to a moss rn mounted on massless springs as described
in p. 145 above, to keep the sheath vibrating in simple harmonic
motion h sin cot, and therefore to do the work of sending out the
two sets of waves with which we have been concerned. Let y
be the required force per unit of velocity of m ; so that ye is
the working force that must be applied to m, at any time when
€ is its displacement from its mean position. Now the springs,
which must act on the sheath with the force P of (72') above,
must react with an equal force on m because they are massless ;

* «* On the Reflexion and Refraction of Light," Phil. Mag. 188S, 2Dd half year.

SPHERICAL WAVES IN ELASTIC SOLID.

215

SO that the equation of motion of m is

d^e ^ de

Molar.

And, by the law of elastic action of the springs, we have

P = c(e--Asinfi)0 (83),

where c denotes what I call the "stiflfness '* of the spring-system.

§ 34. For e, (83) and (72'") give

e = h

■(> - f ) "

sm a>t-{-- cos cot
c

1

,(84);

and with this in (82) we find

nuo^

(' - f) "

sm cot-{-- cos cot
c

]

= (a-{-yco-jsmcot+\b — yco f 1 + - ) sin w^,

which requires that

( 1 + -) mo)' = a + - 7«, and - mco^ = 6 — ( 1 + -) 7« ;

\ C / C C \ C /

by which, solved for two unknown quantities, yco and m<a^, we
find

and

mco^ =

c[a(a + c) + ¥]

.(86).

(a + cy + 6«

If we suppose co and c known, these equations, with (72"), (72"')
for a and b, tell what m/Q must be in order that the force
applied to maintain the periodic motion of the sheath shall vary
in simple proportion to the velocity ; and they give 7, the mag-
nitude of this force per unit velocity.

§ 35. If we denote by E the maximum kinetic energy of m,
we find immediately from (84),

E = yi'^mco'

hij^m

.(87).

And by (73) we have, for the work per period done on the sheath
by p.

TW = iTA'«6 (88).

216 LECTURE XIV.

ioleeoUur. This ought to agree with the work done by ye per period, being
rate ye, which, by (84), is Jty^'A^ [(l + ^V + Q'l ...(89).

The agreement between (89) and (88) is secured by (85).

§ 36. By (87), (89), and to) = 27r ; and by (86), (85), we find

^ _ m _ mo)' _ g (o -h c) 4- 6* .q^v

TW "~ T7 ~ 27r7a) "" 27r6c

which, as we shall see, is a very important result, in respect to
storage of energy in vibrators for originating trains of waves.

§ 37. Remark now that a, b, c are each of the dimensions
of a "longitudinal stiflfness," that is to say Force -^ Length, or
Mass -5- (Time)" ; and for clearness write out the full expressions
for a and 6 from (72") and (72'") as follows ;

^ ~ ^^ fu^ -h 2i;» _ Y /u 4- 2t;y

2w/9t^ 3t^ \ jv /9u* 3?*"^ 11

b = Q

qto

§ 38. Let q<o be very large in comparison with the larger of u
or V (Case I. of § 32). We have

q^or q(o

therefore ? = 0; —=5^ (92).

b rw ZTTC

This case is interesting in connection with the dynamics of
waves in an elastic solid, but not as yet apparently so in respect
to light.

§ 39. Let qto be very small in comparison with the smaller of
w or v (Case IV. of § 32). We have

therefore ^r^^^^^^^'^; ^N?^\

•-.

.(93).

SPHERICAL WAVES IN ELASTIC SOLID. 217

This case is supremely important in respect to molecular sources MoUr.
of light.

§ 40. Let c be very small in comparison with a + f/o. We
have

E . a^b + b . bd" ,. .Q..

§ 41. Let c = 00 . We have

■^ = 2^6' 'y'"^^' "*"*=" ^®^^-

This is the simple case of a rigid globe of mass m embedded
firmly in the elastic solid, and no other elasticity than that of the
solid around it brought into play. It is interesting in respect to
Stokes' and Rayleigh's theory of the blue sky.

§ 42. Let t; = 00 . We have

This case is of supreme interest and importance in respect to
the Dynamical Theory of Light.

§ 43. Take now the particular example suggested in the
addition of March 6, 1899 (p. 188 above), which ia specially
interesting as belonging to cases intermediate between those of
§ 38 and § 39 ; a vast mass of granite with a spherical hollow of
ten centimetres diameter acted on by an internal simple harmonic

vibrator of 1006^ periods per second [being 1000 a/-^)- This

makes a)« = 40 x 10«, jW = 10», o) = 6324, qo) = 31620. Now the
velocities of the equivoluminal and the irrotational waves in
granite* are about 2*2, and 4 kilometres per second; so we have
w = 2-2 X 10», t;= 4 X 10». Hence, and by (80) and (91),

u u^ ,« . u*

= 6-957 ; -f-, = 48-4 ; --: = 2342-56 ;
qo) q^(o^ q*(o^

— =12-649; -^ = 160; -~ =25600;

,...(97).

a

J= 11-96; a=1891xQa)»; 6 = 25-59xQa)^ j = 7-390

♦ Gray and Milne, Phil Mag,, Nov. 18S1.

218 LECTURE XIV.

MoleeoUr. And by (85), (86), (90) ; with for brevity c = 8Qw\

25-59. j^ ^ ,. m [1891 (189'1 -f g) 4- 6548] s'

'^^ (1891 +«)« + 654-8 ^ ^"^ ' Q" " (189-1 +«)» + 654*8

E 7-390 (189-1 + s) + 2559 , , ^^ 226

— = ^^ ■ ^ = 1176 + —

TW zirs 8

(98X

§ 44. As a first sub-case take (§ 41) c = oo : we find by (95),
(97), m= 1891Q; and E/tw = 1176. These numbers show that
the kinetic energy of m at each instant of transit through its
mean position, supplies only 1*176 of the energy carried away in
the period by the outward travelling waves ; though its mass is
as much as 189 times that of grauite enough to fill the hollow.
Hence we see that if the moving force ye were stopped the motion
of m would subside very quickly and in the course of six or seven
times T it would be nearly aunuUed. The not very simple law of
the subsidence presents an extremely interesting problem which is
easily enough worked out thoroughly according to the methods
suitable for § 25 above. Meantime we confine ourselves to cases
in which E/rw is very large.

§ 45. Such a case we have, under ^ 39, 41, if instead of 1006^
periods per second we have only 10065; which makes g'G)' = 1000;
qa> = 10 VIO = 31-620 ; and, by (93), (95) with still our values of u
and V for granite,

a = 1-892 xlVx Q(o^ ; ^ = 7394 ; — = 1177 (99).

b TW ^ ^

Hence the kinetic energy of m in passing through the middle
of its range is nearly 1200 times the work required to maintain
its vibration at the rate of 10065 periods per second : and the
value found for a, used in (95), shows that m, supposed to be a
rigid globe filling the hollow, must be 189 million times as dense
as the surrounding granite, in order that this period of vibration
can be maintained by a force in simple proportion to velocity.
(See § 40.) It is now easy to see that if the maintaining force
is stopped, the rigid globe will go on vibrating in very nearly the
same period, but subsidentially according to the law

A€«"sin^^ (100);

SPHERICAL WAVES IN ELASTIC SOLID. 219

and there will be the corresponding subsidence in the amplitudes Moleoukr.
of the two sets of waves travelling outwards in all directions at
great distances from the origin, when, according to the pro-
pagational velocities u, v, the effects of the stoppage of the
maintaining force reach any particular distance.

It is quite an interesting mathematical problem, suggested at
the end of § 44, to fiilly determine the motion in all future time,
when m is left with no applied force, after any given initial
conditions, with any value, large or small, of m/Q,

§ 46. Returning now to the maintenance of vibrations at
the rate of 1006^ periods per second ; and w, v for granite ; and
q = 5 cm., all as in § 43 ; see (98) and remark that, to make
E/tw very large, 8 must be a very small fractional numeric ; and
this makes m/Q =8. To take a vibrator not differing greatly in
result from the violin string suggested in the addition of March 6
(p. 188), let m be a little ball of granite of ^ cm. diameter. This
makes m = 0/8000, and therefore (§ 40) 8 = 1/8000. Hence by
(98), E/tw = 1808001, from which, with what we know of wave-
motion, we infer that if m be projected with any given velocity, V,
from its position of equilibrium, it will for ever after vibrate,
with amplitude diminishing according to the formula

Ce'm^sin ^^ (101).

T

Thus during 3,616,002 periods the range of m will be reduced in
the ratio of € to 1 (say approximately 2f to 1), by giving away its
energy to be transmitted outwards by the two species of waves,
of which, according to J of (97), the equivoluminal takes twelve
times as much as the irrotational.]

LECTURE XV.

Tuesday, October 14, 6 p.m.

Molecular. RETURNING to our model, we shall have in a short time a state
of things not very diflferent from simple harmonic motion, if we
get up the motion very gradually. We have now an exciting
vibration of shorter period than the shortest of the natural
periods. We must keep the vibrator going through a uniform
range. We are not to augment it, and it will be a good thing
to place something here to mark its range. [This done.] Keep it
going long enough and we shall see a state of vibration in which
each bar will be going in the opposite direction to its neighbour. If
we keep it going long enough we certainly will have the simple
harmonic motion ; and if this period is smaller than the smallest
of the three natural periods, we shall, as we know, have the alternate
bars going in opposite directions. Now you see a longer-period
vibration of the largest mass superimposed on the simple har-
monic motion we are waiting for. I will try to help towards
that condition of afifairs by resisting the vibration of the top
particle. In fact, that particle will have exceedingly little motion
in the proper state of things (that is to say, when the motion is
simple harmonic throughout), and it will be moving, so far as it
has motion at all, in an opposite direction to the particle im-
mediately below it. It is nearly quit of that superimposed
motion now. We cannot give a great deal of time to this, but
I think we may find it a little interesting as illustrating
dynamical principles. Prof. Mendenhall is here acting the part
of an escapement in keeping the vibrator to its constant range.
We cannot get quit of the slow vibration of the particle. A
touch upon it in the right place may do it. A very slight touch

EXCITATION OF SYNCHRONOUS VIBRATOR BY LIGHT. 221

is more than enough. I have set it the wrong way. Now we Molecular,
have got quit of that vibration and you see no sensible motion
of m, at all. These two, (m,, m,), are going in opposite directions,
and the lower one in opposite direction to the exciter. There-
fore this is a shorter vibration than the shortest natural period.
Now I set it to agree with the shortest of the periods, the first
critical position. If we get time in the second lecture to-day,
I am going to work upon this a little to try to get a definite
example illustrating a particle of sodium. Before we enter upon
any hard mathematics, let us look at this a little, and help our-
selves to think of the thing. What I am doing now is very
gradually getting up the oscillation. I am doing to that system
exactly what is done to the sodium molecule, for example, when
sodium light is transmitted through sodium vapour. We may feel
quite certain, however, that the energy of vibration of the
sodium molecule goes on increasing during the passage through
the medium of at least two-hundred thousand waves, instead
of two dozen at the most perhaps that I am taking to get up
this oscillation. But just note the enormous vibration we have
here, and contrast it with the state of things that we had just
before. The upper particle is in motion now and is performing
a vibration in the same period and phase as the lower particle,
only through comparatively a very small range. The second
particle, I am afraid, will overstrain the wire. (By hanging up a
watch, bifilarly, so that the period of bifilar suspension approxi-
mately agrees with the balance wheel, you get likewise a state
of wild vibration. But if you perform such experiments with a
watch, you are apt to damage it.) This, which you see now, is
a most magnificent contrast to the previous state of things when
the period of the exciter was very far from agreeing with any of
the fundamental periods.

We will now return to our molar subject, the elastic solid.
You will see a note in the paper of yesterday to which I have
referred, stating that the thiipsinomic method is more convenient
than the tasinomic for dealing with incompressibility, and in
point of fact it is so.

I explained to you yesterday Bankine's nomenclature of
thiipsinomic and tasinomic coeflBcients, according to which, when
the six stress-components are expressed in terms of strain-
components, the coefficients are called tasinomic; and when the

222 LECTURE XV.

Molar. strain-components are expressed in terms of the stress-components,
the coeiBcients are called thlipsinomic. [Thus, going back to
Lecture XL (p. 132 above), we see, in the six equations (1), 36
tasinoniic coefficients expressing the six stress-components as
linear functions of the six strain -components : and we see, in
virtue of 15 equalities among the 36 coefficients, just 21 indepen-
dent values, being Green's celebrated 21 coefficients. Use now
those six equations to determine the six quantities «,/) jr, a, 6, c in
terms of P, Q, R, flf, T, U. Thus we find

c = (PP)P+(PQ)Q-f(Pi2)iJ + (P^)S + (Pr)r+(Pt7)ir
f^(QP)P + m) Q+{QR)It + (QS)S + (QT)T+ (QU) U
ff^(RP)P-\'{RQ)Q-\'(RR)R+(RS)S'^(RT)T-\-(RU)U
a = (SP)P + (flfQ)Q + (flf72) R -^^ (SS) S -{• (ST) T + (SU) U
b=(TP)P + (TQ)Q-\-{TR) R + (TS)8-^(TT)T+(TU)U
c=^(UP)P + (UQ)Q + {UR)R-h(US)S-\'(UT)T-\-(UU)U'

(1),

where (PP\ (PQ), &c., denote algebraic functions of 11, 12, 22,
&c., found by the process of elimination. This process, in virtue
of the 15 equalities 12 = 21, &c., gives- (PQ) = (QP), &c.; 15
equalities in all. The 21 independent coefficients (PP), (PQ), &c.,
thus found, are what Rankine called the thlipsinomic coefficients.
Taking now from Lecture II., p. 24,

E^^iPe + Qf-^-Rg + Sa-^'Tb+Uc) (2),

and eliminating P, Q, R from this formula by the equations (1)
of Lecture XL, (p. 132), we find the tasinomic quadratic function
for the energy which we had in Lecture II. (p. 23). And eliminat-
ing e, f, g, a, 6, c from it by our present six equations, we find the
corresponding thlipsinomic formula for the energy with 21 inde-
pendent coefficients (PP), (PQ), &c.*] In a certain sense, these
coefficients, both tasinomic and thlipsinomic, may be all called
moduluses of elasticity, inasmuch as each of them is a definite
numerical measurement of a definite elastic quality. I have,
however, specially defined a modulus as a stress divided by a
strain, following the analogy of Young's modulus. If we adhere
to this definition, then the tasinomic coefficients are moduluses,
and the thlipsinomic coefficients are reciprocals of moduluses.

♦ (Compare Thomson and Tait, § 673, (12)— (20).

THLIPSINOMIC TREATMENT OP COMPRESSIBILITY. 223

In the Lecture notes in your hands for today you see the Molar,
thlipsinomic discussion of the question of compressibility or in-
compressibility ; which is much simpler than our tasinomic
discussion of the same subject in which we failed yesterday. You
see that if the dilatation, e +/+5^, be denoted by S, you have

S = [{PP) + {QP) + {RP)\ P + [(PQ) + (QQ) + (72(2)] Q
+ [{PR) + {QR) + {RR)] R + [{PS) + {QS) + {RS)] S
+ [(Pr) + (Qr) + {RT)]T + [{PU) + {QU)'{-{RU)] U (3).

Thus if P is the sole stress, a dilatation [{PP) + {QP) + {RP)] P
is produced ; and if S is the sole stress, a dilatation

[{PS) + {QS) + {RS)]S

is produced. We see therefore that S, a kind of stress which in
an isotropic solid w^ould produce merely distortion, may produce
condensation or rarefaction in an aeolotropic solid. The coefficients
of P, Q, R, S, T, U in our equation for S may be called compressi-
bilities. Their reciprocals are (accoixiing to my definition of a
modulus) moduluses for compressibility. In an isotropic solid,
each of the last three coefficients vanishes ; and the reciprocal of
each of the others is three times what I have denoted by k
(Lecture II., p. 25), and called the compressibility-modulus or the
bulk-modulus, being P/S, where P denotes equal pull, or negative
pressure, in all directions.

An leolotropic solid is incompressible if, and is compressible
unless, each of the six coefficients in our formula for i vanishes.
That is to say, it is necessary and sufficient for incompressibility
that

(PP) + (QP) + (i2P) = 0

(PQ) + {QQ) + {RQ) = 0

{PR)-\-{QR)-\-{RR) = 0
{PS) -h {QS) + {RS) = 0
{PT) -{- {QT) + {RT) == 0
{PU)-\-{QU)'\-{RU)^0'

Thus we see that six equations among the 21 coefficients suffice
to secure that there can be no condensational-rarefactional wave,
or, what is the same thing, that, in every plane wave, the vibration
must be rigorously in the plane of the wave-front ; and therefore
that Qreen was not right when, in proposing to confine himself

(4).

224 LECTURE XV.

olar. '' to the consideration of those media only in which the directions
of the transverse vibrations shall always be accurately in the
front of the wave," he said*, "This fundamental principle of
Fresnels theory gives fourteen relations between the twenty-one
constants originally entering into our function." What Green
really found and proved was fourteen relations ensuring, and
required to ensure, that there can be plane waves with direction
of vibration accurately in the plane of the wave, if there can also
be condensationaUrarefactional waves in the medium. What we
have now found is that not fourteen, but only six, equations suffice
to secure incompressibility and therefore to compel the direction
of the vibration in any actual plane wave to be accurately in the
plane of the wave.

[March 7, 1899. — I have only today found an interesting and
instructive mode of dealing with those six equations of incom-
pressibility, which I gave in this Baltimore Lecture of October 14,
1884. By the first three of them, eliminating {PP\ (QQ), (RR),
and by the other three, (PS), (QT), (RU) from the equation of
energy, we find

2^ = - [(QR) (Q - Ry + (RP) (R -Py + ( PQ) (P - QY]

+ 2{[(QU)U--(RT)T]{Q-R)-\-[(RS)S-(PU)V](R-P)

+ [{PT)T-{QS)8](P-Q)\
-h(SS)S''\-(TT)T^-\-{UU)U^
+ 2[{TU)TU-\'(US)US-\'{ST)ST] (5).

In this, P, Q, R appear only in their diCFerencos, which is an
interesting expression of the dynamical truth that i( P = Q = R,
they give no contribution to the potential energy.

The three dififerences Q — R, R — Py P—Q, arc equivalent to
only two independent variables ; thus if we put P - Q = F and
P^R^ W, we have Q — iZ = TT — F, and the expression for E
becomes a homogeneous quadratic function of the five indepen-
dents V, Wy 8y T, U, with fifteen independent coefficients, which
we may write as follows : —

E= \ [{VV) V^-\-{WW) W^-\-{SS)S''h{TT)T'-\'(UU) U']
+ (VW)VW +{V8)VS -\'(VT)VT'\-(VU)VU
+ {W8)WS ■^{WT)WT+(WU)WU
+ (ST)ST +(SU)SU ^•{TU)TU (6).

* Green, Collected Papers, p. 293.

THLIPSINOMIC TREATMENT OF COMPRESSIBILITY.

225

The diflTerential coeiBcients of this with reference to S, T, CT Molar.
are of course as before, a, 6, c. Denote now by h and t its
differential coefficients with reference to V and W. Thus we find

A= (vv^v-^^ (vw)w+ {VS)S-h (Fr)r+ (vu)u

% = (WV)V + (WW)W-\-{WS)S + (WT)T-\'(WU)U
a= (Sr^V-^- (SW)W-h (SS)S'\- (ST)T-^ (SU)U\'''0)\
6= {TV)V+ (TW)W'^ (TS)S-\- (TT)T+ {TU)U
c= {UV)V+{VW)W-\- (U8)S-\' (UT)T-\-{UU)U}

and we have

E =\{hV + iW + aS -{-hT + cU) (8).

The dynamical interpretation shows that h must represent — /,
and t must represent —g, when V and W represent P — Q,
and P-R.

Solving the five linear equations (7) for F, W, 8, T, U, we
find the tasinomic expressions for the five stress-components in
terms of the five strain-components, which we may write as
follows : —

F==(AA)A + (Ai)i + (Ma + W6 + (Ac)c\

Tr= {ih)h + {ii) i+ (ta) a + (i6) 6+ {ic)c

8 = (ah) h + (ai) i + (aa) a + (ah) b + (ac) c V (9).

T = (bh) h + (bi) i + (ba) a + (bb) b + (be) c

U = (cA) h + (ct) i + (ca) a -h (c6) 6 -h (cc) c^

The algebraic process shows us that

(hi)=:(ih)\ (ha) = (ah); &c (10);

so that we have now found the 15 independent tasinomic coeffi-
cients from the 15 thlipsinomic. Lastly, eliminating by (9)
F, TT, 8, r, U from (8), we find the tasinomic quadratic expressing
the energy.]

As I said in the first lecture, one fundamental difficulty is
quite refractory indeed. In the wave-theory of light the velocity
of the wave ought to depend on the plane of distortion. If you
compare the details of motion in the wave-surface* worked out
for an incompressible aeolotropic elastic solid, with equalities
enough among the coefficients to annul all skewnesses, you will
see that it agrees exactly with FresneFs wave-surface but that
instead of the direction of the line of vibration of the particles as

* Leetme XII., p. 187 ; Lecture XIIL, p. 176.
T. L. 15

226

LECTURE XV.

in Fresnel's construction we have the normal to the plane of distor-
tion as the direction on which the propagational velocity dependa
I see no way of getting over the dilBculty that the return
forces in an elastic solid — the forces on which the vibration
depends — are dependent on the strain experienced by the solid
and on that alone. I have never felt satisfied with the ingenious
method by which Green got over it. Stokes quotes in his report
on Double Refraction, page 265 (British Association 1862) : " In
" his paper on Reflection, Green had adopted the supposition of
"Fresnel that the vibrations are perpendicular to the plane of
" polarization. He was naturally led to examine whether the laws
"of double refraction could be explained on this hjrpothesis.
" When the medium in its undisturbed state is exposed to pressure
" diflfering in diflferent directions, six additional constants are intro-
" duced into the function ^, or three in the case of the existence of
" planes of symmetry to which the medium is referred. For waves
" perpendicular to the principal axes, the directions of vibration
** and squared velocities of propagation are as follows : —

Wave normal

X

y

X

Direction of vibration <

1

X

y

0 + A

Na-B

M+C

N + A

H + D

L + C

z

M + A

L^B

I + C

"Green assumes, in accordance with Fresnels theory, and with
"observation if the vibrations in polarized light are supposed
"perpendicular to the plane of polarization, that for waves
" perpendicular to any two of the principal axes, and propagated
" by vibrations in the direction of the third axis, the velocity of
" propagation is the same."

Let us see what this statement means before considering
whether it may be verified, as Green supposes, by the introduction
of "extraneous pressure." Consider waves having their fronts
parallel to the sides N and W (North and West) of this box, which
are perpendicular to two of the three principal axes of the crystal,
and such having its vibrations in the direction of the third axis (up
and down). Take first the wave that is propagated south as I hold

WAVE VBLOCITT DEPENDS ON LINE OF VIBRATION. 227

the box. There is the plane of the wave (N). The vibration up MoUr.
and down with N held fixed will give a shear like that marked 1,

in which a square becomes a rhombic figure. That represents the
strain in the solid corresponding to this first state of motion. Simi-
larly the wave propagated in the eastward direction will give rise
to a shear of this kind marked 2, the vibration still being upward.
The assumption is that one of these sets of waves is propagated
at the same speed as the other. That is to say, the waves which
have their shear in this west plane have the same velocity as
the waves which have their shear in this north plane. The
essence of our elastic solid is three diflferent rigidities, one for
shearing in this plane W, one for shearing in this plane N, and
one for shearing in the other principal plane (the horizontal plane
of our box). The incongruous assumption is that the velocities of
propagation do not depend on the planes of the shearing strain,
and do depend, simply and solely, on the direction of the vibration.
The introduction by Green (in order to accomplish this) of
what he calls "extraneous force," which gives him three other
coefficients has always seemed to me of doubtful validity. In
the little table above, taken from Stokes, X, Jf, N are the three
principal rigidities, the 44, 55, 66 of our own notation. A, B, C
are the eflfects of extraneous pressure. The table gives the squared
velocities of propagation and waves of different wave-normals and
directions of vibration along the axes. The principal diagonal refers
only to condensational waves, or waves in which the direction
of vibration coincides with the wave-normal. Green's assumption

15—2

228 LECTURE XV.

)lar. makes, for vibrations in the a;-direction, N-^B^M+C; which,
with the two corresponding equations for vibrations in the
directions y, z, gives

[March 7, 1899. Addition. On Caucht's and Gbern's
Doctrine of Extraneous Force to explain dtnamicallt
Fresnel's Kinematics of Double Refraction*.

§ 1. Green's dynamics of polarization by reflexion, and Stokes'
dynamics of the diffraction of polarized light, and Stokes' and
Rayleigh's dynamics of the blue sky, all agree in, as seems to me,
irrefragably, demonstrating Fresnel's original conclusion, that in
plane polarized light the line of vibration is perpendicular to the
plane of polarization; the ** plane of polarization" being defined
as the plane through the ray and perpendicular to the reflecting
surface, when light is polarized by reflexion.

§ 2. Now when polarized light is transmitted through a crystal,
and when rays in any one of the principal planes are examined, it
is found that —

(1) A ray with its plane of polarization in the principal
plane travels with the same speed, whatever be its direction
(whence it is called the " ordinary ray " for that principal plane) ;
and (2) A ray whose plane of polarization is perpendicular to the
principal plane, and which is called the " extraordinary ray " of
that plane, is transmitted with velocity differing for different
directions, and having its maximum and minimum values in two
mutually perpendicular directions of the ray.

§ 3. Hence and by § 1, the velocities of all rays having their
vibrations perpendicular to one principal plane are the same ; and
the velocities of rays in a principal plane which have their direc-
tions of vibration in the same principal plane, differ according to the
direction of the ray, and have maximum and minimum values for
directions of the ray at right angles to one another. But in the
laminar shearing or distortional motion of which the wave-motion
of the light consists, the "plane of the shear f" (or "plane of
the distortion," as it is sometimes called) is the plane through the

♦ Reprinted, with additions, from the Proc, JR. S. JS., Vol. xv. 1887, p. 21, and
Phil Mag,, Vol. xxy. 1888, p. 116.

t Thomson and Tait's Natural Phihtopky, § 171 (or ElemenU, § 150).

CAUGHT AND ORIS£N*S EXTRANEOUS FORCE. 229

direction of the ray and the direction of vibration ; and therefore Molar,
it would be the ordinary ray that would have its line of vibration
in the principal plane, if the ether's difference of quality in
different directions were merely the aeolotropy of an unstrained
elastic solid*. Hence ether in a crystal must have something
essentially different from mere intrinsic aeolotropy ; something
that can give different velocities of propagation to two rays, of
one of which the line of vibration and line of propagation coincide
respectively with the line of propagation and line of vibration
of the other.

§ 4. The difficulty of imagining what this something could
possibly be, and the utter failure of d}mamics to account for double
refraction without it, have been generally felt to be the greatest
imperfection of optical theory.

It is true that ever since 1839 a suggested explanation has
been before the world ; given independently by Cauchy and
Qreen, in what Stokes has called their ''Second Theories of
Double Refraction," presented on the same day, the 20th of
May of that year, to the French Academy of Sciences and the
Cambridge Philosophical Society. Stokes, in his Report on Double
Refractionf, has given a perfectly clear account of this explana-
tion. It has been but little noticed otherwise, and somehow it
has not been found generally acceptable ; perhaps, because of a
certain appearance of artificiality and arbitrariness of assumption
which might be supposed to discredit it. But whatever may have
been the reason or reasons which have caused it to be neglected
as it has been, and though it is undoubtedly faulty, both as given
by Cauchy and by Green, it contains what seems to me, in all
probability, the true principle of the explanation, and which is,
that the ether in a doubly refracting crystal is an elastic solid,
unequally pressed or unequally pulled in different directions, by
the unmoved ponderable matter.

§ 5. Cauchy's work on the wave-theory of light is complicated
throughout, and to some degree vitiated, by admission of the

* The elementary dynamics of elastic solids shows that on this supposition
there might be mazimom and minimom velocities of propagation for rays in
directions at 45° to one another, but that the velocities must eMcntially be equal for
every ttDO directions at 90° to one another^ in the principal plane, when the line of
vibration is in this plane.

t Briti9h Association Bepoft, 1862.

230 LECTURE XV.

Molar. Navier-Poisson false doctrine* that compressibility is calculable
theoretically from rigidity; a doctrine which Green sets aside,
rightly and conveniently, by simply assuming incompressibility.
In other respects Cauchy's and Green's "Second Theories of
Double Refraction/* as Stokes calls them, are almost identical.
Each supposes ether in the crystal to be an intrinsically aeolotropic
elastic solid, having its seolotropy modified in virtue of internal
pressure or pull, equal or unequal in different directions, produced
by and balanced by extraneous force. Each is faulty in leaving
intrinsic rigidity-moduluses (coeflScients) unaffected by the equi-
librium-pressure, and in introducing three fresh terms, with
coeflScients (A, B, C in Green's notation) to represent the whole
effect of the equilibrium-pressure. This gives for the case of an
intrinsically isotropic solid, augmentation of virtual rigidity, and
therefore of wave- velocity, by equal pullf in all directions, and
diminution by equal positive pressure in all directions ; which is
obviously wrong. Thus definitively, pull in all directions outwards
perpendicular to the bounding surface equal per unit of area to
three times the intrinsic rigidity-modulus, would give quadrupled
virtual rigidity, and therefore doubled wave-velocity ! Positive
normal pressure inwards equal to the intrinsic rigidity-modulus
would annul the rigidity and the wave-velocity — that is to say,
would make a fluid of the solid. And, on the other hand, nega-
tive pressure, or outward pull, on an incompressible liquid, would
give it virtual rigidity, and render it capable of transmitting
laminar waves I It is obvious that abstract dynamics can show
for pressure or pull equal in all directions, no effect on any physical
property of an incompressible solid or fluid.

§ 6. Again, pull or pressure unequal in different directions, on
an isotropic incompressible solid, would, according to Green's
formula (A) in p. 303 of his collected Mathematical Papers, cause
the velocity of a laminar wave to depend simply on the wave-
front, and to have maximum, minimax, and minimum velocities

* See Stokes, **0n the Friction of Fluids in Motion and on the Equilibrium and
Motion of Elastic Solids," Camh. Phil. Trans., 1S45, §§ 19, 20; reprinted in Stokes'
Mathematical and Physical Papers, Vol. i. p. 123 ; or Thomson and Tait's Natural
Philosophy, §§ 684, 685; or Elements, §§ 655, 656.

t So little has been done towards interpreting the formulas of either writer that
it has not been hitherto noticed that positive values of Cauchy's G, H, I, or of
Green's A^ B, C, signify pulls, and negative values signify pressures.

STBESS-TUEORY OF DOUBLE REFRACTION. 231

for wave-fronts perpendicular respectively to the directions of Molecular,
maximum pull, minimax pull, and minimum pull; and would
make the wave-surface a simple ellipsoid ! This, which would be
precisely the case of foam stretched unequally in different direc-
tions, seemed to me a very interesting and important result, until
(as shown in § 19 below) I found it to be not true.

§ 7. To understand fully the stress-theory of double refraction,
we may help ourselves effectively by working out directly and
thoroughly (as is obviously to be done by abstract d3mamics)
the problem of § 6, as follows : — Suppose the solid, isotropic
when unstrained, to become strained by pressure so applied to
its boundary as to produce, throughout the interior, homogeneous
strain according to the following specification : —

The coordinates of any point M of the mass which were f , 17, f
when there was no strain, become in the strained solid

fVa. ^V/8, ?V7 (1);

Va, \/)8, V7, or the "Principal Elongations*," being the same
whatever point M of the solid we choose. Because of incompres-
sibility we have

a/37=l (2).

For brevity, we shall designate as (a, /8, 7) the strained condition
thus defined.

§ 8. As a purely kinematic preliminary, let it be required to

find the principal strain-ratios when the solid, already strained

according to (1), (2), is further strained by a uniform shear, <7,

specified as follows ; in terms of x, y, z, the coordinates of still the

same particle, M, of the solid and other notation, as explained

below : —

a? = f Va + apl \

y = i;V/8 + a-pml (3),

z = ^ i^y -^ <rpn ]

where p= OP = Xf Va + /^^ V/8 + j'S'Vy W,

with Z*+m» + n'*=l, X* + /i*+i/» = l (5),

and l\•\^m^l -f nv = 0 (6);

* See chap. iv. of '* Mathematical Theory of Elasticity" (W. Thomson), Trans,
Roy. Soc. Lond, 1856, reprinted in Vol. in. of Mathematical and Physical Papers^
now on the point of being published, or Thomson and Tait's Natural Philosophy,
§§ 160, 164, or EUnients, §§ 141, 158.

232 LECTURE XV.

Molar. \ /i, V denoting the direction-cosines of OP, the normal to the
shearing planes, and i, m, n the direction-cosines of shearing dis-
placement. The principal axes of the resultant strains are the
directions of OM in which it is maximum or minimum, subject to
the condition

P + i7» + ?«=l (7);

and its maximum, minimax, and minimum values are the three
required strain-ratioa Now we have

Oif» = a?» + y" + -2'

=»f»a + V/3 + r7+2(7(ZfVa + mi7V/8 + nCV7)p-hay...(8),

and to make this maximum or minimum subject to (7), we have

where in virtue of (7), and because OM^ is a homogeneous quad-
ratic function of f , 17, 5^

p = 0M\

The determinantal cubic, being

(c5^-p)(<^-p)(^-p)-a"

(S4-p)-b^^-p)-d'(9^-'p)'h2abc = 0 (10),

where .9^ = a (1 + 2al\ + <r»X') ; jr= /3 (1 + 2<rmfi -h o-V) ;

^=7(l + 2<r;ii/ + (7>j/«) (11)

and a = V()87) [o* {rnv + rifi) + a^fiv] ; 6 = V(7a) [o" (wX + Zi/)-|-crVX] ;

c = V(a/3) [o- (i/^ + ^i^') + o-^Xm] ....(12),

gives three real positive values for p, the square roots of which
are the required principal strain-ratios.

§ 9. Entering now on the dynamics of our subject, remark
that the isotropy (§1) implies that the work required of the
extraneous pressure, to change the solid from its unstrained
condition (1, 1, 1) to the strain (a, ^, 7), is independent of the
direction of the normal axes of the strain, and depends solely on
the magnitudes of a, )8, 7. Hence if E denotes its magnitude per
unit of volume; or the potential energy of unit volume in the
condition (a, yS, 7) reckoned from zero in the condition (1, 1, 1) ;
we have

^ = -f(«./8.7) (13),

STRESS-THEORY OF DOUBLE REFRACTION.

233

where ^ denotes a function of which the magnitude is unaltered Molar,
when the values of a, 0, y are interchanged. Consider a portion
of the solid, which, in the unstrained condition, is a cube of unit
side, and which in the strained condition (a, /3, 7), is a rectangular
parallelepiped Va • V)8 • V7. In virtue of isotropy and symmetry,
we see that the pull or pressure on each of the six faces of this
figure, required to keep the substance in the condition (a, /3, 7), is
normal to the face. Let the amounts of these forces per unit area,
on the three pairs of faces respectively, he A, B, C\ each reckoned
as positive or negative according as the force is positive ptdl,
or positive pressure. We shall take

A+B + G = 0 (14),

because normal pull or pressure uniform in all directions produces
no effect, the solid being incompressible. The work done on any
infinitesimal change from the configuration (a, /9, 7) is

A V(/37)d(Va) + B V(72) d (V/9) + C^(a/3) d (V7X
or (because a/87 = 1)

A ^ B ,^ C ,

...(16).

§ 10. Let Sa, 8/8, 87 be any variations of a, yS, 7 consistent
with (2), so that we have

and

(a-f8a)(/8-f8y8)(7 + S7) = l|

a/37=l j

Now suppose Sa, 8/8, S7 to be so small that we may neglect their
cubes and corresponding products, and all higher products. We
have

^ + ?f + ^ + aS/8S7 + /8S7Sa + 7Sa8/8 = 0 (17);

a ^ 7

whence (ty=(^^ + hj,

whence, and by the symmetrical expressions.

2ByS2

^1 /S^_ V_??'\

>

^^-\&-^-fh

(18).

234 LECTURE XV.

ir. § 1 1. Now, if E 4- iE denote the energy per unit bulk of the

solid in the condition

(a + Sa, iS+SA 7 + *y>>
we have, by Taylor's theorem,

where Hi, H^, &c. denote homogeneous functions of So, 8^, Sy of
the Ist degree, 2nd degree, &c. Hence, omitting cubes, Ac, and
eliminating the products from H2, and taking Hi from (15), we
find

where G, H, I denote three coefficients depending on the nature of
the function yfr (13), which expresses the energy. Thus in (19),
with (14) taken into account, we have just five coefficients inde-
pendently disposable. A, B, 0, H, /, which is the right number
because, in virtue of a/3y = 1, -ff is a function of just two indepen-
dent variables.

§ 12. For the case of a = 1, /9 = 1, 7 = 1, we have
A=B==C = 0 and G = H = / = ffi, suppose;
which give S^ = i Gi (Sa« + S^S" + St*).

From this we see that 2Gi is simply the rigidity-modulus of the
unstrained solid ; because if we make S7 = 0, we have 8a = — 8/8,
and the strain becomes an infinitesimal distortion in the plane
{xy), which may be regarded in two ways as a simple shear of
which the magnitude is 8a* (this being twice the elongation in
one of the normal axes).

§ 13. Going back to (10), (11), and (12), let a- be so small
that <7* and higher powers can be neglected. To this degree
of approximation we neglect abc in (10), and see that its three
roots are respectively

6« c'

^-

^-J^ M-S^'

c' a«

^-^--^-^^•> <^«>'

9^ ^' ^'

• Thomson and Tait's Natural Philotophy, § 175, or EUmenU, § 151.

STRESS-THEORY OF DOUBLE REFRACTION.

235

provided none of the diflFerences constituting the denominators is Molar,
infinitely small. The case of any of these diflFerences infinitely
small, or zero, does not, as we shall see in the conclusion, require
special treatment, though special treatment would be needed to
interpret for any such case each step of the process.

§ 14. Substituting now for ^, SS, % a, 6, c in (20), their
values by (11) and (12), neglecting <7* and higher powers, and
denoting by Sa, 8/8, 87 the excesses of the three roots above
a, /8, 7 respectively, we find

Sa = a \2al\ + a« Tx* ^ (nX + Iv)^ - -^ (Ifju -f mXylil

&y = 7 |2(7ni; 4- <r» v" - -^^{mvh UfiY ^ (nX + hy \\

M21),

...(22),

and using these in (19), we find
SJ5 = (7 (Atk + Bmfi + Cnv)

+ L {mv -\- rifiy 4- -M (nX -f hy -\-N{lfi+ inXy]

where

p — y 7 — a a — yS ^

§ 15. Now from (5) and (6) we find

{mv + nfiy = 1 - Z» - X« + 2 {PX.^ - mV - ^V) (24),

which, with the symmetrical expressions, reduces (22) to

SE=^a {AlX + Bmfi + Cnv) -h ^a^ {L -\- M -\- N
+ (il-X)X» + (5-Jlf)/i2 + ((7-i\r)i;«
-ZZ«-ifm«-iVrn«+2[(2(?+X-3f-iV^)Z«X2
+ (2fl' + Jlf-J\r-Z)mV + (2/+i\^-Z-jlf)nV]} j

...(25).

§ 16. To interpret this result statically, imagine the solid
to be given in the state of homogeneous strain (a, /9, 7) through-
out, and let a finite plane plate of it, of thickness A, and of very
large area Q, be displaced by a shearing motion according to the
specification (3), (4), (5), (6) of § 8 ; the bounding-planes of the

JQ

236 LBcruBE xv.

^oUr. plate being unmoved, and all the solid exterior to the plate being
therefore undisturbed except by the slight distortion round the
edge of the plate produced by the displacement of its substance.
The analytical expression of this is

<r=/(p) (26).

where /denotes any function of OP such that

W(i>) = 0 (27).

0

If we denote by W the work required to produce the supposed
displacement, we have

W^QrdpSE + ^ (28),

SE being given by (25), with eveiything constant except o-, a
function of 0P\ and ^l^ denoting the work done <m the solid
outside the boundary of the plate. In this expression the first
line of (25) disappears in virtue of (27) ; and we have

+ (4-L)X«H-(5-Jf)/i« + (a-J\^)i/>
-Zi«-3fm»-J\rn«4-2[(2(? + X-Jlf-J\r)i«X«)^...(29).

+ (2^ + 3f-iVr-(Z)mV

+ (2/ + iV^ - Z - 3f ) nV]} f * dp<r»

When every diameter of the plate is infinitely great in comparison
with its thickness, ^/Q is infinitely small ; and the second
member of (29) expresses the work per unit of area of the plate,
required to produce the supposed shearing motion.

§ 17. Solve now the problem of finding, subject to (5) and
(6) of § 8, the values of I, m, n which make the factor { } of the
second member of (29) a maximum or minimum. This is only
the problem of finding the two principal diameters of the ellipse
in which the ellipsoid

+ [2(2/ + JV-Z- if) i/»-JV]^» = const (30)

s cut by the plane

Xa?-f /Ay + !/-? = 0 (31).

STRESS-THEORY OF DOUBLE REFRACTION. 237

If the displacement is in either of the two directions (2, m, n) thus Molar,
determined, the force required to maintain it is in the direction of
displacement ; and the magnitude of this force per unit bulk of
the material of the plate at any point within it is easily proved
to be

W| (32).

where {if} denotes the maximum or the minimum value of the
bracketed factor of (29).

§ 18. Passing now from equilibrium to motion, we see at
once that (the density being taken as unity)

V*={M}.., (33),

where V denotes the velocity of either of two simple waves whose
wave-front is perpendicular to (X, /i, v). Consider the case of
wave-front perpendicular to one of the three principal planes;
(yz) for instance: we have X=0; and, to make { } of (29) a
maximum or minimum, we see by symmetry that we must either
have

(vibration perpetidicular to principal plane)

1 = 1, m= 0, n = 0> (34).

(vibration in principal plane). . .Z = 0, m = — i/, n = /LtJ

Hence, for the two cases, we have respectively :

Vibration perpendicular to yz ) .^-.

F« = Jlf + iV -h (fi - itf )/i» + (0- iV^) i^j '"^^ '

Vibration inyz...V^ = L-\'Bfi' + Gp^ + ^(H + I -^f^V ...(36).

§ 19. According to FresneFs theory (35) must be constant, and
the last term of (36) must vanish. These and the corresponding
conclusions relatively to the other two principal planes are satisfied
if, and require that,

^ — X/ = jO — JH = G — J}/ (u7),

and H + I=L; I+0 = M) 0 + H=N. (38).

Transposing M and N in the last of equations (37), substituting
for them their values by (23), and dividing each member by /Sy,
we find

238 LECTITBE XV.

f oUr. whence (sum of numerators divided by sum of denominators),

B — C C " A. A. — B /A/\\

7a - ai9 "^ a)8 - /87 "" ^87 — 7a

The first of these equations is equivalent to the first of (37) ; and
thus we see that the two equations (37) are equivalent to one
only ; and (39) is a convenient form of this one. By it, as put
symmetrically in (40), and by bringing (14) into account, we find,
with q taken to denote a coefficient which may be any function of

(a, /3, 7) :

il=g(flf-
where

/37); B = q(8-ya); C=g(S-a/3);l ,
<S = i(/37 + 7« + «/3) r'^ ^'

and using this result in (23), we find

L = q[a(^+y)-S]; M = q[fi {y + a)-S]; \

or L = q{2S-fiy); M ==q(2S-ya); N=q(28-afi)]

By (2) we may put (41) and (42) into forms more convenient for
some purposes as follows : —

A=q{s-ly, 5 = g(s-i); C = q(8-l) ...(43).
Z = 5 (25 - i) ; M=q (iS -^);N=q (28 - ^) ...(44).

where 5 = ^(1+1+1) (45).

Next, we find 0,H,I; by (38), (44), and (45) we have

whence, by (38) and (44).

<? = <?Q-i-s); fl'=?(^-is); / = 9(i-is)...(47).

§ 20. Using (43) and (47) in (19), we have

Sa» SB*

^M^^f^U

.(48).

STRESS-THEORY OF DOUBLE REFRACTION. 239

Now we have, by (2), log (a/87) = 0. Hence, taking the variation Molecular,
of this as far as terms of the second order,

which reduces (48) to

Remembering that cubes and higher powers are to be neglected
we see that (50) is equivalent to

«^ = i58(l + l + i) (51).

Hence if we take q constant, we have

^Ml-'hr') ^'^^

It is clear that q must be stationary (that is to say, Sq = 0) for
any particular values of a, yS, 7 for which (52) holds ; and if (52)
holds for all values, q must be constant for all values of a, /3, 7.

§ 21. Going back to (29), taking Q great enough to allow
^/Q to be neglected, and simplifying by (46), (43), and (44), we
find

f-(M'-^")/v <-)^

and the problem (§ 17) of determining Z, m, n, subject to (5)
and (6), to make fi/a + m'/yS + n^/y a maximum or minimum for
given values of X, /a, v, yields the equations

(k)\ — 0)7 + - = 0; ©/Lt — (k)'m + -^^ = 0; eov — (o'n-h -=^0 (54),

* r* 7

6), ft)' denoting indeterminate multipliers ; whence

^ =-+^+- (^^X

a p y ^ ^

ft)«

=p(,'«i)V^.(,^.i)V^,(,'^iy (56),

^ , / 1 - Z* m» nn ,
/P 1 - m» n'X

y (57).

240 LSCrUBB XY.

MoImt. These formulas are not directly convenient for finding l,ffi,n

from X, fjb, Vf cf. § 33 (the ordinary formulas for doing so need not be
written here) ; but they give \ fj^, v explicitly in terms of I, m, n
supposed known ; that is to say, they solve the problem of finding
the wave-front of the simple laminar wave whose direction of
vibration is (Z, m, n). The velocity is given by

/fi m* n*\ .^^^

It is interesting to notice that this depends solely on the direction
of tho line of vibration ; and that (except in special cases, of
partial or complete isotropy) there is just one wave-firont for any
given line of vibration. These are precisely in every detail the
conditions of Fresnel's Kinematics of Double Refiraction.

§ 22. Qoing back to (35) and (36), let us see if we can fit
them to double refraction with line of vibration in the plane
of polarization. This would require (36) to be the ordinary ray,
and thoreforo re(}uires the fulfilment of (38), as did the other
HUpposition ; but instead of (37) we now have [in order to make
(30) constant]

A^B^C (59),

afid thcnjforo each, in virtue of (14), zero ; and

a = /3 = 7 = l;

so that wo are driven to complete isotropy. Hence our present
form (§ 7) of tho stresH-theory of double refraction cannot be fitted
to givn linc^ of vibration in the plane of polarization. We have
H(H!n (§ 21) that it does give line of vibration perpendicular to the
plane of polariztttion ivith exactly FresneVs form of wave-surface,
whrri ixiivd for tho purpose, by the simple assumption that the
potential (mergy of the strained solid is expressed by (52) with
q constant I It is important to remark that q is the rigidity-
modulus of the unstrained isotropic solid.

§ 23. From (58) we see that the velocities of the waves corre-
sponding to the three cases, i = 1, m= 1, n = I, respectively are
V(9/«)» V(?/^)> V(9/7)- Hence the velocity of any wave whose
vibrations are parallel to any one of the three principal elongations,
multiplied by this elongation, is equal to the velocity of a wave in
the unstrained isotropic solid.

HYPOTHETICAL ELASTIC SOLID. 241

[§§ 24... 34, added March 1899.] Molar

§ 24. To fix and clear our understanding of the ideal solid,
introduced in § 7 and defined in § 20 (52) and in § 22, take a bar
of it of length I when unstrained, and of cross-sectional area A.
For our prasent purpose the cross-section may be of any shape,
provided it is uniform from end to end. Apply opposing forces P
to the two ends ; and, when there is equilibrium under this stress,
let the length and cross-section be x and A', We have A' = lA/x,
because the solid is incompressible. The proportionate shortenings
of all diameters of the cross-section will be equal, as there is no
lateral constraint. Hence if the a, )8, 7 of (52) refer to the length
of our bar and two directions perpendicular to it, we have

' = $■' ^ = y-l (60);

and therefore by (52)

^ = i?(^ + 2f-3)«^ (61).

where E denotes the whole work required to make the change
from 2 to ^ in the length of our bar, of which the bulk is lA.
Hence, as P = dE/dx, we find

p=rqf^^l + ^\lA, and SP=3?^-^Sa: (62).

Hence, denoting by M the Young's modulus for values of x differ-
ing not infinitely little from /, and defining it by the formula

YfQ f,nd

ox-^x

Jf=3<?5 (63).

Hence augmentation of length from I to x diminishes the Young s
modulus, as defined above, in the ratio of l^ to ar^,

§ 25. The property of our ideal solid expressed by the con-
stancy of g, to which we have been forced by the assumption of
FresneFs laws for light traversing a crystal, is interesting in
respect to the extension of theory and ideas regarding the
elasticity of solids from infinitesimal strains, for which alone the
dynamics of the mathematical theory has hitherto been developed,
to strains not infinitesimal. To contrast the law of relation
between force and elongation for a rod of our ideal substance,
as expressed in (62) above, and for a piece of indiarubber cord or
T. L. 16

242 LECTURE XV.

Molar. indiarubber band, as shown by experiment, is quite an instructive
short lesson in the elements of the extended theory.

§ 26. Ten yeai-s ago, because of pressure of other avocations,
I reluctantly left the stress theory of FresneFs laws of double
refraction without continuing my work far enough to find complete
expressions for the equilibrium and motion of the elastic solid for
any infinitesimal deviation whatever from the condition (a, /8, 7).
Here is the continuation which I felt wanting at that time.
Let (iPo, y©, ^q) be the coordinates of a point of the solid in its
unstrained condition (a=l,)8=l,7 = l); and (x, y, z), the co-
ordinates of the same point in the strained condition (a, fi, 7).
If the axes of coordinates are the three lines of maximum, mini-
max, and minimum elongation, (which are essentially* at right
angles to one another), we have

a? = a;oVa; y = yoV)8; z=^Zo^y (64).

Let the matter at the point (a?, y, z) be displaced to (a? + f , y -f 1;,
z-h^); f , 17, f having each, subject only to the condition of no
change of bulk, any arbitrary value for every point (x, y, z)
within some finite space 8, outside of which the medium remains
in the condition (a, /8, 7) undisturbed.

Let now the variation of the displacement (f, rj, ^) from point
to point of the solid be so gi-adual that each one of the nine
ratios

d^ d^ d^ drj drj drj df df d^ .

ch' dy' dz' dx' dy' dz' dx' dy' dz ^ ^^

is an infinitely small numeric. Consider the system of bodily
forces, which must be applied to the solid within the disturbed
region /S, to produce the specified displacement (f, ^, f). Let
Jffl, Ffl, Zfl denote components of the force which must be
applied to any infinitesimal volume, fl, of matter around the point
(a:, y, z). Our directly solvable problem is to determine X, F, Z
for any point, f , 77, f being given for every point.

§ 27. Our supposition as to infinitesimals, which is that the
infinitesimal strain corresponding to the displacement (f, 17, f), is
superimposed upon the finite strain (a, yS, 7), implies that if instead

* 8ee Thomson and Tait, § 164.

FURTHER DEVELOPMENT OF STRESS-THEORY. 243

of (f, 17, f), the displacement be (cf,ci7, cf), where c is any numeric MoUr.
less than unity having the same value for every point of the
disturbed region, the force required to hold the medium in this
condition is such that instead of X, F, Z, we have cX, cY, cZ.
Hence if SE denote the total work required to produce the
displacement (f, 17, 5"), we have

SE = ^jjfdxdydz(X^+Yv-hZ^) (66),

where 1 1 1 dxdydz denotes integration throughout S.
§ 28. Now according to (52) we have

= i7 ///ctrdyd^ (iS-y + 7V + a'iS- - /37 - 7a - a/3))

where a', P, 7' denote the squares of the principal elongations
(§ 7 above) as altered from a, /S, 7, in virtue of the infinitesimal
displacement (f , 17, f).

To find a\ /S', 7', draw, parallel to our primary lines of reference
OX, OY, OZ, temporary lines of reference thn)ugh the point
(^ + ?. y + ^> -2: + f), which for brevity we shall call Q, Let P be
a point of the solid infinitely little distant from Q, and let (/, g, h)
be its coordinates relatively to these temporary reference lines.
Relatively to the same lines, let (/o, g^, ho) be the coordinates of
the position, Po» which P would have if the whole solid were
unstrained. We have

From these we find

{p£\ = Al^ + Bm^ + Cn'' + 2 (amn + bnl + dm) (69),

f fi h

where I, m, n denote p' , -^' , ^"^ ; and

16—2

244

LECTTJRE XV.

IfoUr.

A

B

C

a

...(70).

-{(>-S)'^©'^(§)]

-{(i)'^©^('^S)'}

\dydz \ dy) dz dy\ dzj]

\\ dx) dy dx\ dy) dx dy)

The values of a', ff, y are the maximum, minimax, and
minimum values of (69), subject to the condition Z* + m* + n'= 1 ;
and therefore according to the well-known solution of this problem,
they are the three roots, essentially real, of the cubic

(A-pHB-pHC-p)

- a*iA-p)-b'iB - p)- <^(C ~ p) + 2ahc = 0 (71).

Hence, by taking the coefficient of p in the expansion of this, we
find

^y' +y'tt' + a'^ = BC -{-CA+AB- a* -b'-d'

= (l + F)^ + (l + 0)ya + (l+H)a^.. (72),

where F, 6, H are symmetrical algebraic functions of the fourth
degree of the nine ratios (G5). Oinitting all terms above the
second degree, we have

(73),

^-<|4V(D*-(S'^(|T-(S)'

dy dz dz dy

and sjmametrical expressions for G and H, Going back now to
(67) and remembering that 0/87=! and a')8'7'=l, and that
a, /8, 7 are constant throughout the solid, we find, by (67),

ZE = \q (^jjjdxdydzF-{-UjjdxdydzO-{--jjjd.rdydzH\ ...(74).

FURTHER DEVELOPMENT OF STRESS-THEORY. 245

To find 1 1 1 dxdydzF by (73), remark first that by simple in- Molar,
tegration with respect to y and with respect to z we obtain

jjjdxdydz^^ = 0; jjjdxdydzf^ = 0 (75),

because rj and (f each vauisli through all the space outside the
space S. For the same reason we find by the well-known integra-
tion by parts [Lecture XIIL, equations (7) above]

///«^£|-///'"w.gS w

We thus have

§ 29. Now the condition a'/8V = 1 gives

df dn dt ^ . , .

-^ H- -T -+ -r- = 0 + terms involving powers,

and products of the mtios (78).

Therefore, neglecting higher powers than squares, wc get

dv . dCx" _ (d^

4y

and (77) becomes

jjjda^yd..F=fjjd.d,d. {(g)V (|)V(f )|..(80);

and we have corresponding symmetrical equations for G and H,
With these used in (74), it becomes

.^=i,///^^[i((DV0.(|;)

Taking any one of the nine terms of (81), the third for example,
and applying to it the process of double integration by parts, we
find

(S47=(ay <'»>.

246 LECTURE XV.

Molar. Hence treating similarly all the other terms, we have

8E = -}sqjjjdadi,dz (f ^ + ,^^ + i:^f) (83).

§ 30. This expression for SE must be equal to that of (66)
above, for every possible value of f, rj, f. If all values were
possible, we should therefore have the coefficients of f , ^, f in (66)
equal respectively to the coefficients of f, ^, f in (83). But in
reality only values of f , rj, ^ arc possible which fulfil the condition
of bulk everywhere unchanged ; and Lagrange's method of inde-
terminate multipliers adds to the second member of (83), the
following

ilJI^Mi^i*^' (»*'•

This, treated by the method of double integration by parts,
becomes

l///«^({^./^.!:^-) ,85),

§ 31. We must now, according to Lagrange's splendidly
powerful method, equate separately the coefficients of f, ?/, f
in (66) to their coefficients in (83) with (85) added to it. Thus
we find

as the equations of equilibrium of our solid with every point of
its boundary fixed, and its interior disturbed from the finitely
strained condition (a, /9, 7) by forces X, F, Z, producing infini-
tesimal displacements (f, ^, f) ; subject only to the condition

doj dy dz" ^ ^'

to provide against change of bulk in any part. If this equation
were not fulfilled, the equations (87), with cr given by (86), would
be the solution of a certain definite problem regarding a compres-
sible homogeneous solid, having cerUvin definite quality of perfect
elasticity, in respect to changes of shape and bulk, defined by the
coefficients q and \. From this problem to our actual problem thei*e

FURTHER DEVELOPMENT OF STRESS-THEORY. 247

is continuous transition by making \ everywhere infinitely great, Molar,
and therefore the first member of (88), being the dilatation, zero ;
and leaving «r as a quantity to be determined to fulfil the condi-
tions of any proposed problem. Thus in (87), (88), supposing
Xy Yy Z given, we have four equations for determining the four
unknown quantities cr, f , ^, f.

§ 32. Suppose now that the solid, after having been in-
finitesimally disturbed by applied forces from the (a, /8, 7)
condition, is left to itself According to D'Alcmbert's principle,
the motion is determined by what the ecjuations of equilibrium,
(87), become with — p^, — p^, — p|J substituted for X, Y, Z; p
denoting the density of the solid. Thus we find for the equations
of motion

^dt^'a ^ dx' ^dt^'p ^ dy' ^^dt'^y ^ dz

(89).

§ 33. We are now enabled by these equations to work out
the problem of wave-motion by a more direct and synthetical
process than that by which we were led to the solution in § 21
above. The simplest mathematical expression defining plane
waves in an elastic solid in terms of the notation of (89) is

p^Xx^ fiy-\-vz

^=lAp-vt); i7 = m/(/>-t;0; ^ = nf(p^vt)\ ...(90);

where X, /x, v are the direction -cosines of the wave-normal, and
/, //i, n those of the line of vibration. Our constraint to incom-
pressibility gives

f X H- 7/i/x -h ni/ = 0 (91).

Eliminating |, 1;, f, «r from (89) by (90), we find

/ (pv^ - qa~^) = Xqto ; m {pv^ - ^/8~') = fiqto ; n {piP - 57"*) = vqto

(92),

where

-'-ItiZ '-^

These equations agree with (54) if for pv^ we take qto'.

248 LECTURE XV.

§ 34. Multiplying equations (92) by

\l{pv'-qa-% fiKpv'-q^-'X p/if^-qy-')
respectively, adding, and using (91), we find

^' I f^' 4. "^ =0. .. (94)

pv^ — qar^ pv^ — ql3-^ piP — qy"^

This is a quadratic for the determination of t;', with its two roots
essentially real and positive. Again, by the equation

P + m" + n»=l,
we find from (92)

^'^[[pv'-qa-'J "^ W- 9/9^7 "*" (pi^'^qrV j '"^^^^'

Equations (94) and (95) give us v- and « ; and (92) now gives
explicitly I, m, n, when \, /x, v are given. Thus we complete the
determination of {I, m, n\ the direction of vibration, in terms of
(\, /x, v), the wave-normal, and the constant coefficients

q^~\ q8~\ 97*"^-

Our solution is identical with Fresnels, and implies exactly the
same shape of wave -surface.

[§§ 35... 47, added April 1901.]

§ 35. It will be convenient henceforth to take a, b, C instead
of crK /8~^ 7~*. Thus, according to § 7 (1), if a?o, l/o, ^0 aiid
X, y, z denote respectively the coordinates of one and the same
particle in the unstrained and in the strained solid, we have*,

^■=1"; y = ]f; ^=f (96).

i-a -.i-b l-c ,„„-

«J- e= a ; /= Y ' ^"~r ^ ^'

where 6 = *---^'; /=^7"; ff = '~/' (98).

•*'o yo ^0

From (97) we get

«=rV«; ^=ilr '=1^ ^^'^•

whence (1 + e)(H-/)(l +5r) = ^ = 1 (100).

* It would have been better from the beginniug to have taken single letters
instead of a"*, fi'K 7~*.

FUKTHER DEVELOPMENT OF STRESS-THEORY. 249

The principal elongations to pass from the unstrained to the Molar,
strained solid are a"S b~^ C""' ; and the principal ratios of elonga-
tion from the strained to the unstrained solid are a, b, C. And if
E denote the work per unit volume required to bring the solid
from the unsti^aiued to the strained condition, we have by (52)

^ = k(a' + b' + C'-3) (101).

From this, remembering that abc = 1, we find

P = -(5a» + tsr); Q = -((?b»+t!r); iJ = - (gC»+ tsr) ....(102);

or p=,y^2+|^_^^^.

«=/?(1t/^-^-9'^ (1^3>'

where P, Q, R denote the normal components of force per unit area
(pulling outward when positive) on the three pairs of faces of a
rectangular parallelepiped required to keep it in the state of strain
a, b, C, with principal elongations perpendicular to the pairs of
faces; and w denotes an arbitrary pressure uniform in all direc-
tions. The proof is as follows : — Consider a cube of unit edges in
the unsti-aiued solid. In the strained condition the lengths of its
edges are 1/a, 1/b, 1/C, and the areas of its faces are a, b, C.
Hence the equation of work done to augmentation of energy pro-
duced in changing a, b, C, to a + 8a, b + Sb, C + Bt, is

PaS - + QbS J + iZcS J = (?S^=5(aSa + bSb + cSc) (104);

a u c

and by abC = 1 (constancy of volume) we have

- -^ -r- -\ = 0 (lOo).

abc ^ ^

Hence by Lagrange's method

-^Sa-§Sb-vSt
abc

= (9a + |-) Sa + (^ft + J) 8b + [qt + fj St (106),

where w denotes an "indeterminate multiplier"; and we have
definitively

-P = ga» + tir; ^ Q^qh'-{-^] -R^qC' + m (107),

250 LECTURE XV.

lar. which prove (102). The meaning of v here, an arbitrary magni-
tude, is a pressure uniform in all directions, which, as the solid
is incompressible, may be arbitrarily applied to the boundary of
any portion of it without altering any of the effective conditions.

§ 36. By § 3**] (92), we see that the propagational velocities
of waves whose lines of vibration are parallel to OX, OY, OZ are
respectively

'-JV^s/y V^ <•<*>•

Thus the propagational velocity of a wave whose front is parallel
to the plane YOZ (\ = 1, /a = 0, i/ = 0) is b^^ if its line of

vibration is parallel to OF (i=0, m=l, n = 0), and is C * /-

if its line of vibration is parallel to OZ (i = 0, m = O, n = 1) ;
and similarly in respect to waves whose fronts are parallel to
ZX and XY,

For brevity we shall call ^ a/- » ^ \/ » ^ \/ ^^^ frin-

cipal velocities of light in the crystal, and OJf, OF, OZ its three
principal lines of symmetry. We arc precluded from calling
these lines optic axes, by the ordinary usage of the word axis in
respect to uni-axial and bi-axial crystals.

§ 37. To help us to thoroughly understand the dynamics of
the stress theory of double refraction, consider as an example
aragonite, a bi-axial crystal of which the three principal refractive
indices are 1*5301, 1*0816, 1*6859. If, as according to our stress-
theory, optic ajolotropy is due to unequal extension and con-
traction in different directions of the ether within the crystal
with volume unchanged, the principal elongations being in simple
proportion to the three principal velocities of light within it,
annulment of the extension and contraction would give isotropy
with refractive index 1*6309, being the cube root of the product
of the three principal indices. Hence, if we call V the pro-
pagational velocity corresponding to this mean index, the three
principal velocities in the actual crystal (being inversely as the
refractive indices) are 10659 F, -9687 F, '9679 F, of which the

FURTHER DEVELOPMENT OF STRESS-THEORY.

251

product is V. Hence according to our notation of § 7 and § 35 Molar,
above we have

a = 10659; b = -9687; £=9679.

§ 38. Let the dotted ellipse in the diagi*am represent a cross
section of an elliptic cylindric portion of aragonite having its
axis of figure in the direction of maximum elongation of the
ether. Let the diameter A' A be the direction of maximum

Fio. 1.

contraction, and B!B that of minimax (elongation-contraction),
being in fact in this case a line along which there is elongation.
The circle in the diagram shows the undisturbed positions of the
particles of ether, which in the crystal are forced to the positions
shown by the dotted ellipse. The axes of the ellipse are equal

252 LECTURE XV.

[olftr. reHpectively to 1/10659 and l/'96ii7 of the diameter of the
circle.

First, consider waves whose fronts are parallel to the plane
of the diagram. If their vibrations are parallel to OA, the
direction of maximum contraction, their velocities of propagation
are 1*0059 V. If their vibrations are parallel to OB their
velocities are '9687 V.

Secondly, consider waves whose fronts are perpendicular to
the plane of the diagram. The propagational velocity of all of
them whose vibrations are perpendicular to this plane is '9679 F,
their lines of vibration being all in the direction of maximum
elongation. If their vibrations are in the plane of the diagram
and parallel to OA, their velocity is 1*0659 V. If their vibrations
are parallel to OB, their velocities are '9687 V, The former of
these is the greatest and the latter the len^t of the propagational
velocities of all the waves whose vibrations are in this plane : the
ratio of the one to the other is 1100.

§ 89. It is interesting, on looking at our diagram, to think
how slight is the distortion of the ether required to produce the
double refraction of aragonite. If the diagram had been made
for the plane ZOX containing the lines of vibration for greatest
and least velocities (ratio of greatest to least 1*101), the increase
of ellipticity of the ellipse would not have been perceptible to the
eye. Only about one and a half per cent greater ratio of the
difference of the diameters of the ellipse to the diameter of the
circle would be shown in the corresponding diagram for Iceland
spar (ratio of greatest to least refractivity = 1115). For nitrate
of soda the ratio is somewhat greater still (1*188). For all other
crystals, so far as I know, of which the double refraction has been
measured it is less than that of Iceland spar. With such slight
distoi-tions of ether within the crystal as the theory indicates for
all these known cases, we could scarcely avoid the constancy of
our coefficient q, to the assumption of which we were forced,
§§ 20, 25 above, in order to fit the theory to Fresnels laws for
light traversing a crystal, irrespectively of smallness or greatness
of the amount of double refraction. Hence we see that, in fact,
without any arbitrary assumption of a new property or new pro-
perties of ether, we have arrived at what would be a perfect
explanation of the main phenomena of double refraction, if we

FURTHER DEVELOPMENT OF STRESS-THEORY. 253

could but see how the molecules of matter could so act upon Moleoalar.
ether as to give a stress capable of producing the strain with
which hitherto we have been dealing. Inability to see this has
prevented me, and still prevents me, from being convinced that
the stress-theory gives the true explanation of double refraction.

§ 40. Now, quite recently, it has occurred to me that the
difficulty might possibly be overcome if, as seems to me necessary
on other grounds, we adopt a hypothesis regarding the motion of
ponderable matter through ether, which I suggested a year ago in
a Friday Evening Lecture, April 27th, 1900, to the Royal Insti-
tution (reproduced in the present volume as Appendix B) and
with somewhat ftiU detail in a communication* of last July to the
Royal Society of Edinburgh, and to the Congrfes Internationale
de Physiquef in Paris last August (Appendix A, below). Accord-
ing to this hypothesis ether is a structureless continuous elastic
solid pervading all space, and occupying space jointly with the
atoms of ponderable matter wherever ponderable matter exists;
and the action between ponderable matter and ether consists of
attractions and repulsions throughout the volume of space occupied
by each atom. These attractions and repulsions would be essen-
tially ineffective if ether were infinitely resistant against forces
tending to condense or dilate it, that is to say, if ether were
absolutely incompressible. Hence, while acknowledging that ether
resists forces tending to condense it or to dilate it, sufficiently to
account for light and radiant heat by waves of purely transverse
vibration (equi-voluminal waves as I have called them), it must,
by contraction or dilatation of bulk, yield to compressing or
dilating forces sufficiently to account for known facts dependent
on mutual forces between ether and ponderable matter. I have
suggested J that there may be oppositely electric atoms which
have the properties respectively of condensing and rarefying the
ether within them. But for the present, to simplify our sup-
positions to the utmost, I shall assume the law of force between
the atom and the ether within it to be such that the average
density of the ether within the atom is equal to the density of

• Proe, Roy, Soc. Edin, July, 1900 ; Phil. Mag. Ang. 1900.
t Reports ^ Vol. ii. page 1.

X CongrH Internationale de Phy»ique, Reports^ Vol. ii. p. 19 ; also Phil. Mag.
Sept. 1900.

254

LECTURE XV.

Moleeulur. the undisturbed ether outside, and that concentric spherical sur-
faces within the atom are surfaces of equal density. The forces
between ether and atoms we can easily believe to be enormous in
coin[)anson with those called into play outside the atoms, in virtue
of undulatory or other motion of ether and elasticity of ether;
as for instance in interstices between atoms in a solid body, or
in the space traversed by the molecules of a gas according to
the kinetic theory of gases, or in the vacuum attainable in our
laboratories, or in interstellar space.

§ 41. To fix our ideas let ether experience condensation in
the central part of the atom and rarefaction in the outer part
according to the law explained generally in the first part of § 5,
Ap|>endix A, and represented particularly by the formulas (9) and
(II) of that section, and fully described with numerical and
graphical illustrations for a -particular case in §§ 5 — 8. Looking

Pio. 2.

to cols. 3 and 4 of Table I, Appendix A, we see that, at distance
r = -50 from the centre of the atom, the density of the ether is
ecjual to the undisturbed density outside the atom ; and that from
r = '56 to r = 1 the density decreases to a minimum, "35, at
r= '865, and augments thence to the undisturbed density, 1, at
the boundary of the atom (r = l). In each of the two atoms
represented in fig. 2, the spherical surface of undisturbed density
is indicated by a dotted circle, that of minimum density by a fine
circle, and the boundary of the atom by a heavy circle. Because
the ethereal density decreases uninterruptedly from the centre to

FURTHER DEVELOPMENT OP STRESS-THEORY. 255

the surface of minimum density, the force exerted by the atom on Molecular,
the ether must be towards the centre throughout this spherical
space ; and because the ethereal density increases uninterruptedly
outwards from the surface of minimum density to the boundary of
the atom, the force exerted by the atom on the ether must be
repulsive in every part of the shell outside that surface.

§ 42. Suppose now two atoms to be somehow held together in
some such position as that represented in fig. 2, overlapping one
another throughout a lens-shaped space lying outside the surface
of minimum ethereal density in each atom. The rarefaction of
the ether in this lens-shaped space is, by the combined action of
the two atoms, greater than at equal distances from the centre in
non-overlapping portions of both the atoms. Hence, remembering
that each atom while attracting the ether in its central parLs
repels the ether in every part of it ouTiside the spherical surface of
minimum density, we see that the repulsion of each atom on the
ether in the lens-shaped volume of overlap is less than its repulsion
in the contrary direction on an equal and similar portion of the
ether within it on the other side of its centre. Hence the re-
actions of the ether on the atoms are forces tending to bring them
together; that is to say apparent attractions. These apparent
attractions are balanced by repulsions between the atoms them-
selves if two atoms rest stably as indicated in fig. 2. It seems not
improbable that these are the forces concerned in the equilibrium
of the two atoms of the known diatomic simple gases Ng, O2, Hj.
I assume that the law of force between ether and atoms and the
law of elasticity of the ether under this force to be such that no
part of the ether outside the atoms experiences any displacement
in consequence of the displacements actually produced within the
space occupied separately and jointly by the two atoms. This
implies that the ether drawn away from the lenticular space of
overlap by the extra rarefaction there, is taken in to the central
regions of the two atoms in virtue of the attractions of the atoms
on the ether in those regions. In an addition to Lecture XVIII.
on the reflection and refraction of light, I shall have occasion to
give explanation and justification of this assumption.

§ 43. As a representation of an optically isotropic crystal,
consider a homogeneous assemblage of atoms for simplicity taken

250 LECTURE XV.

Moleonlar. in cubic order as shown in fig. 3. Each one of the outermost
atoms experiences resultant force inwards from the ether within
it, and this force is balanced by repulsion exerted upon it by the
atom next it inside. For every atom except those lying in the
outer faces, the forces which it experiences in different directions
from the ether within it balance one another : and so do the forces
which it experiences from the atoms around it. But each of the
outermost atoms experiences a resultant repulsion from the other

Fio. 8.

atoms in contact with it, and this repulsion outwards is balanced
by a contrary attraction of the ether on the atom. Hence the
outermost atoms all round a cube of the crystal exert an outward
pull upon the ether within the containing cube. In accordance
with the assumption stated at the end of § 42, the position and
shape of every particle of ether outside the atoms is undisturbed
by the forces exerted by the atoms in the spaces occupied by them
separately and jointly; and it is only in these spaces that the
ether is disturbed by the action of the atoms.

§ 44. Suppose now that by forces applied to the atoms as
indicated by the arrow-heads in fig. 4, the distances between the
centres of contiguous atoms are increased and diminished as shown
in the diagram. With this configuration of the atoms the ether
is pulled outwards by the atoms with stronger forces in the
direction parallel to AC, BD than in directions parallel to -45
and CD, Hence as (^ 42, 43 al)ove) the ether is unstressed and
unstrained in the interstices between the atoms, the ether within
the atoms experiences an excess of outward pulling stress in the

FURTHER DEVELOPMENT OF STRESS-THEORY. 257

direction parallel to AC and BD above outward pulling stress in MoleouU
the direction perpendicular to these lines : and therefore, if there
were no ceolotropy of inertia, and if the stress theory which we
have worked out (^ 35, 36) for homogeneous ether is applicable

Fig. 4.

to average action of the whole ether with its great inequalities of
density in the space occupied by the assemblage . of atoms, the
propagational velocity of distortional waves would be greater when
the direction of vibration is parallel to -4 C and BD than when it
is parallel to AB and CD, But alas! this is exactly the reverse
of what, thirteen years ago, Kerr, by experimental research of a
rigorously testing character, found for the bi-refringent action of
strained glass*, the existence of which had been discovered by
Sir David Brewster seventy years previously. We are forced to
admit that one or both of our two " ty s must be denied.

§ 45. It seems to me now worthy of consideration whether
the true explanation of the double refraction of a natural crystal
or of a piece of strained glass may possibly be that given by
Glazebrook in another paper-f* in the same volume of the Philo-
sophical Magazine as Kcit's already referred to. Glazebrook made

* ** Experiments on the Birefringent Action of Strained Glass," Phil. Mag, Oct
1S88, p. 839.

t «*0n the Application of Sir WiUiam Thomson's Theory of a Contractile
Ether to Doable Refraction, Dispersion, MetaUic Reflexion, and other Optical
Problems." Phil Mag, Deo. ISSS, p. 524.

T. L. 17

258 LECTURE XV.

Molar, the remarkable discovery expressed in his equation (14) (p. 525)
that, when the propagational velocity of the condensational-
rarefactional wave is zero, the two propagational velocities due to
* aeolotropic inertia in a distortional wave with same wave-front*
are the same as those given by my equation (94), and originally by
Fresnel; from which it follows that the wave-surface is exactly
Fresners. It is certainly a most interesting result that the wave
surface should be exactly FresneFs, whether the optic ajolotropy is
due to diflFerence of stress in different directions in an incompres-
sible elastic solid, or to seolotropy of inertia in an ideal elastic
solid endowed with a negative compressibility modulus of just
such value as to make the velocity of the condensational-rare-
factional wave zero. In Glazebrook's theory the direction of
vibration is perpendicular to the line of the ray, and in the
vibratory motion the solid experiences a slight degree of change
of bulk combined with pure distortion. In the stress theory of
§§ 7... 34 the line of vibration is exactly in the wave-front or
perpendicular to the wave normal and there is no change of bulk.

§ 46. In Lecture XVIII, when we are occupied with the
reflection and refraction of light, we shall see that Fresnel's
formula for the three rays, incident, reflected, and refracted, when
the line of vibration is in their plane, is a strict dynamical conse-
quence of the assumption of zero velocity for the condensational-
rarefactional wave in both the mediums, or in one of the mediums
only while the other medium is incompressible. But the difficulties
of accepting zero velocity for condensational-rnrefactional wave,
whether in undisturbed ether through space or among the atoms
of ponderable matter, are not overcome.

§ 47. It will be seen, (§§ 3, 4, 7, 21, 22, 24, 25, 33, 35, 37, 38,
39, 40, 45 above,) that after earnest and hopeful consideration of
the stress theory of double refraction during fourteen years,
I am unable to see how it can give the true explanation
either of the double refraction of natural crystals, or of double
refraction induced in isotropic solids by the application of unequal
pressures in different directions. Nevertheless the mathematical
investigations of ^ 7. ..21 and ^ 24.. .34, interesting as they are

* Z, in, n are the direoiion-oosines of the normal to the wave- front in Glaze-
brook's paper, and X, ft, •», in my §§ 18, 21, 33, 34 above.

FURTHER DEVELOPMENT OF STRESS-THEORY. 259

in the abstract dynamics of a homogeneous incompressible elastic Molar,
solid, have an important application in respect to the influence of
ponderable matter on ether. They prove in fact the truth of the
assumption at the end of § 42, however the forces within the atoms
are to be explained ; because any distortion of ether in the space
around a crystal, even so slight as that illustrated in flg. 1, § 38
above, would produce double refraction in air or vacuum outside
the crystal, not quite as intense outside as inside ; not vastly less,
close to the outside ; and diminishing with distance outside but
still quite perceptible, to distances of seveml diameters of the
mass.

17—2

LECTURE XVI.

Wednesday, October 16, 5 p.m.

[This was a double lecture ; but as the substance of the first
part, with amplification partly founded on experimental discoveries
by many workers since it was delivered, has been already repro-
duced in dated additions on pp. 148 — 157 and 176 — 184 above,
only the second part is here given.]

I want now to go somewhat into detail as to absolute
magnitudes of masses and energies, in order that there may be
nothing indefinite in our ideas upon this part of our subject ; and
I commence by reading and commenting on an old article of mine
relating to the energy of sunlight and the density of ether.

[Nov, 20, 1S99,.. March 28, 1901. From now, henceforth till
the end of the Lectures, sections will be numbered continuously.]

Note on the Possible Density of the Luminiferous
Medium and on the Mechanical Value of a Cubic
MiLE*t OF Sunlight.

[From JSdin. Royal Soc. Tratu., Vol. xxi. Part i. May, 1854 ; Phil, Mag,
IX. 1854 ; Comptes Rendus, xxxix. Sept. 1854 ; Art. Lxvii. of Jfath, attd
Phyt, Papers.']

Molar. § 1. That there must be a medium forming a continuous
material communication throughout space to the remotest visible
body is a fundamental assumption in the undulatory Theory
of Light. Whether or not this medium is (as appearsj to

* [Note of Dec. 22, 1892. The brain-wasting perversity of the insular inertia
which still condemns British Engineers to reckonings of miles and yards and feet
and inches and grains and pounds and ounces and acres is curiously iUnstrated by
the title and numerical results of this Article as originally published.]

t [Oct. 18, 1899. In the present reproduction, as part of my Lee. XVI. of
Baltimore, 1884, 1 suggest cubic kilometre instead of " cubic mile " in the title and
use the French metrical system exclusively in the article.]

X [Oct. 18, 1899. — Not so now. I did not in 1854 know the kinetic theory of gases.]

UNDISTURBED DENSITY OP HOMOGENEOUS ETHER. 261

me most probable) a continuation of our own atmosphere, MoUr.
its existence is a fact that cannot be questioned, when the
overwhelming evidence in favour of the undulatory theory is
considered; and the investigation of its properties in every
possible way becomes an object of the greatest interest. A first
question would naturally occur, What is the absolute density of
the luminiferous ether in any part of space ? I am not aware of
any attempt having hitherto been made to answer this question,
and the present state of science does not in fact afford sufficient
data. It has, however, occurred to me that we may assign an
inferior limit to the density of the luminiferous medium in inter-
planetary space by considering the mechanical value of sunlight
as deduced in preceding communications to the Royal Society
[Tran^. R. S, E. ; Mechanical Energies of the Solar System ; re-
published as Art. LXVi. of Math, and Phys. Papers] from Pouillet's
data on solar radiation, and Joule's mechanical equivalent of
the thermal unit. Thus the value of solar radiation per second
per square centimetre at the earth's distance from the sun,
estimated at 1235 cm.-grams, is the same as the mechanical
value of sunlight in the luminiferous medium through a space
of as many cubic centimetres as the number of linear centimetres
of propagation of light per second. Hence the mechanical value
of the whole energy, kinetic and potential, of the disturbance
kept up in the space of a cubic centimetre at the earth's distance

1235 412

from the sun*, is g ^ ^^^ , or j^ of a cm.-gram.

§ 2. The mechanical value of a cubic kilometre of sunlight is
consequently 412 metre-kilograms, equivalent to the work of one
horse-power for 5*4 seconds. This result may give some idea of
the actual amount of mechanical energy of the luminiferous
motions and forces within our own atmosphere. Merely to com-
mence the illumination of eleven cubic kilometres, requires an
amount of work equal to that of a horse-power for a minute;
the same amount of energy exists in that space as long as
light continues to traverse it ; and, if the source of light be

* The mechanical value of sunlight in any space near the son's surface must
be greater than in an equal space at the earth's distance, in the ratio of the square
of the earth's distance to the square of the sun's radius, that is, in the ratio of
46,000 to 1 nearly. The mechanical value of a cubic centimetre of sunlight near

the sun must, therefore, be — s — ttzik— or about '0019 of a cm. -gram.

3 X lO*"

262 LECTURE XVI.

Moleoolar. suddenly stopped, must pass from it before the illumination
ceases*. The matter which possesses this energy is the lumini-
ferous medium. If, then, we knew the velocities of the vibratory
motions, we might ascertain the density of the luminiferous
medium ; or, conversely, if we knew the density of the medium,
we might determine the average velocity of the moving particles.

§ 3. Without any such definite knowledge, we may assign a
superior limit to the velocities, and deduce an inferior limit to the
quantity of matter, by considering the nature of the motions
which constitute waves of light. For it appears certain that the
amplitudes of the vibrations constituting radiant heat and light
must be but small fractions of the wave-lengths, and that the
greatest velocities of the vibrating particles must be very small
in comparison with the velocity of propagation of the waves.

§ 4. Let us consider, for instance, homogeneous plane polarized
light, and let the greatest velocity of vibration be denoted by v ;
the distance to which a particle vibrates on each side of its
position of equilibrium, by A; and the wave-length, by X. Then,
if V denote the velocity of propagation of light or radiant heat,
we have

y=27r-;

and therefore if il be a small fraction of \, r must also be a small
fraction (27r times as great) of V, The same relation holds for
circularly polarized light, since in the time during which a particle
revolves once round in a circle of radius -4, the wave has been pro-
pagated over a space equal to \. Now the whole mechanical value
of homogeneous plane polarized light in an infinitely small space
containing only particles sensibly in the same phase of vibration,
which consists entirely of potential energy at the instants when
the particles are at rest at the extremities of their excursions,
partly of potential and partly of kinetic energy when they are
moving to or from their positions of equilibrium, and wholly of
kinetic energy when they are passing through these positions, is
of constant amount, and must therefore be at every instant equal
to half the mass multiplied by the square of the velocity which the
particles have in the last-mentioned case. But the velocity of any

* Similftrly we find 4140 horse-power for a minute as the amount of work
required to generate the energy existing in a cubic kilometre of light near the sun.

KINETIC AND POTENTIAL ENERGY OF LIGHT. 263

particle passing through its position of equilibrium is the greatest Moleoula:
velocity of vibration. This we have denoted by v; and, there-
fore, if p denote the quantity of vibrating matter contained in a
certain space, a space of unit volume for instance, the whole me-
chanical value of all the energy, both kinetic and potential, of the
disturbance within that space at any time is \pt^. The mechani-
cal energy of circularly polarized light at every instant is (as has
been pointed out to me by Professor Stokes) half kinetic energy
of the revolving particles and half potential energy of the dis-
tortion kept up in the luminiferous medium ; and, therefore, v
being now taken to denote the constant velocity of motion of each
particle, double the preceding expression gives the mechanical
value of the whole disturbance in a unit of volume in the present
case.

§ 5. Hence it is clear, that for any elliptically polarized light
the mechanical value of the disturbance in a unit of volume will
be between \(ytf and pr*, if v still denote the greatest velocity of
the vibrating particles. The mechanical value of the disturbance
kept up by a number of coexisting series of waves of different
periods, polarized in the same plane, is the sum of the mechanical
values due to each homogeneous series separately, and the greatest
velocity that can possibly be acquired by any vibrating particle is
the sum of the separate velocities due to the different series.
Bxactly the same remark applies to coexistent series of circularly
polarized waves of different periods. Hence the mechanical value
is certainly less than half the mass multiplied into the square of
the greatest velocity acquired by a particle, when the disturbance
consists in the superposition of different series of plane polarized
waves ; and we may conclude, for every kind of radiation of light
or heat except a series of homogeneous circularly polarized waves,
that the mechanical value of the disturbance kept up in any space
is less than the product of the ma^s into the square of the greatest
velocity acquired by a vibrating particle in the varying phases of its
motion. How much less in such a complex radiation as that of
sunlight and heat we cannot tell, because we do not know how
much the velocity of a particle may mount up, perhaps even to
a considerable value in comparison with the velocity of propaga-
tion, at some instant by the superposition of different motions
chancing to agree ; but we may be sure that the product of the

264 LBCTURK XVI.

[oleonlftr. mass into the square of an ordinary maximum velocity, or of^the
mean of a great many successive maximum velocities of a vibrat-
ing particle, cannot exceed in any great ratio the true mechanical
value of the disturbance.

§ 6. Recurring, however, to the definite expression for the
mechanical value of the disturbance in the case of homogeneous
circularly polarized light, the only case in which the velocities
of all particles are constant and the same, we may define the
mean velocity of vibration in any case as such a velocity that
the product of its square into the mass of the vibrating par-
ticles is equal to the whole mechanical value, in kinetic and
potential energy, of the disturbance in a certain space traversed
by it ; and from all we know of the mechanical theory of undula-
tions, it seems certain that this velocity must be a very small
fraction of the velocity of propagation in the most intense light
or radiant heat which is propagated according to known laws.
Denoting this velocity for the case of sunlight at the earth's
distance from the sun by v, and calling W the mass in grammes of
any volume of the luminiferous ether, we have for the mechanical
value of the disturbance in the same space, in terms of terrestrial
gravitation unit«,

where g is the number 981, measuring in (c.G.s.) absolute units

of force, the force of gravity on a gramme. Now, from Pouillet's

observation, we found in the last footnote on § 1 above,

1235 X 46000 . ,, i • , , • ...

^ for the mechanical value, m centiraetre-grams,

of a cubic centimetre of sunlight in the neighbourhood of the
sun; and therefore the mass, in grammes, of a cubic centimetre
of the ether, must be given by the equation,

^r_ 981 J< L235 X 46000

If we assume v = - F, this becomes

n

^ 981 X 1235 X 46000 , 981 x 1235 x 46000 ,
W^ V2 X ^'= (S^TTO^ - ^ '''

20-64 ,
= i^ xn»gm.;

RIGIDITY OF ETHER. 265

and for the mass, in grammes, of a cubic kilometre we have Moleeola

20-64

10'

xn\

§ 7. It is quite impossible to fix a definite limit to the ratio
which V may bear to V; but it appears improbable that it could
be more, for instance, than ^, for any kind of light following the
observed laws. We may conclude that probably a cubic centimetre
of the luminiferous medium in the space near the sun contains
not less than 516 x 10~* of a gramme of matter; and a cubic
kilometre not less than 516 x 10~* of a gramme.

§ 8. [Nov, 16, 1899. We have strong reason to believe that
the density of ether is constant throughout interplanetary and
interstellar space. Hence, taking the density of water as unity
according to the convenient French metrical system, the preceding
statements are equivalent to saying that the density of ether in
vacuum or space devoid of ponderable matter is everywhere
probably not less than 5 x 10""".

Hence the rigidity, (being equal to the density multiplied
by the square of the velocity of light), must be not less than
4500 dynes* per square centimetre. With this enormous value
as an inferior limit to the rigidity of the ether, we shall see in an
addition to Lecture XIX. that it is impossible to arrange for a
radiant molecule moving through ether and displacing ether by
its translatory as well as by its vibratory motions, consistently
with any probable suppositions as to magnitudes of molecules
and ruptural rigidity-modulus of ether; and that it is also
impossible to explain the known smallness of ethereal resistance
against the motions of planets and comets, or of smaller ponder-
able bodies, such as those we can handle and experiment upon in
our abode on the earth's surface, if the ether must be pushed
aside to make way for the body moving through it. We shall find
ourselves forced to consider the necessity of some hypothesis for
the free motion of ponderable bodies through ether, disturbing it
only by condensations ^nd rarefactions, with no incompatibility
in respect to joint occupation of the same space by the two
substances.] See PhiL Mag. Aug. 1900, pp. 181 — 198.

* See Math, and Pkys. Paper*, Vol. iii. p. 522 ; and in last line of Table 4,
for " /) > 10-« " substitute " /> < 10-«."

266 LECTURE XVI.

Molar. § 9. I wish to make a short calculation to show how much com-
pressing force is exerted upon the luminiferous ether by the sun s
attraction. We are accustomed to call ether imponderable. How
do we know it is imponderable ? If we had never dealt with air
except by our senses, air would be imponderable to us; but we
know by experiment that a vacuous glass globe shows an increase
of weight when air is allowed to flow into it. We have not the
slightest reason to believe the luminiferous ether to be imponder-
able. [Nov. 17, 1899. I now see that we have the strongest
possible reason to believe that ether is imponderable.] It is just
as likely to be attracted to the sun as air is. At all events the
onus of proof rests with those who assert that it is imponderable.
I think we shall have to modify our ideas of what gravitation is,
if we have a mass spreading through space with mutual gravita-
tions between its parts without being attracted by other bodies.
[Nov. 17, 1899. But is there any gravitational attraction between
different portions of ether? Certainly not, unless either it is
infinitely resistant against condensation, or there is only a finite
volume of space occupied by it. Suppose that ether is given uni-
formly spread through space to infinite distances in all directions.
Any large enough spherical portion of it, if held with its surface
absohitely fixed, would by the mutual gmvitation of its parts become
heterogeneous ; and this tendency could certainly not be counter-
acted by doing away with the supposed rigidity of its boundary
and by the attraction of ether extending to infinity outside it.
The pressure at the centre of a spherical portion of homogeneous
gravitational matter is proportional to the square of the radius,
and therefore, by taking the globe large enough, may be made as
large as we please, whatever be the density. In fact, if there
were mutual gravitation between its parts, homogeneous ether
extending through all space would be essentially unstable, unless
infinitely resistant against compressing or dilating forces. If we
admit that ether is to some degree condeusible and extensible,
and believe that it extends through all space, then we must
conclude that there is no mutual gravitation between its parts,
and cannot believe that it is gravitationally attracted by the sun
or the earth or any ponderable matter; that is to say, we must
believe ether to be a substance outside the law of universal
gravitation.]

WEIGHT OF ETHER, IF GRAVITATIONAL. 267

§ 10. In the meantime, it is an interesting and definite question Molar,
to think of what the weight of a column of luminiferous ether of
infinite height resting on the sun, would be, supposing the sun
cold and quiet, and supposing for the moment ether to be
gravitationally attracted by the sun as if it were ponderable
matter of density 5 x 10~". You all know the theorem for mean
gravity due to attraction inversely as the square of the distance
from a point. It shows that the heaviness of a uniform vertical
column AB, of mass w per unit length, and having its length in a
line through the centre of force C, is

tnw mw mw .^ ^^

cJ-cB' «>-ei'^^^=*'

where m denotes the attraction on unit of mass at unit distance.
Hence writing for mw/CA, viwCA/CA\ we see that the attraction
on an infinite column under the influence of a force decreasing
according to inverse square of distance, is equal to the attraction
on a column equal in length to the distance of its near end from
the centre, and attracted by a uniform force equal to that of
gravity on the near end. The sun's radius is 697 x 10* cma and
gravity at his surface is 27 times* terrestrial gravity, or say
27000 dynes per gramme of mass. Hence the sun's attraction on
a column of ether of a s<juare centimetre section, if of density
5 X 10~" and extending from his surface to infinity, would be
9*4 X 10~' of a dyne, if ether were ponderable.

§ 11. Considerations similar to those of November 1899 in-
serted in § 9 above lead to decisive proof that the mean density of
ponderable matter through any very large spherical volume of
space is smaller, the greater the radius ; and is infinitely small for
an infinitely great radius. If it were not so a majority of the
bodies in the universe woiild each experience infinitely great
gravitational force. This is a short statement of the essence of
the following demonstration.

§ 12. Let V be any volume of space bounded by a closed
surface, S, outside of which and within which there are ponderable
bodies; M the sum of the masses of all these bodies within S\

* This is founded on the foUowing values for the son's mass and radius and the
earth's radius: — sun's mass =824000 earth's mass; sun's radius =697000 kilo-
metres; earth's radius =6S71 kilometres.

268 LECTURE XVI.

Molar, and p the mean density of the whole matter in the volume F.

We have

M^pV (1).

Let Q denote the mean value of the normal component of the
gravitational force at all points of 8. We have

Qfif=47rif=47rpF (2),

by a general theorem discovered by Green seventy-three years
ago regarding force at a surface of any shape, due to matter
(gravitational, or ideal electric, or ideal magnetic) acting according
to the Newtonian law of the inverse square of the distance. It
is interesting to remark, that the surface-integi*al of the normal
component force due to matter outside any closed surface is zero
for the whole surface. If nonnal component force acting inwards
is reckoned positive, force outwards must of course be reckoned
negative. In equation (2) the normal component force may be
outwards at some points of the surface fif, if in some places the
tangent plane is cut by the surface. But if the surface is
wholly convex, the normal component force must be everywhere
inwards.

§ 13. Let now the surface be spherical of radius r. We have

6' = 47r?-^ F = ^.7^'»; V = \rS (3).

Hence, for a spherical surface, (2) gives

^ = -3-^^ = :^ ^*)-

This shows that the average normal component force over the
surface S is infinitely great, if p is finite and r is infinitely great,
which suffices to prove § 11.

§ 14. For example, let

r = 150 . 10«. 2()G . 10« = 3-09 . 10»« kilometres (5).

This is the distance at which a star must be to have parallax one
one-thousandth of a second ; because the mean distance of the
earth from the sun is one-hundred-and-fifty-million kilometres, and
there are two-hundred-and-six-thousand seconds of angle in the
radian. Let us try whether there can be as much matter as a

GRAVITATIONAL MATTER WITHIN VERY LARGE VOLUME. 269

thousand-million times the sun's mass, or, as we shall say for MoUr.
brevity, a thousand-million suns, within a spherical surface of that
radius (5). The sun*s mass is 324,000 times the earths mass; and
therefore oiu: quantity of matter on trial is 3*24 . 10" times the
earth's mass. Hence if we denote by g terrestrial gravity at the
earth's surface, we have by (4)

e = 3-24.10»(|g^J-^,)V = l-37.10-.fir (6).

Hence if the radial force were equal over the whole spherical
surface, its amount would be 137. 10~" of terrestrial surface-
gravity ; and every body on or near that surface would experience
an acceleration toward the centre equal to

1*37 . 10~" kilometres per second per second (7),

because g is approximately 1000 centimetres per second per
second, or '01 kilometre per second per second. If the normal
force is not uniform, bodies on or near the spherical sur&ce will
experience centre ward acceleration, some at more than that rate,
some less. At exactly that rate, the velocity acquired per year
(thii-ty-one and a half million seconds) would be 4*32 . 10"* kilo-
metres per second. With the same rate of acceleration through
five million years the velocity would amount to 2l*C kilometres
per second, if the body started from rest at our spherical surface ;
and the spa<;e moved through in five million years would be
17 . 10" kilometres, which is only '055 of r (5). This is so small
that the force would vary very little, unless through the accident
of near approach to some other body. With the same acceleration
constant through twenty-five million years the velocity would
amount to 108 kilometres per second; but the space moved through
in twenty-five million years would be 4*25 . 10" kilometres, or
more than the radius r, which shows that the rate of acceleration
could not be approximately constant for nearly as long a time as
twenty-five million years. It would, in fact, have many chances
of being much greater than 108 kilometres per second, and many
chances also of being considerably less.

§ 15. Without attempting to solve the problem of finding
the motions and velocities of the thousand million bodies, we can
see that if they had been given at rest* twenty -five million years

* " The potential energy of gravitation may be in reality the ultimate created

270 LECTURE XVI.

Molar, ago distributed uniformly or non-uniformly through our sphere (5)
of 309 . 10" kilometres mdius, a very large proportion of them
would now have velocities not less than twenty or thirty kilometres
per second, while many would have velocities less than that ; and
certainly some would have velocities greater than 108 kilometres
per second; or if thousands of millions of years ago they had
been given at rest, at distances from one another very great in
comparison with r (5), so distributed that they should temporarily
now be equably spaced throughout a spherical surface of radius r
(5), their mean velocity (reckoned as the square ix)ot of the mean
of the squares of their actual velocities) would now be 50*4 kilo-
metres per second*. This is not very unlike what we know of
the stars visible to us. Thus it is quite possible, perhaps pro-
bable, that there may be as much matter as a thousand million
suns within the distance corresponding to parallax one one-
thousandth of a second (3*09 . 10" kilometres). But it seems
perfectly certain that there cannot be within this distance as
much matter as ten thousand million suns ; because if there were,
we should find much greater velocities of visible stars than
observation shows ; according to the following tables of results,
and statements, from the most recent scientific authorities on the
subject.

" antecedent of all the motion, heat, and light at present in the universe." See
Mechanical Antecedents of Motion ^ Heat^ and Light, Art. lxix. of my Collected
Math, and Phys, Papers, Vol. ii.

* To prove this, remark that the exhaustion of gravitational energy

1 /"■*"* /"-♦■« /*-♦■•

Thomson and Tait^s Natural Philosophy, Part II. § 540) when a vast number, iV, of

equal masses come from rest at infinite distances from one another to an equably

spaced distribution through a sphere of radius r in easily found to be 3/10 Fr,

where F denotes the resultant force of the attraction of all of them on a material

point, of mass equal to the sum of their masses, placed at the spherical surface.

Now this exhaustion of gravitational energy is spent wholly in the generation of

1 3

kinetic energy ; and therefore we have 2 ^ niv^ = ts ^'^ »°d ^y iV F=1'S7 , 10-" 2m ;

whence

2mv* 3 , __ -rt-i«
^^=^1-87.10 ».r

which, for the case of equal masses, gives, with (5) for the value of r,

^/

^= ^(1 1-37 . 10-« . 3-09 . 10") = 50-4 kilometres per second.

VELOCITIES OF STARS.

271

From the Annuaire du Bureau des Longitudes (Pam, 1901). Molar.

Magni-
tude

0-7
6-8
6 1
1-4
8-2
7-9
7-5
0-5
9-0
6 5
8-5
4-7
3-6
02
9 0

0
5
4
2
1
7
4
0

9
2
5
4
0
0
1
2

2-2

Name of Star

a Centauri

21185 Lalande

61Cygni

Sinus

18609 Arg.-(Eltzen...

34 Groombridge

9352Lacaille

Procyon

11677 Arg.-(Eltzen...

1643 Fedorenko

21258 Lalande

<r Draoonis

If GassiopeisB

a AurigaB

17415 Arg.-(£ltzen...

a AquilflB

e "Indien"

o'&idani

/3 CassiopeisB

a Tauri..

1831 Fedoreuko

p' Ophiuchi

Vega

a Urs. Min. (Polaris)

Distance

from earth,

in million

million

Kilometres

Annual

proper

motions

Parallax

Velocities
perpendicular

to line of
sight, in kilo-
metres per
second

Stars which have largest of observed Velocities in the Line of
Sight. (Extract by the Astronomer Royal from an Article
in the Astraphysiccil Journal for 1901, January, by W. W.
Campbell, Director of Lick Observatory.)

Magni-
tudes

Star

B. A.

Deo.

Velocity

4-6
4*2

c AndromedsB

h. m.

0 33

1 0

5 47

6 50
21 17
18 8

+ 28 46
+ 54 20
-20 54
-11 55
+ 19 23
-21 1

- 84 km. per sec.
-97 ., ..
+ 95 „
+ 96 „
-76 ,. „
-76 „ „

u Cassiopeis

3 Leporis

d Canis Majoris

t Peoasi

! 4-1

ii Sainttarii

The + sign denotes reoesoion, the - sign approach.

272

LECTURE XVI.

Molar. Motions of Stars in the Line of Sight determined at Potsdam
Observatory, 1889-1891. (Communicated by Professor Becker,
University Observatory, Glasgow.)

Star

a Andromeds ...

/SCassiopeiflB

a Caasiopeiie

7 CaBsiopein

/9 AndroinedaB ...
a Ureas Minoris ...
7 AndromediD . . .

a Arietis

/SPersei

a Persei

a Tauri

a Aurigs

p Orionis

7 Orionis

/5 Tauri

9 Ononis

e Orionis

^Ononis

aOrionis

/9 Aurigffi

7 Geminornm ...
a Canis Majoris...
a Geminorum ...
a CanisMinoris...
p Geminorum . . .
a Ijeonis

*«S^ relative to
*°^® the Sun

20

21

var.

20

2-3

20

2-4

20

var.

20

10

10

10

20

20

2-6

2-0

20

var.

2-0

2-3

1-0

2-8

10

1-3

1-8

km.
+ 4-6
+ 6-2
-15-2

- 3-6
+ 11-2
-26-9
-12-9
-14-7

- 1-6
-10-8
+ 48-6
+ 24-6
+ 16-4
+ 9-2
+ 8-0
+ 0-9
+ 26-5
+ 14-8
+ 17-2
-281
-16-6
-16-6
-29-7

- 9-2
+ M

- 91

Star

7Leoni8

/9 UrsfB Majoris...
a UrsfB Majoris ...

8 Leonis

/3 Leonis

7 Ursas Majoris...
e UrsfB Majoris ...

a Virginis

j'UrsflB Majoris ...
17 Ursie Majoris...

a Bootis

c Bootis

fi UrssB Minoris...

/9 Librie

a Coronas

a Serpentis

/3 Herculis

a Ophiuchi

a LyraB

a Aquilas

-yCygni

a Cygni

e Pegasi

p Pegasi

a Pegasi

Magni-
tude

Velocity

relatiye to

the Sun

km.

20

-88-5

2*3

-29-3

20

-11-9

2-3

-14-4

20

-12-2

2-3

-26-6

2-0

-30-3

10

-14-8

21

-31-2

20

-26-2

10

- 7-7

20

-16-3

20

+ 14-2

20

- 9-6

20

+ 3S0

2-3

+ 22-3

2-3

-35-3

20

+ 19-2

10

-lo-3

1-3

-36-9

2-4

- 6-4

1-6

- 80

2-3

+ 8-0

var

+ 6-7

2-0

+ 1-3

The velocity of the sun relatively to stars in general according
to Kenipf and Risteen is probably about 19 kilometres per second*.
In respect to greatest proper motions and velocities Sir Norman
Lockyer gives me the following information : — "The star with
" the greatest known proper motion (across the line of sight) is
"243 Cordoba = 8"*7 per annum. Velocity in kilometres not
" known.

" 1830 Groombridge has a proper motion of 7"0 per annum
" and a parallax of 0"089, from which it results that the velocity
" across the line of sight is 370 kms. per second. Various esti-
" mates of the parallax, however, have been made and this velocity
"is somewhat uncertain. The star with the greatest known
"velocity in the line of sight is f Herculis, which travels at
" 70 kms. per second.

* See footnote on § 10 of Appendix B.

VELOCITIES OF STARS. 273

" The dark line component of Nova Persei was approaching Molar,
"the earth with a velocity of over 1100 kms. per second." This
last-mentioned and greatest velocity is probably that of a ton-ent
of gas due to comparatively small particles of melted and evaporat-
ing fragments shot out laterally from two great solid or liquid
masses colliding with one another, which may be many times
greater than the velocity of either before collision ; just as we see
in the trajectories of small fragments shot out nearly horizontally
when a condemned mass of cast-iron is broken up by a heavy
mass of iron falling upon it from a height of perhaps twenty feet
in engineering works.

§ 16. Newcomb has given a most interesting speculation
regarding the very great velocity of 1830 Groombridge, which he
concludes as follows: — "If, then, the star in question belongs to
"our stellar system, the masses or extent of that system must
" be many times greater than telescopic observation and astro-
"nomical research indicate. We may place the dilemma in a
" concise form, as follows : —

" Either the bodies which compose our universe are vastly
" more massive and numerous than telescopic examination seems
" to indicate, or 1830 Groombridge is a runaway star, flying on a
" boundless course through infinite space with such momentum that
" the attraction of all the bodies of the universe can never stop it.

" Which of these is the more probable alternative we cannot
" pretend to say. That the star can neither be stopped, nor bent
"far from its course until it has passed the extreme limit to
"which the telescope has ever penetrated, we may consider
" reasonably certain. To do this will require two or three millions
"of years. Whether it will then be acted on by attractive forces
"of which science has no knowledge, and thus carried back to
" where it started, or whether it will continue straightforward for
" ever, it is impossible to say.

" Much the same dilemma may be applied to the past history
" of this body. If the velocity of two hundred miles or more per
" second with which it is moving exceeds any that could be pro-
" duced by the attraction of all Ihe other bodies in the universe,
" then it must have been flying forward through space from the
" beginning, and, having come from an infinite distance, must be
" now passing through our system for the first and only time."
T.L. 18

274 LECTURE XVI.

Molar. § 17. In all these views the chance of passing another star at

some small distance such as one or two or three times the sun's
radius has been overlooked; and that this chance is not excessively
rare seems proved by the multitude of Novas (collisions and their
sequels) known in astronomical history. Suppose, for example,
1830 Groombridge, moving at 370 kilometres per second, to chase
a star of twenty times the sun's mass, moving nearly in the same
direction with a velocity of 50 kilometres per second, and to
overtake it and pass it as nearly as may be without collision. Its
own direction would be nearly reversed and its velocity would be
diminished by nearly 100 kilometres per second. By two or three
such casualties the greater part of its kinetic energy might be
given to much larger bodies previously moving with velocities of
less than 100 kilometres per second. By supposing reversed, the
motions of this ideal history, we see that 1830 Groombridge may
have had a velocity of less than 100 kilometres per second at some
remote past time, and may have had its present great velocity
produced by several cases of near approach to other bodies of much
larger mass than its own, previously moving in dii'ections nearly
opposite to its own, and with velocities of less than 100 kilometres
per second. Still it seems to me quite possible that NewcomVs
brilliant suggestion may be true, and that 1830 Groombridge is a
roving star which has entered our galaxy, and is destined to travel
through it in the course of perhaps two or three million years, and
to pass away into space never to return to us.

§ 18. Many of our supposed thousand million stars, perhaps
a great majority of them, may be dark bodies ; but let us suppose
for a moment each of them to be bright, and of the same size and
brightness as our sun ; and on this supposition and on the further
suppositions that they are imiformly scattered through a sphere
(5) of radius 3*09 . 10** kilometres, and that there are no stars out-
side this sphere, let us find what the total amount of starlight
would be in comparison with sunlight. Let n be the number per
unit of volume, of an assemblage of globes of radius a scattered
uniformly through a vast space. The number in a shell of radius
q and thickness dq will be n . ifirq^dq, and the sum of their
apparent areas as seen from the centre will be

—J n . 4>irq'dq or n . 4f7rWdq,

TOTAL APPARENT AREA OF STARS. 276

Hence by integrating from 5 = 0 to 5 = r we find Molar.

n.^TT^ah- (8)

for the sum of their apparent areas. Now if N be the total number
in the sphere of radius r we have

n = J\r^(^r») (9).

Hence (8) becomes -AT. Stt [- j ; and if we denote by a the ratio of
the sum of the apparent areas of all the globes to 4^ we have

«-¥©■ ('»>

(1 — a)/a, very approximately equal to 1/a, is the ratio of the
apparent area not occupied by stars to the sum of the apparent
areas of all their discs. Hence a is the ratio of the apparent
brightness of our star- lit sky to the brightness of our sun's disc.
Cases of two stars eclipsing one another wholly or partially would,
with our supposed values of r and a, be so extremely rare that
they would cause a merely negligible deduction from the total of
(10), even if calculated according to pure geometrical optics. This
negligible deduction would be almost wholly annulled by diflfraction,
which makes the total light from two stars of which one is eclipsed
by the other, very nearly the same as if the distant one were seen
clear of the nearer.

§ 19. According to our supposition of § 18, we have N = 10*,

a = 7.10* kilometres, and therefore rja = 4*4 . 10^®. Hence

by (10)

a = 3-87.10-^» (11).

This exceedingly small ratio will help us to test an old and
celebrated hypothesis that if we could see far enough into space
the whole sky would be seen occupied with discs of stars all of
perhaps the same brightness as our own sun, and that the
reason why the whole of the night-sky and day-sky is not as
bright as the sun's disc is that light suflFers absorption in
travelling through space. Remark that if we vary r keeping
the density of the matter the same, N varies as the cube of r.
Hence by (10) a varies simply as r\ and therefore to make a
even as great as 3*87/100, or, say, the sum of the apparent

18—2

276 LECTURE XVI.

Molar. areas of discs 4 per cent, of the whole sky, the radius must be
10". r or 309.10^ kilometres. Now light travels at the rate
of 300,000 kilometres per second or 9*45.10*' kilometres per year.
Hence it would take 3*27.10" or about 3 J. 10** years to travel
from the outlying suns of our great sphere to the centre. Now
we have irrefragable dynamics proving that the whole life of
our sun as a luminary is a very moderate number of million
years, probably less than 50 million, possibly between 50 and 100.
To be very liberal, let us give each of our stars a life of a hundred
million years as a luminary. Thus the time taken by light to
travel from the outlying stars of our sphere to the centre would
be about three and a quarter million times the life of a star.
Hence, if all the stars through our vast sphere commenced
shining at the same time, three and a quarter million times the
life of a star would pass before the commencement of light
reaching the earth from the outlying stars, and at no one
instant would light be reaching the earth from more than an
excessively small proportion of all the stars. To make the whole
sky aglow with the light of all the stars at the same time the
commencements of the different stars must be timed earlier
and earlier for the more and more distant ones, so that the time
of the arrival of the light of every one of them at the earth
may fall within the durations of the lights at the earth of all
the others ! Our supposition of uniform density of distribution
is, of course, quite arbitrary; and (§§ 13, 15 above) we ought, in
the greater sphere of § 19, to assume the density much smaller
than in the smaller sphere (5) ; and in fact it seems that there
may not be enough of stars (bright or dark) to make a total of
star-disc-area more than 10~" or 10~" of the whole sky. See
Appendix D, " On the Clustering of Gravitational Matter in any
** part of the Universe."

§ 20. To understand the sparseness of our ideal distribution
of 1000 million suns, divide the total volume of the supposed
sphere of radius r(5) by 10*, and we find 123*5.10*^ cubic kilo-
metres as the volume per sun. Taking the cube root of this
we find 4*98 . 10*^ kilometres as the edge of the corresponding
cube. Hence if the stars were arranged exactly in cubic order
with our sun at one of the eight comers belonging to eight
neighbouring cubes, his six nearest neighbours would be each

NUMBER OF VISIBLE STARS. 277

at distance t*98.10" kilometres; which is the distance corre- Molar,
spending to parallax 0" 02. Our sun vseen at so great a distance
would probably be seen as a star of something between the first
and second magnitude. For a moment suppose each of our
1000 million suns, while of the same mass as our own sun, to
have just such brightness as to make it a star of the first magni-
tude at distance corresponding to parallax 1"'0. The brightness
at distance r (5) corresponding to parallax 0"'001 would be one
one-millionth of this, and the most distant of our assumed
stars would be visible through powerful telescopes as stars of the
sixteenth magnitude. Newcomb {Papular Astronomy, 1883,
p. 424) estimated between 30 and 50 million as the number of
stars visible in modem telescopes. Young {General Astronomy^
p. 448) goes beyond this reckoning and estimates at 100 million
the total number of stars visible through the Lick telescope.
This is only the tenth of our assumed number. It is never-
theless probable that there may be as many as 1000 million
stars within the distance r (5); but many of them may be
extinct and dark, and nine-tenths of them though not all dark
may be not bright enough to be seen by us at their actual
distances.

§ 21. I need scarcely repeat that our assumption of equable
distribution is perfectly arbitrary. How far from being like the
truth is illustrated by Herschel's view of the form of the universe
as shown in Newcomb's Popular Astronomy y p. 461). It is quite
certain that the real visible stars within the distance r (5) from
us are very much more crowded in some parts of the whole
sphere than in others. It is also certain that instead of being
all equally luminous as we have taken them, they diflfer largely
in this respect from one another. It is also certain that the
masses of some are much greater than the masses of others;
as will be seen from the following table, which has lieen compiled
for me by Professor Becker from Andre's Traiti d'Astronomie
SteUaire, showing the sums of the masses of the components of
some double stars, and the data from which these have been
determined.

278

LECTTTBE XVI.

i Major axis

Period,

in
years

in nnits

of the

sun*s mass

1

r

1

1

Parallax

in seconds

in terms of
semi-major

axis of
earth's orbit

a Centauri

6'75

1817
29-48
8-31
5-84
6-72
8-20
4B0

25

84

20
0-6
3-2
6-3
0-9
4-3
3-6

61 Cygni

Sirius

Procyon

0*44
0-39
0-27

68 : 783
24 52
4 40
28 176
39 1 190
30 88

1

o' Eridani

019
015
015

17 CasfliopeisB

p Ophinohi

•y Virginis

005 1.
002 1

3-99
1-98

79*
102*

194
407

150
6-5

7 Leonis

§ 22. There may also be a large amount of matter in many
stars outside the sphere of 3.10^' kilometres radius, but however
much matter there may be outside it, it seems to be made highly
probable by §§ 11 — 21, that the total quantity of matter within
it is greater than 100 million times, and less than 2000 million
times, the sun's mass.

I wish, in conclusion, to express my thanks to Sir Norman
Lockyer, to the Astronomer Royal Mr Christie, to Sir Robert
Ball, and to Prof. Becker, for their kindness in taking much
trouble to give me information in respect to astronomical data,
which has proved most useful to^me in §§ 11 — 21 above.

* From spectroscopic observations by Belopolsky of Poulcowa, combined with
elements of orbit.

t Parallax calculated from dynamical determinations of ratio of semi-major
axis of double-star's orbit to semi-major axis of earth's orbit.

LECTURE XVIL

Thursday, October 16, 3.30 p.m. Altered (1901, 1902) to

extension of Lee, XVI,

§ 23. Hitherto in all our views we have seen nothing of abso- Molecula
lute dimensions in molecular structure, and have been satisfied to
consider the distance between neighbouring molecules in gases,
or liquids, or crystals, or non-crystalline solids to be very small in
comparison with the shortest wave-length of light with which we
have been concerned. Even in respect to dispersion, that is to
say. difference of propagational velocity for diflferent wave-lengths,
it has not been necessary for us to accept Cauchy's doctrine that
the spheres of molecular action are comparable with the wave-
length. We have seen that dispersion can be, and probably in
fact is, truly explained by the periods of our waves of light being
not infinitely great in comparison with some of the periods of
molecular vibration; and, with this view, the dimensions of
molecular structure might, so fai* as dispersion is concerned, be as
small as we please to imagine them, in comparison with wave-
lengths of light. Nevei-theless it is exceedingly interesting and
important for intelligent study of molecular structures and the
dynamics of light, to have some well-founded understanding in
respect to probable distances between centres of neighbouring
molecules in all kinds of ponderable matter, while for the present
at all events we regard ether as utterly continuous and structure-
less. It may be found in some future time that ether too has a
molecular structure, perhaps much finer than any structure of
ponderable matter ; but at present we neither see nor imagine
any reason for believing ether to be other than continuous and
homogeneous through infinitely small contiguous portions of
space void of other matter than ether.

§ 24. The first suggestion, so far as we now know, for estimat-
ing the dimensions of molecular structure in ordinary matter was

280 LEOTURE XVII.

[oleoular. given in 1805 by Thomas Young*, as derived from his own and
Laplace's substantially identical theories of capillary attraction.
In this purely dynamical theory he found that the range of the
attractive force of cohesion is equal to ST/K; where T denotes
the now well-known Young's tension of the free surface of a
liquid, and K denotes a multiple integral which appears in
Laplace's formulas and is commonly now referred to as Laplace's
K, as to the meaning of which there has been much controversy
in the columns of Naturae and elsewhere. Lord Rayleigh in his
article of 1890, ** On the Theory of Surface Forces f," gives the
following very interesting statement in respect to Young's estimate
of molecular dimensions : —

§ 25. " One of the most remarkable features of Young's treatise
" is his estimate of the range a of the attractive force on the basis
" of the relation T—^aK, Never once have I seen it alluded to ;
" and it is, I believe, generally supposed that the first attempt of
" the kind is not more than twenty years old. Estimating K at
" 23000 atmospheres, and T at 3 grains per inch, Young finds that
" ' the extent of the cohesive force must be limited to about the
"*250 millionth of an inch [10~® cm.]'; and he continues, 'nor is
" * it very probable that any error in the suppositions adopted can
"'possibly have so far invalidated this result as to have made it

"'very many times greater or less than the truth' Young con-

" tinues : — ' Within similar limits of uncertaiuty, we may obtain
"'something like a conjectural estimate of the mutual distance
" ' of the particles of vapours, and even of the actual magnitude
" ' of the elementary atoms of liquids, as supposed to be nearly in
" ' contact with each other ; for if the distance at which the force
" ' of cohesion begins is constant at the same temperature, and if
" ' the particles of steam are condensed when they approach within
' this distance, it follows that at 60"" of Fahrenheit the distance
'of the particles of pure aqueous vapour is about the 250
" * millionth of an inch ; and since the density of this vapour is
" * about one sixty thousandth of that of water, the distance of the
" ' particles must be about forty times as great ; consequently the
"* mutual distance of the particles of water must be about the

* "On the Cohesion of Fluids," PhiL Trans, 1805; Collected Works, Vol. i. p. 461.
t Phil. Mag, Vol. xxx. 1890, p. 474.

It

I<

(I

EARLIEST ESTIMATES OF MOLECULAR DIMENSIONS. 281

"*ten thousand millionth of an inch* [025 x 10~* cm.]. It isMoleculi
" * true that the result of this calculation will diflfer considerably
'according to the temperature of the substances compared —
* This discordance does not however wholly invalidate the general
*' * tenour of the conclusion... and on the whole it appears tolerably
"'safe to conclude that, whatever errors may have affected the
"'determination, the diameter or distance of the particles of
"'water is between the two thousand and the ten thousand
" * millionth of an inch* [between 125 x 10"^ and 025 x 10-^ of a
"cm.]. This passage, in spite of its great interest, has been so
" completely overlooked that I have ventured briefly to quote it,
" although the question of the size of atoms lies outside the scope
" of the present paper."

§ 26. The next suggestion, so fia,r as I know, for estimating the
dimensions of molecular structure in ordinary matter, is to be
found in an extract from a letter of my own to Joule on the
contact electricity of metals, published in the Proceedings of the
Manchester Literary and Philosophical Societyf, Jan. 21, 1862,
which contains the following passage : — " Zinc and copper con-
" nected by a metallic arc attract one another from any distance.
" So do platinum plates coated with oxygen and hydrogen respec-
" tively. I can now tell the amount of the force, and calculate
" how great a proportion of chemical affinity is used up electrically,
** before two such discs come within 1/1000 of an inch of one
" another, or any less distance down to a limit within which
" molecular heterogeneousuess becomes sensible. This of course
" will give a definite limit for the sizes of atoms, or rather, as I do
" not believe in atoms, for the dimensions of molecular structures."
The theory thus presented is somewhat more fully developed in a
communication to Nature in March 1870, on "The Size of Atoms J,"
and in a Friday evening lecture§ to the Royal Institution on the

* Yooiig here, curiously ineeusible to the kinetic theory of gases, supposes the
molecules of vapour of water at 60° Fahr. to be within touch (or direct mutual
action) of one another; and thus arrives at a much finer-grainedness for liquid
water than he would have found if he had given long enough free paths to molecules
of the vapour to account for its approximate fulfilment of Boyle's law.

t Reproduced as Art. 22 of my EUctrwtatics and Magnetism,

X Republished aa Appendix (F) in Thomson and Tait's Natural Philosophy,
Part II. Second Edition.

§ Republished in Popular Lectures and Addresses^ Vol. i.

282 LECTURE XVII.

bioleonlar. same subject on February 3, 1883; but to illustrate it, infoiination
was wanting regarding the heat of combination of copper and
zinc. Experiments by Professor Roberts- Austen and by Dr A.
Gait, made within the last four years, have supplied this want ;
and in a postscript of February 1898 to a Friday evening lecture
on " Contact Electricity," which I gave at the Royal Institution
on May 21, 1897, I was able to say " We cannot avoid seeing
" molecular structures beginning to be perceptible at distances of
" the hundred-millionth of a centimetre, and we may consider it
" as highly probable that the distance from any point in a molecule
" of copper or zinc to the nearest corresponding point of a neighbour-
"ing molecule is less than one one-hundred-millionth, and greater
"than one one-thonsand-millionth of a centimetre"; and also to
confirm amply the followiug definite statement which I had
given in my Nature article (1870) already referred to: — "Plates
" of zinc and copper of a three hundred-millionth of a centimetre
" thick, placed close together alternately, form a near approxima-
"tion to a chemical combination, if indeed such thin plates could
" be made without splitting atoms."

§ 27. In that siime article thermodynamic considerations in
stretching a fluid film against surface tension led to the following
result: — "The conclusion is unavoidable, that a water-film falls
" oflF greatly in its contractile force before it is reduced to a thick-
'*ness of a two hundred-millionth of a centimetre. It is scarcely
" possible, upon any conceivable molecular theory, that there can
** be any considemble falling oft* in the contnictile force as long as
" there are several molecules in the thickness. It is therefore
" probable that there are not several molecules in a thickness of a
" two-hundred-millionth of a centimetre of water." More detailed
consideration of the work done in stretching a water-film led me
in my Royal Institution Lecture of 1883 to substitute one one-
hundred-millionth of a centimetre for one two-hundred-millionth
in this statement. On the other hand a consideration of the
large black spots which we now all know in a soap-bubble or soap-
film before it bursts, and which were described in a most interest-
ing manner by Newton*, gave absolute demonstration that the
film retains its tensile strength in the black spot "where the

* Newton's Optic$, pp. 187, 191, Edition 1721, Second Book, Part i.: qaoted in
my Boyal Institution Lecture, Pop, Lectures and Addressesy Vol. i. p. 175.

REINOLD AND RUCKER, RONTGEN, RAYLEIGH. 283

"thickness is clearly much less than 1/60000 of a centimetre, Moleoula
"this being the thickness of the dusky white" with which the
black spot is bordered. And further in 1883 Reinold and
Rticker's* admirable application of optical and electrical methods
of measurement proved that the thickness of the black film in
Plateau's "liquide glycdrique" and in ordinary soap solution is
between one eight-hundred-thousandth of a centimetre and one
millionth of a centimetre. Thus it was certain that the soap-film
has full tensile strength at a thickness of about a millionth of a
centimetre, and that between one millionth and one one-hundred-
millionth the tensile strength falls oflF enormously.

§ 28. Extremely interesting in connection with this is the
investigation, carried on independently by Rontgen -f- and Ray-
leighj and published by each in 1890, of the quantity of oil
spreading over water per unit area required to produce a sensible
disturbance of its capillary tension. Both experimenters ex-
pressed results in terms of thickness of the film, calculated as if
oil were infinitely homogeneous and therefore structureless, but
with very distinct reference to the certainty that their films were
molecular structures not approximately homogeneous. Rayleigh
found that olive oil, spreading out rapidly all round on a previously
cleaned surface of water from a little store carried by a short
length of platinum wire, produced a perceptible effect on little
floating fragments of camphor at places where the thickness of
the oil was 106 x 10""* cm., and no perceptible effect where the
thickness was 8*1 x 10~* cm. It will be highly interesting to
find, if possible, other tests (optical or dynamical or electrical or
chemical) for the presence of a film of oil over water, or of films
of various liquids over solids such as glass or metals, demonstrat-
ing by definite effects smaller and smaller thicknesses. Rontgen,
using ether instead of camphor, found analogous evidence of layers
5*6 X 10~* cm. thick. It will be very interesting for example to
make a thorough investigation of the electric conductance of a
clean rod of white glass of highest insulating quality surrounded
by an atmosphere containing measured quantities of vapour of

* *♦ On the Limiting Thickness of Liquid Films,** Roy. Soc. Proc, April 19, 1883;
Phil. Trans. 1883, Part n. p. 645.
t Wied, Ann. Vol. xli. 1890, p. 321.
t Proe. Roy. Soc. Vol. xLvn. 1890, p. 364.

284 LECTURE XVII.

loleoular. water. When the glass is at any temperature above the dew-
point of the vapour, it presents, so far as we know, no optical
appearance to demonstrate the pressure of condensed vapour of
water upon it : but enormous diflferences of electric conductance,
according to the density of the vapour surrounding it, prove the
presence of water upon the surface of the glass, or among the
interstices between its molecules, of which electric conductance
is the only evidence. Rayleigh has himself expressed this view in a
recent article, " Investigations on Capillarity*' in the Philosophical
Magazine* From the estimates of the sizes of molecules of argon,
hydrogen, oxygen, carbonic oxide, carbonic acid, ethylene (CjH4),
and other gases, which we shall have to consider (§ 47 below), we
may judge that in all probability if we had eyes microscopic
enough to see atoms and molecules, we should see in those thin
films of Rayleigh and Rontgen merely molecules of oil lying
at greater and less distances from one another, but at no part
of the film one molecule of oil lying above another or resting
on others.

§ 29. A very important and interesting method of estimating
the size of atoms, founded on the kinetic theory of gases, was
first, so far as I know, thought of by Lo8chmidt"f" in Austria and
Johnstone Stoney in Ireland. Substantially the same method
occurred to myself later and was described in Nature, March 1870,
in an article J on the " Size of Atoms '* already referred to, § 26
above, from which the quotations in ^ 29, 30 are taken.

" The kinetic theory of gases suggested a hundred yeara ago
" by Daniel Bernoulli has, during the last quarter of a century,
"been worked out by Hcrapath, Joule, Clausius, and Maxwell
"to so great perfection that we now fiud in it satisfactory ex-
"planations of all non-chemical" and non-electrical "properties of
" gases. However difficult it may be to even imagine what kind
" of thing the molecule is, we may regard it as an established
" truth of science that a gas consists of moving molecules dis-
" turbed from rectilinear paths and constant velocities by collisions
" or mutual influences, so rare that the mean length of nearly

♦ Phil. Mag. Oct. 1899. p. 337.

f Bitzungsberichte of the Vienna Academy, Oct. 12, 1865. p. 395.
X Beprinted as Appendix (F) in Thomson and Tait's Natural Philosophy,
Part n. p. 499.

LOSCHMIDT, JOHNSTONE STONEY, KELVIN. 285

it

rectilinear portions of the path of each molecule is many times Moleool
greater than the average distance from the centre of each
" molecule to the centre of the molecule nearest it at any time.
" If, for a moment, we suppose the molecules to be hard elastic
"globes all of one size, influencing one another only through
" actual contact, we have for each molecule simply a zigzag path
" composed of rectilinear portions, with abrupt changes of direc-

"tion But we cannot believe that the individual molecules

"of gases in general, or even of any one gas, are hard elastic
" globes. Any two of the moving particles or molecules must act
" upon one another somehow, so that when they pass very near
" one another they shall produce considerable deflexion of the
" path and change in the velocity of each. This mutual action
" (called force) is different at different distances, and must vary,
"according to variations of the distance, so as to fulfil some
" definite law. If the particles were hard elastic globes acting
*'' Upon one another only by contact, the law of force would be
"...zero force when the distance from centre to centre exceeds
" the sum of the radii, and infinite repulsion for any distance less
" than the sum of the radii. This hypothesis, with its ' hard and
" * fast* demarcation between no force and infinite force, seems to
" require mitigation." Boscovich's theory supplies clearly the
needed mitigation.

§ 30. To fix the ideas we shall still suppose the force absolutely
zero when the distance between centres exceeds a definite limit, X ;
but when the distance is less than X, we shall suppose the force
to begin either attractive or repulsive, and to come gradually to
a repulsion of very great magnitude, with diminution of distance
towards zero. Particles thus defined I call Boscovich atoms. We
thus call ^X the radius of the atom, and X its diameter. We
shall say that two atoms are in collision when the distance
between their centres is less than X. Thus "two molecules in
" collision will exercise a mutual repulsion in virtue of which the
" distance between their centres, after being diminished to a mini-
" mum, will begin to increase as the molecules leave one another.
" This minimum distance would be equal to the sum of the radii,
"if the molecules were infinitely hard elastic spheres; but in
"reality we must suppose it to be very different in different
" collisions."

286 LECTURE XVII.

Moleoalar. § 31. The esseDtial quality of a gas is that the straight line
of uniform motion of each molecule between collisions, called
the free path, is long in comparison with distances between centres
during collision. In an ideal perfect gas the free path would
be infinitely long in comparison with distances between centres
during collision, but infinitely short in comparison with any length
directly perceptible to our senses ; a condition which requires the
number of molecules in any perceptible volume to be exceedingly
great. We shall see that in gases which at ordinary pressures
and temperatures approximate most closely, in respect to com-
pressibility, expansion by heat, and specific heats, to the ideal
perfect gas, as, for example, hydrogen, oxygen, nitrogen, carbon-
monoxide, the free path is probably not more than about one
hundred times the distance between centres during collisions,
and is little short of 10~'cm. in absolute magnitude. Although
these moderate proportions suffice for the well-known exceedingly
close agreement with the ideal gaseous laws presented by those
real gases, we shall see that large deviations from the gaseous
laws are presented with condensations sufficient to reiluce the free
paths to two or three times the diameter of the molecule, or to
annul the free paths altogether.

§ 32. It is by experimental detenninations of difiusivity that
the kinetic theory of gases affords its best means for estimating
the sizes of atoms or molecules and the number of molecules
in a cubic centimetre of gas at any stated density. Let us
therefore now consider carefully the kinetic theory of these
actions, and with them also, the properties of thermal conductivity
and viscosity closely related to them, as first discovered and
splendidly developed by Clausius and Clerk Maxwell.

§ 33. According to their beautiful theory, we have three
kinds of diflFusion ; diflFusion of molecules, diftusion of energy, and
diflFusion of momentum. Even in solids, such as gold and lead,
Roberts- Austen has discovered molecular diflFusion of gold into
lead and lead into gold between two pieces of the metals when
pressed together. But the rate of diflFusion shown by this ad-
mirable discovery is so excessively slow that for most purposes,
scientific and practical, we may disregard wandering of any
molecule in any ordinary solid to places beyond direct influence of

CLAUSTUS, MAXWELL. 287

its immediate neighbours. In an elastic solid we have diflFusion Moleonlai
of momentum by wave motion, and diffusion of energy consti-
tuting the conduction -of heat through it. These diffusions are
effected solely by the communication of energy from molecule to
molecule and are practically not helped at all by the diffusion of
molecules. In liquids also, although there is thorough molecular
diffusivity, it is excessively slow in comparison with the two other
diffusivities, so slow that the conduction of heat and the diffusion
of momentum according to viscosity are not practically helped by
molecular diffusion. Thus, for example, the thermal diffusivity*
of water (002, according to J. T. Bottomley's first investigation,
or about '001 5 -f- according to later experimenters) is several
hundred times, and the diffusivity for momentum is from one to
two thousand times, the diffusivity of water for common salt, and
other salts such as sulphates, chlorides, bromides, and iodides.

§ 34. We may regard the two motional diffusivities of a liquid
as being each almost entirely due to communication of motion from
one molecule to another. This is because every molecule is always
under the influence of its neighbours and has no free path. When
a liquid is rarefied, either gradually as in Andrew's experiments
showing the continuity of the liquid and gaseous states, or
suddenly as in evaporation, the molecules become less crowded
and each molecule gains more and more of freedom. When the
density is so small that the straight free paths are great in com-
parison with the diameters of molecules, the two motional diffu-
sivities are certainly due, one of them to carriage of energy, and
the other to carriage of momentum, chiefly by the fi*ee rectilinear
motion of the molecules between collisions. Interchange of
energy or of momentum between two molecules during collision
will undoubtedly to some degree modify the results of mere
transport; and we might expect on this account the motional
diffusivities to be approximately equal to, but each somewhat
greater than, the molecular diffusivity. If this view were correct,
it would follow that, in a homogeneous gas when the free paths
are long in comparison with the diameters of molecules, the
viscosity is equal to the molecular diffusivity multiplied by the

* Math, atid Phys, Paperg^ Vol. iii. p. 226. For explanation regarding diffusi-
vity and viscosity see same Yolnme, pp. 428—435.

t See a paper by Milner and Chattock, Phil, Mag. Vol. xlviii. 1899.

288 LECTURE XVIL

bioleonlar. density, and the thermal conductivity is equal to the molecular
diffusivity multiplied by the thennal capacity per unit bulk,
pressure constant : and that whatever deviation from exactness of
these equalities there may be, would be in the direction of the
motional diti'usivities being somewhat greater than the molecular
diffusivity. But alas, wc shall see, § 45 below, that hitherto
experiment does not confirm these conclusions: on the contrary
the laminar diffusivities (or diffusivities of momentum) of the
only four gases of which molecular diffusivities have been de-
termined by experiment, instead of being greater than, or at
least equal to, the density multiplied by the molecular diffusivity,
are each somewhat less than three-fourths of the amount thus
calculated.

§ 35. I see no explanation of this deviation from what
seems thoroughly coiTCct theory. Accurate experimental deter-
minations of viscosities, whether of gases or liquids, are easy by
Graham s transpirational method. On the other hand even roughly
approximate experimental determinations of thermal diffusivities
are exceedingly difficult, and I believe none, on correct experi-
mental principles, have really been made*; certainly none un-
vitiated by currents of the gas experimented upon, or accurate
enough to give any good test of the theoretical relation between
thermal and material diffusivities, expressed by the following
equation, derived from the preceding verbal statement regarding
the three diffusivities of a gas,

P

where 6 denotes the thermal conductivity, fi the viscosity, p the
density, Kp the thermal capacity per unit bulk pressure constant,
K the thermal capacity per unit mass pressure constant, c the
thermal capacity per unit mass volume constant, and k the ratio
of the thermal capacity pressure constant to the thermal capacity
volume constant. It is interesting to remark how nearly theo-

* So far aR I know, all attempts hitherto made to determine the thermal oon-
duotivities of gaHes have been founded on observations of rate of communication of
heat between a thermometer bulb, or a stretched metallic wire constituting an
electric resistance thermometer, and the walls of the vessel enclosing it and the gas
experimented upon. See Wiedemann's ^^Nmi^N, 1888, Vol. xxuv. p. 023, and 1891,
Vol. xLiv. p. 177. For other referenoes, see 0. E. Meyer, § 107.

MOLECULAR DIFFUSIVITY. 289

retical investigators* have come to the relation O^^kcfi; Clausius Moleoali
gave 0 = ^c/jl; 0. E. Meyer, 0 = 16027 Cfi, and Maxwell, 0=^CfA.
Maxwell's in fact is 0 = kcfi for the case of a monatomic gaa

§ 36. To understand exactly what is meant by molecular
diffusivity, consider a homogeneous gas between two infinite
parallel planes, 000 and RRR, distance a apart, and let it be
initially given in equilibrium ; that is to say, with equal numbers
of molecules and equal total kinetic energies in equal volumes,
and with integral of component momentum in any and every
direction, null. Let N be the number of molecules per unit
volume. Let every one of the molecules be marked either green
or red, and whenever a red molecule strikes the plane OOOy let its
marking be altered to green, and, whenever a green molecule
strikes RRR, let its marking be altered to red. These markings
are not to alter in the slightest degree the mass or shape or elastic
quality of the molecules, and they do not disturb the equilibrium
of the gas or alter the motion of any one of its particles; they
are merely to give us a means of tracing ideally the history of any
one molecule or set of molecules, moving about and colliding with
other molecules according to the kinetic nature of a gas.

§ 37. Whatever may have been the initial distribution of the
greens and reds, it is clear that ultimately there must be a regular
transition from all greens at the plane 000 and all reds at the
plane RRR, according to the law

g = N-] r^N"^—^ (1),

where g and r denote respectively the number of green molecules
and of red molecules per unit volume at distance x from the
plane RRR, In this condition of statistical equilibrium, the
total number of molecules crossing any intermediate parallel
plane from the direction 000 towards RRR will be equal to the
number crossing from RRR towards 000 in the same time ; but
a larger number of green molecules will cross towards RRR than
towards 000, and, by an equal diflference, a larger number of red
molecules will cross towards 000 than towards RRR. If we
denote this diflference per unit area per unit time by QN, we have

* See the last ten lines of O. E. Meyer's book.
T. L. 19

290 LECTURE XVII.

Moleonlar. for what I call the material diffusivity (called by Maxwell, " co-
efficient of diffusion"),

D = Qa (2).

We may regard this equation as the definition of diffusivity.
Remark that Q is of dimensions ZIP'S because it is a number per
unit of area per unit of time (which is of dimensions Ir^T^^)
divided by iV, a number per unit of bulk (dimensions X~'). Hence
the dimensions of a diffusivity are XT"^; and practically we
reckon it in square centimetres per second.

§ 38. Hitherto we have supposed the 0 and the R particles
to be of exactly the same quality in every respect, and the dif-
fusivity which we have denoted by J9 is the inter-diffusivity of
the molecules of a homogeneous gas. But we may suppose G and
fi to be molecules of different qualities; and assemblages of 0
molecules and of R molecules to be two different gases. Every-
thing described above will apply to the inter-diffusions of these
two gases ; except that the two differences which are equal when
the red and green molecules are of the same quality are now not
equal or, at all events, must not without proof be assumed to
be equal. Let us therefore denote by QgN the excess of the
number of 0 molecules crossing any intermediate plane towards
RRR over the number crossing towards GOO, and by QrN the
excess of the number of R molecules crassing towards 000 above
that crossing towards RRR. We have now two different diffusi vities
of which the mean values through the whole range between the
bounding planes are given by the equations

one of them, J9^, the diffusivity of the green molecules, and the
other, Dr, the diffusivity of the red molecules through the hetero-
geneous mixture in the circumstances explained in § 37. We
must not now assume the gradients of density of the two gases
to be uniform as expressed by (1) of § 37, because the homogeneous-
ness on which these equations depend no louger exists.

§ 39. To explain all this practically*, let, in the diagram, the
plaues OOOf and RRR, be exceedingly thin plates of dry porous
material such as the fine unglazed earthenware of Graham's experi-

* For a practical experiment it might be necessary to aUow for the difference of
the proportions of the O gas on the two sides of the RRR plate and of the R gas
on the two sides of the GGQ plate. This would be exceedingly difficult, though
not impossible, in practice. The difficulty is analogous to that of allowing for the

INTER-DIFFUSION OF EQUAL AND SIMILAR MOLECULES. 291

ments. Instead of our green and red marked molecules of the same Molecular,
kind, let us have two gases, which we shall call 0 and -B, supplied in
abundance at the middles of the two ends of a non-porous tube of

i

A

p?^s--3s.>j:«a«ffawt^5;jCWftS4\v?»«WL^

o

'.iiKAW/«^*s«»Ka5[TO»»W««SS®^??fitt^

ft

R

t

eleotrio resistanoes of the oonneotions at the ends of a stoat bar of metal of whioh
it is desired to measure the electric resistance. But the simple and accurate
"potential method'* by which the difficnlty is capily and thoroughly overcome
in the electric case is not available here. I do not, however, put forward the
arrangement described in the text as an eligible plan for measuring the inter-
diffusivity of two gases. Even if there were no other difficulty, the quantities of the
two pure gases required to realize it would be impracticably great.

19—1

292 LECTURE XVII.

Moleoalar. glass or metal, and guided to flow away radially in contact with
the end-plates as indicated in the diagram. If the two axial
supply-streams of the two pure gases are sufficiently abundant,
the spaces 000, RRR, close to the inner sides of the porous
end-plates will be occupied by the gases 0 and R, somewhat nearly
pure. They could not be rigorously pure even if the velocities of
the scouring gases on the outer sides of the porous end-plates
were comparable with the molecular velocities in the gases, and
if the porous plates were so thin as to have only two or three
molecules of solid matter in their thickness. The gases in contact
with the near faces of the porous plates would, however, probably
be somewhat approximately pure in practice with a practically
realizable thinness of the porous plates, if a, the distance between
the two plates, is not less than five or six centimetres and the
scouring velocities moderately, but not impracticably, great.
According to the notation of § 37, Qg is the quantity of the 0 gas
entering across 000 and leaving across RRR per sec. of time per
sq. cm. of area ; Qr is the quantity of the R gas entering across
RRR and leaving across 000 per sec. of time per sq. cm. of area ;
the unit quantity of either gas being that which occupies a cubic
centimetre in its entry tube. The equations

where g and r are the proportions of the 0 gas at R and of the R
gas at 0, define the average diffusivities of the two gases in the
circumstances in which they exist in the different parts of the
length a between the end-plates. This statement is cautiously
worded to avoid assuming either equal values of the diffusivities
of the two gases or equality of the diffusivity of either gas through-
out the space between the end-plates. So far as I know difference
of difiusivity of the two gases has not been hitherto suggested by
any writer on the subject. What is really given by Loschmidt's
experiments, § 43 below, is the arithmetic mean of the two
diffusivities Dg and Dr.

§ 40. In 1877 0. E. Meyer expressed the opinion on theoreti-
cal grounds, which seem to me perfectly valid, that the inter-
diffusivity of two gases varies according to the proportions of the
two gases in the mixture. In the 1899 edition of his Kinetic
Theory of Gases* he recalls attention to this view and quotes
results of various experimenters, Loschmidt, Obermayer, Waitz,

r

* Baynes' translation, p. 264.

INTER-DIFFUSIVITY OF TWO GASRS. 293

seeming to support it, but, as he says, not quite conclusively. On Molecular
the other hand, Maxwell's theory (§ 41 below) gives inter-diflfu-
sivity as independent of the proportions of the two gases; and
only a single expression for diflfusivity, which seems to imply that
the two diffusivities arc equal according to his theory. The
subject is of extreme difficulty and of extreme interest, theoretical
and practical; and thorough experimental investigation is greatly
to be desired.

§41. In 1873 Maxwell* gave, as a result of a theoretical
investigation, the following formula which expresses the inter-
diflFusivity ( Aa) of two gases independently of the proportion of the
two gases in any part of the mixture: each gas being supposed to
consist of spherical Boscovich atoms mutually acting according to
the law, force zero for all distances exceeding the sum of the radii
(denoted by «ia) and infinite repulsion when the distance between
theii* centres is infinitely little less than this distance :

where Wi, w^ are the masses of the molecules in the two gases
in terms of that of hydrogen called unity ; V is the square root of
the mean of the squares of the velocities of the molecules in
hydrogen at 0° C; and N is the number of molecules in a cubic
centimetre of a gas (the same for all gases according to Avogadro s
law) at O^'C. and standai*d atmospheric pressure. I find the
following simpler formula more convenient

^-=2vsk;.^<'''+''-"' <^>-

where F,', F/ are the mean squares of the molecular velocities of
the two gases at t)° C, being the values of 3p//> for the two gases,
or three times the squares of their Newtonian velocities of sound,
at that temperature. For brevity, we shall call mean molecular
velocity the square root of the mean of the squares of the
velocities of the molecules. The same formula is, of course,
applicable to the molecular diflfusivity of a single gas by taking
Fi = Fj = F its mean molecular velocity, and i^u = 8 the diameter
of its molecules ; so that we have

"" = 2^-1 <'^>-

* " On Losohmidt's Experiments on Diffusion in relation to the Kinetic Theory
of Gases," Nature, Aog. 1873; ScieiUific Papers, Vol. n. pp. 343—350.

294

LECTURE XVIL

MoleouUr. § 42. It is impossible by any direct experiment to find the
molecular diffusivity of a single gas, as we have no means of
marking its particles in the manner explained in § 36 above; but
Maxwell's theory gives us, in a most interesting manner, the
means of calculating the diffusivity of each of three separate
gases fix)m three experiments determining the inter-difinsivities
of their pairs. From the inter-diffusivity of each pair determined

by experiment we find, by (2) § 41, a value of «ii\/(2 VStt-BT) for
each pair, and we have «„ = ^ («j + «g)* whence

^1 — ^19 "T ^U *« ; ^ — ^ij + 5j| "" ^u ; ^3 — ^13 + S^i — 5ij.

.(1).

Calculating thus the three values of 8 V(2 v^STriV), and using
them in (3) § 41, we find the molecular diflfusivities of the three
separate gases.

§ 43. In two communications"!" to the Academy of Science
of Vienna in 1870, Loschmidt describes experimental determin-
ations of the inter-diffusivities of ten pairs of gases made, by
a well-devised method, with great care to secure accuracy. In
each case the inter-diflFusivity determined by the experiment
would be, at all events, somewhat approximately the mean of the
two diffusivities, § 39 above, if these are unequal. The results
reduced to 0° C. and standard atmospheric pressure, and multi-
plied by 278 to reduce from Loschmidt's square metres per hour
to the now usual square centimetres per second, are as follows : —

Table of inter-diffusivities D.

1

Pairs of gases .^^^

D
ms. per sec.

H„ 0,

7214

H„ CO

6422

Ha. CO,

5558

Oj, CO

1802

0„, COj

1409

CO, CO3

1406

COg, Air

•1423

CO2, NO

0984

COa, CH4

1587

SOj, Ha

•4809

* This agrees with MaxweU's equation (4), but shows his equation (6) to be
incorrect.

t ''Experimental-Untersuchungen Uber die Diffusion von Gasen ohno porose
Soheidewande," SiU. d, k, Akad. d. WUtciuch,, March 10 and May 12, 1870.

LOSCHMIDT's determinations of INTER-DIFFUSIVITIES. 295

In the first six of these, each of the four gases Ha, O,, CO, COj Molecula
occurs three times, and we have four sets of three inter-diflFusivities
giving in all three determinations of the diflFusivity of each gas as
follows : —

Pairs of gases

D,

'(12, 13, 23) 1-32

(12, 14, 24) 1-35

(13, 14, 34) 1-26

Gases

Pairs of gases D,

(12, 13, 23) 193

(12, 14, 24) 190

(23, 24, 34) 183

H,
O,
CO.
CO,.

(1)
.(2)
.(3)
.(4)

Mean 131

Mean 189

^8

^4

(12, 13, 23) 169

(13, 14, 34) 175

(23, 24, 34) 178

\

Mean 174

(12, 14, 24) 106

(13, 14, 34) Ill

(23, 24, 34) 109

Mean -109

Considering the great difficulty of the experimental investiga-
tion, we may regard the agreements of the three results for each
separate gas as, on the whole, very satisfactory, both in respect to
the accuracy of Loschmidt s experiments and the correctness of
Maxwell's theory. It certainly is a very remarkable achievement
of theory and experiment to have found in the four means of the
sets of three determinations, what must certainly be somewhat
close approximations to the absolute values for the four gases,
hydrogen, oxygen, carbon-monoxide, and carbon-dioxide, of some-
thing seemingly so much outside the range of experimental
observation, as the inter-ditfusivity of the molecules of a separate
gas.

§ 44. Maxwell, in his theoretical writings of different dates,
gave two very distinct views of the inner dynamics of viscosity in
a single gas, both interesting, and each, no doubt, valid. In one*,
viscous action is shown as a subsidence from an " instantaneous
rigidity of a gas." In the other"!", viscosity is shown as a diffu-
sion of momentum : and in p. 347 of his article quoted in § 41

* Trans, Roy, So€,<, May 1866 ; Scientific Papers, Vol. ii. p. 70.

t *' Molecules," a lecture delivered before the Brit. Assoc, at Bradford, Scientific
Papers, Vol. n. p. 878. See also O. £. Meyer's Kinetic Theory of Oases (Baynes'
trans. 18d9), §§ 74—76.

296

LECTURE XVII.

Molecular, above he gives as from " the theory," but without demonstra-
tion, a formula (5), which, taken in conjunction with (1), makes

.(1);

^ = i)

p

p denoting the density, fi the viscosity, and D the molecular
diffusivity, of any single gas. On the other hand, in his 1866
paper he had given formulas making*

.(2).

^=-648i)

P

§ 45. Viewing viscosity as explained by diffusion of momentum
we may, it has always seemed to me (§ 34 above), regard (1) as
approximately true for any gas, monatomic, diatomic, or polyatomic,
provided only that the mean free path is large in comparison with
the sum of the durations of the collisions. Unfortunately for this
view, however, comparisons of Loschmidt's excellent experimental
determinations of diffusivity with undoubtedly accurate determin-
ations of viscosity from Graham's original experiments on trans-
piration, and more recent experiments of Obermayer and other
accurate observers, show large deviations from (1) and are much
more nearly in agreement with (2). Thus taking -0000900,
•001430, -001234, 001974 as the standard densities of the four
gases, hydrogen, oxygen, carbon-monoxide, and carbon-dioxide,
and multiplpng these respectively by the diffusivities ftx)m
Loschmidts experiments and Maxwell's theory, we have the
following comparison with Obermayer s viscosities at 0° C. and
standard pressure, which shows the discrepance from experiment
and seeming theory referred to in § 34.

Col. 2

Col. 3

Col. 4

Viscosity calculated

by Maxwell's theory

from Loschmidt's

diffusivities

fi=pD

•000119
•000269
*000212
•000218

Viscosities according
to Obermayer

•0000822
•0001873
•0001630
•0001414

Katio of values in

Col. 3 to those

in Col. 2

•691
•696
•769
•649

♦ The formala for viscosity (Set. Paper*, Vol. n. p. 68) taken with the formula
for molecular diffusivity of a single gas, derived from the formula of inter-diffusivity

maxwell's theory. 297

§ 46. Leaving this discrepance unexplained, and eliminating MolecuL
D between (1) of § 44 and (3) of § 41, we find as Maxwell's latest
expression of the theoretical relation between number of molecules
per cubic centimetre, diameter of the molecules, molecular velocity,
density, and viscosity of a single gas,

Ns'^ \ 1^=1629^ (1).

The number of grammes and the number of molecules in a cubic
centimetre being respectively p and N, pjN is the mass of one
molecule in grammes ; and therefore, denoting this by m, we have

m=2V37r^»^=6140^«2 (2).

In these formulas, as originally investigated by Maxwell for the
case of an ideal gas composed of hard spherical atoms, 8 is
definitely the diameter of the atom, and is the same at all
temperatures and densities of the gas. When we apply the
formulas to diatomic or polyatomic gases, or to a monatomic gas
consisting of spherical atoms whose spheres of action may over-
lap more or less in collision according to the severity of the
impact, 8 may be defined as the diameter which an ideal hard
spherical atom, equal iu mass to the actual molecule, must have to
give the same viscosity as the real gas, at any particular tem-
perature. This being the rigorous definition of «, we may call it
the proper mean shortest distance of inertial centres of the mole-
cules in collision to give the true viscosity ; a name or expression
which helps us to understand the thing defined.

§ 47. For the ideal gas of hard spherical atoms, remembering

that V is independent of the density and varies as ^^ (^ denoting
absolute temperature), § 46 (2) proves that the viscosity is inde-
pendent of the density and varies approximately a&t^. Rayleigh's
experimental determinations of the viscosity of argon at different
temperatures show that for this monatomic gas the viscosity varies
as t'^ ; hence § 46 (2) shows that «' varies as ^— "•, and therefore
8 varies as ^"*". Experimental determinations by Obermayer* of
viscosities and their rates of variation with temperature for car-
bonic acid, ethylene, ethylene-chloride, and nitrous oxide, show

u A
of two gases of eqaal densities, gives —= = ^ i which is equal to '648 aocording to

the values of ^| and A^ shown in p. 42 of Vol n., Sci, Papers,
* Obermajer, Wien. Akad. 1S76, Mar. 16th, Vol. 73, p. 433.

298

LECTURE XVII.

Molecular, that for these the viscosity is somewhat nearly in simple propor-
tion to the absolute temperature : hence for them s^ varies nearly
as ir'^. His determinations for the five molecularly simpler gases,
air, hydrogen, carbonic oxide, nitrogen, and oxygen show that the
increases of fi, and therefore of «"*, with temperature are, as
might be expected, considerably smaller than for the more complex
of the gases on which he experimented. Taking his viscosities
at 0" Cent., for carbonic acid and for the four other simple gases
named above, and Rayleigh's for argon, with the known densities
of all the six gases at 0° C. and standard atmospheric pressure,
we have the following table of the values concerned in § 46 (1):

Ck)l. 1

! Gm

COa

Ha
CO

Argon

Col. 2

in U*niit of
icnunmcs {Hsr
cubic centi-
metre

•001974

•0000900

•001234

•001257

•00143

•001781

Col. 3

M

in terms

of dynes

per square

centimetre

•<K)01414
•0000822
•0001630
•0001636
•0001873
•0002083

Col. 4

Col. 6

y

in terms uf
centimetres
I>er secuud

39200
184000
49600
49200
46100
41300

in terms of
(centimetre) -*

89200
32800
61200
61600
57300
57600

Col. 6

Hence taking
A'=10*'(§60)

we hare
g at 0' Cent
in terms uf
centimetres

2-99

' 1-81

• 247

I 2-48

2-39

2-40

10-8

It
»»

Col. 7

wo liare

m

in terms of

grammes

1974. 10-«

•90
12-34
12-57
14-3
17^81

»»

»»
t«
it

Col. 8

Mean free
paths ac-
cording to
Maxwell's
formula*

VS.rA'j*

in terms of
centimetres

Col. 9

I

t

i Katio of
I volume oc
I cupied by
I molecules t
! whole Tolun

! K"

1

1

1 2o2 .

io-«

1-390.10

16^89

•311 „

13-68

•792 „

3-66

•800 „

, 3-93

•719 „

3-91

•722 „

§ 48. The meaning of "«," the diameter, as defined in § 46, is
simpler for the monatomic gas, argon, than for any of the othei's ;
and happily we know for argon the density, not only in the gaseous
state (-001 781) but also in the liquid state (l-42)t. The latter
of these is 797 times the former. Now, all things considered, it
seems probable that the crowd of atoms in the liquid may be
slightly less dense than an assemblage of globes of diameter 8 just
touching one another in cubic order ; but, to make no hypothesis
in the first place, let qs be the distance from centre to centre
of a cubic arrangement of the molecules 797 times denser than
the gas at 0° C. and standard atmospheric pressure ; q will be
greater than unity if the liquid is less dense, or less than unity

* Maxwell's CoUecUd Papers, Vol. ii. p. 348, eqn. (7). The formula as printed
in this paper contains a yeiy embarrassing mistake, tj2T for ;iJ2 . t.
t See Baly and Donan, Chem, Jour., July, 1902.

NUMBER OF MOLECULES IN A CUBIC CENTIMETRE OF GAS. 299

if the liquid is denser, than the cubic arrangement with molecules, Molecul
regarded as spherical of diameter «, just touching. We have

797^^= iKqsY (3),

and for argon we have by § 46 (1),

i\r«« = 57500 (4).

Eliminating s between these equations we find

i\r=797«.57500'g« = 121.10^.g« (5).

If the atoms of argon were ideal hard globes, acting on one
another with no force except at contact, we should almost cer-
tainly have 9^1 (because with closer packing than that of cubic
order it seems not possible that the assemblage could have
sufficient relative mobility of its parts to give it fluidity) and
therefore N would be ^ 1-21 . 10«>.

§ 49. For carbonic acid, hydrogen, nitrogen, and oxygen, we
have experimental determinations of their densities in the solid or
liquid state ; and dealing with them as we have dealt with argon,
irrespectively of their not being monatomic gases, we find results
for the five gases as shown in the following table :

CoLl

Gu

CO,

N,

Argon

CoL2

Solid or liquid density

SoUd 1-58

Uquid at 17° absolute ... -OIK)

\Uquid 1047

jaoUd 1-400

liquid at its freoziug pt. 1*27

(Uquid at 84° absolute . . . 1-42

soUd 1-396*

Col. 3

Ratio of

solid or

liquid

density to

standard

gaseous

density

800
1000
833
1114
888
797
784

Col. 4

Number of

molecules per

cubic centimetre

of gas at standard

density

; 4-65 .10*>.g«
, -352

I 1-62

I 2-90

1
1
1

49
21
17

Col. 5

Values of </(§ 48)

according to

7-* =1-21 for argon

(liquid compared

with gas at 0* and

atmospheric

pressure)

•777
1191
•923
•837
•936
•969
•974

In this table, q denotes the ratio to 8 of the distance from
centre to centre of nearest molecules in an ideal cubic assemblage
of the same density as the solid or liquid, as indicated in cols. 3
and 2.

* From informatioii oommunicated by Prof. W. Ramsay, July 23, 1901,

300 LECTURE XVII.

(oleonlar. § 50. According to Avogadro's doctrine, the number of mole-
cules per cubic centimetre is the same for all " perfect " gases at
the same temperature and pressure; and even carbonic acid is
nearly enough a " perfect gas " for our present considerations. Hence
the actual values of (f are inversely proportional to the numbers
by which they are multiplied in col. fS of the preceding table.
Now, as said in § 48, all things considered, it seems probable that
for argon, liquid at density 1*42, q may be somewhat greater, but
not much greater, than unity. If it were exactly unity, N would

be 1-21 . 10"; and I have chosen 5= (1-21)-* or 969, to make N
the round number 10**. Col. 6, in the table of § 47 above, is
calculated with this value of iV; but it is not improbable that the
true value of N may be considerably greater than 10"*.

§ 51. As compared with the value for argon, monatomic, the
smaller values of g for the diatomic gases, nitrogen and oxygen,
and the still smaller values for carbonic acid, triatomic, are quite
as might be expected without any special consideration of law of
foi*ce at different distances between atoms. It seems that the
diatomic molecules of nitrogen and oxygen and still more so the
triatomic molecule of carbonic acid, are effectively larger when
moving freely in the gaseous condition, than when closely packed
in liquid or solid assemblage. But the largeness of 5 for the
diatomic hydrogen is not so easily explained : and is a most in-
teresting subject for molecular speculation, though it or any other
truth in nature is to be explained by a proper law of force accord-
ing to the Boscovichian doctrine which we all now accept (many

* Maxwell, judging from *'moleoalar volumes*' of chemical elements estimated
by Lorentz, Meyer and Kopp, unguided by what we now know of the densities
of liquid oxygen and liquid hydrogen and of the liquid of the then undiscovered gas
argon, estimated ^='19 . 10** (Maxwell's Collected Papers, Vol. 11. p. 350) which is
rather less than one-fifth of my estimate 10^. On the same page of his paper
is given a table of estimated diameters of molecules which are about 3*2 or 3*3 times
larger than my estimates in coL 6 of the table in § 47. In a previous part of
his paper (p. MS) Maxwell gives estimates of free paths for the same gases, from
which by his formula (7), corrected as in col. 8 of my table in § 47, 1 find values of
N ranging from 605 . 10»« to 6-96 . 10'' or about one-third of -19 . 10». EQs
uncorrected formula tj2T (instead of ,^2 . ir) gives values of N which are ^w times,
or 1 *77 times as great, which are still far short of his final estimate. The discrepance
is therefore not accoimted for by the error in the formula as printed, and I see no
explanation of it. The free paths as given by Maxwell are about 1*3 or 1*4 times
as large as mine.

rayleioh's theory of the blue of the sky. 801

of us without knowing that we do so) as the fundamental hypo- Moleonl
thesis of physics and chemistry. I hope to return to this question
as to hydrogen in a crystallographic appendix.

I am deeply indebted to Professor Dewar for information
regarding the density of liquid hydrogen, and the densities of
other gases, liquefied or frozen, which he has given me at various
times within the last three years.

§ 52. A new method of finding an inferior limit to the number
of molecules in a cubic centimetre of a gas, very different from
anything previously thought of, and especially interesting to us
in connection with the wave-theory of light, was given by Lord
Rayleigh*, in 1899, as a deduction from the djnaamical theory of
the blue sky which he had given 18 years earlier. Many previous
writers, Newton included, had attributed the light from the sky,
whether clear blue, or hazy, or cloudy, or rainy, to fine suspended
particles which divert portions of the sunlight from its regular
course; but no one before Rayleigh, so far as I know, had published
any idea of how to explain the blueness of the cloudless sky.
Stokes, in his celebrated paper on Fluorescence "f", had given the
true theory of what was known regarding the polarization of the
blue sky in the following "significant remark" as Rayleigh calls it :
" Now this result appears to me bo have no remote bearing on the
" question of the directions of the vibrations in polarized light.
" So long as the suspended particles are large compared with the
" waves of light, reflection takes place as it would from a portion of
" the surface of a large solid immersed in the fluid, and no con-
" elusion can be drawn either way. But if the diameter of the
" particles be small compared with the length of a wave of light, it
" seems plain that the vibrations in a reflected ray cannot be per-
"pendicular to the vibrations in the incident ray"; which implies
that the light scattered in directions perpendicular to the exciting
incident ray has everywhere its vibrations perpendicular to the
plane of the incident ray and the scattered ray; provided the
diameter of the molecule which causes the scattering is very small
in comparison with the wave-length of the light. In conversation
Stokes told me of this conclusion, and explained to me with

♦ Rayleigh, Collected Papers^ Vol. i. Art. vin. p. 87.

t "On the Change of Refrangibility of Light," Phil Trans. 1852, and Collected
Papers, Vol. m.

302 LECTURE XVIL

oleoalar. perfect clearness and completeness its d}mamical foundation ;
and applied it to explain the polarization of the light of a cloud-
less sky, viewed in a direction at right angles to the direction
of the sun. But he did not tell me (though I have no doubt he
knew it himself) why the light of the cloudless sky seen in any
direction is bine, or I should certainly have remembered it.

§ 53. Rayleigh explained this thoroughly in his first paper
(1871), and gave what is now known as Rayleigh's law of the blue
sky; which is, that, provided the diameters of the suspended particles
are small in comparison with the wave-lengths, the proportions of
scattered light to incident light for different wave-lengths are
inversely as the fourth powers of the wave-lengths. Thus, while
the scMtered light has the same colour as the incident light
when homogeneous, the proportion of scattered light to incident
light is seven times as great for the violet as for the red of the
visible spectrum ; which explains the intensely blue or violet
colour of the clearest blue sky.

§ 54. The dynamical theory shows that the part of the light
of the blue sky, looked at in a direction perpendicular to the
direction of the sun, which is due to sunlight incident on a single
particle of diameter very small in comparison with the wave-
lengths of the illuminating light, consists of vibrations perpen-
dicular to the plane of these two directions: that is to say, is
completely polarized in the plane through the sun. In his 1871
paper*, Rayleigh pointed out that each particle is illuminated, not
only by the direct light of the sun, but also by light scattered
from other particles, and by earth-shine, and partly also by sus-
pended particles of dimensions not small in comparison with the
wave-lengths of the actual light ; and he thus explained the
observed fact that the polarization of even the clearest blue sky
at 90° from the sun is not absolutely complete, though it is very
nearly so. There is very little of polarization in the light from
white clouds seen in any direction, or even from a cloudless sky
close above the horizon seen at 90" from the sun. This is partly
because the particles which give it are not small in comparison
with the wave-lengths, and partly because they contribute much
to illuminate one another in addition to the sunlight directly
incident on them.

* Collected Papers, Vol. i. p. 94.

rayleioh's theory of the blue of the sky. 303

§ 55. For his dynamical foundation, Rayleigh definitely Moleculi
assumed the suspended particles to act as if the ether in their
places were denser than undisturbed ether, but otherwise unin-
fluenced by the matter of the particles themselves. He tacitly
assumed throughout that the distance from particle to particle
is very great in comparison with the greatest diameter of each
particle. He assumed these denser portions of ether to be of the
same rigidity as undisturbed ether; but it is obvious that this
last assumption could not largely influence the result, provided
the greatest diameter of each particle is very small in comparison
with its distance from next neighbour, and with the wave-lengths
of the light : and, in fact, I have found from the investigation of
§§ 41, 42 of Lecture XIV. for riffid spherical molecules embedded
in ether, exactly the same result as Rayleigh's ; which is as follows

where X denotes the wave-length of the incident light supposed
homogeneous ; T the volume of each suspended particle ; D the
undisturbed density of the ether ; U the mean density of the
ether within the particle; n the number of particles per cubic
centimetre ; and k the proportionate loss of homogeneous incident
light, due to the scattering in all directions by the suspended
particles per centimetre of air traversed. Thus

l-€-** (2)

is the loss of light in travelling a distance x (reckoned in centi-
metres) through ether as disturbed by the suspended particles.

It is remarkable that D' need not be uniform throughout the
particle. It is also remarkable that the shape of the volume T
may be anything, provided only its greatest diameter is very
small in comparison with \, The formula supposes T{iy'-D)
the same for all the particles. We shall have to consider cases in
which differences of T and D' for different particles are essential to
the result; and to include these we shall have to use the formula

"-^'^l D J ^•^^•

where 2 — n " denotes the sum of
the particles in a cubic centimetre.

D

foi" all

304 LECTURE xvir.

Molecular. § 56. Supposing now the number of suspended particles per
cubic wave-length to be very great, and the greatest diameter
of each to be sraall in comparison with its distance from next
neighbour, we see that the virtual density of the ether vibrating
among the particles is

D + lT{iy-D) (4);

and therefore, if u and u be the velocities of light in pure ether,
and in ether as disturbed by the suspended particles, we have
(Lecture VIII. p. 80)

„,.... [,,^n^] (.).

Hence, if fi denote the refractive index of the disturbed ether,
that of pure ether being 1, we have

,.[,+S?<?^^)]' (6,;

and therefore,

._,.^n^-Di

H'

D

§ 57. In taking an example to illustrate the actual trans-
parency of our atmosphere, Rayleigh says* ; " Perhaps the best
" data for a comparison are those afforded by the varying bright-
"ness of stars at various altitudes. Bouguer and others esti-
" mate about '8 for the transmission of light through the entire
" atmosphere from a star in the zenith. This corresponds to 8*3
"kilometres (the "height of the homogeneous atmosphere" at
" 10° Cent.) of air at standard pressure." Hence for a medium of
the transparency thus indicated we have €~****^ = '8; which gives
1/A: = 3720000 centimetres =37*2 kilometres.

§ 58. Suppose for a moment the want of perfect trans-
parency thus defined to be wholly due to the fact that the
tdtimate molecules of air are not infinitely small and infinitely
numerous, so that the " suspended particles " hitherto spoken of
would be merely the molecules N,, Oa; and suppose further
{U — D)T to be the same for nitrogen and oxygen. The known

♦ Phil. Mag, April, 1899, p. 382.

NUMBER OF MOLECnLES PER CUBIC CENTIMETRE OP GAS. 305

refractivity of air (/i — 1 = '0003), nearly enough the same for MoleonUr.
all visible light, gives by equation (7) above, with n instead of 2,

n(iy-D)T

D

Using this in (1) we find

= 0006.

*~n^^^o* ^^>'

for what the rate of loss on direct sunlight would be, per centi-
metre of air traversed, if the light were all of one wave-length, X.
But we have no such simplicity in Bouguer's datum regarding
transparency for the actual mixture which constitutes sunlight:
because the formula makes Ar~* pmportional to the fourth power
of the wave-length ; and every cloudless sunset and moonset and
sunrise and moonrise over the sea, and every cloudless view of
sun or moon below the horizon of the eye on a high mountain,
proves the transparency to be in reality much greater for red
light than for the average undimmed light of either luminary,
though probably not so much greater as to be proportional to the
fourth power of the wave-length. We may, however, feel fairly
sure that Bouguer's estimate of the loss of light in passing
vertically through the whole atmosphere is approximately true
for the most luminous part of the spectrum corresponding to
about the D line, wave-length 5*89 . 10~' cm., or (a convenient
round numBer) 6 . 10"* as Rayleigh has taken it. With this value
for X, and 3-72 . W centimetres for kr-\ (8) gives n = 8-54 . 10" for
atmospheric air at 10° and at standard pressure. Now it is quite
certain that a very large part of the loss of light estimated by
Bouguer is due to suspended particles ; and therefore it is certain
that the number of molecules in a cubic centimetre of gas, at
standard temperature and pressure, is considerably greater than
8-54 . 10".

§ 59. This conclusion drawn by Rayleigh from his dynamical
theory of the absorption of light from direct rays through air,
giving very decidedly an inferior limit to the number of molecules
in a cubic centimetre of gas, is perhaps the most thoroughly well
founded of all definite estimates hitherto made regarding sizes or
numbers of atoms. We shall see (§§ 73... 79, below) that a much
larger inferior limit is found on the same principles by careful

T. L. 20

306 LECTURE XVII.

Moleoolar. consideration of the loss of light due to the ultimate molecules
of pure air and to suspended matter undoubtedly existing in all
parts of our atmosphere, even where absolutely cloudless, that is
to say, warmer than the dew-point, and therefore having none of
the liquid spherules of water which constitute cloud or mist.

§ 60. Go now to the opposite extreme from the tentative
hypothesis of § 58, and, while assuming, as we know to be true,
that the observed refractivity is wholly or almost wholly due to
the ultimate molecules of air, suppose the opacity estimated by
Bouguer to be wholly due to suspended particles which, for
brevity, we shall call dust (whether dry or moist). These par-
ticles may be supposed to be generally of very unequal magni-
tudes : but, for simplicity, let us take a case in which they are all
equal, and their number only 1/10000 of the 8'54.10", which
in § 69 we found to give the true refractivity of air, with
Bouguer s degree of opacity for X = 6.10"*. With the same
opacity we now find the contribution to refractivity of the
particles causing it, to be only 1/100 of the known refractivity
of air. The number of particles of dust which we now have is
8*54.10" per cubic centimetre, or 184 per cubic wave-length,
which we may suppose to be almost large enough or quite large
enough to allow the dynamics of § 56 for refractivity to be
approximately true. But it seems to me almost certain that
8*54.10" is vastly greater than the greatest number of dust
particles per cubic centimetre to which the well-known haziness
of the clearest of cloudless air in the lower regions of our
atmosphere is due ; and that the true numbers, at different times
and places, may probably be such as those counted by Aitken*
at from 42500 (Hyferes, 4 p.m. April 5, 1892) to 43 (Kingairloch,
Argyllshire, 1 p.m. to 1.30 p.m. July 26, 1891).

§ 61. Let us, however, find how small the number of par-
ticles per cubic centimetre must be to produce Bouguer s degree
of opacity, without the particles themselves being so large in
comparison with the wave-length as to exclude the application of
Rayleigh's theory. Try for example r=10~'.X' (that is to say,
the volume of the molecule 1/1000 of the cubic wave-length, or

♦ Trans. R. 8, E, 1894, Vol. xxxvn. Part ni. pp. 675, 672.

HAZE DUE TO PARTICLES TOO LARGE FOR RAYLEIGH'S THEORY. 307

roughly, diameter of molecule 1/10 of the wave-length) which Molecular,
seems small enough for fairly approximate application of Ray-
leigh's theory; and suppose, merely to make an example, ZX to be
the optical density of water, D being that of ether ; that is to say,
ly/Drr (1-3337)"= 1-78. Thus we have {U -D) T/D^imiSX*:
and with X=6.10-», and with J!r-»=3-72 . 10«, (1) gives n = 1-485. 10«,
or about one and a half million particles per cubic centimetre.
Though this is larger than the largest number counted for natural
air by Aitken, it is interesting as showing that Bouguer's degree of
opacity can be accounted for by suspended particles, few enough
to give no appreciable contribution to refractivity, and yet not too
large for Rayleigh's theory. But when we look through very
clear air by day, and see how far from azure or deep blue is the
colour of a few hundred metres, or a few kilometres of air with
the mouth of a cave or the darkest shade of mountain or forest,
for background ; and when in fine sunny weather we study the
appearance of the grayish haze always, even on the clearest da3n3,
notably visible over the scenery among mountains or hills ; and
when by night at sea we see a lighthouse light at a distance of
45 or 50 kilometres, and perceive how little of redness it shows ;
and when we see the setting sun shorn of his brilliance sufficiently
to allow us to look direct at his face, sometimes whitish, oftener
ruddy, rarely what could be called ruby red ; it seems to me that
we have strong evidence for believing that the want of perfect
clearness of the lower regions of our atmosphere is in the main
due to suspended particles, too large to allow approximate ful-
filment of Rayleigh's law of fourth power of wave-length.

§ 62. But even If they were small enough for Rayleigh's
theory the question would remain. Are they small enough and
numerous enough to account for the refractivity of the atmo-
sphere? To this we shall presently see we must answer un-
doubtedly "No"; and much less than Bouguer's degree of opacity,
probably not as much as a quarter or a fifth of it, is due to the
ultimate molecules of air. In a paper by Mr Quirino Majorana in
the Transactions of the R. Accademia dei Lincei (of which a
translation is published in the Philosophical Ma^gazine for May,
1901), observations by himself in Sicily, at Catania and on Mount
Etna, and by Mr Gaudenzio Sella, on Monte Rosa in Switzerland,
determining the ratio of the brightness of the sun's surface to the

20—2

308 LECTURE XVII.

MoleooUur. brightness of the sky seen in any direction, are described This
ratio they denote by r. One specially notable result of Mr
Majorana's is that " the value of r at the crater of Etna is about
five times greater than at Catania." The barometric pressures
were approximately 53*6 and 76 cms. of mercury. Thus the atmo-
sphere above Catania was only 1*42 times the atmosphere above
Etna, and yet it gave five times as much scattering of light by its
particles, and by the particles suspended in it. This at once
proves that a great part of the scattering must be due to sus-
pended particles; and more of them than in proportion to the
density in the air below the level of Etna than in the air above it.
In Majorana's observations, it was found that '* except for regions
" close to the horizon, the luminosity of the sky had a sensibly
" constant value in all directions when viewed from the summit of
"Etna." This uniformity was observed even for points in the
neighbourhood of the sun, as near to it as he could make the
observation without direct light from the sun getting into his
instrument. I cannot but think that this apparent uniformity
was only partial. It is quite certain that with sunlight shining
down from above, and with light everywhere shining up ftx)m
earth or sea or haze, illuminating the higher air, the intensities
of the blue light seen in different directions above the crater
would be largely different. This is proved by the following
investigation ; which is merely an application of Rayleigh's theory
to the question before ua. But from Majorana's narrative we
may at all events assume that, as when observing from Catania,
he also on Etna chose the least luminous part of the sky {PhU.
Mag., May 1901, p. 561), for the recorded results (p. 662) of his
observations.

§ 63. The diagram, fig. 1 below, is an ideal representation of
a single molecule or particle, T, with sunlight falling on it indi-
cated by parallel lines, and so giving rise to scattered light seen
by an eye at E. We suppose the molecule or particle to be so
massive relatively to its bulk of ether that it is practically un-
moved by the ethereal vibration ; and for simplicity at present we
suppose the ether to move freely through the volume T, becoming
effectively denser without changing its velocity when it enters
this fixed volume, and less dense when it leaves. In ^ 41, 42, of
Lecture XV. above, and in Appendix A, a definite supposition,

DYNAMICS OF BLUE SKY.

309

attributing to ether no other property than elasticity as of an Moleoalai
utterly homogeneous perfectly elastic solid, and the exercise of
mutual force between itself and ponderable matter occupying
the same space, is explained : according to which the ether within

Fig. 1,

the atom will react upon moving ether outside, just as it would
if our present convenient temporary supposition of magically
augmented density within the volume of an absolutely fixed
molecule were realized in nature. For our present purpose, we
may if we please, following Rayleigh, do away altogether with
the ponderable molecule, and merely suppose T to be not fixed,
but merely a denser portion of the ether. And if its greatest
diameter is small enough relatively to a wave-length, it will
make no unnegligible diflference whether we suppose the ether in
T to have the same rigidity as the surrounding free ether, or
suppose it perfectly rigid as in ^ 1 — 46 of Lecture XIV. dealing
with a rigid globe embedded in ether.

§ 64. Resolving the incident light into two componeuts having
semi-ranges of vibration «r, p, in the plane of the paper and
perpendicular to it; consider first the component in the plane

310 LECTURE XVII.

Molecular, having vibrations symbolically indicated by the arrow-heads, and
expressed by the following formula

trsin — — ,

At

where w is the velocity of light, and X the wave-length. The
greater density of the ether within T gives a reactive force on
the surrounding ether outside, in the line of the primary vibration,
and against the direction of its acceleration, of which the magni-
tude is

D \ "^^"xT ^^^•

This alternating force produces a train of spherical waves spread-
ing out from T in all directions, of which the displacement is, at
greatest, very small in comparison with «r; and which at any
point E at distance r from the centre of T, large in comparison
with the gi'eatest diameter of T, is given by the following ex-

pression*

2ir

with ? = iHr ^^-p -"cos 5 (10),

where 6 is the angle between the direction of the sun and the
line TE, Formula (10), properly modified to apply it to the other
component of the primary vibration, that is, the component per-
pendicular to the plane of the paper, gives for the displacement at
E due to this component

t; cos — {at — r),

^ith , = /,^___^ (11).

Hence for the (juantity of light falling from T per unit of time,

* This formula is readily found from §§ 41, 42 of Lecture XIV. The complexity
of the formulas in §§ 8— iO is due to the inclusion in the investigation of forces and
displacements at small distances from 7, and to the condition imposed that T
is a rigid spherical figure. The dynamics of §§ 33 — 36 with c=0, and the details of
§§ 37—89 further simplified by taking v=ao , lead readily to the formulas (10) and
(11) in our present text.

BRIGHTNESS OF BLUE SKY. 311

on unit area of a plane at E, perpendicular to ET, reckoned in Molecu
convenient temporary units, we have

(iHT^ cos' ^ 4- p") (12).

§ 65. Consider now the scattered light emanating from a large
horizontal plane stratum of air 1 cm. thick. Let T of fig. 1 be one
of a vast number of particles in a portion of this stratum sub-
tending a small solid angle H viewed at an angular distance /3
from the zenith by an eye at distance r. The volume of this
portion of the stratum is H sec )8 r* cubic centimetres ; and there-
fore, if 2 denotes summation for all the particles in a cubic centi-
metre, small enough for application of Rayleigh's theory, and
q the quantity of light shed by them from the portion ft sec ^ r*
of the stratum, and incident on a square centimetre at E^ per-
pendicular to ET^ we have

?=^2r?^^^-^?ft8eci8(iir«cos'^ + pO (l^)'

Summing this expression for the contributions by all the
luminous elements of the sun and taking

h

to denote this summation, we have instead of the factor

tsr* cos' 0 + p\

cos' 0 I tsr' -f I p' :

and we have Iw' = |p' = iS (14),

where 8 denotes the total quantity of light from the sun falling
perpendicularly on unit of area in the particular place of the
atmosphere considered. Hence the summation of (13) for all the
sunlight incident on the portion ft sec 13 1^ of the stratum, gives

Q = ~Sr?^^^^^Tftsec/8(icos'^ + i)^ (^^>-

§ 66. To define the point of the sky of which the illumination
is thus expressed, let ^ be the zenith distance of the sun, and '^
the azimuth, reckoned from the sun, of the place of the sky seen

312

LECTURE XVII.

[oleoaUr. along the line ET, This place and the sun and the zenith are at
the angles of a spherical triangle SZT, of which ST is equal to 0.
Hence we have

cos ^ = cos fcos/S-fsin ^amficosy^ (16).

Let now, as an example, the sun be vertical: we have f=0,
0 = 0, and (15) becomes

Q = Jsp^P^^]°fl.i(co8i8+8ec/3)S (17).

This shows least luminosity of the sky around the sun at the
zenith, increasing to oo at the horizon (easily interpreted). The
law of increase is illustrated in the following table of values of
i (cos /8 -f sec i8) for every 10° of 0 from 0° to 90°.

1

p

4(co8/S+Bec/S)

^

J (coB/5+seo/5)

0'

1000

50'

1099

10**

1000

60'

1-250

20*

1002

70'

1-633

30**

1010

80'

2-966

40'

1036

90'

00

§ 67. Instead now of considering illumination on a plane
perpendicular to the line of vision, consider the illumination by
light from our one-centimetre-thick great* horizontal plane
stratum of air, incident on a square centimetre of horizontal
plane. The quantity of this light per unit of time coming from
a portion of sky subtending a small solid angle SI at zenith
distance /8 is Qcos/8. Taking ft = sin/8rf/9d'^ and integrating,
we find for the light shed by the one-centimetrc-thick horizontal
stratum on a horizontal square centimetre of the ground,

T(i)'-i))l«

D

V

(18).

Now each molecule and particle of dust sheds as much light
upwards as downwards. Hence (18) doubled expresses the

* We are neglecting the curvature of the earth, and supposing the density and
composition of the air to be the same throughout the plane horizontal stratum to
distances from the zenith very great in comparison with its height above the
ground.

SKY-LIGHT OVER ETNA AND MONTE ROSA. 313

quantity of light lost by direct rays from a vertical sun in cross- Molecula
ing the one-centimetre-thick horizontal stratum. It agrees with
the expression for A: in (1) of § 55, as it ought to do.

§ 68. The expression (15) is independent of the distance of
the stratum above the level of the observer s eye. Hence if H
denote the height above this level, of the upper boundary of an
ideal homogeneous atmosphere consisting of all the ultimate
molecules and all the dust of the real atmosphere scattered uni-
formly through it, and if 8 denote the whole light on unit area of
a plane at E perpendicular to ET, from all the molecules and dust
in the solid angle £1 of the real atmosphere due to the sun's direct
light incident on them, we have

J = ir sec /8 ^ 2 [?^-— ^T a i(cos«^ + l) (19);

provided we may, in the cases of application whatever they
may be, neglect the diminution of the direct sunlight in its
actual course through air, whether to the observer or to the
portion of the air of which he observes the luminosity, and neglect
the diminution of the scattered light fix)m the air in its course
through air to the observer. This proviso we shall see is prac-
tically fulfilled in Mr Majorana's observations on the crater of
Etna for zenith distances of the sun not exceeding 60"", and in
Mr Sella's observation on Monte Rosa in which the sun's zenith
distance was 50''. But for Majorana's recorded observation on
Etna at 5.50 a.m. when the sun s zenith distance was 81°'71, of
which the secant is 6*928, there may have been an important
diminution of the sun's light reaching the air vertically above the
observer, and a considerably more important diminution of his
light as seen direct by the observer. This would tend to make
the sunlight reaching the observer less strong relatively to the
sky-light than according to (19); and might conceivably account
for the first number in col. 8 being smaller than the first number
in col. 4 of the Table of § 69 below ; but it seems to me more
probable that the smallness of the first two numbers in col. 3,
showing considerably greater luminosity of sky than according to
(19), may be partly or chiefly due to dust in the air overhead,
optically swelled by moisture in the early morning. Indeed the
largeness of the luminosity of the sky indicated by the smallness

314

LECTURE XVII.

MolecaUr. of the first three numbers in coL 3, in comparison with the corre-
sponding numbers of col. 4, is explained most probably by gray
haze in the early morning.

§ 69. The results of Majorana's observations from the crater
of Etua are shown in the following Table, of which the first and
third columns are quoted from the Philosophical MagcLzine for
May, 1901, and the second column has been kindly given to me
in a letter by Mr Majorana. The values of Sjs shown in col. 4
are calculated from § 68 (19), with the factor of sec P (cos* ^ + 1)
taken to make it equal to Majorana's r for sun's zenith distance
44'''6, on the supposition that the region of sky observed was in
each case (see § 62 above) in the position of minimum luminosity
as given by (21). It is obvious that this position is in a vertical

Col. 1

Col. 2

CoL 3 Col. 4

Col. 5

Time

Zenith

distance

of sun.

Batioof

laminosity of

8nn*8 disc to

luminosity of

Ay.

r

S

8

Zenith distance

of least lami-

noos part of

sky.

5.50 A.M.
7
8
9
11

8r7

68-0
561
44-6
29-9

2570000
3125000
3650000
3930000
3760000

4820000
4610000
4310000
3930000
3370000

5-5
14-4
21-7
27-8
33-6

great circle through the sun, and on the opposite side of the
zenith from the sun; and thus we have ^ = f+/8. Hence (19)
becomes

J = ^ ^ 2 [^ ^^^^'^O, . i sec ^ [cos' (C + iS) + 1]. . .(20).
To make (20) a minimum we have

^/3=|«^\<f±/>, (21).

3-i-cos2(/9-f ?)

The value of /9 satisfying this equation for any given value of
f is easily found by trial and error, guided by a short preliminary

SKY-LIGHT OVER ETNA. 315

table of values of ^ for assumed values of /8 + f. Col. 5 shows Molecu
values of /3 thus found approximately enough to give the values of
'Sjs shown in col. 4 for the several values of ^.

§ 70. Confining our attention now to Majorana's observations
at 9 a.m. when the sun's altitude was about 44°*6 ; let e be the
proportion of the light illuminating the air over the crater of
Etna which at that hour was due to air, earth, and water below ;
and therefore \—e the proportion of the observed luminosity of
the sky which was due to the direct rays of the sun, and expressed
by § 68 (19). Thus, for /S = 27°-8, f = 44**-6, and ^= 72''-4, we have
Sja = 3930000/(1 - e\ instead of the Sja of col. 4, § 69. With this,
equation (20) gives

z mf^zSi

^*^^ ^\418.10-« (22).

mi

Here, in order that the comparison may be between the whole
light of the sun and the light from an equal apparent area of the
sky, we must take

a = 7r/214-6«* = 1/14660,

being the apparent area of the sun's disc as seen from the earth.
As to ^, it is what is commonly called the " height of the homo-
geneous atmosphere" and, whether at the top of Etna or at
sea-level, is

7-988 . 10» f 1 + ^r^ centimetres ;

where t denotes the temperature at the place above which H is
reckoned. Taking this temperature as 15° C, we find

H = 8*43 . 10* centimetres.
Thus (22) becomes

2 r?%^_^)y ^ X* (1 - e) . -728 . 10-».

.(23).

§ 71. Let us now denote by / and 1 — / the proportions of
(23) due respectively to the ultimate molecules of air and to
dust. We have

n r?L(^~-^)? = v/(l -«). -728. 10-* (24);

* The son's distance from the earth is 214-6 times his radius.

316 LECTURE XVII.

IColeeolar. where n denotes the number of the ultimate molecules in a cubic
centimetre of the air at the top of Etna ; and T (IX — D)/D relates
to any one of these molecules ; any difference which there may be
between oxygen and nitrogen being neglected. Now assuming
that the refractivity of the atmosphere is practically due to the
ultimate molecules, and that no appreciable part of it is due to the
dust in the air, we have by § 56 (7),

•0002 = n^^-^ (2.5),

the first number being approximately enough the refractivity of
air at the crater of Etna (barometric pressure, 53*6 centimetres of
mercury). Hence

[^^^H^--<- <^«).

and using this in (24) we find

220 ,„^,

«=xy(r^) <2^>-

Here, as in § 58 in connection with Bouguer s estimate for loss
of light in transmission through air, we have an essential un-
certainty in respect to the effective wave-length; and, for the
same reasons as in § 58, we shall take X = 6 . 10~* cm. as the
proper mean for the circumstances under consideration. With
this value of \, (27) becomes

n=2^~ .liid.W (28).

f(l - c)

§ 72. In Mr ISella's observations on Monte Rosa the zenith
distance of the sun was 50°, and the place of the sky observed was
in the zenith. He found the brightness of the sun's disc to be
about 5000000 times the brightness of the sky in the zenith.
Dealing with this result as in §§ 70, 71, with /S=0 in (20),
aud supposing the temperature of the air at the place of obser-
vation to have been 0° C, we find

^i'=^,^l^_^^2-34.10^» (29),

where e\ f\ and n' are the values of e, /, and n, at the place of
observation on Monte Rosa. Denoting now by N the number of
molecules in a cubic centimetre of air at 0° C. and pressure

8KY-LIOHT OVER MONTE ROSA.

317

75 centimetres of mercury, we have, by the laws of Boyle and Moleenl:
Charles, on the supposition that the temperature of the air was
15° on the summit of Etna, and 0° on Monte Rosa

„ 75 /, 15 \ ,75
^="53^l^ + 273J = "49'

or iV = l-48» = l-53n (30).

From these, with (28) and (29), we find

iV = ^

2-50

/(I - e)

. 10" =

3-58

/'(i-O

.10".

(31).

§ 73. To estimate the values of e and e' as defined in §§ 70, 72,
consider the albedos* of the earth as might be seen from a balloon
in the blue sky observed by Majorana and G. Sella over Etna and
over Monte Rosa respectively. These might be about '2 and '4,
the latter much the greater because of the great amount of snow
contributing to illuminate the sky over Monte Rosa. With so
much of guess-work in our data we need not enter on the full
theory of the contribution to sky-light by earth-shine from below
according to the principle of §§ 67, 68, interesting as it is ; and we
may take as very rough estimates '2 and '4 as the values of e and
e\ Thus (3 1 ) becomes

,(32).

N^^ 10'» = ®^ 10"

/ /'

§ 74. Now it would only be if the whole light of the sky were
due to the ultimate molecules on which the refractivity depends
that / or /' could have so great a value as unity. If this were
the case for the blue sky seen over Monte Rosa by G. Sella in

* Albedo is a word introdnoed by Lambert 150 years ago to signify the ratio of
the total light emitted by a thoroughly unpolished solid or a mass of cloud to the
total amount of the incident light. The albedo of an ideal perfectly white body
is 1. My friend Professor Becker has kindly given me the following table of albedos
from Miiller's book Die Photometrie der Oeatime (Leipsic, 1897) as determined by
observers and experimenters.

Mercury 0*14

Uranus 0*60

Venus 0-76

Neptune 0*52

Moon 0-34

Snow 0*78

Mars 0-22

White Paper 0*70

Jupiter 0-62

White Sandstone 0-24

Saturn 072

Damp Soil 008

318 LECTURE XVII.

llblecular. 1900, we should have /' = 1 , and therefore N = 5*97 . 10". But it
is most probable that even in the very clearest weather on the
highest mountain, a considerable portion of the light of the sky
is due to suspended particles much larger than the ultimate
molecules N,, Oa, of the atmosphere ; and therefore the observa-
tions of the luminosity of the sky over Monte Rosa in the summer
of 1900 render it probable that N is greater than 5*97 . W\ If
now we take our estimate of § 50, for the number of molecules in
a cubic cm. of air at 0°, and normal pressure, JV= 10* we have
1 — /= "688 and 1 — /' = •403 ; that is to say, according to the
several assumptions we have made, '688 of the whole light of the
portion of sky observed over Etna by Majorana was due to dust,
and only '403 of that observed by Sella on Monte Rosa was due
to dust. It is quite possible that this conclusion might be exactly
true, and it is fairly probable that it is an approximation to the
truth. But on the whole these observations indicate, so far as
they can be trusted, the probability of at least as large a value as
10« for N.

§ 75. All the observations referred to in ^ 57 — 74 are vitiated
by essentially involving the physiological judgment of perception
of difference of strengths of two lights of difl*erent colours. In
looking at two very differently tinted shadows of a pencil side by
side, one of them blue or violet cast by a comparatively near
candle, the other reddish-yellow cast by a distant brilliantly white
incandescent lamp or by a more distant electric arc-lamp, or by
the moon ; when practising Rumford's method of photometry; it is
quite wonderful to find how unanimous half-a-dozen laboratory
students, or even less skilled observers, are in declaring This is
the stronger! or. That is the stronger! or, Neither is stronger
than the other! When the two shadows are declared equally
strong, the declaration is that the differently tinted lights from
the two shadowed places side by side on the white paper are,
according to the physiological perception by the eye, equally
strong. But this has no meaning in respect to any definite
component parts of the two lights; and the unanimity, or the
greatness of the majority, of the observers declaring it, only proves
a physiological agreement in the perceptivity of healthy average
eyes (from which colour-blind eyes would no doubt differ wildly).
Two circular areas of white paper in Sellas observations on

PRISMATIC ANALYSIS OF SKY-LIGHT. 319

Monte Rosa, a circle and a surrounding area of ground glass in Moleonli
Majorana's obsci*vations with his own beautiful sky-photometer
on Etna, are seen illuminated respectively by diminished sunlight
of unchanged tint and by light from the blue sky. The sun-lit
areas -seem reddish-yellow by contrast with the sky-lit areas which
are azure blue. What is meant when the two areas differing so
splendidly are declared to be equally luminous? The nearest
approach to an answer to this question is given at the end of
§71 above, and is eminently unsatisfactory. The same may be
truly said of the dealing with Bouguer's datum in § 57, though
the observers on whom Bouguer founded do not seem to have
been disturbed by knowledge that there was anything indefinite
in what they were trying to define or to find by observation.

§ 76. To obtain results not vitiated by the imperfection of
the physiological judgment described in § 75, Newton's prismatic
analysis of the light observed, or something equivalent to it, is
necessary. Prismatic analysis was used by Rayleigh himself for
the blue light of the sky, actually before he had worked out
his dynamical theory. He compared the prismatic spectrum of
light from the zenith with that of sunlight diffused through
white paper; and by aid of a curve drawn from about thirty
comparisons ranging over the spectrum from C to beyond F,
found the following results for four different wave-lengths.

c

Wave-length 656-2

Observed brightness ... 1
Calculated according to \"* . 1

On these he makes the following remarks: — "It should be
"noticed that the sky compared with diffused light was even
" bluer than theory makes it, on the supposition that the diffnsed
" light through the paper may be taken as similar to that whose
"scattering illuminates the sky. It is possible that the paper
"was slightly yellow; or the cause may lie in the yellowness
"of sunlight as it reaches us compared with the colour it
"possesses in the upper regions of the atmosphere. It would
"be a mistake to lay any great stress on the observations in
" their present incomplete form ; but at any rate they show that
"a colour more or less like that of the sky would result from

D

*.

F

589-2

617-3

486-2

1-64

2-84

3-60

1-54

2-52

3-34

320 LBCTTRK XVIL

ioleeiilmr. " taking the elements of white light in qoantities proportional
''to X~^. I do not know how it mar strike others; but in-
** dividoally T was not prepared (otr so great a diffnenoe as the
" observations show, the ratio (osr F being more than three times
"as great as for C" For myself I thorooghlT agree with this
last sentence of Rayleigh's. There can be no doobt of the
tmstwiHthinesB of his observational results; bat it seems to me
most probable, or almost certain, that the yellowness, <»* orange-
colour, of the sunlight seen through the paper, caused by larger
absorption of green, blue, and violet rays, explains the extreme
relative richness in green blue, and violet rays which the results
show for the zenith blue sky observed.

§ 77. An elaborate series of researches <m the blue of the
sky on twenty-two days from July, 1900, to February, 1901, is
described in a very interesting paper, "Ricerche sul Bleu del
Cielo," a dissertation presented to the Royal University of Rome
by Dr Giuseppe Zettwuch, as a thesis for his degree of Doctor
in Physics. In these researches, prismatically analysed light frx>m
the sky was compared with prismatically analysed direct sunlight
reduced by passage through a narrow slit; and the results were
therefore not vitiated by unequal absorptions of direct sunlight
in the apparatus. A translation of the author's own account
of his conclusions is published in the Philosophical Magazine
for August, 1902; by which it will be seen that the blueness
of the sky, even when of most serene azure, was always much
less deep than the true Rayleigh blue defined by the X"* law.
Hence, according to Rayleigh's theory (see § 53 above) much of
the light must always have come from particles not exceedingly
small in proportion to the wave-length. Thus in Zettwuch's
researches we have a large confirmation of the views expressed
in §§ 54, 58, 61, 74 above, and §§ 78, 79 below.

§ 78. Through the kindness of Professor Becker, I am now
able to supplement Bouguer's 170-year old information with the
results of an admirable extension of his investigation by Professor
Muller of the Potsdam Observatory, in which the proportion
(denoted by p in the formula below) transmitted down to sea-level
of homogeneous light entering our atmosphere vertically is found
for all wave-lengths from 4*4. 10~* to 6*8.10-*, by comparison of

ATMOSPHERIC ABSORPTION ANALYSED BY MOLLER AND BECKER, 321

the solar spectrum with the spectrum of a petroleum flame, for Molecular
different zenith distances of the sun. It is to be presumed,
although I do not find it so stated, that only the clearest
atmosphere available at Potsdam was used in these observations.
For the sake of comparison with Rayleigh's theory, Professor
Becker has arithmetically resolved Mtiller's logarithmic results
into two parts ; one constant, and the other varying inversely as
the fourth power of the wave-length. The resulting formula*,
modified to facilitate comparison with §§ 57 — 59 above, is as
follows :

p = g-(08W+(»7ar-)=.9i52€-onar-* (33)^

where -^ = \ -7- 6 . 10~'. In respect to the two factors here shown,
we may say roughly that the first factor is due to suspended
particles too large, and the second to particles not too large, for
the application of Rayleigh's law. For the case of \ = 6 . 10^
(z= 1) this gives

p = -9152 . -9258 = -847 (34).

§ 79. Taking now the last term in the index and the last
factor shown in (34) and dealing with it according to §§ 57 — 59
above; and still, as in § 55, using k to denote the proportionate
loss of light per centimetre due to particles small enough for
Rayleigh's theory, whether "suspended particles" or ultimate
molecules of air, or both ; we have e~®*^** = '9258 which gives
kr^ = 10*76 . 10* cms. Hence if, as in § 08, we suppose for a
moment the want of perfect transparency thus defined to be
wholly due to the ultimate molecules of air, we should have, by

the dynamics of refractivity, n - — ^c ^ = *0006 ; and thence by

(1) of § 55 with \=6.10~* we should find for the number of
molecules per cubic centimetre n = 2*47 . 10*^. But it is quite
certain that a part, and most probable that a large part, of the want
of transparency produced by particles small enough for Rayleigh's
theory is due to " suspended particles " larger than the ultimate
molecules: and we infer that the number of ultimate molecules
per cubic centimetre is greater than, and probably very much
greater than, 2*47 . 10^'. Thus from the surer and more complete
data of Miiller regarding extinction of light of different wave-

• Miiller, Die Photometrie der Gestimey p. 140.
T.L. 21

322

LECTUKK XVn.

Molecnlftr. lengths traversing the air, we find an inferior limit for the number
of molecules per cubic centimetre nearly three times as great as
that which Rayleigh showed to be proved from Bouguer's datum.

§ 80. Taking, somewhat arbitrarily, as the result of §§ 23 — 77
that the number of molecules in a cubic centimetre of a perfect
gas at standard temperature and pressure is 1(P, we have the
following interesting table of conclusions regarding the weights
of atoms and the molecular dimensions of liquefied gases, of water,
of ice, and of solid metals.

1

Maaoffttom

1

NinlMror

tecmtMtf
r»Dcedtec«bic
1 ordcrvttli
MSoaldenrity

' Substoiiee

oroTH/)
ingrammet

Demhy

of iBolecvIci

1

H

0-45 . 10-«*

liquid at 17^ absolute

•090

200 . 10»»

1-71 . 10-»

0

715 „

„ „ freezing point

1-27

178 „

1-78 „

H.O

8-05 „

water

1-00

124 „

2-00 „

H,0

8-05 „

ice

-917

114 ..

206 „

H,0

805 „

Tapour at 0° C. -487 . 10"*

605 . 10>»

118-2

6-29 „

liquid

1-047

166.10"

1-82 „

Argon

17-81 „

»»

1-42

79-7,.

2-SS „

Gold

88-52 „

solid

19*32

218 „

l-^ »

Silver

48-47 „

»t

10-53

217 „

1-66 „

Copper

28-43 „

tt

8-95

315 „

1-47 „

Iron

2515 „

>t

7-86

313 „

1-47 „

Zine

29-30 „

»»

715

245 „

1-60 „

«

€€

§ 81. In the introductory Lecture (p. 14) we considered the
question " are the vibrations of light perpendicular to, or are they
*'in, the plane of polarization ? — defining the plane of polarization
as the plane through the incident and reflected raj-s, for light
polarized by reflection." We are now able to answer perpen-
dicular to the plane of polarization, with great confidence founded
on two experimental proofe both given by Stokes, and each of
them alone sufficient, I believe, for the conclusion.

I. The observed feet that a large proportion of the light of
the blue sky looked at in any direction perpendicular to the
direction of the sun, and not too nearly horizontal, is polarized in
the plane through the sun ; interpreted according to the dynamics
of Lecture VIII., pp. 88, 89, 90, and of Lecture XVII., §§ 52, 54,
63.

II. Observation of the change of the plane of polarization ex-
perienced by plane polarized light when diffi-actionally changed in

STOKES, HOLTZMANN, LORENZ, RAYLEIGH. 323

direction through a large angle by passage across a Fraunhofer Molar,
grating ; described and interpreted in Part II. of his great paper
" On the Dynamical Theory of Diffraction*/' In a short Appendix,
pp. 327, 328, added to his Reprint, Stokes notices experiments
by Holtzniann f, published soon after that paper, leading to
results seemingly at variance with its conclusions, and gives a
probable explanation of the reason for the discrepance; and he refers
to later experiments agreeing with his own, by Lorenz of Denmark.

Thus we find in II. confirmation of the conclusion, first drawn
by Stokes from I., that the vibrations in plane polarized light
are perpendicular to the plane of polarization.

§ 81'. Finally, we have still stronger confirmation of Stokes's
original conclusion, in fact an irrefragable independent demonstra-
tion, that the vibrations of light are perpendicular to the plane of
polarization, by Rayleigh; founded on a remarkable discovery,
made independ^tly by himself and Lorenz of Denmark, regarding
the reflection of waves at a plane interface between two elastic
solids of different rigidities but equal densities : That, when the
difference of rigidities is small, and when the vibrations are in
the plane of incidence, there are two angles of incidence (tt/S
and 37r/8), each of which gives total extinction of the reflected
light. That is to say ; instead of one " polarizing-angle," tSLU"^ fi;
there are two " polari zing-angles " 22°*5 and 67°-5 ; which is utterly
inconsistent with observation. The old-known fact (proved in § 123
below) that, if densities are equal and rigidities unequal, vibrations
perpendicular to the plane of incidence give reflected light obeying
Fresnel's " tangent " law and therefore vanishing when the angle
of incidence is tan~*/Lt, had rendered very tempting, the false
supposition that light polarized by reflection at incidence tan~* fi
has its vibrations in the plane of incidence ; but this supposition
is absolutely disproved by the two angles of extinction of the
reflected ray which it implies for vibrations in the plane of
incidence and reflection.

We may consider it as one of the surest doctrines through the
whole range of natural philosophy, that plane-polarized light consists
of vibrations of ether perpendicular to the plane of polarization.

• Reprint of Mathematical and Physical Papers, Vol. n. pp. 290 — 327.
t PoggendorfiTs AnnaUn, Vol. xciz. (1S56), p. 446, or Philosophical Magazine,
Vol. xin. p. 135.

21—2

LECTUKE XVin.

Thubsday, October 16, 5 pjl Written afresh 1902.

Reflection of Light.

iColar. § 82. The subject of this Lecture when originally given was
" Reflection and Refraction of Light." I have recently found it
convenient to omit "Refraction" from the title because (§130
below), if the reflecting substance is transparent, and if we
know the laws of propagation of ethereal waves through it, we
can calculate the amount and quality of the refracted light for
every quality of incident light, and every angle of incidence,
when we know the amount and quality of the reflected light.
When the reflecting substance is opaque there is no '* refracted "
light; but, co-periodic with the motion which constitutes the
incident light, there is a vibratory motion in the ether among
the matter of the reflecting body, diminishing in amplitude
according to the exponential law, c"**^, with increasing distance
D from the interface. When mr^ is equal to 10,000 wave-lengths,
say half-a-centimetre, the amplitude of the disturbance at one-
half centimetre inwards from the interface would be €""* of the
amplitude of the entering light, and the intensity would be €~*,
or 1/7*39, of the intensity of the entering light. The substance
might or might not, as we please, be called opaque ; but it would
be 80 far from being perfectly opaque that both theoretically
and experimentally we might conveniently deal with the case
according to the ordinary doctrine of " reflected " and " refiucted "
light. The index of refraction might be definitely measured by
using very small prisms, not thicker than 1 mm. in the thickest
part. But, on the other hand, when the opacity is so nearly
perfect that mr^ is ten wave-lengths, or two or three wave-
lengths, or less than a wave-length, we have no proper application
of the ordinary law of refraction, and no modification of it can
conveniently be used. Omission to perceive this negative truth
has led some experimenters to waste precious time and work in
making, and experimenting with, transparent metal prisms of

TRANSLUCENCE OF METALLIC FILMS. * 325

thickness varyiDg from as nearly nothing as possible at the edge, Molar,
to something like the thickness of ordinary gold-leaf at distance
of two or three millimetres from the edge.

§ 83. A much more proper mode of investigating the pro-
pagation of ethereal vibrations through metals is that followed
first, I believe, by Quincke, and afterwards by other able experi-
menters ; in which an exceedingly thin uniform plate of the solid
is used, and the retardation of phase (found negative for metals
by Quincke !) of light transmitted through it is measured by the
well-known interferential method. Some mathematical theorists
have somewhat marred their work by holding on to " refraction,"
and giving wild sets of real numbers for refractive indices* of
metals (different for different incidences !), in their treatment of
light reflected from metals; after MacCullagh and Cauchy had
pointed out that the main observed results regarding the reflection
of light from metals can be expressed with some approach to
accuracy by Fresnel's formulas with fi = ip -{-q, where p and q
are real numbers, and p certainly iiot zero : q very nearly zero for
silver, or quite zero for what I propose to call ideal silver (§§ 150,
151 below). It was consideration of these circumstances which
led me to drop " Refraction " from the title of Lecture XVIII.

§ 84. The theory of propagation of the ethereal motion
through an opaque solid, due to any disturbance in any part of
it, including that produced by light incident on its surface, is a
most important subject for experimental investigation, and for
work in mathematical dynamics. The theory to be tried for
(§ 159) is that of the propagation of vibratory motion through
ether, when under the influence of molecules of ponderable matter,
causing the formulas for wave-motion to be modified by making
the square of the propagational velocity a real negative quantity,
or a complex. We shall see in § 159 below that a new molecular
doctrine gives a thoroughly satisfactory dynamical theory of
what I have called ideal silver, a substance which reflects light
without loss at every incidence and in every polarizational
azimuth; its quality in wave-theory being defined by a real
negative quantity for the square of the propagational velocity
of imaginary waves through ether within it. Liquid mercury,

* /A is essentially the ratio of the propagational velocities in the two mediums,
one of which may, with good dynamical reason, be a complex imaginary. See
144—162.

328 LECTURE XVIII.

Molar, placed anywhere not in the course of the properly reflected light
from sun or sky. That is to say, the whole room being dark, and
the screen around the hole perfectly black, the whole mirror wonld
seem perfectly black when looked at by an eye so placed as not to
see an image (of the hole, and therefore) of any part of the sky or
sun. A condition of polish very nearly^ but not quite, optically
perfect, would be shown by a faint violet-blue light from the
surface of the mirror instead of absolute blackness. The tint of
this light would be the true Rayleigh \~* blue, if the want of per-
fectness of the polish is due to craters small in comparison with
the wave-length (even though large in comparison with distances
between nearest molecular neighbours). When this condition is
fulfilled, the blue light due to want of perfectness in the polish,
seen on the surface if viewed in any direction perpendicular to the
direction of the incident beam, would be found completely polar-
ized in the plane of these two directions. Everything in § 86 is
applicable to reflection from any kind of surface, whether the
reflecting body be solid or liquid, or metallic, or opaque with any
kind of opacity, or transparent. Experimental examination of the
polish of natural faces of crystals will be interesting.

§ 87. Valuable photometric experiments with reference to
reflection of light by metals have been made by Bouguer, Biot,
Brewster, Potter, Jamin, Quincke, De Senarmont, De la Provostaye
and Desains, and Conroy*. But much more is to be desired, not
only as to direct reflection from metals, but as to reflection at
all angles of incidence from metals and other opaque and trans-
parent solids and liquids. In each case the intensity of the
whole reflected light should be compared with the intensity of
the whole incident light ; in the first place without any artificial
polarization of the incident, and without polarizational analysis
of the reflected, light. It is greatly to be desired that thorough
investigation of this kind should be made. It would be quite an
easy kind of work"f because roughly approximate photometry is

• Bouguer, Traits d*Optique, 1760, pp. 27, 131: Biot, Ann. de Chimie, 1S16,
Vol. xciv. p. 209: Brewster, A Treatise on new Philosophical InstrumenU, Edin.
1S13, p. 347; Phil Tram. 1830, p. 69: Potter, Edin. Jour, of Science, 1830: De la
Provostaye and Desains, Ann. de Chim. et de Phys. [3], 1849, Vol. xxvii. p. 109, and
1850, Vol. XXX. pp. 159, 276: Conroy, Proc. R. S. ; see § 84 above. See Masoart,
TraiU d'Optique, Vol. n. 1891, pp. 441—459.

t Bouguer, TraitS d'Optique, 1760, pp. 27, 131 : Arago, (Euvres Competes, Vol. x.
pp. 150, 185, 216, 468. See Masoart, Traiti d'Optique, Vol. n. 1891, pp. 441—501.

reflectivities; fresnel, potter, jamin, and conroy. 329

always easy, except when rendered impossible by difference ofUoIur.
colour in the lights to be compared.

Absolute determinations of reflected light per unit of incident
light cannot be made with great accuracy because of the inherent
difficulty, or practical impossibility, of very a/xurate photometric
observations, even when, as is largely the case for reflected lights,
there is no perceptible diflerence of tint between the lights to be
compared.

§ 88. The accompanying diagram, fig. 1, shows, for angles of
incidence from 10° to 80°, the reflectivities* of silver, speculum-

-"

^

"yF

d

—

-^

__

~\

it

s

.,

/ 1

I/.

n

M

y

^r

J>

S^

C€

'

-J

—

K

^

J

A

^

r^

w

r

in-,

^

/

w

jf

T'

/

1

^

',

1

rt"

_

/

1

/

/

y

/

jj

y

tLi.

ill

Fig. 1.

* I propow the word "reflectivity" to deBi^nste the ratio of the whole refleoted
light to the whole inoideat light, whether the incident light is anpolarized as in
flg. 1; or pluie polarized, either in or perpendicaUr to the plane of inoidence,
u in flg*. i — 7; and whether it is ordinary while light, or homogeneoaa light, or
light of anj mixed tint.

330 LECTURE XVIII.

Molar, metal, steel, and tin, according to the observations of Potter,
Conroy, and Jamin; and for all incidences from 0° to 90° the
reflectivities of diamond, flint-glass, and water, calculated from
Fresnel's admirable formulas. These are, for almost all trans-
parent bodies even of so high refiuctivity as diamond, (g 96, 100
below), probably very much nearer the truth than any photo-
metric observations hitherto made, or possible to human eyes.
The ordinates represent the reflected light in percentage of the
incident light, at the angles of incidence represented by the
abscissas. The reflectivities thus given for the three transparent
bodies arc the means of the reflectivities given in figs. 4, 5, 6 of § 102
below for light polarized in, and perpendicular to, the plane of
incidence and reflection. Jamin's results* for steel are given by
himself as the means of the reflectivities for light polarized in,
and perpendicular to, the plane of incidence and reflection ; each
determined photometrically by comparison with reflectivities of
glass calculated from FresneFs formulas. The six curves for metals,
of this diagram (fig. 1), show I believe all the reflectivities that
have hitherto been determined by observation; except those of
Jam in f for normal incidence of homogeneous lights from red to
violet on metals, and Rayleigh'sJ for nearly normal incidence of
white light on mercury and glass. All the curves meet, or if
ideally produced meet, in the right-hand top comer, showing total
reflection at grazing incidence.

§ 89. The accompanying sketches (figs. 2 a, 2 6) represent
what (on substantially the principle of Bouguer, Potter, and
Conroy) seems to me the best and simplest plan for making
photometric determinations of reflectivity. The centres of lamps,
mirror, and screen, are all in one horizontal plane, which is taken
as the plane of the drawings. The screens and reflecting face of
the miiTor are vertical planes. R is the reflecting surface, shown
as one of the faces of an acute-angled prism. X, Z' are two as
nearly as may be equal and similar lamps (the smaller the
horizontal dimensions the better §, provided the light shed on the

• Mascart, Traits d'Optique, Vol. ii. pp. 634, 636. f Ibid, p. 644.

X Scientific Papers, Vols. u. and nr. ; Proc. R, S. 1886. Phil. Mag, 1892.

§ The best non-electrio light I know for the purpose is the ordinary flat-wicked
paraffin -lamp, with a screen having a narrow vertical slot placed close to the lamp-
glass, with its medial line in the middle plane of the flat flame. The object of the
screen is to cut off light reflected from the glass funnel, without intercepting any
of the light of the flame itself; the width of the slot should therefore be very

EXPERIMENTAL METHOD FOR REFLECTIVITIES.

331

mirror is sufficient). L is too far from R to allow its true position Molar,
to be shown on the diagram, but its direction as seen from R is
indicated by a broken straight line. TF is an opaque screen coated

W

H

i

t

Fig. 2 6.

Fig. 2 a.

slightly greater than the thickness of the flame. This arrangement gives a much
finer and steadier line of light than any unscreened candle, and vastly more light;
and it gives more light than comes through a slot of the same width from a
round- wicked paraffin -lamp. An electric lamp of the original Edison ** hair-pin"
pattern, with a slotted screen to cut off reflected light from the glass, would be
more convenient than the paraffin-lamp, and give more light with finer shadows.

332 LECTURE XVIIL

Molar, with white paper on the side next E, the eye of the observer.
Fig. 2 a and fig. 2 b show respectively the positions for very
nearly direct incidence, and for incidence about 60°. Each
drawing shows, on a scale of about one-tenth or one-twentieth,
approximately the dimensions convenient for the case in which
the mirror is a surface of flint-glass of refractive index 1*714;
and, as according to Fresnel's theory, represented by the mean of
the ordinates of curves 1 and 2 of fig. 6 (§ 102 below), having re-
flectivities 1/14 and 1/4 at incidences 0° and 60°. The eye at E
sees a narrow portion of the white screen next the right-hand edge
illuminated only by Z, the image of L in the mirror; and all of the
screen on the left of that portion, illuminated by the distant lamp
L', The distant lamp is moved nearer or farther till the light is
judged equal on the two sidas of the border line between the
illumination due to I, the image of Z, and the direct illumination
due to L\ By shifting L slightly to the right, or slightly to the
left, we may arrange to have either a very narrow dark space, or
a space of double brightness, between the two illuminations ; aad
thus the eye is assisted in judging as to the perfect equality of
the two. When, as in fig. 2 6, the angle of incidence exceeds
45° a dark screen DD is needed to prevent the light of L from
shining directly on the white screen, W, The method, with the
details I have indicated, is thoroughly convenient for reflecting
solids, whether transparent or metallic or of other qualities of
opacity; provided the mirror can be made of not less than two
or three centimetres breadth. But, as in the case of diamond,
when only a very small mirror is available, modification of the
method to allow direct vision of the lights to be compared (with-
out projection on a white screen) would be preferable or necessary.
For liquids, of course, modification would be necessary to suit it
for reflection in a vertical plane.

§ 90. As remarked at the end of § 87, absolute determinations
of reflectivities cannot be made with great accuracy, because of the
imperfectness of perceptivity of the eye in respect to relative
strengths of light, even when the tints are exactly the same. On
the other hand, a very high degree of accuracy is readily attainable
by the following method*, when the problem is to compare, for
any or every angle of incidence, the reflected lights due to the

* Given originally by MaoOuUagh. See his ColUcUd Works, p. 239; also
Stokes, Collected Works, Vol. m. p. 199.

POLABIZATIONAL ANALYSIS OF REFLECTED LIGHT. 333

incidence of equal quantities of light vibrating in, and vibrating Molar,
perpendicular to, the plane of incidence.

Use two NicoFs prisms, which, for brevity, I shall call Ni and
jftTa, in the course of the incident and reflected light respectively.
Use also a Fresnels rhomb (F) between the reflecting surface
and J^,. Set N^, and keep it permanently set, with its two
principal planes at 45° to the plane of reflection, but with facilities
for turning from any one to any other of the eight positions thus
defined, to secure any needful accuracy of adjustment. For brevity,
I shall call the zeras of N^ and F, positions when their principal
planes are at angles of 45° to the plane of incidence and
reflection. In the course of an observation JVi and F are to be
turned through varying angles, n and /, from their zeros, till
perfect extinction of light coming through N^ is obtained.

§ 91. To begin an observation, with any chosen angle of
incidence of the light on the reflecting surface, turn N^ to a
position giving as nearly as possible complete extinction of light
emerging fi-om N^. Improve the extinction, if you can, by turning
F in either direction, and get the best extinction possible by
alternate turnings of Ni and F, Absolutely complete extinction
is thus obtained at one point of the field if homogeneous light
is used, and if the Nicols and rhomb are theoretically perfect
instruments. The results of the completed observation are the
two angles (w, /) through which Ni and F must be turned from
their zero positions to give perfect extinction by N^. From f
thus found we calculate by two simple formulas, (7) of § 93,
the ratios of the vibrational amplitudes of the two constituents
defined below, (c^, g), of the reflected vibrations in the plane of
reflection, to the vibrational amplitude (C) of the reflected vibra-
tions perpendicular to that plane. The definite constituents here
referred to are (c^ vibrations in the same phase as C; and (g)
vibrations in phase advanced by a quarter-period relatively to C.
It is clear that if ^ = 0, complete extinction would be had without
turning F from its zero position, and would be found by the
same adjustment of Ni as if there were no Fresnel's rhomb in the
train. {C corresponds to the C of §§ 117, 123, below.)

§ 92. Figs. 3a, 36, are diagrams in planes respectively perpen-
dicular to the reflected, and to the incident, light. Let 0 be a point
in the course of the light between the reflecting surface and the

334

LECTURE XVIII.

Molar. Fresners rhomb. Let 00 be the plane of reflection of the light from
the reflecting surface, and let OZ be perpendicular to that plane.

21

Let ON 2 be the vibrational plane* of the second Nicol; and OP
its " plane of polarization" (being perpendicular to ON^). Let OF
be the plane through the entering ray, and perpendicular to the
facial intersections f of the Fresnel's rhomb ; shown in the diagram
as turned through an angle /= NfiF from the zero position
OJV,. Let OK be perpendicular to OF. In respect to signs we
see by § 158^ (1) that in fig. 3a, for reflected, and 36 for incident,
vibrations, 00 is positive.

Considering the light coming from the reflecting surface and
incident on the Fresnel, let C sin at be the component along OZ,
and c^ sin tot—g cos wt be the component along 00, of the dis-
placement at time ^ of a particle of ether of which 0 is the
equilibrium position. cf/C and gjC are two functions of the angle
of incidence to be determined by the observation now described.
By proper resolutions and additions for vibrational components
in the principal planes of the Fresnel, we find as follows:

sin iot Tc^cos (^ -/) - Csin (i -f\\ along OF ^

sin 6)^ c^sin

U«' g -/)

+ Ocos

a -/)]

i*

OK

• •■(!).

and

— cos (ot g cos ( T — /) along OF

— cos (ot g sin i-z — /) „ OK

(2).

* I use this expression for brevity to denote the plane of the vibrations of light
transmitted through a Nicol.

t An expression used to denote the intersections of the traversed faces and the
reflecting surfaces. See § 15S' below.

POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 335

By the two total internal reflections at the oblique faces of the Molar.
Fresnel, vibrational components in the plane OF are advanced a
quarter-period relatively to the vibrational components perpen-
dicular to OF, Hence to find the vibrational components at time
^ at a point in the course of the light emerging from the Fresnel,
we must, in the components along OF of (1) and (2), change the
sin (Dt into cos at, and change the cos tot into — sin tot ; and leave
unchanged the components along OK. Thus we find for the
vibrational components at the chosen point in the course of the
light from the Fresnel towards the second Nicol, as follows :

sino)^

sm
and

g cos r^ -/j along OF '

in a>t \ c^sin (^^fWc cos (j -/)1 „ OK J

...(3),

co&tot c^cos (t -/) - C^sin (t -/) alo"g OF
— cos (ot ^r sin f -r — /j „ OK

§ 93. For extinction by the second Nicol, the sum of all the
vibrational components parallel to OiVa of the light reaching it
must be null; and therefore, by the proper resolutions and
additions, and by equating to zero separately the coefficients
of sin (ot and cos tot thus found, we have

- r^sin {^ -/) + Ocos (j^ -/)J sin/+flrcos (^ -/) cos/=0

(5),

pcos (^ -/) - Csin Q -/)j COS/-I- (/sin (^ -/) sin/=0

(6).

Solving these equations for <f and g, we find

^^ltco_8_4/^ 28iny

2 4-sm4f "^ 2 4-8in4f ^ '

§ 94. Go back now to the light emerging from the first
Nicol and incident on the reflecting surface. Let / be its

336

LECTURE XVIII.

Molar. vibratioDal amplitude. The vibratioDal amplitudes of its com-
ponents in, and perpendicular to, the plane of incidence are

I cosl-j'-nj y and — /sinf— — nj,

when the Nicol is turned from its zero position, OQ, through the
angle n, as indicated in fig. 36 ; ONi being the vibrational plane of
the first Nicol.

G

§ 94'. Let S be the ratio of reflected to incident vibrational
amplitude for vibrational component perpendicular to the plane
00. Considering next the component of the incident light having
vibrations in the plane 00 ; let T and E be the ratios of the vibra-
tional amplitudes of two particular constituents of its reflected
light both vibrating in the plane 00, to the vibrational amplitude
of the incident component; these two constituents being respec-
tively in the same phase as the component of the reflected light
vibrating perpendicular to 00, and in phase behind it by a quarter-
period*. We have

(7=-/sin(J-w)iS, cf'=/cos('j-n)r, ^ = 7 cos (^ -n) ^...(8).

From these and (7) we have
T_ _ 1-f cos4/ ^_ (IT ^\ E -2 sin 2/

S 2 + sin 4/

tan

'TT \ E

2 -f sin 4/

tan

(i-»)

.(9).

* When the reflecting body is glass, or other transparent isotropic solid or
liquid, Fresnel's prophecy (we cannot caU it physical or dynamical theory) declares

sin (i - ,i) ^ _ tan (i - ,i)

s= -

E=0.

8in(i + ,i)' * tan (i+,i)

The notation in the text is partially borrowed from Bayleigh {Scientific Papers,
Vol. III. pp. 496 — 512) who used S, and T, to denote respectively the *' sine-formula,"
and the 'tangent- formula," of Fresnel. What I have denoted in the text by E is,
for all transparent solids and liquids, certainly very small ; and though generally

POLAHIZATIONAL ANALYSIS OF REFLECTED LIGHT. 337

Thus our experimental result (/, n) gives the two constituents Molar.
(T, E) of the reflected vibrations in the plane of reflection, when
the single constituent (S) of the reflected vibrations perpendicular
to the plane of reflection, is known.

§ 95. Going back to § 90, note that there are four independent
variables to be dealt with : — the angle of incidence, the orientations
of the two Nicol's prisms, and the orientation of the Fresnel's
rhomb. Any two of these four variables may be definitely chosen
for variation while the other two are kept constant ; to procure,
when homogeneous light is used, extinction of the light which
enters the eye from the centre of the field ; that is to say, to
produce perfect blackness at the central point of the field. Theo-
retically there is, in general, just one point of the field (one point
of absolute blackness) where the extinction is perfect ; and always
before the desired adjustment is perfectly reached a black spot is
conceivably to be seen, but not on the centre of the field. By
changing any one of the four independent variables the black spot
is caused to move; and generally two of them must be varied
to jcause it to move towards the centre of the field for the desired
adjustment. What in reality is generally perceived is, I believe,
not a black spot, but a black band ; and this is caused to travel
till it passes through the middle of the field when the nearest
attainable approach to the desired adjustment is attained. When
faint or moderate light, such as the light of a white sky, is used,
the whole field may seem absolutely dark, and may continue so
while any one of the four variable angles is altered by half a
degree or more. For more minutely accurate measurements, more
intense light must be used; a brilliant flame; or electric arc-
light; or lime-light; or, best of all, an unclouded sun as in
Rayleigh's very seai'ching investigation of light reflected from
water at nearly the polarizing angle, of which the result is given
in § 105 below.

§ 95'. An easy way to see that just two independent vaiiables are
needed to obtain the desired extinction, is to confine our attention
to the centre of the field, and imagine the light reaching the eye

believed to be perceptible for snbstanoes of high refrangibility such as diamond,
Bayleigh has qaestioned its existeoce for any of them, and saggested that its non-
nullity may be dae to extraneous matter on the reflecting surface. I believe, however,
that Airy, and Brewster, and Stokes who called it the adamantine property, and
Fresnel himself though it was outside his theory, were right in believing it real.
Bee § 158, below.

T. L. 22

338 LECTURE XVUI. -

Molar, from it when the extinctioD is not perfect, to be polarizationally
resolved into two components having their vibrational lines
perpendicular to one another. The desired extinction requires the
annulment of each of these two components, and nothing else.
Proper change of the two chosen independent variables deter-
•minately secures these two annulments when what is commonly
called homogeneous light, that is, light of which all the vibrations
are of the same period, is used.

§ 95". In the detailed plan of ^ 90 — 94 the two independent
variables chosen are the orientation of the first Nicol, and the
orientation of the Fresnel's rhomb. This is thoroughly convenient if
Ni is mounted on a proper mechanism to give it freedom to move
in a plane perpendicular to its axis, and to keep its orientation
round this axis constant. The Fresnel should be mounted so as
to be free to move round an axis fixed in the direction of the light
entering it: and it should carry a short tube round the light
emeiging from it into which N^ fits easily. Thus, when the
Fresnel is turned round the line of the light i-eflected fix)m the
mirror, it carries N^ round in a circle (as it were with a hollow
crank-pin), so that it is always in the proper position for carrying
the emergent light to the eye of the observer.

Two other choices of independent variables, each, I believe,
as well-conditioned and as convenient as that of §§ 90 — 94, but
simpler in not wanting the special mechanism for carrying N^,
are described in § 98 below.

§ 96. Nothing in ^ 90 — 95 involves any hypothesis : and we
have, in them, an observational method for fully, without any
photometry, determining T/S and E/8; which are, for incident
vibrations at 45° to the plane of incidence, the intensities of the
two constituents of the reflected light vibrating in the plane of
incidence, in terms of the intensity of the component of the re-
flected light vibrating perpendicularly to that plane. Fully carried
through, it would give interesting and important information, for
transparent liquids and solids, and for metals and other opaque
solids, through the whole range of incidence from O"" to 90^ It can
give extremely accurate values of T/S for transparent liquids and
solids; and it will be interesting to find how nearly they agree

cos il "^ l^

with the formula - '/ which Fresnel's "tangent-law" and

cos (I ^— .* )

" sine-law '* imply.

POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 339

§ 96'. Fresuel himself used it* for reflection from water Molar.

and from glass with incidences from 24'" to 89° (not using the

T

Fresners rhomb), and found values for tan~* -^ differing from

cos 0/ '\' xS
tan~^ — 7-^-^^ by sometimes more than 1°; but it has been
cos (^ — ,%)''

supposed that these differences may be explained by the im-
perfection of his apparatus, and by the use of white lightt.
Similar investigation was continued by Brewster J, on several
species of glasses and on diamond. With, for example, a glass of

T

refractive index 1*4826, his observed results for tan~* ^ differed

cos C% '^ %\
from tan""^ — )-, — ~ by + 1° 4': which he considered micfht be
cos(i-,i) -^ - ' o

within the limits of his observational errors. For diamond, he
found greater deviations which seemed systematic, and not errors
of observation. With a Fresnel's rhomb used according to the
method of ^ 90 — 94 he might probably have found the definite
correction on Fresnel's formula, required to represent the polari-
zational analysis of reflection from diamond. Can it be that
both Fresnel and Brewster underestimated the accuracy of theii*
own experimcuts, and that even for water and glasses, devia-

cos (.1 '\' %\

tions which they found from Fresnels — 7^ — \ naay have been

•^ cos (i — ft)

real, and not errors of observation ? The subject urgently demands

full investigation according to the method of §§ 90 — 98, with all

the accuracy attainable by instruments of precision now available.

§ 97. The reader may And it interesting to follow the formulas
of § 94 through the whole range of incidences from 0° to 90°.
For the present, consider only the case of the angle of incidence
which makes 7=0. This, being the incidence which, when the
incident light is polarized in any plane oblicjue to the plane of
incidence, gives 90° difference of phase for the components of the
reflected light vibrating in that plane and perpendicular to it, has
been called by Cauchy, and I believe by all following writers, the
principal incidence. We shall see presently (§§ 97", 99, 105) that,
for every transparent substance, observation and dynamics show one
incidence, or an odd number of incidences, fulfilling this condition.

* Fresnel, (Euvres^ VoL i. p. 646.

t Masoart, TraiU d^Optique, VoL 11. p. 466.

t Brewster, Phil Trans, 1S30. See also Masoart, Vol. 11. p. 466.

22—2

340 LECTURE XVIII.

Molar. By observatioD and dynamics we learn (§81 above), that it is
fulfilled 110^ by the vanishing of S, but by the vanishing of T.
To make T = 0 we have by (9) for the case of principal incidence,
1 + cos 4/"= 0, and therefore /= ± 45^ This, by (9), makes (if we
takey*= + 45°) for Piincipal Incidence,

iF/S = tan(n-7r/4) = i (10>

§ 97'. The k here introduced is Jamin s notation, adopted
also by Rayleigh. It is the ratio of the vibrational amplitude of
reflected vibrations in the plane of incidence to the vibrational
amplitude perpendicular to it, when the incident light is polarized
in a plane inclined at 45° to the plane of incidence, and when the
angle of incidence is such as to make the phases of those two com-
ponents of the reflected light differ by 90°. k is positive or
negative according as the phase of the reflected vibration in the
plane of incidence lags or leads by 90° relatively to the component
perpendicular to it. It is positive when (as in every well assured
case* whether of transparent or of metallic mirrors) the obser-
vation makes n > 45° ; it would be negative if n < 45°.

§ 97". The angle n — 7r/4 found by our observation of § 97 is
called the ** Principal Azimuth." See § 158*^" below. It has been
the usage of good writers regarding the polarization of light, par-
ticularly in relation to reflection and refraction, to give the name
" azimuth f" to the angle between two planes through the direction
of a ray of light ; for instance, the angle between the plane of in-
cidence and the plane of vibration of rectilineally polarized light.
A "Principal Azimuth," for reflection at any polished surface, I
define as the angle between the vibrational plane of polarized light
incident at Principal Incidence, and the plane of the incidence,
to make the reflected light circularly polarized. There is one, and
only one, Principal Incidence for every known mirror: except
internal reflection in diamond and other substances whose refrac-
tive indices exceed 2*414; these have three Principal Incidences
(§ 158'" below). The number is essentially odd: on this is founded
the theory of the polarization of light by reflection.

• See Jamin, Cours de Physique^ Vol. n. pp. 694, 696 ; also below, §§ 105, 164,
168^, ISr', 179. 192.

t This, when understood, is very convenient ; though it is not strictly correct.
Azimuth in astronomy is essentiaUy an angle in a horizontal plane, or an angle
between two vertical planes. A reader at all conversant with astronomy would
naturally think this is meant when a writer on optics uses the expression
"Principal Azimuth," in writing of reflection from a horizontal mirror.

PRINCIPAL INCIDENCE AND PRINCIPAL AZIMUTH. 341

§ 98. The vibrational plane of the incident light is inclined Molar,
to the plane of incidence at an angle of n — 45** ; which, for the
Principal Incidence, is such as to render the two components
of the reflected light equal; and therefore to make the light
circularly polarized. However a Fresnel's rhomb is turned, circu-
larly polarized light entering it, leaves it plane polarized. In
the observation, with the details of ^ 90 — 94, it is turned so
that the vibrational plane of the light emerging from it is per-
pendicular to the fixed vibrational plane of N^. Hence it occurs
to us to think that a useful modification of those details might
be; — to fix the Fresnel's rhomb with its principal planes at 45**
to the plane of reflection, and to mount j^s so as to be free
to turn round the line of light leaving the Fresnel's rhomb.
Alternate turnings of N^ and Ni give the desired extinction for
any angle of incidence. Take /= 0 in (1), (2), (3), (4); which
makes 00^^=45** in fig. 3 a. Let ii^ be the angle (clockwise in
the altered fig. 3 a) from OF to 0N^\ and put w = Jtt + a in fig. 3 6.
Eliminating ^ and g from (3), (4), by (8) and proceeding as in
§ 93, we find

r//S = tan a cos 2w, ; JS?/fif = - tan a sin 2wa ; ^/r=-tan2i?a...(100.

If a is positive observation makes n^ negative when taken acute.
For principal incidence it is —45°, and jEy/S> = tana. The phasal
lag of {E, T) behind {S) is — 2n2. Negative is anti-clockwise in
fig. 3 a. SeeglSS'^^

§ 99. The following table shows, for six different metals,
determinations of principal incidences and principal azimuths
which have been made by Jamin and Conroy, experimenting
on light from diflerent parts of the solar spectrum. This table
expresses, I believe, practically almost all that is knov^n from
observations hitherto made as to the polarizational analysis of
homogeneous light reflected from metals. The differences between
the two observers for silver are probably real, and dependent on
differences of condition of the mirror-surfaces at the times of
the experiments, as modified by polishing and by lapses of time.
It will be seen that for each colour Jamin's h^ is intermediate
between Conroy's two values for the same plate, after polishing
with rouge and with putty powder respectively. On the other
hand, each of Conroy's Principal Incidences for silver is greater

.^42

LECn^RE XVIII.

pS "^ *- v^

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CO

POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 343

than Jamin's ; and by greater differences for the yellow and blue Molar,
light than for the red.

Experimental determinations of TjS and EjS, (9) or (lOO,
through the whole range of incidence below and above the
Principal Incidence are still wanting.

§ 100. As for transparent solids and liquids, we may consider
it certain that Fresnels laws, giving

_^^am(i-^^ _!Z'=*^t4 E = 0 (11).

sm(i + ,i) tan(i4-,i)

are very approximately true through the whole range of incidence
from O"" to 90**; but, as said in § 95, it is still much to be desired
that experimental determinations of TjS should be made through
the whole range ; in order either to prove that it differs much less

cos (,% "f" %\

from — 7-; — ^ than found experimentally by Fresnel himself and
cos(i — ,i) ^ '^ ^

Brewster; or, if it differs discoverably from this formula, to deter-
mine the differences. It is certain, however, that at the Principal
Incidence the agreement with Fresnel's formula (implying iF= 0 in
the notation of § 94') is exceedingly close ; but the very small
deviations from it found experimentally by Jamin and Rayleigh
and represented by the values of h shown in the table of § 105
below, are probably real. An exceedingly minute scrutiny as to
the agreement of the Principal Incidence with tan~y, Brewster's
estimate of it ; — a scrutiny such as Rayleigh made relatively to
the approach to nullity of k for purified water surfaces ; is still
wanted; and, so far as I know, has not hitherto been attempted
for water or any transparent body. See ^ 180, 182 below.

Hitherto, except in §§ 81, 84, 86, we have dealt exclusively
with what may be called the natural history of the subject, and
have taken no notice of the dynamical theory ; to the considera-
tion of which we now proceed.

§ 101. Green's doctrine* of incompressible elastic solid with
equal rigidity, but unequal density, on the two sides of an interface,
to account for the reflexion and refraction of light, brings out for
vibrations perpendicular to the plane of incidence (§ 123 below)
exactly the sine-law which Fresnel gave for light polarized in the
plane of incidence. On the other hand, for vibrations in the

* Camh. Phil, Soc, Dec. 1887 ; Oreen^s Collected Papers^ pp. 246, 268, 267, 268.

342

LECTURE XVIII.

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POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 343

than Jamin's ; and by greater differences for the yellow and blue Molar,
light than for the red.

Experimental determinations of T/S and E/8, (9) or (lO^),
through the whole range of incidence below and above the
Principal Incidence are still wanting.

§ 100. As for transparent solids and liquids, we may consider
it certain that Fresnels laws, giving

_S = !!liiW). .T^^-4, E = 0 (11),

sin(i + ,i)' tan(i + ,*) ^ '

are very approximately true through the whole range of incidence
from O"" to 90°; but, as said in § 95, it is still much to be desired
that experimental determinations of T/S should be made through
the whole range ; in order either to prove that it differs much less

cos a "f" t^

from — 7-; — '-^ than found experimentally by Fresnel himself and

C08(l — ,i) '^ '^ '^

Brewster; or, if it differs discoverably from this formula, to deter-
mine the differences. It is certain, however, that at the Principal
Incidence the agreement with Fresnel's formula (implying E=Oin
the notation of § 94') is exceedingly close ; but the very small
deviations from it found experimentally by Jamin and Rayleigh
and represented by the values of k shown in the table of § 105
below, are probably real. An exceedingly minute scrutiny as to
the agreement of the Principal Incidence with tan""^, Brewster's
estimate of it ; — a scrutiny such as Rayleigh made relatively to
the approach to nullity of k for purified water surfaces ; is still
wanted; and, so far as I know, has not hitherto been attempted
for water or any transparent body. See §§ 180, 182 below.

Hitherto, except in §§ 81, 84, 86, we have dealt exclusively
with what may be called the natural history of the subject, and
have taken no notice of the dynamical theory ; to the considera-
tion of which we now proceed.

§ 101. Green's doctrine* of incompressible elastic solid with
equal rigidity, but unequal density, on the two sides of an interface,
to account for the reflexion and refraction of light, brings out for
vibrations perpendicular to the plane of incidence (§ 123 below)
exactly the sine-law which Fresnel gave for light polarized in the
plane of incidence. On the other hand, for vibrations in the

• Camh. Phil. Soc, Dec. 1887 ; Green's Collected Papers^ pp. 246, 268, 267, 268.

344 LECTURE XVIll.

Molar, plane of incidence it gives a formula (§§ 104, 105, 146 below)
which, only when the refractive index differs infinitely little from
unity, agrees with the tangent-law given by Fresnel for light
polarized perpendicular to the plane of incidence ; — but differing
enormously from Fresnel, and from the results of observation, in
all cases in which the refractive index differs suflBciently from
unity to have become subject of observation or measurement.

§ 102. The accompanying diagrams, figs. 4, 5, 6, illustrate,
each by a single curve (CuiTe 1), the perfect agreement between
Green and Fresnel for the law of reflection at different incidences
when the vibrations are perpendicular to the plane of incidence ;
and by two other curves the large disagreement when the vibra-
tions are in the plane of incidence.

Curve 1 in each diagram shows for vibrations perpendicular to
the plane of incidence the ratio of the reflected to the incident
light according to FresneFs sine-law

rsin(i-,i)T
[8in(i-h,i)J

dynamically demonstrated by Green on the hypothesis of equal
rigidities and unequal densities of the two mediums.

Curve 2 shows, for vibrations in the plane of incidence, the
ratio of the reflected to the incident light according to FresneFs
tangent-law,

[tan (i -f- ,i) J '

Curve 3 shows, for vibrations in the plane of incidence, the
ratio of reflected to incident activity (rate of doing work) per
unit area of wave-plane, rigorously demonstrated by Green (§ 146
below) for plane waves incident on a plane interface between
elastic solids of different densities but the same rigidity; on the
supposition that each solid is absolutely incompressible, and that
the two are in slipless contact at the interface.

In each diagram abscissas from 0° to 90° represent " angles of
incidence,'* that is to say, angles between wave-normals and the
line-normal to the interface, or angles between the wave-planes
and the interfiaice.

§ 103. Curves 2 and 3 of fig. 4 show for water a seemingly fair
agreement between Fresnel and Green for vibrations in the plane

FRESNEL, GREEN, AND TRUTH.

345

of reflection. But the scale of the diagram is too Htnall to show MoIbt.
iraportant differencea for iocideiiceH lesa thau 60' or 65°, especially
in the neighbourhood of the polarizing angle, .53°1 ; this want is
remedied by the larger scale diagram fig. 7 showing Curves 2 and
3 of fig. 4; on a scale of ordinates 48-5 times as large, in which,
for vibrations in the plane of incidence, the unit for intensity
of light is the reflected light at zero angle of incidence, instead

1

1

1

Ij

jl

1

1

1

I

■'

/

■3

/

,1*

y

t.-*

/

^/

^

_

^

,>•■

^A

i"

_

■ —

—

_i

■j

^

I

^

a

?^

3

0"

4

tf

6

^

6

0°

7

w~

s

?"

9

tf

Fig. i

Wftter. (,u = l'334)

of the incident light at incidence i as in the other diagrams.
Curve 2 in figs, 4 and 7 shows, for water, the absolute extinction
at angle of incidence ta.n'-'ft, given by Fresnel's formula. Curve 3
(Qreen's formula) shows, for a slightly smaller angle of incidence
(oO^O instead of 53°'l), a minimum intensity equal to '295 of
that of directly reflected light ; that is to say Green's formula

346 I.ECTURB XVIII.

Uolar. makes the directly reflected light from water only 3J times
as strong as the light reflected at the angle which gives least
of it.

§ 104. To test whether Fresnel or Green is more nearly right,
take a black japanned tray with water poured into it enougH to
cover its bottom, and look through a Nicol's prism at the image

"

q

1

"

1

a

I

it

ir

u

n

ti

-. t

7- t

^A

/

It

s/

tt

f

1

,

9

t'l

_,

^

t

—

"

L--

>^

A

0-

K

2

<r

-5

iv

*

s=-^

fi

r

H

n°

7

0* 80' 9C

Pig. 6. Flint Glaw. (/■= 1-714)

of a candle in the water-surface. Yon will readily find in half-a-
minute's trial a proper inclination of the light and orieatatiou of
the Nicol to give what seems to you extinction of the light. To
test the approach to completeness of the extinction let an assistant
raise and lower alternately a piece of black cloth between the candle
and the water sur&ce, taking care that it is lowered sufficiently
to eclipse the image of the candle when it is not extinguished by

PRBSHRL, ORKEN, AND TRUTH.

347

the Nicol. By holding the Nicol very steadily in your hand, and HoIbt.
turninfi; to give the best extinction you can produce by it, you
will find no difference in what you see whether the screen is down
or up, which proves a very close approach to perfect extinction.
Judging by the brilliance of the im^i^ of the candle when viewed'
through the Nicol at ne&riy normal incidence, and distances at

i-U

t

Ji

7t

LI

4^-

tl--

-j t

t r __

-/^ 4

* L

■A «^' ^

J^ / -:

y^ y i

^■^ ^^ t

— -=^=— — ■ "" L

"""^^ 7

30° stf^ So' 6^ eo" Ttf^ Bo^ eff

Fig. 6. Diamond. U=2«4)

which this image can be seen, I think we may safely guess that the
light reflected from the water at normal incidence was at least 500
(instead of Green's 3J) times as strong as the imperceptible light
of the nearest approach to extinction which the Nicol gave at the
polarizing angle of incidence. And from Rayleigh's accurate experi-
ments{§ lOo below) we know that it is 2o,000,000 times, when the
water surface is uncontaminated by oil or scum or impurity of any

348

LECTURE XVIir.

Holar. kin(). It would be nnty 3^ timesifGreen'sdynamics were applicable
(without change of hypotheses) to the physical problem. Hence,
looking at fig. 7, we see that, for water, Green's theory (Curve 3)
differs enormously from the truth, while Fresnel's formula
(Curve 2) shows perfect agreement with the truth, at the angle of
incidence 53°'l. Lookiug at fig. 4 we see still greater differences

•vx

» ^^ ;

c

^s;

v^

-^^

v^

3^ A

^ 4

tv i '^

A^ 1 ^

4 l_^ L-.'

X^ ^ i

4 ^Z r

'

C

A

^ 1

^ t

N_/

10* ao* 3ff 40" SO" 6ff 70* 80- 90*

Fig. 7. Water,

between Curves 2 and 3 even at as high angles of incidence as
75°, though there is essentially a perfect concurreoce at 90°.

§ 105. Looking at Curve 3 in figs. 5 and 6 we see that for flint
glass Green's theory gives scaively any diminution of reflected
light, while for diamond it actually gives increase, when the angle
of incidence is increased from zero to tan-'/i, Brewster's polariziDg

FRESNEL, GREEN, AND TRUTH. 349

angle. Yet even a hasty observation with no other apparatus Molar,
than a single Nicol's prism shows, both for flint glass and diamond,
diminution from the brightness of directly reflected light to what
seems almost absolute blackness at incidence tan~y, when the
reflection is viewed through the Nicol with its plane of polariza-
tion perpendicular to the plane of reflection (that is to say, its
transmitted light having vibration in the plane of reflection).

One readily and easily observed phenomenon relating to
polarization by reflection is: — with a Nicol turned so as to transmit
the vibrations in the plane of reflection, diminution of reflected
light to nearly zero with angle of incidence increasing to tan~y ;
and after that increase of the reflected light to totality when the
angle of incidence is farther increased to 90°.

Another is: — for light reflected at a constant angle of incidence,
diminution from maximum to minimum when the Nicol is turned
90° round its axis so as to bring the direction of vibration of the
light transmitted by it from being perpendicular to the plane of
reflection to being in this plane. This diminution from maximum
to minimum is the difference between the ordinates of two curves
representing, respectively for vibrations perpendicular to the plane
of incidence and vibrations in the plane of incidence, the intensity
of the reflected light at all angles of incidence. These two curves,
if drawn with absolute accuracy, would in all probability agree
almost perfectly with Fresnels sine-law and Fresnels tangent-law
(Curves 1 and 2 of the diagrams) for all transparent substances.
We have, however, little or no accurately measured observatioual
comparisons except for light incident at very nearly the polariza-
tional angle. By a very different mode of experimenting from
that indicated in § 95 above, Jamin* found for eight substances
having refractive indices from 2*454 to 1*334, results shown in
the following table; to which is added a measurement made by
Rayleigh for water with its surface carefully purified of oil or
scum or any other substance than water and air.

* Ann, de Chimie et de Phygique, 1850, Vol. xxiz., p. 303, and 1851, Vol. xxxi.,
p. 179 ; and p. 174 corrected by pp. 180, 181. (Ck>nfuBion between Cauohy's c and
Jamin's k is most bewildering in Jamin*B papers. After mnoh time sadly spent in
trying to find what was intended in matually contradictory Tables of results,
I have given what seems to me probably a correct statement in the Table belonging
to §105.)

350

LECTUBE XVIU.

1

• 1

Angle of

Substance | m i ^"""'^""^

1 tan""* fi

i
1

BatioB of vibrational ampli-
tudes, and of strengths, of
i reflected lights due to equal
1 incident lights vibrating in
and perpendicular to the
plane of incidence

"Sulfure d'arsenic trans-
parent " (Realgar) . . .

"Blende ti-ansimrente "
(zinc sulphide)

Diamond

1

1 !

1

' 2-454 67'*-8

; 2-371 G7*'-l
2-434 67'*-7

k
+ 0850

+ -0420
-»- -0190
+ -0180
-»- -0060

- -0084
+ -00208

- -00577

1/138

1/567

1/2770

1/3086

1/27778

1/14172

1/231160

1/30030

1/25,000,000

Flint Glass

1-714 i 59'-7
1-487 56'-l
1-441 ; 55''-2
1-366 53°-8
1-334 1 53'*-l

1-334 53*'-l

1

"Verre"

Fluorine

Absolute Alcohol

Water ^Jamin)

Water, with specially puri-
fied surface (Rayleigh)

-»- -0002

Molar. The greatest numeric in the last column, 1/138, would be barely
perceptible on the diagram, fig. 7 ; and none of those for the other
seven substances would be perceptible at all without a large
magnification of the scale of ordinates.

§ 106. Although these results are related only to the ratio of
the ordinates of Curve 1 to those of Curve 2 for one angle of
incidence in each case, and do not touch the absolute values of the
reflection due to unit quantities of incident light, we may infer as
almost absolutely certain, or at all events (^ 100, 125) highly
probable, that Curve 1 (Fresnel's sine-law) and Curve 2 (Fresnels
tangent-law) ai*e each of them about as nearly correct at all other
incidences as at the critical incidences for which the observations
were made. We shall see in fact (§§ 125, 138 below) in the dy-
namical theory to which we proceed, that the sine-law is absolutely
accurate for vibrations perpendicular to the plane of incidence, on
the supposition that the rigidities of the two mediums are equal
and their densities unequal; and that the only correction of Green's
dynamical postulates which can procure approximate annulment,
for the reflected light, of vibrations in the plane of incidence at
the angle tan~*/i, gives correspondingly close agreement with
Fresnel's tangent-law throughout the whole i-auge of incidences
from 0° to 90'' !

DYNAMICAL THEORY OF REFLECTION AND REFRACTION. 351

(The following, §§ 107... Ill, is quoted from a paper written Molar,
by myself in September — October, 1888, and published in the
JPhilosophical Magazine for 1888, second half year.)

§ 107. " Since the first publication of Cauchy's work on the
" subject in 1830, and of Green's in 1837, many attempts have
''been made by many workers to find a dynamical foundation
" for Fresnels laws of reflection and refraction of light, but all
"hitherto ineffectually. On resuming my own efforts since the
"meeting of the British Association at Bath, I first ascertained
"that an inviscid fluid permeating among pores of an incom-
" pressible, but otherwise sponge-like, solid does not diminish, but
"on the contrary augments, the deviation from Fresnel's law of
"reflection for vibrations in the plane of incidence. Having
" thus, after a great variety of previous efforts which had been
"commenced in connexion with preparations for my Baltimore
"Lectures of this time four years ago, seemingly exhausted
"possibilities in respect to incompressible elastic solid, without
"losing faith either in light or in dynamics, and knowing that
" the condensational-rarefactional wave disqualifies* any solid of
" positive compressibility, I saw that nothing was left but a solid
" of such negative compressibility as should make the velocity of
" the condensational-rarefactional wave, zero or small. So I tried
" it and immediately found that, with other suppositions unaltered
" from Green 8, it exactly fulfils FresneFs * tangent-law ' for vibra-
" tions in the plane of incidence, and his ' sine-law * for vibrations
''perpendicular to the plane of incidence. I then noticed that
" homogeneous air-less foam, held from collapse by adhesion to
"a containing vessel, which may be infinitely distant all round,
" exactly fulfils the condition of zero velocity for the condensational-
" rarefactional wave ; while it has a definite rigidity and elasticity
"of form, and a definite velocity of distortional wave, which
''can be easily calculated with a fair approximation to absolute
" accuracy.

§ 108. " Green, in his original paper * On the Reflexion and
" Effraction of Light * had pointed out that the condensational-
" rarefactional wave might be got quit of in two ways, (1) by its
" velocity being infinitely small, (2) by its velocity being infinitely

* Green's Collected Papen, p. 246.

(i

«

(C

352 LECTURE XVIII.

Molar. '' great. But he curtiy dismissed the former aud adopted the
"latter, in the following statement: 'And it is not difficult to
'** prove that the equilibrium of our medium would be unstable
'unless AjB>\j"i, We are therefore compelled to adopt the
'latter value of AJB*,' (oo) *and thus to admit that in the
* luminiferous ether, the velocity of transmission of waves pro-
"'pagated by normal vibrations is very great compared with that
"'of ordinary light/ Thus originated the 'jelly-' theory of ether
"which has held the field for fifty years against all dynamical
"assailants, and yet has hitherto failed to make good its own
" foundation.

§ 109. " But let us scrutinize Green s remark about instability.
•* Every possible infinitesimal motion of the medium is, in the
" elementary dynamics of the subject, proved to be resolvable into
"coexistent equi-voluminal wave-motions, and condensational-
" rarefactional wave-motions. Surely, then, if there is a real
" finite propagational velocity for each of the two kinds of wave-
" motion, the equilibrium must be stable ! And so I find Green's
"own formulaf proves it to be provided we either suppose the
" medium to extend all through boundless space, or give it a fixed
'^containing vessel as its boundary. A finite portion of Green's
" homogeneous medium left to itself in space will have the same
"kind of stability or instability according as AlB>4/S, or
" A/B < 4/3. In fact A — f JB, in Green's notation, is what I have
"called the * bulk-modulus'* of elasticity, and denoted by k
" (being infinitesimal change of pressure divided by infinitesimal
"change from unit volume produced by it: or the reciprocal of
"what is commonly called 'the compressibility'). B is what
"I have called the 'rigidity,' as an abbreviation for 'rigidity-
" modulus,' and which we must regard as essentially positive.
" Thus Green's limit A/B > 4/3 simply means positive compres-
" sibility, or iK)8itive bulk-modulus : and the kind of instability
"that deterred him from admitting any supposition of -4/5 < 4/3,
" is the spontaneous shrinkage of a finite portion if left to itself

* A and B are the squares of velocities of the condensational and distortional
waves respectively ; supposing for a moment the density of the medium unity.

t CoUeeted Papers, p. 253 ; formula (G).

X Enofelopatdia Britannica, Article *' Elasticity" : reproduced in Vol. ui. of my
Collected Papers.

VOLUMINAL INSTABILITY AND STABILITY. 353

"in a volume infinitesimally less, or spontaneous expansion if Molar,
"left to itself in a volume infinitesimally greater, than the
"volume for equilibrium. This instability is, in virtue of the
"rigidity of the medium, converted into stability by attaching
" the bounding surface of the medium to a rigid containing vessel.
" How much smaller than 4/3 may AjB be, we now proceed to
"investigate, and we shall find, as we have anticipated, that for
" stability it is only necessary that A be positive.

§ 110. "Taking Greens formula (C); but to make clearer the
" energy-principle which it expresses (he had not even the words
"'energy,' or *workM), let W denote the quantity of work re-
" quired per unit volume of the substance, to bring it fi:x)m its
"unstressed equilibrium to a condition of equilibrium in which
" the matter which was at (a?, y^ z) is at (a? + f , y + *?, -8^ + ?) >
" f , i;, f being functions of x, y, z such that each of the nine
"differential coefficients d^/dx, d(/dy, ... drj/dx .,, etc, is an in-
" finitely small numeric; we have

2 ( \dx dy dz)

\\dy dz) \dz dx) \dx dy) J

-,B{^Jif^ff^fdnS\ (1).

\dy dz dz dx dx dyj)

"This, except difference of notation, is the same as the formula
"for energy given in Thomson and Tait's Natural Philosophy y
" § 693 (7).

§111. "To find the total work required to alter the given
"portion of solid firom unstrained equilibrium to the strained

"condition (f, 17, f) we must take llldxdydzW throughout the

"rigid containing vessel. Taking first the last line of (1);
"integrating the three terms each twice successively by parts
"in the well-known manner, subject to the condition f =0, rj = 0,
" f = 0 at the boundary ; we transform the factor within the last
"vinculum to

JJJ ^ \dz dy dx dz dy dx) "
T. L. 23

354

LECTURE XYIIL

MoUr. " Adding this with its factor — 4£ to the other terms of (1)
"under jljdxdydz, we find finally

^'[(i-S)'-(S-S)'^(S-Dl} -'^^

"This shows that positive work is needed to bring the solid to
"the condition (^, 7f, ^) from its unstrained eqailibriam, and
" therefore its unstrained equilibrium is stable, if A and B are
" both positive, however small be either of them."

§ 112. The equations of motion of the general elastic solid
taken direct from the equations of equilibrium, with p to denote
density, are, as we found in Lecture 11. pp. 25, 26

^^dP dU dT\
^ dt^ dx dy dz

dPv^dU dQ dS
d^ dx dy dz

d^^dT dS dR
^dt^ dx'^ dy^ dz )

where f , rj, f denote (as above from Green) displacements ; P, Q, U,
normal components of pull (per unit area) on interfiu^es respec-
tively perpendicular to x, y, z\ and S, T, U respectively the
tangential components of pull as follows : —

^ J = pull parallel to y on face perpendicular to z '
1= >* ,» z „ „ „ „ y

(3);

-{-.

'*' }

f w

tf i

if

z ,

i ft

ft

X

X ,

i it

19

z

^ >

i ft

»»

y

y

i ft

»

X

•■ •••v^j«

§ 113. For an isotropic solid we had in Lecture XTV. (p. 191,
above),

aENERA.L SOLUTION FOB MOTION OF ELASTIC SOLID. 366

MoUr.

iJ = (A-fn)S + 2ng

^y'[....(,6):

where S = f + p + f^ (7).

ax ay dz

Using these values of S, T, U, P, Q, 12 in (3) we find

(8).

§ 114. Taking djdx of the first of equations (8), d/dy of the
second, and d/dz of the third, and adding we find

^S=^^'« W'

where A — k + ^n (10),

this being Green's "il" as used in §§ 108, 111 above.

Put now

which implies ^ + ^' + ^ = 0 (12):

•^ ax ay dz

and we find, by (8),

Equations (9), (12), and (13) prove that any infinitesimal dis-
turbance wTiatever is composed of specimens of the coudensa-
tional-rarefactional wave (9), and specimens of the distortional
wave (13), coexisting; and they prove that the displacement in
the condensational-rarefactional wave is irrotational, because we
see by (11) that an absolutely general expression for its com-
ponents, f - fi, 17 - %, ?- ?i, if denoted by fj, i/j, f,, is

^^=d^' ^»=d^' ^'=dJ <^*>'

23—2

l"'

\nl\ LECTCRE XVm.

MfJiir wlM>M^ wliKfi X In known, ^ in detennined bjr

V«I^ == «• ,15 1

llf«ni'i\ fiM X MjitiHfif'H (9), we have

p ^ ^AV*^ a«^:

fffffl wn ntHt, liriiLlly. (hat Um* moHt general soladoa of ih^ eq:
Iff ifiDnititniiiiiil iiinf.ionM Ih given by

f "fi »^fa, V^Vi-^Vif K-Kz-^Kt •!*> =

|iM'vii|i.i| (-,, f;,. (;, niainfy (12) and (13); and f,, %. ^ sasi^fl4>
<ifi«l (10;

§ I ITi TliM KMMnriil HoIiitionH of (11) and (12) for plane eqai-
voliiMiifMil wtiviiA. imkI of (14) and (16) for plane
iMiorfif lioniil witvt'n fU'«< fiH followH (easily proved by
l.loriM)

t-_»;a_ti _ ///-ff . «»a? -f )9ay + 7,x\ I

V />

wlimii // in jt I'oiiMliinli, (M|ual to tho diHpIacement in the con-
ilrijiiiiUofMil hiin|jLri.iiiiHi.| wiLvn w)i(*n y= 1 ; Ay B, C, are constants
i'i|Miil Ui Mm- >f> . // « s niiii|M)rirn1«M of the displacement, due to
Ijjis f:i|ii)voltiiiiiniil wiivi* wlii'h / 0; (a,, fiu 7i)i («s, )9a, 7*) are
IliM iliiKi'l.i'fn I'onini'h of Um noriniilH to the wave-planesf of the

* l'n\nwin'» Willi liiiimji fiiiKliiinniiiiil tliiMmaii, in the elementary mathematiea
nl fiiiiHi vfiijriiiH itivifiMiiy nn tint Nqtmnt of tiiu diiitanoe, telU ns that when 5 ia
liiiifwii, tit Kivmi MiiiiLrarily tiiriMiffli nil Hpoce, V-*5 is determinate; being the

poiiiiiLiai tit an iilital liiiitritiiitioii of niattiir, of which the density ie eqnal to j- .

t My " wavii platin" of a piano wave I mean any plane passing through partioles
all in onii piiaMt of tnotion. Kor «ixaniple, in sinusoidal plane waves the "wave-
plan«i " may \m Ukfu as oim of the planes containing particles having no diaplaoe-
mont but maximum velocity, or it may be taken as one of the planes having
mazimnm displacument and no velocity. For an arbitrary impulsive wave (as
ezpretfed in the text with / an arbitrary function through a finite range, and zero
for all values of the argument on either side of that range) the *' wave-plane ** may

REFLECTION AND REFRACTION AT INTERFACE BETWEEN SOLIDS. 357

two waves ; and w, v are the propagational velocities of the equi- MoUur.
voluminal and eondensatioual-rarefactional waves respectively. In
the condensational-rarefactional wave in an isotropic medium, the
displacement or line of vibration is in the direction (a,, )9,, 7,),
normal to the wave-plane.

§ 116. For the problem of reflection and refraction at an
interface between two mediums, which (following Green) we shall
call the upper medium and the lower medium respectively, let the
interface be XOZ, and let this plane be horizontal. Let the
wave-planes be perpendicular to XOY, This makes 71 = 0, 7,5=0
in (18) and (19). For brevity we shall frequently denote by P,
the plane of incidence and reflection.

§ 117. Beginning now with vibrations perpendicular to P,

we have -4 = 0, B = 0; and (18) becomes, for an incident wave as

represented in fig. 8,

?=C/(f-cui? + 6y) (20).

where

a = sin i/u, and b = cos t/u, with u= I -

.(21).

Fig. 8.

OZ is perpendicular to the diagram towards the eye.

The wave-planes of incident (/), reflected (/'), and refracted
(,/), waves are shown in fig. 8 for the particular case of inci-
dence at 30°, and refractive-index for flint-glass = 1*724; which
makes ,i = 16°*9.

be taken as any plane through particles all in the same phase of motion. In this
case we have a wave-front and a wave-rear; in the sinusoidal wave we have no
front and no rear. I have therefore introduced the word *' wave-plane " in prefer-
ence to the generaUy used word *' wave-front."

358 LECTURE XVnL

MoUr. For the reflected and refracted waves we may take respectively
ISC' — t, aud ,% instead of t, and C\ ,C instead of C. We have

sin .%

^'» = ^".with,« = y^ (22).

and C and ,C by equations (28) below. Thus for the displace-
ments in the two mediums, due to the three waves, we have

f + f = Cf(t - cue + 6y) + Cy (t - cur -by) in upper medium. . .(23),

and

,f = ,Cf (i — cm: + ,by) in lower medium. . .(24),

where ^ = cos,t/,tt, with,u= /^ (26);

V ,p

f> and ;u being both positive. Remember that y is negative in

the lower medium. Remark that, by (21), (22), (25), we have

6« = M-«-a', ,6« = ,t4-«-a« (25)'.

The sole geometrical condition to be fulfilled at the interfisu^e is

?+?' = ,? when y=0, which gives C + C" = ,C ...(26).
The sole dynamical condition is found by looking to § 113 (5). It is

S = ,S when y = 0; giving w6(C-C") = ,»,6,C ...(27).
From these we find

C" = ^LzA?c, ,(7=^2^,-C (28).

,b,n + bn ' ' fi,n + bn ^ ^

§ 118. The interpretation of these formulas is obvious when

the quantities denoted by the several symbols are all real. But in

an important and highly interesting case of a real incident wave,

expressed by the first term of (23) with all the symbols real,

imaginaries enter into (24) and (25) by ,b being imaginary ;

which it is when

a-»<,w (29);

or, in words, when the velocity of the trace of the wave-planes on

the interface is less than the velocity of the wave in the lower

medium. In this case a' — ,ti~' is positive: denoting its value by g*

we may put ,6 = — t^^*, where q is real; positive to suit notations

in § 119. Thus (28) becomes

^, _ bn -h iq,n ^ _ hhi? — q^,n^ + 2iqbn,n ^

bn — iq/n "" IW + (f,n?

Q^ 26n f^_2bn{bn + iq,n)
' bn-iq.n 6*n* + 3»,n«

* See footnote on § 15S'' below.

(30).

TOTAL REFLECTION OF EQUI-VOLUMINAL WAVEa 359

We have also in (24) an imaginary, ,6 = — 2^, in the argument of/ Molar,
for the lower medium.

§ 119. To get real results we must choose / conveniently to
make /{t — ax— iqy) ^F^ lO, where F and 6 are real. We may
do this readily in two ways, (31), or (32), by taking t, an arbitrary
length of time, and putting

^/* . J. \ 1 t-ax + by-iT

j\ -r ^/ ^_^^i^^^^ (t - ax -{' byf + 1^

This makes

/(^ - cur + ,6y) =

1 __ f — go?— ^(t — gy)

r-ax-~2qy^Yr " {t - axf -{- {r - qyf

...(31),

}

or

f(t-ax + by) = €"»<«-«*+6y) = cos ft)(^- cur + 6y) + A sin ft) (f - oo? + by)
f(t - ax-^-fby) =€i«<«-fl»-i«y) = e-^' [cos a)(t - cue) + * sin a> (^ - ax)]

(32).

The latter of these is the proper method to show the results
following the incidence of a train of sinusoidal waves ; the former,
which we shall take first, is convenient for the results following
the incidence of a single pulse.

§ 120. With (23), (24), and (30), it gives for the incident wave
in the upper medium

t-axVby^-j^
^-(t^ax + byy+i^^ ^^"^^^

and for the reflected wave in the upper medium

_ [(fc'n»-g^w»)(^ - ax-'by)+2qbn,nT] + 1 [2qbn,n (t-ax- 6y)-(6»n«-g';n»)T] ^
^^ ' (6«n« + }>») [(^ - cur - 6y)» + T«]

(34),

and for the disturbance in the lower medium

2 [[bW (t-ax) + qbn,n (t - gy)] + 1 [qbn,n (t - ax) -bW(T- qy)]] „
'^ (i'tt' + q\n*) [(t - cm:)» + (t - qyf]

(35).

The real parts of these three formulas represent a certain form
of arbitrarily given incident wave : and the consequent reflected
wave in the upper medium, and disturbance (a surface-wave.

360 LECTCHE xvm.

folar. analogous to a forced sea-wave) in the lower mediom. The
imaginary parts with i removed, represent another form of incident
wave and its consequences in the upper and lower mediomsi In
neither case does anv wave travel into the lower medium away
from the interface, and therefore the whole activity of the incident
wave is in each case carried on by the reflected wave in the ii]qper
medium : that is to say, we have total reflection. It is interesting
to see that in this total reflection, the reflected wave in each case
differs in character from the incident wave, except for direct
incidence ; and it differs by being compounded of two constitaents,
one of the same character as the incident wave for that case,
and the other of the same character as the incident wave for
the other case. This corresponds to the change of phase (§§ 152
and 158'^, below) by total internal reflection of waves of vibratimi
perpendicular to P.

§ 121. The accompanying diagram (fig. 9) shows the characters
of the two forms of incident wave, and of two constituents of the
forced sur£skce-wave in the lower meilium, referred to in § 120.
The two curves represent, fix)m f = — x to f = +x, the part of
the displacement of any particle in the upper medium, due to
one or other alone of the two forms of incident wave. The
abscissa in each curve is t. The ordinates of the two curves
are as follows : —

Curve 1, --^^, and Curve 2, ^^.

The unit of ordinates in each case is r^K

The disturbance in the lower medium is a forced wave, of
character represented by a combination of these two curves,
travelling under the interface at speed a'

i—i

§ 122. All the words of § 120 apply also to the total reflection
of sinusoidal waves, with this qualification, that the two char-
acters of incident wave are expressed respectively by a cosine
and a sine, and the •* difference" becomes simply a difference of
phase. Thus having taken the real parts of the formulas we
get nothing new by taking the imaginary parts. The real parts
of (23) and (24), with (32) for / and irith (30) for C and ,C,
give us

Incident wave, Cco6«(f-flur-|-fry) (36);

TOTAL REFLECTION OF EQCI-VOLOmSAL PUI^ES. 361

.-f

- t

. t

4

-- U

; *" T

r ^

■^ L

W- . ^

^ %

^^ - ^

^-'^ ^

^ ___ -^ « . 1 .1 w

■^--^^

I i^

5 :^ ,

V \

3 J „

\_ 1

4 t'-

1

L °

t- -

4-

4 =^

43

362 LECTURE XVIU.

Molar. fReflected wave, \

'Forced wave in lower medium, \

^ 2bn e-*^ [bti cos co (^ - oar) - q,n sin <o{t- ax)] \ ^^^^

Thus we see that the amplitude of the resultant reflected wave
is C ; that its phase is put forward tan""* [2qbn,n/(bW — 5*,n*)] ;
and that the phase of the forced wave-train in the lower medium
is before that of the incident wave by tan""* (q,n/bn).

§ 123. Leaving for § 158', the case of total reflection, and
returning to reflection and refraction, that is to say, b and Jb both
real, we see that equations (28) give for the case of equal rigidities

— C _ ,6 — 6 _ cot ,i — cot t _ sin {% — ,{) .

C "",6 + 6"" cot,i + cott~8in(i + ,t) ^ ^*

which is Fresnel's "sine-law"; and, belonging to it for the
refracted ray,

. ,0 _ 2 cot t _ 2 cos i sin ,%
iJ cot ,t + cot » sin (,t + ») ^ '

In the case of equal densities and unequal rigidities we have
w/,n = sin'i/sin* ,t and equations (28) give

cos i sin ,t

C _ cos ,i sin i __ sin 2i — sin 2,i _ tan (i — f) .

C cost sin,i sin2i+8in2,t tan(i + ,i) ^ ^'

cos ,1 sin I

which is Fresnel's "tangent-law"; and, belonging to it for the
refracted ray,

fC ^ 2 sin i cos i _ sin 2t ...

C sm ,t cos ,1 + sm t cos % sm 2,t + sm 2% '

The third member of (41) is in some respects more convenient
than Fresnel's beautiful " tangent-formula."

§ 124. These formulas, valid for pulses as well as for trains
of sinusoidal waves, show, without any hypothesis in respect to
compressibility or non-compressibility of ether, that, if the densities

DISPLACEMENTS PERPENDICULAR TO PLANE OF INCIDENCE. 363

of the mediums are equal on the two sides of the interface, MoUr.
Fresnel's "tangent-law" is fulfilled by the reflected waves, if the
vibrations are parallel to the interface ; and therefore the reflected
light vanishes when the angle of refraction is the complement of
the angle of incidence. Hence non-polarized light incident at
the angle which fulfils this condition would give reflected light
consisting of vibrations in the plane of the incident and reflected
rays. Now (§ 81) we have seen from Stokes' dynamical theory of
the scattering of light from particles small in comparison with
the wave-length, as in the blue sky, and also from his grating
experiments and their theory, and (§ 81') confirmation by Rayleigh
and by Lorenz of Denmark, that, in light polarized by reflection,
the vibrations are perpendicular to the plane of polarization
defined as the plane of the incident and reflected rays ; that is to
say the vibrations in the reflected ray are parallel to the interface.
Hence it is certain that the densities of the mediums are not
equal on the two sides of the interface; and that the densities
and rigidities must be such as not to give evanescent reflected ray
for any angle of incidence, when, as in § 123, the vibrations are
parallel to the interface.

§ 125. On the other hand, formulas (39), (40) show that the
supposition of equal rigidities not only does not give evanescent
ray for any angle of incidence, but actually fulfils Fresnels
"sine-law" of reflection for all angles of incidence when the vibra-
tions are parallel to the interface. Looking back to (28) and (39),
we see that Fresnel's "sine-law" is expressed algebraically by

,6,n-h6n",6-h6 ^ ^'

If this equation is true for any one value of 6/,6 other than 0
or X , we must have nl,n = 1. Hence if, with vibrations parallel
to the interface, Fresnel's "sine-law" is exactly true for any one
angle of incidence other than 90°, the rigidities of the mediums
on the two sides are exactly equal, and the "sine- law" is exactly
true for all angles of incidence ; a vei-y important theorem.

§ 126. Going back now to the more diflScult case of vibrations
in the plane of incidence, let us still take the interface as XOZ
and horizontal; and the wave-planes perpendicular to XOY.
Instead of the single displacement-component, ^, and the single

364

LECTCJRE XVm.

MoUtf. surface-pull-component, S, of §§ 117 — 125, we have now two dis-
placement-components, ^, 17 ; and two surface-pnll-components, Q,
perpendicular to the interface, and U parallel to the direction X
in the interface. The interfiEicial conditions are that each of these
four quantities has equal values on the two sides of the inter-
face. We have now essentially the further complication of two
sets of waves, equivoluminal and condensational-rare£su!tional,
which are essentially to be dealt with. We might suppose the
incident waves in the upper medium to be simply equivoluminal
or simply condensational-rare&ctional ; but incidence on the inter-
face between two mediums of different densities or different
rigidities would give rise to reflected waves of both classes in the
upper medium, and to refracted waves of both classes in the
lower medium. It will therefore be convenient to begin with
incident waves of both classes in the upper medium.

§ 127. Going back now to § 115, according to the details chosen
in § 126, and taking J for the angle of incidence of condensational-
rare£eu;tional waves, we have 7i = 0, (7= 0, 7, = 0 ; and we may put

ai = sint\ il = Gcosi, a. = sinj, 1

h ... (44).
ft = — cost; 5 = Gsini; ^, = — cosJ.J ^

Thus instead of (18) and (19) we now have

f.

cos

with

t Bint V

X sin t — y cos
tt

')•)

(«X

sinj cosj •' \ V I \

with

(*6).

as the two constituents for the incident waves. Let now ,u, ,(7, ,i,
and ,v, fl, j be the values of the constants for the waves in the
lower medium.

For the reflected waves in the upper medium we have — cos i,
— cos J, instead of cos i, coaj] while sin t, sin J are the same as for
the incident waves. Let — G\ + H\ be the constants for the
magnitudes of the reflected rays corresponding to G, H, for the

DISPLACEMENTS IN PLANE OF INCIDENCE. 365

incident rays. Thus for the total displacement-components due Molar,
to the four waves in the upper medium we have

•. -m^l^ arsini — yco8i\ . . .ttj^I± xain j — y cos jV
S^co8%0/[t ^-^— j-^-smjH/^t ^ ^j

'm,rl. a:sini + vcosix . ^mj^l^ a? sin j + v cos A

+co8i(?y^< ^^— j + smjjyy(^< •^ — ^j

i7 = 8miG/ [t ^ J -C08jJ5r/(^< ^^ -"j

(47),

and for total displacement-components in the lower medium the
same formulas with ,i, j, ,w, ,v, ,G, ,H, 0, 0, in place of t, j, u, v,
0, H, 0', H' respectively.

§ 128. Taking from these formulas the resultants of the ({, 17)
components of the displacements in the several waves, we have as
follows : —

Equivoluminal wave^

Incident, g/(f -^ ^ *' ^ ^^^^ *)

Reflected. 0/ (« - ^^5_L±y£2!J)

Refracted. ,gy (<-^ "" '* " ^ °^'*)

^Incident. Hf (t - ^fl^l^li?^")

Coudensational-rare-,
&ctional wave ^

Reflected, gy^^.^sm .; + ycojjj
Refracted, ,g/ (f - ^ »'° '•? - y '^ '•?) .

§ 129. Putting now y = 0 to find displacement-components at
the interface, and equating values on the two sides, we see in the
first place that the arguments of/ with y zero must be all equal ;
and that the coefiicient of x is the reciprocal of the velocity of the
trace of each of the four waves on the interface ; and if we denote
this velocity by a~* we have

• •

sin 1 sin ,1 sm 7 sin , 7 ,^^^

U ,U V »v

366 LECTURE XVIIL

MoIat. The last three of these equations express the laws of refraction
of waves of either class in the lower medium consequent on waves
of either class in the upper medium. They show that the sines of
the angles of incidence and refraction are inversely as the propa-
gational velocities in the two mediums, whether the two waves
considered are of the same species or of the two different species.
For brevity in what follows we shall put

cosi , cos.i , cos; cos,?' ,-^v

— =6, — ^=A — - = c, ^ = ,c (49),

frt)m which we find

...(50).

n' --'^ '^ ^--hj,n-

] (^^)'

§ 130. Putting now f(t - cur) = i^, we find by (47) with y = 0

V = [au (G -G') + cv (- H^'H')'\ ^

as the displacement-components of the upper medium at the
interface.

Using now (47) to find Q and U by (6) and (5) of § 113, and
putting y = 0 we find, as the components of surface-pull of the
upper medium at the interface

(52) ;

U=n\gu{G-G')-\'2acv{H'-H')]^]
re 5r = 6«-a», \

nh = {k- |n) v-* + 2nc» = p + 2/i (c* - tr*) = p - 2na*J ""

Taking the terms of (51) and (52) which contain G, and H,
and in them substituting ,G, ,H, ,6, ,c, ,5r, X /^. /^» /^' for G, fT, 6, c,
g. A, », n, k, we find for the two refracted waves in the lower
medium the displacement-components, and the surface-pull-com-
ponents, at the interface, as follows: —

,ff^(a,u,G'',c,v,H)ylr

,Q = ,n[2afi,u,G^MB]yjr{

SOLUTION FOR DISPLACEMENTS IN PLANE OF INCIDENCE. 367

Equating each component for the two sides, as said in § 126, Molar,
we find the following four equations for determining the four
required quantities, ,0, ,H, 0\ H\ in terms of the two given
quantities G, H,

bu(0+0') +ai;(fi^-hfr') = fi,u,0+ a,v,H \

n [grw ((? - (?0 + 2act; {H-H')] = ,n {,g,u,0 + 2a,c,v,H)

§ 131. Considering for the present 0 — 0' and H+ H' as the
known quantities, and 0-\'0\ H — H\ ,0, ,H, as unknown; first
find two values of 0 + 0' from the first and third of equations (55),
and two values of H — H' from the second and fourth of (55).
Thus we have

H-H' = --{a,u,0-,c,v,H) + — {G-0')

CV III/ ^y \ I

= 2£i;<'^'«'^ + 2a,c,t,,5)-^(G-G')

The equalities of the second and third members of these two
double equations may be taken as two equations for the deter-
mination of ,(?, fH in terms of 0 + 0', and H + H\ With some
simplifying reductions they become

2a,6(n - ,n\ufi + [,p + 2(n - ,n)a%v,H = pv {H+H')\
\jP + 2{n- ,n) a'] fU,0 - 2a fi (n - ,n) ,v,H = pxi (0 - 0'))

Finding ,0 and ,H from these, and using the results in either
the firsts or the seconds of the pairs of equations (56), we have
(? + 6' and fl'- iT' in terms of 0-0' and H + H\ These last
two equations may be used to find 0' and H' in terms of 0 and H,
Thus we have finally 0\ H\ ,(?, ,H in terms of the, given
quantities 0, H. This, of course, might have been found directly
fix)m the four equations (55) by forming the proper determinants.
The algebraic work is somewhat long either way.

...(56).

368 LECTURE XVIIL

Molar. § 132. But the process we have followed in § 131 has the
advantage of giving us the two intermediate equations (57);
with the great simpUfication which they present in the case
n = ,n, for which they become

,p,v,H^pv{H-\-H'\ ,p,u,0 = pu{0-&) (58).

Let us now work this case out to the end, with the further
simplification ^" = 0; because the particular problem which we
wish to solve is to find the two reflected waves (G', H*) and the
two refracted waves (,G, ,H) due to a single incident equi-
voluminal wave {O). Eliminating ,G, ^H from the first two of (55),
by (58) with J? = 0; and then finding G' and H' irom the two
equations so got, we have

-"■-i'ifyA'' <'»>•

-H'-^^tl'llt-G (60).

,pb '{■pfi + Lv

where L = ^Jp^Zf^^" (61).

,pc + p,c

Lastly (58) gives ,0= 2 ^^,/^^^^^ 0 (62^

P^ I

_,F = 2%4^«G> (63).

,ph ■\- p,h ■¥ L ,v ^ '

This completes the theory of the reflection and refraction of
waves at a plane interface of slipless contact between two ordinary
elastic solids, with any given bulk-moduluses and rigidities.

§ 133. Remark now that by equations (49), (48), we have

,6 II COS,! sinicos,f .

1-= .= -" — -' i (64);

0 ,ucost sm,icosi ^ ^'

and by (50) with n^^n, pl,p^ ,u*lu^ (65);

whence, by (64), g^^ ^ sin ,i cos ,»

•^ ^ ,ph smicosi ^ ^

Hence, we see that if Z = 0 we have

— (?'_ sin 2i--sin2,i tan(i — ,t) {a>T\.

G sin 2t + sin 2,1 tan(i4-,t)

which is Fresnel's formula for reflection when the vibrations are in
the plane of incidence.

.TRIAL OF UNEQUAL RIGIDITIES.

369

Looking to (61) we see that this could only be the case for Molar,
other than direct incidence by either c, or ,c, or both being
infinitely great ; because, § 129 (48), a can only be zero for direct
incidence (i= 0). By (50) we see that to make c or ,c very great,
we must have v or ,v very small. Returning to this suggestion in
(Lect. XIX. § 167), we shall find good physical reason to leave, for
undisturbed ether, t; = oo as Green made it : and to let fV be small
enough, to make L not absolutely zero but as small as required
to give the closeness of approximation to truth which observation
proves for Fresnel's formulas for the great majority of transparent
liquids and solids, including optically isotropic crystals; and yet
large enough to give modified foimulas representing the devia-
tions from Fresnel found in diamond, sulphide of zinc, sulphide of
arsenic, etc. (see § 105 above and § 182 below).

§ 134. Meantime we shall consider some interesting and
important characteristics of the general problem of § 130 without
the limitation n^,n: and of its solution for the case n = ,7i
expressed by § 123 (39), (40) and § 132 (59)... (63), with the
modification due to t; = oo in space containing no ponderable
matter ; and with the special further modification regarding waves
or vibrations of ether in the space occupied by metals (solid or
liquid) to account for the observationally discovered facts of
metallic reflection.

§ 135. Consider first direct incidence, whether of an equi-
voluminal or of a condensational-rarefactional wave. For this we
have in g 129, 130,

* = ,»=i = /7 = 0; a = 0; c = tr»; ,c^,v-^;]

n

.n

These details reduce (55) to

6 + G'= ,0

H-H'^ fl

pv{H + H')=,p,v,H

* ••••••••••••••••••

pu(0-G') = ,P,u,0 ,

.(68).

The first and fourth of these give (?' and ,0 in terms of 0 ; the
T. L. 24

870 LECTURE XVIII.

Molar, second and third give J?' and ,Hin terms of H. Thus our formulas
verify what is obvious without them ; that a directly incident wave,
if equivoluminal, gives equivoluminal reflected and transmitted
waves ; and if condensational-rarefactional, gives condensational-
rarefactional reflected and transmitted waves. For direct incidence,
remark that in the equivoluminal waves the displacement is parallel
to the reflecting surface, and in the x direction ; in the conden-
sational-rarefactional waves it is perpendicular to the reflecting
surface, which is the y direction. For ratios of these displace-
ments we find, as follows, from (68) ; —

(Equivoluminal) -!^' = ^ei^^:i^; '^=-?^^ (69);

^ ^ ' G ,P,u-\-pu' G ,p,u-hpu ^ "

(Condensational-rarefactional) , 7- = — , '-r? = —- -

U iP,v-\-pv H fPfV-^-pv

(70).

In respect to the general theory of the reflection of waves at
a plane interface between two elastic solids of diCFerent quality,
it is interesting to see that, provided only they are sliplessly
connected at the interface, we have, for the case of direct incidence,
the same relations between the displacements of the reflected and
transmitted waves and of the incident wave in terms of densities
and propagational velocities, for equivoluminal waves of transverse
vibration, as for condensational-rarefactional waves (vibrations in
the line of transmission). If, on the other hand, the connection
between the two solids were merely by normal pressure, and if
the surfaces of the two solids at the interface were perfectly
frictionless, and allowed perfect freedom for tangential slipping ;
the reflection in the case of directly incident waves of transverse
vibration would be total, and there would be no transmission of
waves into the other solid.

§ 136. In respect to physical optics the solution (69) is ex-
ceedingly interesting. If we eliminate p, ,p from it by

p = nM-», ,p = ,n,u-^,

we find

— G' ,nu — n,u ,G _ 2n,u

,(71).

G ,vu + n,u^ G ,nu-hn,u

Hence, denoting u/,u by fi (the refractive index), we see that,

DIRECT REFLECTIVITY PROVES EQUAL RIGIDITIES OR DENSITIES. 371

for the reflected wave, in the two cases of equal rigidities and Molar,
equal densities, we have respectively,

(?' l-/i , 0' Ai-1 ,^^.

rr'=''rT^f and 7^ = ^- — ^ (72).

Thus G'\G is equal but with opposite signs in the two cases;
aud therefore, as the ratio of the intensity of the reflected light
to the intensity of the incident light is equal to {G*jGy, we see
that it is equal to

> - 1\'

U + U

.(73),

and is the same in the two cases of equal densities and equal
rigidities; an old known and very important result in physical
optics. It was, I believe, first given by Thomas Young; it is
also found by making i = 0 in FresneFs formulas for reflection of
polarized light at any incidence. But full dynamical theory
proves that, if the refractivity /it — 1 is produced otherwise than
by either equal rigidities or equal densities, the ratio of reflected
light to incident light would not be exactly equal to (73). This
dynamical truth was referred to in my introductory Lecture
(pp. 15, 16, above); and I had then come to the conclusion from
Professor Rood's photometric experiments that the observed
amount of reflected light from glass agrees too closely with (73)
to allow any deviation from either equal rigidities or equal
densities, sufiicient to materially improve Green's dynamical
theory of the polarization of light by reflection. This conclusion
is on the whole confirmed by Rayleigh's very searching investi-
gation of the reflection of light from glasses of diCFerent kinds*;
but the great differences of reflectivity which he found in the
surface of the same piece of glass in diCFerent states of polish,
rendered it impossible to get thoroughly satisfactory results in
respect to agreement with theory; as we see by the following
statement which he gives as a summing-up of his investigation.

" Altogether the evidence favours the conclusion that recently
'' polished glass surfaces have a reflecting power differing not more
" than 1 or 2 per cent, from that given by Fresnel's formula ; but

* " On the IntenBity of Light Reflected from certain snrfaceB at nearly Perpen-
dicolar Incidence," Proc. R, S., xix pp. 276 — 294, 18S6; and Scientific Papers, ii.
pp. 522—542.

24-2

<(

«

372 LECTURE XVIII.

Molar. " that after some months or years the reflection may tall off from
'' 10 to 30 per cent., and that without any apparent tarnish.

" The question as to the cause of the falling off, I am not in
" a position to answer satisfactorily. Anything like a disint^;rati<m
of the surface might be expected to reveal itself on close in-
spection, but nothing of this kind could be detected. A super-
" ficial layer of lower index, formed under atmospheric influence,
"even though no thicker than 1/100000 inch, wonld explain a
'' diminished reflection. Possibly a combined examination of the
"lights reflected and transmitted by glass surfaces in varioas
" conditions would lead to a better understanding of the matter.
" If the superficial film act by difl'usion or absorption, the trans-
" mitted light may be expected to fall off. On the other hand,
" the mere interposition of a transparent layer of intermediate
"index would entail as great an increase in the transmitted as
" falling off* in the reflected light. There is evidently room here
" for much further investigation, but I must content mjrself with
" making these suggestions."

§ 137. Consider next grazing incidence (t = 90®, 6 = 0, a = ti~')
of an equivoluminal wave of vibrations in the plane of incidence,
and therefore very nearly perpendicular to the reflecting surface.
We see immediately that equations (55) are satisfied by

H = 0, H' = 0, ,H==0, ,(? = 0, G-(?' = 0.

This shows that, whether for equal or unequal rigidities, we have
approximately total reflection, and that the phase of the reflected
light corresponds to G' = G. Looking now to §136(71), we see
that in the case of equal rigidities of the two mediums G'/G is
negative for direct incidence, while we now find it to be positive
for grazing incidence. Hence if it is real for all incidences it
must be zero for one particular incidence. This is fundamental
in the dynamics of polarization by reflection and of principcU
incidences.

§ 138. On the other hand, for equal densities and unequal
rigidities, look to § 135 (69); and we see that G'/G is positive for
direct incidence: and by § 137 it is +1 for grazing incidence.
Hence, it cannot vanish just once or any odd number of times
when i is increased from 0° to 90°; but it can vanish twice or
an even number of times. This is essentially concerned in the

OPACITY AND REFLECTIVITY OF METALS. 373

explanation of the remarkable discovery of Lorenz and Rayleigh, Molar,
referred to in § 81 above.

§ 138'. Lastly; going back to § 123, we see that our notation
has secured that, for direct incidence, C'jG is negative or positive
just as is G'jG, according as the rigidities or the densities of the
two mediums are equal. But at grazing incidence, C'/C, while
still negative for equal rigidities, is positive for equal densities.

§ 139. So far everything before us, dynamical and experi-
mental, confirms Green's original assumption of equal rigidities
and different densities to account for light reflected from and
transmitted through transparent bodies. Before going on in
Lecture XIX. to the promised reconciliation between Fresnel and
dynamics for transparent substances, let us, while keeping to the
supposition of equal rigidity of ether throughout vacant space and
throughout space occupied by ponderables of any kind, briefly
consider what suppositions we must make in respect to our
solution of § 132 [(59) — (63)], to explain kqown truths regarding
the reflection of light from metals or other opaque bodies.

§ 140. The extremely high degree of opacity presented by all
metals for light of all periods, from something considerably longer
than that of the extreme red of the visible spectrum (2*5 .10~" sees,
for A line) to something considerably less than that of the extreme
violet (13 .10"" sees, for H line), is the most definite of the visible
characteristics of metals: while the great brilliance of light re-
flected from them, either directly or at any angle other than
grazing incidence*, compared with that reflected from glass or

* As obliquity of incident light is increased to approach more and more nearly
grazing incidence, the brilliance of the reflected light approaches more and more
nearly to equality with the incident light. At infinitely nearly grazing incidence
we find theoretically (§ 137) total reflection from every polished surface; polish
being defined as in § S6 above. Indeed we find observationally in surfaces such as
sooted glass, which could scarcely be called polished according to any definition, a
manifest tendency towards total reflection when the angle of incidence is increased
to nearly 90°.

A striking illustrative experiment may be made by placing, on a table covered
with a black cotton -velvet tablecloth, two pieces of plate glass side by side, with an
arrangement of light and screen as indicated in the accompanying sketch. Zi is a
lamp which may be held by hand, and raised or lowered slightly at pleasure. 00'
is an opaque screen. PP* is a screen of white paper resting on the table. It would
be startUng, if we did not expect the result, to see how much light is reflected from

f%^

374 LECTURE XTIIL

yio\ki. crv«ftAl<i or trorn the rn*>st brilliantlr poKslied of eommovilj kik>vn
and -T^^on non-rrprtallic bodies, is their most obvious and best knovn
'inality. The ofjacity of thin metal pUles hhheito tested for all
visible lightri from red to viulet has been fbond seemii^r perfect
for all thickneTJ!f;.s exceeding 3.10"' cm. <i>r half the vave-leogth
of yellow light in air). When, in the process of gold-beating.
the thickn^rH- of the gold-leaf is reduced to aboat 2.10^ cm. lor
aU/iit one-third of the wave-length of yellow light) it begins to
U; |M:rr;eptibly translucent, tiansmittin^ faint green light when
illuminates] by strong white light on one side. The thinnest k{
onJinarj' gold-leaf (7.10"' cm., or aboat one-eighth of the wave-
l^ngth of yellow light) is quite startlingly tianslocent, giving a
•strong green tinge to the trun>*mitted light. Silver foil l"5.10^cni.
thiek ^':/in5^iderably thinner than translacent gold-leaf) is qoite
h\9iu\nti to the electric light so &r as our eyes allow as to jndge ;

t)M! R/y/t K\tti\'t: the \fonTiAs%Ty of the flhjidov of Cm/. The
<till riion; nXrxk'xn^ hy jiU^sin;; a flat pUte of polishcl fflvcr bcslie the two
piftU'St, and ^*-*t'\n7, how n^arlj >iOth the sooted and the clean gUis plitef eotne in
rivalry with th<; *\Ut:t platr; in respect to totality of reflection, vhcn L is lowerad to

P'. d

P'

:■ i AL

tWftH and iriore nearly v^ta/au)!, incidence of its light. It is interesting Also lo take
away the paji^r iiorf^cn, and vi<:w the three plates and the lamp hj an eje plaeed in
\Hmiitiu% Ut r*^A!iv*: rcflocti'^l ht^ht from the three mirrors, sooted, polished glaaa,
and poliN}i'!«l nilvar.

Another int';ref»tin(; experiment may he made by looking TerticallT downwards
thr';rj(^j a Nir/il at the three 8urfac4^;B, or at a clean snrfiace of mercnzy or water,
illuminaU:^! by light from L at nearly grazing incidence in a dark room. A snr-
priitingly large amount of light is seen from the sooted snrCaoe, and is found
Ut )tti ahnoHt wholly fiolarize/l Uf vibrations perpendicular to the plane throogh L
and the line of viaion. If the glans and silver are very well polished and ekan,
little or no light will lie seen frr;m them ; unless L is very intense, when probably
a faint blue light, jfolarized just as is the light from the soot, will be visible,
indicating want of molecular perfectness in the polish, or a want of optical
l>erfeetneMi in the most perfect polinh pomible for the molecular constitution of
the solid or liquid, according to the principles indicated in j 86 above.

ELECTROMAGNETIC SCREENING AND TRANSPARENCY. 375

but it is transparent to an invisible violet light through a small Molar,
range of wave-length from about 3'07.10~* to 3*32. 10"' cm.*
(periods from 102. 10"*' to 1*11.10"" of a second).

§ 141. The extreme opacity of metals is quite lost for Roentgen
rays (which are probably light of much shorter period than 10"" of
a second); sheet aluminium of thicknesses up to two or three
centimetres being transparent for them. For some qualities of
Roentgen rays even thick sheet lead is not perfectly opaque.

§ 142. We have no experimental knowledge in respect to the
opacity of exceedingly thin metallic films for radiant heat of
longer periods than that of the reddest visible light. It seems
not improbable that through the whole range of periods up to
il0~" of a second, through which experiments on the refractivity
and reflectivity of rock-salt and sylvin have been made by Langley,
Rubens, Paschen, Rubens and Nicols, and Rubens and Aschkinaas
(see above, Lecture XII. p. loO), the opacity of gold-leaf and other
of the thinnest metal foils may be as complete, or nearly as com-
plete, as it is for visible light.

§ 143. But, when we go to very much longer periods, we
certainly find these thin metal foils quite transparent for variations
of magnetic force. Thus, sheet copper of thickness 7. 10"* cm.
(about a wave-length of orange light in air), though almost per-
fectly opaque to variation of magnetic force of period one-third or
one-fourth of 1/(8.10*) of a second, is almost perfectly transparent
for periods of magnetic force of three or four times 1/(8.10®) of
a second, and for all longer periods.

Taking greater thicknesses, we find a copper plate two milli-
metres thick, almost a perfect screen, that is, almost perfectly
opaque, in respect to the transmission of the magnetic influence
of a little bar-magnet rotating 8000 times per second f; somewhat
opar^ue, but not wholly so, when the speed is 100 times per second;
almost, but not perfectly, transparent, that is to say, very slightly

* See pp. 185, 186 of Popular Lecture* and Addresses^ Vol. i., Ed. 1891 (Friday
Evening Royal Institution Lecture, Feb. 3, 1883), where experiments illustrating the
reflectivity and the transparency of some exceedingly thin films of platinum, gold,
and silver, supplied to me through the kindness of Professor Dewar, are described.

t See § 3 of Appendix K, ** Variational Electric and Magnetic Screening,"
icprinted from Proc, R. S, Vol. xlix. April 9, 1891 ; also Math, and Phys, Papers^
Vol. III. Art. cii. "Ether, Electricity, and Ponderable Matter," § 35.

376 LECTUKK XVUL

ioUr if perceptibly obstructive, when the speed is onoe per seeood ; not
perceptibly obstructive, when the period is 10 seconds or more.
Now, according to what is without doubt remlly Tmlid in the
80-calIed electro-magnetic theory of light, we may regard as a
lamp, a bar-magnet rotating about an axis perpendicolar to its
length, or having one pole caused to vibrate to and firo in a straight
line. We may regard it as a lamp emitting light of period equal
to the period of the rotation or of the vibration. F<m* the light
of this lamp, sheet copper two millimetres thick is almost per-
fectly transparent if the period is anything longer than one second ;
but it is almost perfectly opaque if the period is anything less than
1/8000 of a secoml down to- one eight hundred million millionth ot
a sec^ind (the f>eriod of extreme violet light); and is probably quite
opar|uc for still smaller periods down to those of the Roentgen
rays, if we regard these rays as due to vibrators giving after each
shock a sufficient number of subsiding vibrations to allow a period
to be reckoneil Whatever the distinctive characteristic of the
Roentgen light, sheet copper two millimetres thick is perceptibly
translucent to it, and sheet aluminium much more sa

§ 144. We may reasonably look for a detailed and satis&ctoiy
investigation, mathematical and experimental intelligence acting
together, by which we shall thoroughly understand the continuous
relation between the reflection and translucence of metals and
transparent bodies, and the phenomena of electric and magnetic
vibrations in insulating matter, in non-magnetic metals, in soft
iron, and in hardened steel, for all vibrational periods from those of
the Ricntgen rays to ten or twenty seconds or more. The in-
vestigation must, of course, include non-periodic motions of ether
and atoms. It cannot but show the relation between the electric
conductivity of metals and their opacity. It must involve the
consideration of molecular and atomic structures. Maxwells
electro-magnetic theory of light was essentially molar*; and there-
fore not in touch with the dynamics of dispersion essentially

* ** Suppose, however, that we leap over this difficulty [regarding electrolysis] by
'* simply asserting the Cact of the constant value of the molecular chaige, and that
**we call this constant molecular charge, for convenience in deeeription, cme
** molecule of electricity.

•<Thi8 phrase, gross as it is, and out of harmony with the rest of this treatise,
"will enable us at least to state clearly what is known about electrolysis, and to
"appreciate the outstanding difficulties.*' Maxwell, Electricity and Magnetism,
YoL I. p. 812.

green's dynamics of reflection and refraction. 377

involved in metallic reflection and translucency; though, outside Molar,
his electro-magnetic theory, he was himself one of the foremost
leading molecularists of the nineteenth century: witness his
kinetic theory of gases; and his estimates of the sizes and
weights of atoms ; and his anticipation of the Sellmeier-Helmholtz
molecular dynamics of ordinary and anomalous dispersion, in a
published* Cambridge Examination question.

§ 145. First, however, without any molecular hypothesis, and
without going beyond Green's purely molar theory of infinite
resistance to compression, and equal effective rigidities of ether in
all bodies and in space void of ponderable matter, let us try how
nearly we can explain the high reflectivity and the great opacity
of metals. Either great rigidity, or great density, or both great
rigidity and great density, of ether in metals would explain these
two properties : but we have agreed not to assume differences of
rigidity, and there remains only great density. It is interesting
to remark however, that infinite rigidity would give exactly the
same law as infinite density; because each extreme hypothesis
would simply keep the ether at the interface absolutely unmoved ;
and this even if we allow the ether to be compressible within
liquids and solids, as we are going to do later on.

§ 146. Go back now to § 132 (59)— (63), and, following

Green, make t; = x , and ,v= cc. This makes, by (50), c = ,c = '-ai'f;

and reduces (61) and (59) to

L^Kai (74),

and

-G-^pb^^^^^

where kJ'^^^^, <^ = tan-^_^, ^ = tan--.=:^|

(75).

* Rayleigh, in a footnote appended in 1899 to the end of his paper "On the
Reflexion and Refraction of Light by Intensely Opaque Matter" {Phil. Mag. 1872;
republished as Art. xvi. of his Scientific Papers^ Vol. i.), writes as follows:

"I have lately discovered that Maxwell had (earlier than Sellmeier) considered
*' the problem of anomalous dispersion. His results were given in the Mathematical
** Tripos Examination, Jan. 21, 1869 (see Cambridge Calendar for that year)."

In this examination question the vi»cou» term, subsequently given by Helmholtz,
is included.

t Take -at (not +ai), a being positive, in order that, in § 128, H, of the upper
medium may have extinctional factor e'*^^, and, H, of the lower, may have
extinctional factor c*'^^. See footnote on § 158". Bee also § 128 and (49), and (54).

•^78 LEcrruRE xviii.

Molar. This agrees with Grctii's result*; and the square of the first

- 0'
factor of the final expression for -^ - is the formula by which

curve 3 of figs. 4, 5, 6, for water, flint-glass, and diamond, in
§§ 102, 103 above, were calculated.

§ 147. To realize this solution in the most convenient way for
physical optics, put

/(^) = €*-• = cos ©^ + A sin ©^ (76);

and take

s
for incident wave d = t (77),

with same continued for reflected wave, where 8 denotes a space
in the path of the incident ray, continued in the reflected ray.

Use these in § 128, and for the real problem take the real part
of each expression so found. We thus have, for the vibrational
displacements,

Incident wave, G cos to It — ]

Reflected wave, \ . . .(78).

§ 148. Let now for a moment ,plp = x . This makes
Kjfp = 1 ; ^ = -^ = i ; and gives, for vibrational displacement in
the reflected wave

- Gsin L (^-^) - 2il = Gsin L («-^) + 7r- 2il .

Thus the formulas show, for vibrations in the plane of inci-
dence, that, at every incidence, the reflection is total (which we
know without the mathematical investigation, because there is
no loss of energy) ; and that the reflected ray is advanced in phase
TT — 2t relatively to the incident ray. The former proposition is
proved for vibrations perpendicular to the plane of incidence by
the formula (39) of § 123; but with just ir for phasal change.
Hence incidence at 45° instead of the 70'' to 76'' of metals (§ 99
above), gives 90° of phasal diflference ; and it is easy to see fi-om

* Green, Mathematical Papers^ p. 2G7.

QUASI WAVE-MOTION WITH XEOATIVK OR IMAGINARY DENSITY. ^79

the formulas that values of ,p/p large enough to give the Molar,
brilliance of metallic reflections could not give any approach
to the elliptic and circular polarizations which observations show
(§ 99 above) in all metallic reflections. Thus, though our trial
hypothesis of great eflFective density of the ether in the substance
could give reflections, and therefore general appearances, un-
distinguishable to the naked eye from what we see in real metals,
it fails utterly to explain the qualities of the reflected light
discovered by polarizational analysis. Seeing thus that no real
positive effective density in the substance can explain the qualities
of metallic reflection, we infer that the effective density is negative
or imaginary ; and thus we are led by strictly dynamical reasoning
to the brilliant prevision of MacCullagh and Cauchy that metallic
reflection is to be explained by an imaginary refractive index.

§ 149. In § 159 below we shall find, by a new molecular
theory to which I had been led by consideration of very different
subjects, a perfectly clear and simple dynamical explanation of a
real negative quantity for effective density of ether traversed
by light-waves of any period within certain definite limits, in a
space occupied by a solid or liquid or gas. We shall also find
definite molecular and dimensional conditions which may possibly
give us a sure molecular foundation for an imaginary effective
density; though we are still very far from a thorough working-
out of the full dynamical theory.

We conclude the present Lecture with a short survey of the
quasi wave-motion which can ^xist in an elastic medium having
a definite negative or imaginary effective density ,p ; and of the
reflection of light at a plane surface of such a medium.

§ 150. Let the rigidity be n, real ; and the density ,p, an
unrestricted complex as follows

fp = «r (cos <^ — A sin <^) (79),

where m and ^ are real, and for simplicity may be taken both
positive. This gives for propagational velocity of equivoluminal
waves

fU=^w (cos i<^ + A sin \^)\ ,ir^ = w~^ (cos i^ — t sin J<^),
where w = a / — , a real velocity

380 LECTURE XVllI.

Molar. For displaccmeDt at time t, considering only equivoluminal wave-
motion, let 17 + ii;' denote displacement in a wave-plane perpen-
dicular to OX ; and choosing for our arbitrary function

we have 17 + 417'= €"•<'-*''•*> (81);

whence, taking ,u-^ from (80), we find, as a real solution,

17 =€--«"*♦*/«' cos ft) (e- cos i<^a:/t(;) (82).

This shows th^t the wave-plane travels in the -h a?-direction with
velocity W8ec^<l>, and with vibrational amplitude diminishing
according to the exponential law e"**"*"*^'^**. It is interesting to
remark that the propagational velocity of this subsidential wave is,
in virtue of the factor sec J<^, essentially greater than the velocity
in a medium of real density equal to the modulus, «r, of our
imaginary density; and is infinite when ^^=90°. This illustrates
Quincke's discovery of greater velocity of light through a thin
metallic film than through air.

§ 151. Remark now that, according as the real part of the

complex, ,/), is positive or negative, ^ is <90°, or >90'' ; that, in the

extreme case of ,p real positive, <f> is zero ; and that, in the other

extreme case (,p a real negative quantity), ^ is 180^ In every

case between those extremes, i^<f> is between 0° and 90°, and

therefore both cos^*^ and sin J<^ are positive. This implies loss

of energy in the inward travelling wave: except in the second

extreme case, ^<^ == 90°, when its propagational velocity is infinite ;

and (82) becomes

V = €—*/«' sin a)t (83),

which represents standing vibrations of ether in the substance,
diminishing inwards according to the exponential law ; and
therefore proves no continued expenditure of energy. We con-
clude that, for our ideal silver, the effective quasi-density of
ether in the metal is essentially real negative: but that for all
metals of less than perfect reflectivity it must be a complex, of
which the real part may be either negative or positive.

§ 152. Confining our attention for the present to ideal silver,
and to merely molar results of the molecular theory promised for
Lecture XIX., let us put, in §§ 123, 132,

-fjL' = p' (84),

INFINITE VELOCITY DISPROVED FOR CONDENSATIONAL WAVES. 381

where i/' denotes a real positive numeric ; and let lis find the Molar
difference of phase between the two components, given respec-
tively by §§ 123 and 132, due to incident light polarized in
any plane oblique to the plane of incidence. That diflFerence of
phase is 90** for Principal Incidence (see § 97 above). I have
gone through the work with Green's supposition v = (x> , ,t; « oo ,
in § 132 ; and using the formula for O'/O so given in § 147, 1 have
found that for all values of v* from 0 to x it makes the Principal
Incidence between 0° and 45**. Now observation gives the
Principal Incidences for all colours of light and all metals,
between 45** and 9^ (see table of § 99, showing for all cases
of metallic reflection hitherto made, so far as generally known.
Principal Incidences ranging from 66° to 78°). Hence polarizational
analysis of the reflected light as thoroughly disproves, for metallic
reflection, Green's assumption of propagational velocity infinite
for condensational-rarefactional waves within the metal, as it was
disproved for transparent substances in §§ 104, 105 by mere
observation of unanalysed reflectivities.

§ 153. Hence, anticipating Lecture XIX. as we did in § 133
above, let us, while still keeping t; = x in the ether outside the
metal, now make ,v small enough, inside the metal to practically
annul L This reduces (39) and (59) of § 123 and § 132, to

C ",6 + 6' G '',pb + p,b ^ ^•

Putting now in (39) and in this

-,-= T) ^ = — 1;*; — = — £1/* (86),

6 lU p ,u ^ ^

^hcre r^'^.^'^-^'^^^t^ (87),

COS! 1/COSi

^^fi°d __ = ___ = ^^^ (88),

and ^= *!— ^ (89).

O v + ir ^

* The sign minus is chosen here in order that €^ not e~^ may be the reducing
factor ot ,0; p"^ being a positive length. See foot-notes on §§ 146, 15S".

382 LECTURE XVlll.

1 r

Molar. Put now tan"* — = ^, tan-*-=a (00).

vr V

This reduces (88) and (89) to

-(7' cos ^-4 sin ^ .

-PT = — n—. ^ — B = cos2^-4Sin2^ (91i

C cos ^ + 4 sm ^ ^ "^

and -rT = — K . i- w = cos2&-Asin 2^ (92).

G cos^+ASin^ ^ '

These formulas express total reflection for the two cases re-
spectively of vibrations perpendicular to the plane of incidence,
and vibrations in this plane ; and they give us 2d and 2^, which
we may, for brevity, call " the phases of the vibrations." Thus,
calling P the plane of incidence and reflection, we find

Phase of vibration perpendicular to P - phase of vibration in P

= 2(^-d) (93)

This means that the vibration perpendicular to P precedes the
other by 2 (^ - 6),

Whatever positive value i/* has, this ditference is essentially
zero for i = 0 ; and we find that it increases through + 90** to
+ 180° when i is increased from 0° to 90"*; as illustrated in the
accompanying tables for the two cases, i/*=o, i/*= 10.

,•

vr

^=tan-i-
p

i

2 (^ - ^)

o

o

o

o

^

54-7

54-7

OO

10

53-5

55-9

4-8

20

501

591

180

30

45-0

63-4

36-8

40

38-7

68-2

590

50

31-7

72-9

82-4

63-2

29-3

74-3

90O

60

241

77-4

106-6

70

16-2

81-7

1310

80

8-2

86-9

165-4

90

OO

90-0

180O

1

i;»=-5.

PRINCIPAL INCIDENCE FOR IDEAL SILVER.

383

■

(

e-ian-^ ^

1

2(^-6)

vr

"

o

o

o

o

0

17-5

17-5

0-0

10

17-3

17-8

1-0

20

16-5

18-7

4-4

30

151

20-3

10-4

. 40

13-4

22-8

18-8

50

11-2

26-8

31-2 1

60

8-7

33-3

49-2 '

70

5-9

44-0

76-2

73-8

4-8

49-7

89-8

80

3-0

62-3

118-6

90

1

OO

90-0

180-0

Molar.

i/» = 10.

§ 154. The angle of incidence (53^-2 for v* = -5, 73°-8 for
1/* = 10) which gives the 90** advance of phase is what has been
defined as the Principal Incidence (§ 97 above). The reflectivities
for the two polarizational components of incident light being
perfect, the azimuth which, for the "Principal Incidence," gives
circular polarization for the reflected light, that is, the " Principal
Azimuth," is 45''. For all real metals observation shows the phase-
diflFerence 2 (^ - ^) to be positive, that is to say, the vibration
in the plane of reflection to lag behind the vibration perpendicular
to it ; as we find it for ideal silver by dynamical theory.

§ 155. The easiest mathematical method for finding the
Principal Incidence, /, for any given value of v^ is algebraic, as
follows. For Principal Incidence we have

2(5-^) = 90" (94).

This gives tan 2^ tan 25 = — 1, or in algebra, from (90),

(iA^^IKT^^T^)"""^ ^^^^'

whence,

r*-r«(i/» + 4 + !;-«) + 1 = 0 (96).

Taking the greater root of this regarded as a quadratic for r",
and substituting it for r* in the general expression

tan'i = i;»(r»- l)/(i/»+ 1) (97)

given by (87) for % in terms of v and ?•, we find tan'/ for tlic

384

LECTURE XVIII.

Molar. Principal Incidence. (The less root of the quadratic is rejected,
because it makes tan'* negative.)

The following table has been thus calculated directly, to show
for fourteen values of v or v-, the Principal Incidence, /.

y

y^

tan I
1-000

/

0-00

0-00

45°-0

0-50

0-25

1-193

60-0

0-71

0-50

1-333

531

1-00

IW

1-554

57-2

1-41

2-00

1-887

621

2-00

4-00

2-387

67-2

2-45

6-00

2-786

70-3

316

10-00

3-438

73-8

3-74

14-00

3-982

75-9

4-00

lC-00

4-223

76-6

4-47

20-0

4-680

77-9

C-00

36-0

6-160

80-8

10-00

100-0

10-10

84-3

14-00

196-0

14-07

85-9

20-00

400-0

20-05

871

§ 156. The converse problem of finding i^ for any given
Principal Incidence, by (96) and (97), yields a cubic equation
for 1/*. The table of § 155 proves that this cubic equation has
one, and only one, real positive root for every value of / between
45** and 90° ; and no real positive root for values of / between
0** and 45''. For our present purpose it is most easily solved by
trial and error, aided by the table of § 155. I have thus foimd for
the three Principal Incidences measured by Conroy for red,
yellow, and blue light respectively, incident on his silver film
polished with putty powder (table of § 99 above), the following
values of v and v*.

Red...
Yellow
Blue...

76" 29'
74° 37'
IV 33'

1
1

p

1

y^

1

f

1

3-9

! 15-2

,

3-3

10-9

1
1

2-7

7-29

1

§ 157. The diagram of § 88 shows that Conroy's silver film,
polished as he polished it with putty powder, may be regarded as
almost our ideal silver: this view is confirmed by his three

"ADAMANTINISM" and unequal RIGIDITIKS IN METALS? 386

Principal Azimuths, 43° 51', 43°52', 43° 0', being each as nearly Molar,
a good approximation to 45° as it is. (See § 159" below.) But
their shortcomings of from one to two degrees below 45° are no
doubt real, and point to the correction of the real values of i^
by the addition of small purely imaginary terms. Thus, to fit
the formulas of § 150 to Conroy's silver, we may, keeping m a
positive quantity, take ^ = 180° — ^ 5 where ;^, which would be
zero for ideal silver, may for real silver, have some small value of
a few degrees.

§ 158. It would be interesting to pursue the subject further,
and include with silver, other metals for which we have the
experimental data such as those shown in the table of § 99.
To do this, we may conveniently in (87), (88), (89), put

j/» = p(cosx-*8inx) (98);

and use the experimental data of Principal Incidence and
Principal Azimuth to determine in each case the two unknown
quantities pcos^ and psin^: but time forbids. This would
be, in fact, a working out of the theory, or empirical formula, of
MacCullagh and Cauchy, to comparison with observational results
regarding metallic reflection, such as has been done by Eisenlohr,
Jamin, Conroy, and others*. The agreement of the theory with
observation has been found somewhat approximately, but not
wholly, satisfactory. Stokes, as communicator of Conroy's paper,
No. III. {Proc. R.S.f Feb. 15, 1883), comments on this want of
perfect agreement; and suggests that it is to be accounted for by
the inclusion in respect to metallic reflection of what he proposes to
call "the adamantine property" of a substance ; being the property
required to explain the deviation, from Fresnel's law of reflection
of light by transparent bodies, discovered more than eighty years
ago by Airy, in diamond. This adamantine property, as we
shall see in Lecture XIX., § 173, is to be explained dynamically
by assuming a small imaginary iq, for the velocity of condensa-
tional-rarefactional waves in the substance ; not small enough to
utterly annul L in our formulas of § 132. Its magnitude is
measured by the ratio g/w, which I propose to call adamantinism.
Some deviation from exact equality between the effective rigidities
of ether in the two mediums might also be invoked to aid in
procuring agreement between dynamical theory and observation,

* Basset, Physical Optics, §§ 871, 380; Mascart, Traiti d'Optique, Vol. ii.
T. L. 25

t^f<^ LECTURE XVIIL

MoUr. for reflection of light from all substances^ tnuisparentv <^ metallic

or otherwise opaque.

§ 158'. Meantime, I finish this Lectoie XVIIL <» the Re-
flection of Light with an application of § 123 (39X and § 132 (59)
with Z = 0, to the theory of Fresnel's rhomb.

Let the medium in which u is the velocity of light be a tnms-
parent solid or liquid ; and let the other mediam be undistiiibed
ether. We cannot now continue Green's convenient usage ad<qpted
in § 116, and call the former "the upper" and the latter ''the lower."
We now have
n>u; p>,p: ,{>i] ,uIu=il\ pl,9^y?\ sin,t=/i«ni ; 6=coBt/tf ; ,ft=oos,t7,«...(98'!

For convenience, to suit the case of ,i>t, write as SdUowSi
the equations cited above, —

Ch^y G^pJb + ^A ^*^^»

the sign minus being transferred from their first members because,
in virtue of ,% > t, (7/(7 is now positive through the whole zaiige of
non-total reflection, [t = 0 to t = sin"''(l//A) giving ,t = 90*]; and
fj G is positive through the range up to the Brewsterian angle of
zero reflected ray [i = 0 to i = tan~* (1/ft)]. From i = tan"* (l//i) to
t = sin"^(l », G'/G decreases fix)m 0 to —1; and it is imaginaiy-
complex of modulus 1, through the whole range of total reflection

[i = sin-»(l/Ai) to 1 = 90'].

§ 158". When i is between 8in-»(l» and 90^ ^ is real
negative ; and, denoting its value by — 9* (as in § 118), we have

,& = -«/•; where j = 6^^*'^*'r^> (98">

fA cost

Remark that q increases from 0 to x when t is increased from

8in-» (l/fi) to 90". Using (98'") in (98") we find,

C^b^ig, G'_p?q-ib
C b'-iq' G'p}q+ib ^^ ^'

whence, by De Moivre's theorem,

4, =cos2^ + isin 2-^; y? = cos 2;^ — i sin 2;^ (98^),

where '^ = tan-*2; ;^=ttan-*-p (98^).

* The aign mimu is here ohoeen in order that c^^, not c'^^v, maj be the
redneing factur of ,C and ,G for the diatnrbanoe in the ether outside, see J 188 ;
y being zero at the reflecting surface, and negatiTe outside.

PHASAL CHANGES BY INTERNAL REFLECTION. 387

By (98^*) and (98'") we see that i|r increases from 0** to 90° and Molar.
X decreases from 90** to 0"*, when i is increased from

sin-H/*"') to 90^
§ 158'". Putting for a moment qlb = x, we find

Ek)uating this to zero we find, for i^ + x) ^ minimum,

q/b = fi-\ which makes ^r = ^ = tan"' (fi"') (98^«) ;

and to make q/b = fi-\ we have tanH' = 2/(jj} - 1). We infer that
when i is increased from sin~^ (/a^O ^^ tan""* V[2/(m' — 1)], ('^ + x)
decreases from 90"* to 2 tan""* (/i~*) and then increases to 90*
again, when i is increased to 90"*. This will be useful to us
in § 158^'.

§ 158^ Realising now, as in § 147, we find real solutions as
follows, for vibrations perpendicular to, and in, the plane (P) of
incidence and reflection : —

Vibrations Incident wave C cosod (t j

perpendi- -^ -...(98*'');

cular to P Reflected wave C cos I© f « - -] + 2^

Incident wave 0 cos mlt j

Reflected wave (?cos talt J — 2^

Vibrations
inP

Thus we see that the reflected vibrations in P are set back in
phase by 2 (-^ + x) relatively to the reflected vibrations perpendi-
cular to P. By § 158'" we see that this back-set is 180'' at the
two incidences, i = sin""* (/a~0» ^^^ * = 90° (^^^ limits of total
reflection); and that it decreases to a minimum, 4 tan""* (/a"*), at
the intermediate angle of incidence i = tan~* V[2/(/*' — 1)].

The back-sets of the phase through the whole range of total
reflection for four refractive indices, 146, 1*5, lol, and 1*6 are
shown in the Table II. of § 158^ below.

Column 1 represents angles of incidence.

25—2

388

LECTURE XVIII.

Molar. Columns 2 and 3 of Table II. represent the absolute back-sets
of phase by reflection respectively in, and perpendicular to, P,

Column 4 represents the back-set of phase of vibrations in P,
relatively to vibrations perpendicular to P, produced by a single
reflection.

The numbers in Column 4, doubled, are the total back-sets by
two reflections, at the same angle of incidence i, of vibrations in,
relatively to vibrations perpendicular to, P. Each of these numbers
is between 180"* and SeO"*; and therefore it indicates virtually a
setting-forward of the phase of vibrations in P, relatively to the
phase of those perpendicular to P, by the amount to which each
number falls short of 360"*. Hence the numbers in Column 6 of
Table II. represent virtual advances of phase of the vibrations in,
relativdy to ike vibrations perpendicular to, P, after two reflections.

§ 158\ It will be seen that all the phasal differences shown
in Column 4 of Table II. are obtuse angles : with minimum values
shown in Table I., which is extended to include zinc sulphide.

Total Internal Reflection. Table I.

BrewHterian

angle
tan-i (1//U)

Limiting inci-
dence for total

Incidence for

Minimum

Refractive index

internal

reflection

8in-i (1/m)

minimum phasal
difference

tan-\/[2/(^«-l)]

phasal

difference

4 tan-i (11 fi)

■
137° 38'

1-46

34° 24'-5

43° 14'

53° 3'

1-60

33 41-25

41 49

51 40

134 45

1-61

33 30-75

41 28

51 20

134 3

1-60

32 0-25

38 41

48 33

128 1

(Zino Sulphide) 2-371

22 52

24 57

33 20

91 28

M= cot 22<»4 = 2-414

22 30

24 28

32 46

90 0

(Diamond) 2-434

22 20

24 15

32 31

89 20

(Bealgar) 2-454

22 10

24 3

32 15

88 40

diamond, and realgar. By Table I. we see that the phasal diflFer-
ences for internal reOection in glasses, and all known transparent
bodies of refractive index less than 2*414, are obtuse for all angles
of incidence through the whole range of total internal reflection.
This conclusion was very startling to myself, because for eighty
years we have been taught that, for total internal reflection in
glass, the phasal diflference was an acute angle in a single re-
flection ; and that it was 45"* for each reflection in the Frcsnel

REFLECTION OF CIRCULARLY POLARIZED LIGHT. 889

rhomb, instead of 1 35° which we now know it ia How it is so, Molar,
we can easily see by the following simple considerations.

(1) Circularly polarized light in every case of direct incidence
and reflection (metallic or vitreous, external or internal), has its
circular orbital motions in the same absolute directions in the
direct and the reflected waves. Hence if the orbital motion ideally
seen by an eye receiving the incident light is anti-clockwise, it is
clockwise to an eye receiving the reflected light. (Hence the
arrangement in respect to signs in figs. 3 a and 3 6 of § 92 above
is explained thus: — First imagine the two diagrams as corre-
sponding to normal incidence : for the incident light, fig. 36, with
its face down, looked up to; and for the reflected light, fig. 3a,
with its face up, looked down on. Then incline the two planes
continuously to suit all incidences from i = 0, to i = 90°.)

(2) Taking first, external reflection of light, of circular anti-
clockwise orbits, incident on glass of negligible adamantinism*;
increase the angle of incidence from 0"* to the Brewsterian tan"*/*.
The motion constituting the reflected light is in clockwise elliptic
orbits, of increasing ellipticity, with long axes perpendicular to
the plane of reflection, till at tan~*ft it is rectilineal. Increase
now the angle of incidence to 90° : the orbital motions are elliptic
anti'Clockunse with diminishing ellipticity (long axes still perpen-
dicular to the plane of reflection); and become infinitely nearly
circular when the incidence is infinitely nearly grazing.

(3) Proceed now as in (2), but with internal instead of
external reflection at a glass surface: the incident circularly
polarized light being anti-clockwise. With increasing incidences
from 1= 0, the reflected light is clockwise elliptic with increasing
ellipticity, till it becomes rectilineal at the Brewsterian tan~*(l//i).
Increase i farther : the reflected light is now anti-clockwise elliptic,
with diminishing ellipticity till it becomes circular at the limit of
total internal reflection, sin"'^(l//A). Increase i farther up to 90°:
the reflected light is elliptic, still anti-clockwise, with ellipticity
increasing to a maximum when i = sin"^ V[2/(/a' — 1)] and
diminishing till the orbit becomes again infinitely nearly circular
when i is infinitely little less than 90°.

* For the change from the present statement, required by adamantinism when
perceptible, see below, §§ 17S — 1S2.

390 LECTURE XVlll.

Molar. (4) With, instead of glass, an ideal substance whose refrac-
tive index is 2*414; anti-clockwise circularly polarized light,
incident at angles increasing from 24" 28', gives total reflection,
and anti-clockwise circular orbits, becoming elliptic, with ellip-
ticities increasing to rectilineal vibrations when i=32°46'.
Farther increase of i up to 90° gives elliptic orbits of ellipticities
diminishing to circularity at % = 90°: all an ti -clockwise.

(3) Diamond and realgar, and all other substances having
fi> 2*414, have three principal incidences for internal reflection;
one with non-total reflection at the Brewsterian angle tan~*(l//A);
and two within the range of total reflection ; one less, the other
greater, than the incidence which makes the phasal difference
a minimum. Anti- clockwise circularly polarized light, incident
at any angle between the last-mentioned two principal incidences,
gives clockwise elliptic orbits for the reflected light; presenting
a minimum deviation from circularity (minimum ellipticity we
may call it) at some intermediate angle.

(6) It is interesting to remark that in every case, according
as the phasal difference is obtuse or acute, the reflected light is
anti-clockwise or clockwise, when the incident light is circular
anti-clockwise. This with § 158*^** gives a simple means for ex-
perimentally verifying the statements (1), (2), (3), (4), (5). The
experimental proof thus given of the truth, that the phasal
difference in every case of approximately grazing incidence is 180°,
is very easy and clear; as it can be arranged to show simul-
taneously to the eye the incident and the reflected light ; both
extinguished simultaneously by the same setting of the analysing
fresnel (or quarter- wave plate) and nicol.

(7) The rule of signs referred to in (1) of the present section,
and illustrated in figs. 3 a and 3& of §92, must be taken into
account to interpret the meaning of phasal difference of two recti-
lineal components of reflected light, due to incident rectilineally
vibrating light. Thus we see that if the vibrational plane of the
incident light is turned amii-clockwise from P, the orbital motions
of the reflected light are anti-clockwise, or clockwise, according as
the component vibrating in, is set back or advanced, relatively to
the component perpendicular to P. See § 158*^" below.

Total Internal Reflection. Table II.'

CoLl

Co]. 3

CoL8

CoL4

Col. 5

to tb« onuwt min.11^

^'Jgi^^fr-J

Ti^f;^'*

»«!>( In (be CMB (X

™nw or vibislion*

tl»lr.qi»Ut]r

Sr"'™

iD. ntellTel; to il-
tioiiB

*♦

*X

^=1-50

41° 48'-6

0° ff

180° 0-

180* 0"

0^ 0'

41

8 GO

160 18

169 6

31 18

43

33 14

133 18

164 83

60 56

44

SO 33

117 86

147 68

64 4

45

86 G3

106 14

148 6

73 48

46

43 30

97 36

140 6

79 48

47

47 84

90 28

138 a

83 66

48

53 16

84 30

186 36

86 48

49

66 40

79 0

186 40

88 40

50

60 50

74 16

186 6

89 18

51

6i 48

70 0

184 48

90 21

61 40-S

67 33-8

67 23-8

184 45-6 (min.)

90 28-eini&i.j

6S

SB 36

66 10

134 46

90 28 '

68

72 18

63 88

184 66

go 8

58 13

90 0

64

76 54

69 22

186 16

89 38

55

78 4

67 36

186 30

89 0

56

88 46

53 33

136 18

87 24

57

86 6

60 62

136 68

86 4

G8

89 33

48 34

187 46

84 38

SO

B3 34

46 3

188 86

83 48

eo

96 44

43 48

189 83

80 56

70

126 32

36 68

161 14

57 83

SO

168 4

13 8

166 18

29 86

90

180 0

0 0

180 0

0 0

fi = l-46

43 18-8

0 0

180 0

180 0

0 0

44

IS 18

143 6

160 34

39 13

46

37 66

124 33

163 18

65 21

46

33 8

113 0

147 8

65 44

47

41 14

103 84

143 18

72 31

48

46 38

94 63

141 80

77 0

49

51 84

88 18

139 52

80 16

SO

66 10

83 88

138 48

83 24

51

60 33

77 86

188 8

S3 11 ,

52

64 40

73 6

187 46

81 88 1

63

68 38

69 1

137 89

81 43 -

53 3-08

68 490

68 490

187 880(01111.)

84 44 (max.)

64

72 36

66 16

137 43

84 36

66

76 8

61 50

137 58

84 1

66

79 44

68 38

138 33

83 16

57

83 16

66 40

138 66

83 8

58

66 40

53 48

139 28

81 4

90 3

60 14

140 16

79 38

60

93 SO

47 46

Ul 6

77 48

70

134 0

38 0

162 0

68 0

80

153 36

13 8

165 84

38 53

00

180 0

0 0

180 0

0 0

Total Internal Reflection. Table II.

CONTINUED.

Col.1

Col. a 1 Col. 8

001.4

Col. 5

<» and ;t cklcnlktBd only
lo the lusmt toinule,
eiwpl in the cue of

"ji^i'Ssp";:!

r,sJ17sy"S

'

tbrir eqiialitj

tiiclf Id vibnUoTK

in. relmli.el7 lo ri-

W

ix

tioni

,.=1-61

*1° 88'-B

0° 0'

180° 0'

180^ 0"

0° 0"

43

14 40

147 16

161 66

36 8

43

36 6

126 8

161 14

57 82

44

33 30

112 16

H6 16

69 38

45

38 38

102 46

141 24

77 13

46

44 0

94 42

138 42

82 86

47

48 64

87 42

136 36

86 48

48

58 30

83 4

135 34

88 63

49

67 46

76 68

134 44

90 33

50

01 60

73 26

134 16

91 38

61

65 41

68 20

134 4

91 53

El 3011

67 1-7

67 1-7

134 8-4 (miD.)

91 63-3 (IMI.)

52

60 SO

64 36

134 6 '

91 48 '

S3

73 6

61 13

134 18

91 24

54

76 38

68 3

134 40

90 40

54 37

W 0

65

80 6

55 6

136 12

69 36

56

83 2G

52 22

136 48

68 34

67

86 44

49 48

136 32

86 56

68

SB 58

47 33

137 20

85 20

69

93 8

46 4

138 13

83 86

60

96 16

43 64

139 10

81 40

70

125 40

26 33

151 3

57 6G

eo

154 4

11 33

166 36

38 48

90

180 0

0 0

180 0

0 0

,.=i-eo

38 40-B

0 .

180 0

180 0

0 0

39

10 46

163 48

163 36

32 48

40

33 12

136 40

148 63

63 16

41

29 86

111 60

141 26

77 8

4a

36 38

101 6

136 44

86 82

43

40 66

93 S6

133 33

93 66

44

45 43

85 40

131 33

97 16

45

60 4

79 48

129 62

100 16

46

54 18

74 38

128 56

103 8

47

58 14

70 6

128 20

103 30

48

62 0

66 4

128 4

103 53

48 3S-0

64 0-6

64 0-6

128 1-2 (min.)

103 57-6 (mM.)

49

65 38

63 34

12« 2

103 56

60

69 6

128 12

103 36

61

73 34

66 8

128 86

102 48

63

76 64

63 14

129 8

101 44

03

79 6

60 38

139 44

100 82

64

83 IS

48 10

130 28

99 4

55

86 24

45 63

131 16

97 38

66

88 28

43 43

132 10

95 40

67

91 30

41 4U

133 10

93 40

58

94 30

89 43

134 12

91 36

58 44

90 0

59

97 26

87 63

185 19

89 34

60

100 30

86 6

136 36

87 8

70

138 3

21 84

149 86

60 48

BO

164 IB

10 IS

164 SO

31 0

90

ISO 0

0 0

180 0

0 0

fresnel's rhomb.

393

§ 158^*. That most exquisite invention, Fresnel's rhomb, Molar,
produces a difference of phase of 90"" between vibrations in, and
vibrations perpendicular to, P. By § 158'" we see that this can
only be with glass of refractive index greater than 1'496. Thus,
the table for refractive index 1*46 shows 84° 44' for the maximum
value in Column 5. The other three tables show, for refiuctive
indices 1 '5, 1*51, and 1*6 respectively, maximum values in Column 5,
90°2§'-8, 91°53'-2, and 103°57'-6. In each of these cases, there
are of course two incidences which make the difference of phase
exactly 90^. Fresnel pointed out* that the larger of the two is to
be preferred to the smaller; because, with the larger, the differ-
ences of refractivity, for the different constituents of white light,
make less error from the desired 90° difference of phase. The
larger of the two incidences is to be preferred, also, because small
differences from it make less error on the 90° phasal differences,
than equal differences from the other. For the same reason, the
le^ the refractive index exceeds 1*496, the better. Fresnel's first
rhomb was made of glass, of refractive index 1*51, from the
factory of Saint Oobain : our tables show that 1*5 would reduce
the phasal errors to about half those given by 1*51, for equal

Fig. y.

small errors in the direction of the ray. It would thus give an
appreciably better instrument than Fresnel's own. The accom-
panying diagram, figure 9', represents a section perpendicular to

• CotUcUd Papers, Vol. i. p. 792; original date 1828.

394 LECTURE xvin.

.raloM the traversed faces and the reflecting surfaces of the rhomb, for
refractive index 1*51, as cut by Fresnel; nvith acute angles of
54*" 37^ so that the rays for its proper use may enter and leave
perpendicularly to the traversed faces. If p denote the perpen-
dicular distance between the two long sides of the parallelogram,
the length of the long side is 2ptant (or p. 2*816 with i=s 54® 37'):
the length of the short side is p . cosec % (or p.l '226). The
perpendicular distance between the entering and the leaving rays
is p. 2 sini (or p. 1-6306).

§ 158'". The length of the long side is chosen such that a
ray, entering perpendicularly* through the centre of one tguce,
passes out perpendicularly through the centre of the parallel &ce.
Fresnel's diagram is far from fulfilling this condition; and so are
many of the diagrams in text-books, and scientific papers on the
subject. The last diagram I have seen shows only about a quarter
of normally entering light to suficr two reflections; and rather
more than half of it to pass straight through without any reflection
at all. Airy's and Jamin's diagi*ams are correct and very clear.
Fresnel must, in 1823, have given clear instructions to his work-
man (with or without a diagram); and, to this day, opticians make
Fresnel's rhombs of right proportions to fulfil the proper condition
in respect to the enteriug and leaving rays. But twenty years
after Fresners invention we learn from MacCuUaghf that Fresnel
rhombs were made by Dollond (and probably also by other
opticians ?), with Fresnel's 54° 37' for the acute angle, but with
refractive indices difibring from his 1*51. I am assured that
some opticians of the present day make the acute angle correct
according to the refractive index of the glass, to give exactly 90°
phasal difierence of the components of the normal ray. I do not
know if they realize the importance of having glass of refractive
index as little above 1*496 as possible.

§ 158'"*. MacCullagh appreciated the beauty and value of
FrcsneFs rhomb, and as early as 1837 had begun using it for
research. But he was at first much perplexed by unexpectedly
large errors, until he found means of taking them into account,

• This, for brevity, I caU a ** normal ray."

t *' On the Laws of Metallic Reflection and the mode of making experiments upon
Elliptic Polarization,'' Proe. Royal Irith Academy, May 8, 1B4.S. MacCuUagh's
ColUcUd Papert, p. 240.

ERRORS FOUND BY MACCULLAGH.

395

''and of making the rhomb itself serve to measure and to eliminate Molar,
"them." He good-naturedly adds; — "The value of the rhomb as
"an instrument of research is much increased by the circumstance
"that it can thus determine its own effect, and that it is not at all
"necessary to adapt its angle exactly to the refractive index of the
"glass." This proves a very forgiving spirit: perplexity and loss
of time in his research gratefully accepted in consideration of his
having been led to an enlarged view of the value of the Fresnel
rhomb as an accurate measuring instrument !

§ 158*^ MacCuUagh had two rhombs from the same maker
each "cut at an angle of 54^** as prescribed by Fresnel." He found
one of them wrong phasally by 3**, the other by S"* ! He does not
say whether the errors were of excess or defect ; but we see that
they must have been excess above 90°, because no refractive index
greater than 1 '5, with i = 54** 37', gives as large a defect below 90°,
as 3°. This we see by looking at the following Table, calculated

M

•

t

2^
78° 4'

2x

2(^ + X)

360°

-4(^ + X)

Excess
above 90°

1-5

54® 37'

57° 28'

135° 32'

88° 66'

-1° 4'

1-61

**

78 48

56 12

135 0

90 0

0 0

1-6

}i

84 14

46 44

130 58

98 4

+ 8 4

1-7

«»

88 32

39 4

127 36

104 48

14 48

1-8

»»

91 44

33 20

125 4

109 52

19 52

according to § 158^ with i=54° 37', for four refractive indices,
other than Fresnel's 1*51. We also see that the refractive indices
of the rhombs, supplied by Dollond to MacCuUagh, must have
been between 1*51 and 1'6.

§ 158*. Notwithstanding MacCullagh's good-natured remark,
it is important that the acute angle of a Fresnel rhomb should be
made, as accurately as possible, such that the phasal difference
shall be exactly 90°, for light of definitely specified period (sodium
light for example), when the direction of the ray is exactly per-
pendicular to the entering and leaving faces. But however
trustworthy may be the instrument-maker's work, MacCullagh's
principle of determining the error in the practical use of the
instrument, and eliminating it if it is perceptible, is highly im-
portant and interesting. It may be carried out, either, as he did

396 LECTURE XVIIL

Molar, it himself, by means of observations on metallic reflection : or,
as he suggested, by observations on total reflection at a sepa-
rating surface of glass and air; a much simpler subject than
metallic reflection. Or, as simplest method when two Fresnel's
rhombs are available, we may place them with leaving face of
one, and entering face of the other, parallel and close together,
and mount the two rhombs so as to turn independently round
a common axis perpendicular to these surfaces through their
centres. The arrangement is completed by mounting two nicols
so as to turn independently, one of them round the central
ray entering the first rhomb, and the other round the central ray
leaving the second rhomb : and providing three graduated circles,
by which differences of angles turned through between the first
nicol and first iresnel; between the two fresnels; and between the
second fresnel and the second nicol ; may be measured. The
mechanism to do this is of the simplest and easiest. The experi-
ment consists in letting light; entering through the first nicol
and traversing the two fresnels and the second nicol ; be viewed
by an eye seeing through the second nicol. The best approach to
extinction that can be had, is to be produced by varying the
three measured angles. For simplicity we may suppose that the
three zeros of the three pointers, on the three circles, are set so
that, using for brevity facial intersections to denote intersections
of the traversed faces and reflecting surfaces of a fresnel, we have
as follows : —

(1) When the first index is at zero on the first circle, the
vibrational lines of the light emerging from the first nicol are
perpendicular to the facial intersections of the first fresnel.

(2) When the second pointer is at zero on its circle, the facial
intersections of the first fresnel are parallel to those of the second.

(3) When the third index is at the zero of the third circle,
the facial intersections of the second fresnel are perpendicular to
the vibrational lines of light entering the second nicol.

Thus if ni, a, 71, denote the readings on the three circles ; r?i is
the inclination of the vibrational lines of the first nicol to the
reflectional plane* of the first fresnel ; a is the inclination of the
facial intersections of the first fresnel to those of the second ; and

* For brevity I call "refleotional plane" of a fresnel, the plane of reflection of a
normal ray. It is perpendicular to the facial intersections.

ERRORS OF TWO FRESNELS DETERMINED.

397

72, is the iuclination of the reflectional plane of the second iresnel to Molar,
the vibrational lines of the light emerging from it. The diagram,
figure 9", shows these angles all on one circle, ideally seen by an eye
looking through the second nicol. OVi, OF^ OF^, OV^, represent
planes through the central emergent ray respectively parallel to the
vibrational lines of the first nicol, the reflectional plane of the

Fig. 9".

first fresnel, the reflectional plane of the second fresnel, and the
vibrational lines of the second nicol set to let pass all the light
coming from the second fresnel. OiT,, OK2, represent planes
through the common axis perpendicular to OFi and OF^

§ 158*'. Let now sin t denote the displacement at time t, of a
point of ether in the light passing from the first nicol to the first
fresnel. This implies a special unit of time, convenient for the
occasion, according to which the period of the light is 27r ; and it
takes as unit of length, the maximum displacement of the ether
between the first nicol and the first fresnel. Following the light
through the apparatus; — first resolve the displacement into two
components, cos n^ sin t, sin rii sin t, parallel to OFi and OKi. The
phase of the former of these is advanced 90° + ei relatively to the
latter by the two total reflections in the first fresnel, if Bi denote
its error, which ought to be zero. Thus, at any properly chosen
point in the space traversed by the light between the two fi'esnels,
the displacement of the ether at time t is

cos Wi cos (t + 61) parallel to OFi ; sin rij sin t parallel to OKi.

398 LECTURE XVIII.

Molar. Resolving each of these into two components, we have, for the
light entering the second fresnel,

cos a cos Th cos (^ + 61) — sin a sin ni sin ^ parallel to OF^ . . .(98**) ;
sin a cos Ui cos (^ + 61) + cos a sin rii sin t parallel to O-ffj. . .(98*").

The former of these is advanced 90® + «b relatively to the latter,
by the two reflections in the second fresnel, e^ being its error ; and
thus, at any properly chosen point in the light passing from the
second fresnel to the second nicol, the displacement of the ether
at time t is

— cos a cos rii sin (^ + 61 4- eg) — sin a sin tii cos (^ 4- e,),

parallel to Oi^a... (98*"*);

sin a cos Wj cos (^ + ei) 4- cos a sin ni sin t, parallel to OK^ . . .(98**^).

We have now to find the condition that the resultant of these
shall be rectilineally vibrating light. This simply implies that
(98**") and (98**'') are in the same phase ; and, 0 F, being the line
of the resultant vibration, we have

A B

-^, = ^ = -00111, (98*^),

where -4, B and A\ R denote the coefficients found by reducing
(98**") and (98**^ to the forms il cos ^ + J5 sin e, and ^' cos ^ 4- 5' sin e.
Thus, performing the reductions, and putting

tanni = Ai, tan 71, = A,, tana=j (98*^),

we find

— 1 _ — sin (ei 4- e>) — h^ cos e^ _ — cos (61 4- ej) 4- AJ sin e,

A, j cos 61 " — jsin6i4-Ai

(98*^*).

m

These are two equations for finding the two unknowns, 61,62] when
f^iy Kiji are aU known; one of them being chosen arbitrarily, and
the other two determined by observation, worked to produce
extinction of the emergent pencil (compare ^ 95, 95', above). 0 V^
in the diagram (fig. 9") is perpendicular to the position of the
vibrational line of the second nicol when set for extinction.

§ 158***. The experimenter will be guided by his mathematical
judgment, or by trial and selection, to get the best conditioned
values of Ai, A,, j', for determining ei, 62. We must not take Ai = 1
(that is to say, we must not set the entering nicol fixed with its
vibrational lines at 45^ to the principal planes of the first fresnel)

SIMPLIFICATION WHEN ERRORS ARE SMALL. 399

because in the case 61 = 0 (the first iresnel true), the light entering Molar,
the second fresnel would be circularly polarized, and therefore the
turning of the second fresnel would have no effect on the quality
of the emergent light ; which would be plane polarized if, and only
if, this fresnel, like the first, is true (compare § 98 above). We
must not fix ^ = 0, because this would render the first fresnel
nugatory. It occurs therefore to take Ui = 22J** which makes

hr'-hi^2 and Ai = -4142 (98^*)-

Probably this may be found a good selection if, for any reason, it
is thought advisable to fix hi and leave j variable.

If $1,6% are so small that we may neglect ei\ ef, «i^, (98*^")
becomes

-;i=«n^.A^ = zi.+^i^ (98^);

whence we have two simple equations for 61, e,

e, + <^=i(^-A.); J'^-A,e.=^(^^-l) (98").

From these we see that, as ei and e^ are small, the observation must
make h^ and A, each very nearly unity unless y is taken either very
small or very great. It may be convenient to fixy= 1, (a= 45®),
and to find w^, n, by experiment.

§ 158*"'. When both the fresnels are perfectly true (ei =0, Ca=0),
formula (98") shows that A, = Ai , if j = 0 ; and ^2 = l/^i » if i = 00 :
but if j has any value between 0 and 00 , we must have 711 = 11^ = 45",
while a may have any value. That is to say, the first fresnel
produces exactly circular orbits and the second rectifies them.
When rtj is any angle between 0 and 90**, we have elliptically
or circularly polarized light, represented by (98^) and (98^^*) with
61=^0. This is light leaving the first and entering the second
fresnel. In passing through the second fresnel it becomes con-
verted into rectilineally vibrating light, represented by (98^") and
(98^^) with ^1 = 0 and e, = 0. Thus we see that g 158*— 158'«,
with Ci = 0 and «, = 0 passim, expresses simply and fully the theory
of the conversion of rectilineally vibrating light into elliptically
or circularly polarized light, and vice versd ; by one true Fresnel's
rhomb. A diagram with rules as to directions in the use of
Fresners rhomb is given below in § 158"". §§ 90 — 98 above

398 LECTURE XVIII.

Molar. Resolving each of these into two components, we have, for the
light entering the second fresnel,

cos a cos Th cos (^ + 61) " ^^Q ^ ^^^ ^ 3^ ^ P^^^ll^l ^ 0^,...(98**);
sin a cos Ui cos (< 4- ^) + cos a sin rii sin t parallel to OK^, . .(98***).

The former of these is advanced 90** + «b relatively to the latter,
by the two reflections in the second fresnel, e^ being its error ; and
thus, at any properly chosen point in the light passing from the
second fresnel to the second nicol, the displacement of the ether
at time t is

— cos a cos rii sin (^ + Ci 4- eg) — sin a sin tii cos (t 4- ^),

parallel to Oi;... (98*"*);

sin a cos Wj cos (t + ^i) 4- cos a sin rii sin t, parallel to OK^ . . .(98**^).

We have now to find the condition that the resultant of these
shall be rectilineally vibrating light. This simply implies that
(98^^") and (98**^) are in the same phase ; and, OF, being the line
of the resultant vibration, we have

A B

2> = 5> = -cot??. (98'^),

where -4, B and A\B! denote the coefficients found by reducing
(98^") and (98**0 to the forms il cos ^ + 5 sin t, and ^' cos * 4- i' sin ^.
Thus, performing the reductions, and putting

tanni = Ai, tan 71, = A,, tana=j (98*^),

we find

— 1 _ — sin ifii 4- 62) — ^ij cos e, _ — cos ( 61 4- ea) 4- h^ sin e,

A, j cos 61 — j*sin6i4-Ai

(98*^).

These are two equations for finding the two unknowns, ei,CB; when
Kffhfjf ^^ ^U known; one of them being chosen arbitrarily, and
the other two determined by observation, worked to produce
extinction of the emergent pencil (compare ^ 95, 95', above). 0 Fi
in the diagram (fig. 9") is perpendicular to the position of the
vibrational line of the second nicol when set for extinction.

§ 158***. The experimenter will be guided by his mathematical
judgment, or by trial and selection, to get the best conditioned
values of Ai, h^,j, for determining Ci, e,. We must not take Ai = 1
(that is to say, we must not set the entering nicol fixed with its
vibrational lines at 45"^ to the principal planes of the first fresnel)

SIMPLIFICATION WHEN ERRORS ARE SMALL. 399

because in the case ei = 0 (the first fresnel true), the light entering Molar,
the second fresnel would be circularly polarized, and therefore the
turning of the second fresnel would have no effect on the quality
of the emergent light ; which would be plane polarized if, and only
if, this fresnel, like the first, is true (compare § 98 above). We
must not fix Ai = 0, because this would render the first fresnel
nugatory. It occurs therefore to take ni = 22J° which makes

hr'-k^2 and Ai = -4142 (98"^").

Probably this may be found a good selection if, for any reason, it
is thought advisable to fix hi and leave j variable.

If 01, ei are so small that we may neglect 6i", e^^ Ci^, (98*^*)
becomes

-U-^+i?-A, = :^±^*^ (98^);

whence we have two simple equations for 6i, 62

e. + e.=j{l-h); ^'^ - A.e. = 1. (| - 1) (98-).

From these we see that, as ei and e^ are small, the observation must
make hi and A, each very nearly unity unless y is taken either very
small or very great. It may be convenient to fixy= 1, (a = 45°),
and to find n^ , n, by experiment.

§ 158'"'. When both the fresnels are perfectly true (ei =0, ea=0),
formula (98**) shows that A, = Ai , if j = 0 ; and h^^^l/hi.if j — oo:
but ifj has any value between 0 and oo , we must have rii = n, = 45",
while a may have any value. That is to say, the first fresnel
produces exactly circular orbits and the second rectifies them.
When r?i is any angle between 0 and 90°, we have elliptically
or circularly polarized light, represented by (98**) and (98**') with
01 = 0. This is light leaving the first and entering the second
fresnel. In passing through the second fresnel it becomes con-
verted into rectilineally vibrating light, represented by (98**") and
(98**^) with 01 = 0 and 63 = 0. Thus we see that g 158*— 158*",
with 61 = 0 and 0, = 0 passim, expresses simply and fully the theory
of the conversion of rectilineally vibrating light into elliptically
or circularly polarized light, and vice versd ; by one true Fresnel's
rhomb. A diagram with rules as to directions in the use of
FresneFs rhomb is given below in § 158*^". §§ 90 — 98 above

400 LECTURE XVIII.

Molar, contain the application of the theory to ordinary reflection at
the surfaces of transparent solids, or liquids, or metals : the first
fresnel of § 158* being done away with, and the reflection substi-
tuted for it. The corresponding application to total internal
reflection is much simpler, because the intensities of incident
and reflected components are equal. It allows the diflei*ence of
phase, produced by the reflection, to be measured with great
accuracy ; and a careful experimental research, thus carried out,
would no doubt prove the difference of phase, produced between
the two components by the reflection, to agree very accurately
with the obtuse angles calculated for different incidences according
to (98^) and § 158'".

§ 158^^ About a year ago, in making some preliminary
experiments by aid of a FresneFs rhomb, to illustrate ^ 90 — 100,
152, 153, I interpreted the phasal difference of the rhomb ac-
cording to Airy's Tracts ; but found error, or confusion, in respect
to phasal change by one internal reflection in glass, and by metallic
reflection. I looked through all the other books of reference
and scientific papers accessible to me at the time; and I have
continued the inquiry to the present time, by aid of the libraries
of the Royal Society of London, and the University of Glasgow ;
but hitherto without success, in trying to find an explicit state-
ment as to which of the two components is advanced upon the
other in the Fresnel's rhomb. I have therefore been obliged to
work the problem out myself mathematically for the Fresners
rhomb ; and with the knowledge thus obtained, to find by experi-
ment which of the two components is advanced on the other by
metallic reflection. Of all the authors I have hitherto had the
opportunity of studying, only Airy in respect to Fresners rhomb,
and Jamin, and Stokes*, and Basset in respect to metallic re-
flection, have explicitly stated which component of the light is
advanced in phase.

In Airy*s Tracts, 2nd edition (1831), page 364, I find a
thoroughly clear statement, agreeing with Fresnel s own, regard-
ing the FresneUs rhomb and total internal reflection: — "If the
"light be twice reflected in the same circumstances and with the
" same plane of reflection, the phase of vibrations in the plane of
" incidence is more accelerated than that of the other vibrations

* Mathematical and Physical Papers y Vol. ii. p. 360.

EIGHTY years' ERROR REGARDING TOTAL REFLECTION. 401

" by 90**." In truth, the vibrations in the plane of incidence are Molar,
not advanced 90°, but set back 270**. They are therefore virttudly
advanced 90°, relatively to the vibrations perpendicular to the
plane of incidence.

In Basset's Physical Optics (Cambridge 1892), page 339, 1 find,
stated as a law arrived at by Jamin by experiment, with reference
to metallic reflection, the following : —

" (1) The wave which is polarized perpendicularly to the plane
** of incidence, is more retarded than that which is polarized in the
"plane of incidence.*'

By experiment I have verified this, working with a Fresnel's
rhomb, interpreted according to Airy, and § 158^* above. In reading
Jamin's experimental paper I had felt some doubt as to his mean-
ing because his expression " vibrations polaris^es dans le plan de
rincidence " (I quote from memory) may have signified, not that
the plane of polarization*, but that the line of vibration, was
in the plane of incidence. That Basset's interpretation was
correct is however rendered quite certain by a clear statement in
Jamin's Cours de Physique, Vol. ii. page 690, describing relative
advance of phase of vibrations perpendicular to the plane of inci-
dence of light reflected from a polished metal. This phasal relative
advance he measured by a Babinet's compensator, and found it to
increase from zero at normal incidence, to 90° at the principal
incidence, /; and up to 180° at grazing incidence. He would
have found not advance, but back-set, if he had used a Fresnel's
rhomb interpreted according to the mathematical theory given (by
himself as from Fresnel) in pages 783 to 787 of the same volume,
with the falsified formulas (98^»).

§ 158'^ The origin of the long standing mistake regarding
the Fresners rhomb is to be found in FresneFs original paper,
"M^moire sur la loi des modifications que la reflexion imprime
"k la lumifere polaris^e**: reproduced in Volume I. of Collected
Papers, Paris 1876, pages 767 to 799. In page 777 we find

* Considering the inevitable liability to ambiguity of this kind, I have abandoned
the designation ''plane of polarization*'; and have resolved always to specify or
describe with reference to vibrational lines. Abandant examples may be found in
the earlier parts of the present volume illustrating the inconvenience of the desig-
nation "plane of polarization.'* In fact *' polar** and '* polarization ** were, as is
now generally admitted, in the very beginning unhappily chosen words for
differences of action in different directions around a ray of light. These difiPerences
are essentially not according to what we now understand by polar quality.

T. L. 26

402 LECTURE XVIII.

Molar* Fresnel's own celebrated formulas, for reflected vibrational ampli-
tudes and their ratio, correctly given as follows : —

tan (i - ,i) sin (i - ,i) ^ cos (i - ,i) ^ ^^^

tan (i 4- ,0 ' sin {i + ,i) ' cos (i 4- ,i) * *

It is obvious that the first two of these expressions must have
the same sign, because at very nearly normal incidences the
tangents are approximately equal to the sines, and at normal
incidences, the two formulas mean precisely the same thing;
there being, at normal incidence, no such thing as a difference
between vibrations in, and vibrations perpendicular to, a plane of
incidence. Yet, notwithstanding the manifest absurdity of giving
different signs to the " tangent formula ** and the " sine formula "
of Fresnel, we find in a footnote on page 789 (by Verdet, one of
Fresnel's editors), the formulas changed to

tan (i — ,i) sin (t — /i)

tan (i + ,i) * sin (i 4- ,i)

.(98*^),

in consequence of certain " considerations " set forth by Fresnel
on pages 788, 789. I hope sometime to return to these "con-
siderations " and to give a diagram showing the displacements of
ether in a space traversed by co-existent beams of incident and
reflected light, by which Fresnel's " petite diflScult6 " of page 787
is explained, and the erroneous change from his own originally
correct formulas is obviated. The falsified formulas (98"") have
been repeated by some subsequent writers; avoided by others.
But, so far as I know, no author has hitherto corrected the conse-
quential error, which gives an acute angle instead of an obtuse
angle for phasal difference in one total internal reflection; and gives
90* phasal difference instead of 270° in the two reflections of the
Fresnel's rhomb; and gives 90° back-set, instead of the truth
which is 90° virtual advance, of vibrations in the plane of incidence,
relatively to vibrations perpendicular to the plane of incidence, in
a Fresnel's rhomb. The serious practical error, in respect to which
of the two components experiences phasal advance in the Fresnel's
rhomb, does not occur in any published statement which I have
hitherto found. Airy accidentally corrected it by another error.
All the other authors limit themselves to saying that there is a
phasal difference of 90° between the two components, without
saying which component is in advance of the other.

ORBITAL DIRECTION IN LIGHT FROM ANY MIRROR. 403

_§ ISS'^^*. In Fresnel (page 782), and Airy (page 362), where Molar.
J—l is first introduced with the factor V(/A"sin"i — 1), the positive
sign is taken accidentally, without reference to any other con-
sideration. A reversal of sign, wherever J—l occurs in the
subsequent formulas, would have given phasal advance instead
of phasal back-set, or back-set instead of advance, in each con-
clusion. If the authors had included the refracted wave (Aiiy,
page 358) in the new imaginary investigation so splendidly dis-
covered by Fresnel, they would have found it necessary, either to

reverse the sign of V-1 throughout, or to reverse the interpreta-
tion of it in respect to phasal difference, given as purely con-
jectural by Fresnel, and by Airy (page 363) quoting from him :
because with this interpretation, and with the signs as they stand,
they would have found for the " refracted wave," a displacement
of the ether increasing exponentially, instead of diminishing
exponentially, with distance from the interface (see footnote on
§ 158"' above). It is exceedingly interesting now to find that an
accidentally wrong choice of signs, in connection with V— 1,
served to correct in the result of two reflections, the practical
error of acute instead of obtuse for the phasal difference due
to one reflection, which is entailed by the deliberate choice of
a false sign in the real formulas of (98^^") : and that, thus led.
Airy gave correctly the only statement hitherto published, so far
as I know, as to which of the two compon^nts experiences phasal
advance in FresneFs rhomb.

§ 158*^". Note on circular polarization in metallic or vitreous
or adamantine reflection. Referring to Basset's Physical Optics,
page 329, edition 1892, 1 find Principal Azimuth defined as the
angle between the plane of polarization of the reflected light,
and the plane of the reflection, when the incident light is circularly
polarized light incident at the angle of principal incidence. This
really agrees with the, at first sight, seemingly different definition
of Principal Azimuth, given in § 97" above : because it is easily
proved that when rectilineally vibrating light, is converted into
circularly polarized light by metallic or other reflection; the
azimuth of the vibrational plane of the incident light, is equal to
the azimuth of the plane of polarization of the reflected light
when circularly polarized light is converted into rectilineally
vibrating light by reflection on the same mirror. The fact that k

26—2

404 LECTURE xvm.

Molar, is positive /or every case of principal incidence (§ 97') is included
and interpreted in the following statement : —

Consider the case of plane polarized light incident on a
polished metallic, or adamantine, or vitreous, reflector. For sim-
plicity of expressions let the plane of the reflector be horizontal.
To transmit the incident light let a nicol, mounted with its axis
inclined to the vertical at any angle i, carry a pointer to indicate
the direction of the vibrational lines of the light emerging from it,
and incident on the mirror. First place the pointer upwards
in the vertical plane through the axis; which is the plane of
incidence of the light on the mirror. The reflected light is of
rectilineal vibrations in the same plane. Now turn the pointer
anti-clockwise through any angle less than 90°. The reflected
light consists of elliptic, or circular, anti-clockwise orbital motions.

If i = i, the principal incidence; the two axes of each elliptic
orbit are, one of them horizontal and perpendicular to the plane
of incidence and reflection; the other in this plane.

To avoid any ambiguity in respect to " clockwise " and " anti-
clockwise," the observer looks at the nicol, and at the circle in
which its pointer turns, from the side towards which the light
emerges afber passing through it: and he looks ideally at the
orbital motion of the reflected light from the side towards which
the reflected light travels to his eye. See § 98 above.

In external reflection of rectilineally vibrating light by all
ordinary transparent reflectors, including diamond (but not realgar),
the deviation from rectilineality of the reflected light is small,
except for incidences within a few degrees of the Brewsterian
\Avr^ fjL. See diagram of §178 below (figures 11 and 12), for
diamond.

§ 158*^"*. The rules for directions in elliptic and circular
polarization by a Fresnels rhomb are represented by the annexed
diagram, figure 9'". (7 and 0 are the centres of the entering and
exit faces. OK is a line parallel to the facial intersections. OF
represents the plane of the reflections in the rhomb, being a plane
perpendicular to the four optically effective faces.

Cases tti, Oa, a,. Plane polarized light enters by 0', OFi,
OV^y OFi, are parallels through 0 to the vibrational lines of the
entering light; they are of equal lengths, to represent the displace-
ments as equal in the three cases. The orbits of the exit light in

Ri;i.ES FOR ORBITAL DtRECTlUN,

405

the three casvH are repreaciitod by (1) ellipse, (2) circle, (3) ellipse; Holar.
being drawn acconliog to the geometrical construction indicated
in one quadrant of the diagram. '

Cases 6i, &,, 6,. The orbits in three cases of equally strong
circularly and elliptically polarized entering light, having axes
along OF and OK, shown with their centres transferred to 0, are
represented by (1) ellipse, (2) circle, (3) ellipse. OV,', 07,', 07,',
represent the displacements in the exit light, which in each of
these three cases consists of rectilineal vibrations.

Pig. r:

The direction of the orbital motions of the exit light in cases a,
and of the entering light in cases b, is, in each of the six cases,
anti-clockwise, as indicated by the arrowheads on the ellipsea and
circle.

Thus we have the following two rules for directions : —

Rule {a). When the vibrational pUne of entering light must

406 LECTURE XVllI.

Molar, be turned anti-clockwise to bring it into coincidence with OK,
the orbital motions in the exit light are anti-clockwise.

Rule (6). When the orbital motion of circularly or elliptically
polarized entering light, having axes parallel to OK and OF^ is
anti-clockwise, the vibrational plane of the exit light is turned
anti-clockwise from OK,

These rules, and the diagram, hold with a "quarter-wave
plate " substituted for the fresnel, provided homogeneous light of
the corresponding wave-length is used. The principal plane of
the quarter-wave plate in which are the vibrations of waves of
greatest propagational velocity, is the OF of the diagram.

§ 159. The beautiful discovery made seventy years ago by
MacCuUagh and Cauchy, that metallic reflection is represented
mathematically by taking an imaginary complex to represent the
refractive index, /i, still wants dynamical explanation. In 1884
we saw (Lee. XII. pp. 155, 156) that — /i' is essentially real and
positive through a definite range of periods less than any one of
the fundamental periods, according to the unreal illustrative mech-
anism of unbroken molecular vibrators constituting the ponderable
matter; embedded in ether, and acting on it only by resistance
between ether and the atoms against simultaneous occupation of
the same space. This gave us a thoroughly dynamical foundation
for metallic reflection in the ideal case of no loss of light, and for
the transmission of light through a thin film of the metal with
velocity (as found by Quincke), exceeding the velocity of light in
void ether. It however gave us no leading towards a dynamical
explanation of the manifestly great loss of light suffered in reflection
at the most perfectly polished surfaces of metals other than silver
or mercury (see § 88 above). But now the new realistic electro-
ethereal theory set forth in Appendices A and E, and in §§ 162... 168
below, while giving for non-conductors of electricity exactly the same
real values of fj?, negative and positive, as we had from the old
tentative mechanism, seems to lead towards explaining loss of
luminous energy both in reflection from, and in transmission
through, a substance which has any electric conductivity, however
small. In App. E § 30, J. J. Thomson's theory of electric conduction
through gases is explained by the projection of electrions out of
atoms. If this never took place, the electro-ethereal theory would,
like our old mechanical vibrator, give loss of energy from trans-
mitted light only in the exceedingly small proportion due to tho

PERFECT OPACITY WITH PERFECT REFLECTIVITY. 407

size of the atoms according to Rayleigh's theory of the blue sky Moleculai
(§ 58 above). On the other hand the crowds of loose electrions
among under-loaded atoms, throughout the volume of a metal, by *
which its high electric conductivity is explained in App. E § 30, may
conceivably give rise to large losses of energy from reflected
light ; losses spent in heating a very thin surface, layer of the
metal by irregular motions of the electrions. This seems to me
probably the true dynamical explanation of the imaginary term
in i;» of § 158 (98). See also § 84 above.

§ 159'. In every case of practically complete opacity (that is
to say, no perceptible translucency from the most brilliant light
falling on a plate of any thickness greater than three or four
wave-lengths, or about 2 . 10~* cm.), measurement of the principal
azimuth, which can be performed with great accuracy by aid of
two nicols and a FresnePs rhomb (§§ 97 — 97" above), gives an
interesting contribution to knowledge regarding loss of luminous
energy in the reflection of light. The notation of § 94' gives

fif» sin^ a + (T» + £») cos» a (98«*)

as the reflectivity for light vibrating in azimuth a from the
plane of incidence (§ 88, foot-note); so that in every case of
practical opacity,

l-[S«sin»a + (T» + ^'»)cos>a] (98"»*)

represents loss of luminous energy by conversion into heat in a
thin surface stratum of less than 2 . 10~^ cm. thickness.

§ 159". For perfect reflectivity (98"**) must be zero for every
value of a, and for every incidence. Hence, as S, T, E are inde-
pendent of a, we have, at every incidence,

S« = r« + £> = 1 (98"*").

Hence for the incidence making 7=0 (that is the principal
incidence), /S' = -£" = 1 ; which makes Ej8 = ± 1 in every case of
perfect reflectivity, in total internal reflection for instance. There-
fore the principal azimuth, being ta,n''^ (E/8) for principal inci-
dence, is ± 45°, if the reflectivity is perfect. (See § 157 above.)

§, 159'". Observation and mathematical theory agree that the
principal azimuth is positive in every case : for interpretation of
this see § 158^^** above. They also agree that in every case short
of perfect reflectivity the principal azimuth is < 45®, not > 45°.

LECTURE XIX.

Friday, October 17, 3.30 p.m., 1884. Written afresh, 1903. •

Reconciliation between Fresnel and Oreen.

Molecular. THIS Lecture or "Conference" began with the consideration
of a very interesting report, presented to us by one of our twenty-
one coefficients, Prof. E. W. Morley, describing a complete solu-
tion, worked out by himself, of the dynamical problem of seven
mutually interacting particles, which I had proposed nine days
previously (Lecture IX. p. 103 above) as an illustration of the
molecular theory of dispersion with which we were occupied.
His results are given in the following Table.

Solution for Fundamental Periods, Displacement, and Energy
Ratios of a System of Spring-connected Particles. By Edward
W. Morley, Cleveland, Ohio.

m=l, 4, 16, 64, 266, 1024, 4096.
0=1, 2, 3, 4, 6, 6, 7, 8.

Fundamental Periods corresppnding to outer ends of

Springs 1 and 8 held fixed.

T> =

•2889

•9952

3350

11^862

89*12

137-89

680-2

1

$•4618

1^0048d

0-29849

0*0880078

00255607

0-0072564

00OU701

-I'-
ll

1-860

1^002

•546

•296

•169

•0851

•0383

A PROBLEM OF SEVEN VIBRATIONAL MODES SOLVED. 409

Displacement Ratios, or values of (— ).

Moleoula

*1

1

1

1

^2

-•231

1000

1-361

^.1

•014

-•341

1047

^4

-•ra27

•026

-•431

^8

•vl6

-•in50

•033

^6

- •vui 26

•v30

-•m68

^7

•XI 13

-•vin61

•v39

1

1466
1-689
1129
-•oil
•040
-•inSl

1

1

1^487

1496

1-761

1^813

1^787

1-997

1223

I960

-•681

1322

•045

-•628

Energy Ratios, or values of — l-l .

triiXi

1

1-499
1829
2066
2216
2208
1-717

Wh^l'

1

1

1

1

1

1

1

'Va'

•213

3-998

730

8^48

8-86

8-96

8-99

wtyCj^

•n33

1-864

1764

40-41

49-64

62-58

63-54

^W

•v47

•039

11-88

81-66

20436

266-34

273-14

^6^1?

•1x61

•IV 66

•28

66-73

382-71

98310

1167-62

^W

•XIV 7

•VIII 9

•ni47

1-62

34660

1788^18

4968-41

WyV

•XX 7

•XII 1

•vn63

•002

8-42

1616-99

1208004

Sum

1-216

6^902

380004

199-90

100067

4706-10

18642-64

At present our subject is the dynamical reconciliation between Molar.
Fresnel and Green, not ooly in respect to reflection and refrac-
tion at an interface between two isotropic transparent bodies,
as promised in Lecture XVIII, §§ 133, 139, but also in respect to
the propagation of light through a transparent crystal (double
refraction) as promised in Lecture XV. § 45.

§ 160. In my paper on the reflection and refraction of light
{Phil Mag. 1888, 2nd half-year), an extract from which is quoted
in ^ 107—111 of Lecture XVIIL, I have shown (§§ 109—111)
that a homogeneous portion of an elastic solid with its boundary

410 LECTURE XIX.

Molar, held fixed is stable if its rigidity (n) is positive ; even though its
bulk-modulus (it) is negative if not less than — fn. And in § 115
it is shown that the propagational velocity (v) of condensational-
rarefactional waves in any homogeneous elastic solid (or fluid if
n = 0) is given by the equation

•VH*-"

.(99).

In virtue of this equation it had always been believed that the
propagational velocity of condensational-rai*efactional waves in an
elastic solid was essentially greater than that of equivoluminal

waves, which is^/-. The Navier-Poisson doctrine, upheld by

many writers loDg after Stokes showed it to be wrong (see
Lecture XI. pp. 123, 124 above, and Appendix I below), made
it s §n (see Lecture VI. p. 61 above), and therefore the velocity of
condensational-rarefactional waves = iJS times that of the equi-
voluminal wave. The deviations of real substances, such as metals,
glasses, india-rubber, jelly, to which Stokes called attention, were
all in the direction of making the resistance to compression greater
than according to the Navier-Poisson doctrine ; but it was pointed
out in Thomson aud Tait (§ 685) that cork deviates in the opposite
direction aud is, in proportion to its rigidity, much less resistant
to compression than according to that doctrine. In truth, without
violating any correct molecular theory, we may make the bulk-
modulus of an elastic solid as small as we please in proportion to
the rigidity provided only, for the sake of stability, we keep it
positive. A zero bulk-modulus makes the velocity of condensational-
rarefactional waves equal to Vf times that of equivoluminal waves.
But now, to make peace between Fresnel and Green, we want for
ether ; if not all ether, at all events ether in the space occupied by
ponderable matter, a negative bulk-modulus just a little short of

— |n, to make the velocity (v) very small in comparison to a /-•

And now, happily (§ 167 below), a theory of atoms and electrions
in ether, to which I was led by other considerations, gives us
a perfectly clear and natural explanation of ether through void
space practically incompressible, as Qreen supposed it to be; while
in the interior of any ordinary solid or liquid it has a large enough
negative bulk-modulus to render the propagational velocity of

RETURN TO HYPOTHESIS OF 1887 WITH MODIFICATION. 411

condensational-rarefactional waves exceedingly small in comparison Molar,
with that of equivoluminal wavea

§ 161. In my 1888 Article I showed that if t; is very small,
or infinitely small, in proportion to the propagational velocity of
equivoluminal waves in the two mediums on the two sides of a
reflecting interface, or quite zero, Fresnel's laws of reflection and
refraction are very approximately, or quite exactly, fulfilled. About
fourteen years later I found that, as said in § 46 of Lecture XY.,
it is enough for the fulfilment of FresneFs laws that the velocity
of the condensational-rare&ctional waves in one of the two mediums
be exceedingly small. During those fourteen years, I had been
feeling more and more the great difficulty of believing that the
compressibility-modulus of ether through all space could be nega-
tive, and so much negative as to make the propagational velocity
of condensational-rarefactional waves exceedingly small, or zero.
One chief object of the long mathematical investigation regarding
spherical waves in an elastic solid, added to Lecture XIV. (pp. 191
— 219 above), was to find whether or not smallness of propaga-
tional velocity of condensational-rarefactional waves through ether
void of ponderable matter could give practical annulment of energy
carried away by this class of waves from a vibrator constituting a
source of light. I found absolute proof that the required practical
annulment was not possible ; and I therefore felt forced to the
conclusion stated in § 32, p. 214: "This, to my mind, utterly
"disproves my old hypothesis of a very small velocity for irro-
" tational wave-motion in the undulatory theory of light." Now,
most happily, seeing that it is enough for the dynamical verification
of FresneUs laws that the velocity of the condensational-rare-
factional wave be exceedingly small for either one or other of the
mediums on the two sides of the interface, I can return to my old
hypothesis with a confidence I never before felt in contemplating
it. It is, I feel, now made acceptable by assuming with Green
that ether in space void of ponderable matter is practically incom-
pressible by the forces concerned in waves proceeding from a source
of light of any kind, including radiant heat and electro-magnetic
waves; while, in the space occupied by liquids and solids, it has a
bulk-modulus largely enough negative to render the propagational
velocity of condensational-rarefactional waves exceedingly small in
comparison with that of equivoluminal waves in pure ether. We

412 LECTURE XIX.

Molar, have now no difficulty with respect to getting rid of the condensa-
tional-rarefactional waves generated by the incidence of light on a
transparent liquid or solid. They may, with very feeble activity,
travel about through solids or liquids, experiencing internal re-
flections, almost total, at interfaces between solid or liquid and air
or vacuous ether; but more probably (§172 below) they will be
absorbed, that is, converted into non-undulatory thermal motion,
among the ponderable molecules, without ever travelling as waves
through more than an exceedingly small space containing ponder-
able matter (solid, or liquid, or gas). Certain it is that neither
ethereal waves, nor any kind of djmamical action, within the body,
can give rise to condensational-rarefactional waves through void
ether if we frankly assume void ether to be incompressible.

(oleeular. § 162. So far, we have not been supported in our faith by
any physical idea as to how ether could be practically incom-
pressible when undisturbed by ponderable matter ; and yet may be
veiy easily compressible, or may even have negative bulk-modulus,
in the interior of a transparent ponderable body.

Do moving atoms of ponderable matter displace ether, or do
they move through space occupied by ether without displacing itt
This is a question which cannot be evaded: when we are concerned
>vith definite physical speculations as to the kind of interaction
which takes place between atoms and ether; and when we
seriously endeavour to understand how it is that a transparent
body takes wave-motion just as if it were denser than the
surrounding ether outside, and were otherwise undisturbed by
the presence of the ponderable matter. It is carefully considered
in Appendix A, and in Appendix B under the heading " Cloud I."
My answer is indicated in the long title of Appendix A, *' On the
"Motion produced in an Infinite Elastic Solid by the motion
" through the space occupied by it, of a body acting on it only by
" Attraction or Repulsion." This title contradicts the old scholastic
axiom, Two different portions of matter cannot simultaneously
occupy the same space, I feel it is impossible to reasonably gain-
say the contradiction.

§ 163. Atoms move through space occupied by ether. They
must act upon it in some way in order that motions of ponderable
matter may produce waves of light, and in order that the vibratory

REAL MOLECULAR EXPLANATION. 413

motion of the waves may act with force od ponderable matter. Moleculai
We know that ether does exert force on ponderable matter in
producing our visual perception of light; and in photographic
action ; and in forcibly decomposing carbonic acid and water by
sunlight in the growth of plants; and in causing expansion of
bodies warmed by light or radiant heat. The title of Appendix A
contains my answer to the question What is the character of the
action of atoms of matter on ether? It is nothing else than
attraction or repulsion.

§ 164. But if ether were absolutely incompressible and in-
extensible, an atom attracting it or repelling it would be utterly
ineffectual. To render it effectual I assume that ether is capable
of change of bulk ; and is largely condensed and rarefied by large
positive and negative pressures, due to repulsion and attraction
exerted on it by an atom and its neutralising quantum of electrions.
As in Appendix A, §§ 4, 5, I now for simplicity assume that an
atom void of electrions repels the ether in it and around it with
force varying directly as the distance from the centre for ether
within the atom, and varying inversely as the square of the
distance for ether outside. I assume that a single separate
electrion attracts ether according to the same laws; the radius
of the electrion being very small compared with the radius of any
atom. I assume that all electrions are equal and similar, and
exert equal forces on ether.

§ 165. While keeping in view the possibility referred to in
§ 6 of Appendix E, I for the present assume that the repulsion of
a void atom* on ether outside it is equal to an integral number of
times the attraction of one electrion on ether; same distances
understood. An atom violating this equation cannot be unelec-
trified. I continue to make the same assumptions as in Appendix D
in respect to mutual electric repulsion between void atoms and
void atoms; attraction between void atoms and electrions; and
repulsion between electrions and electrions. And as in Appendix A
an " unelectrified atom " is an atom having its saturating quantum
of electrions within it.

* For brevity I use the expression *' void atom *' to signify an atom having no
electrion within it,

414 LBCTURE XIX.

Mokr. § 166. I asBume that the law of compressibility of ether is
sach as to make the mean density of ether within any space which
contains a large namber of anelectrified atoms, exactly eqaal to
the natural density of ether andistnrbed by ponderable matter.
If there were just one electrion within each atom this assumption
would exactly annul displacement of the ether outside an atom by
the repulsion of the atom and the attraction of the electrion ; and
would very nearly annul it when there are two or m<H« electrions
inside. Thus the ether within each atom is somewhat rarefied
from the surface inwards: and farther in, it is condensed round each
electrion. In a mono-electrionic atom, the spherical snr&oe of
ncurroal density between the outer region of rare&ction and the
central region of condensation, I call for brevity, the sphere of con-
densation. In a poly-electrionic atom the density of the ether
decreases from enormous condensation around the centre of each
electrion to exactly normal value at an enclosing sur&ce,the space
within which I shall call the electrion's sphere of condensation.
It is very approximately spherical except when, in the course of
some violent motion, two electrions come very nearly together.

§ 167. I assume that the law of compressibility of ether is
such as to make the equilibrium described in § 166 stable ; but so
nearly unstable that the propagational velocity of condensational-
rarefactional waves travelling through ether in space occupied by
ponderable matter, is very small in comparison with the propa-
gational velocity of equivoluminal waves through ether undisturbed
by ponderable matter.

§ 168. Lastly I assume that the effective rigidity of ether in
space occupied by ponderable matter is equal to that of pure ether
undisturbed by ponderable matter. This is not an arbitrary
assumption: we may regard it rather as a proposition proved
by experiment, as explained in Lecture I. pp. 15, 16 and §§8r, 136
of Lectures XVII. and XVIII. But, as will be shown in Lee. XX.,
§ 237 below, it is somewhat satisfactory to know that it follows
directly from a natural working out of the set of assumptions of
g 163—167.

§ 169. Return now to § 132. This is merely Green's wave-
theory extended to include not only the equivoluminal waves but

DYNAMICAL THEORY OP ADAMANTINISM. 415

also the condensational-rarefactional waves resulting from the Molar,
incidence of plane equivolumiDal waves on a plane interface
between two sliplessly connected elastic solids of equal rigidities,
each capable of condensation and rarefaction. Let us suppose
that in either one or other of the two mediums the propagational
velocity of the condensational-rarefactional wave is very small.
This makes L very small as we see by using (48) and (50) to
eliminate a, c, and ,c from (61) which gives

l^ Cp"P)'8in«tu-^ ...(100).

fP 's/{v'~* — sin* iu~^) + p »J{,v~^ — sin* t it"*) * * *^

§ 170. If either the upper or the lower medium be what we
commonly call vacuum, (in reality ether void of ponderable matter),
the upper for example, we have v = oo , and (100) becomes

X — _%^zf:/«-i_. (101).

fpL sm ttt ' + p »J(ffr^ — sm* %u^) ^ '

whence when ,v is very small

£ i kZL£^ sin« tM-»,t) (102),

where N denotes approximate equality. If v and ,v are each very
small (100) becomes

L'AjPjzPl^^'^:!!^ (103),

thus by (100), (102) and (103), we see not only, as said above in
§ 133, that Fresnel's Laws are exactly fulfilled if either t; or ,t; is
zero. We see farther how nearly, to a first approximation, they are
fulfilled if i; = 00 , and ,i; is very small ; also if v and ,t; are both
very small.

Probably values of ,t;, or of v and ,t;, as small as w/500 or iz/lOOO
may be found small euough, but w/100 not small enough, to give
as close a fulfilment of Fresnel's Laws as is proved by observation.
See § 182 below.

§ 171. Thus so far as the reflection and refraction of light are
concerned, the reconciliation between Fresnel and Green is com-
plete : that is to say we have now a thoroughly realistic dynamical
foundation for those admirable laws which Fresners penetrating
genius prophesied eighty years ago from notoriously imperfect
dynamical leadings. (See §§ 106, 107 above.)

416 LECTURE XnL

Mobr. § 172. The only hitherto known deriation from absolate
rigour in Fresnels Laws for the reflection and refraction of light
at the sur&ce of a transparent body in air or void ether, or at the
interfiice between two transparent bodies liquid or solid, is that
which was discovered by Sir George Airy for diamond in air more
than eighty years ago, and many years afterwards was called by
Sir George Stokes the adamantine property. (See § 158 above.)
It is shown in the Table of § 105 above along with correspimding
deviations discovered subsequently by Jamin in other transparent
bodies solid and liquid. It now appears by (102) and (103) that
the explanation of these deviations from Fresnel, which even in
diamond are exceedingly small, is to be found by giving very small
imaginary values to ,v, or to v and ,v. To suit the case of light
travelling through vacuum or through air and incident on a trans-
parent solid or liquid, we should take v = x if the incident light
travels through vacuum ; or if through air, v perhaps = x , but
certainly very great in comparison with ,r. Hence in either case
the proper approximate value of L is given by (102).

§ 173. Looking to (59) and (67) of § 132 we see that if ,v have
a small real value (positive of course) the reflected ray, of vibrations
in the plane of incidence, will vanish for an angle of incidence
slightly less than the Brewsterian angle tan"' /i. If we give to ,v
a complex value ,p + i^ with ,p positive, we have the adamantine
property, and principal incidence slightly less than with ,j[> = 0.
Hitherto we have no very searching observations as to the perfect
exactness of tan~'^ whether for the polarizing angle when the
extinction is seemingly perfect or for the principal incidence in
cases of perceptible adamantine property. For the present there-
fore we have no reason to attribute any real part to ,v ; and we

may take

,t;»=-9»; ,v = iq (104),

where q denotes a real velocity. Using this in the last formula of
§ 128 and eliminating sin ,j/,v by (48) and cos ,j/,v and ,c by (49)
and (50) we find for the displacement in the refracted conden-
sational-rarefactional wave

^/P-.cM:-iV(l+aV)y/g] .(105).

Looking now to (63) and (102) we see that in fH/0 there is an
imaginary part which is exceedingly small in comparison with the

IMAGINARY VELOCITY OF CONDENSATIONAL WAVES. 417

real part, because, by (102), Lj^v is real and not small, while L is Molar,
imaginary and exceedingly small. For our present approximate
estimates we suppose G to be real and therefore ,H to be real.
Taking now /(f) = 6**' in (105) and taking the half-sum of the
two imaginary expressions as given with ± t, we find for the real
refracted condensational-rarefactional displacement the following
expression ; with approximate modification due to qa being a very
small fraction ;
D^,H cos (o{t-ax) €«^^(i+«V)!f/« ^ fl cos © (e - cw?) e*^'^. . .(106).

Hence q is positive. (See footnote on § 168".) The direction of
this real displacement is exceedingly nearly OY' of § 117, that is
to say the direction of Y negative, because the imaginary angle ,j
differs by an exceedingly small imaginary quantity from 90°. It
would be exactly 90° if /v were exactly zero; and it would be
less than 90° by an exceedingly small real quantity if /v were an
exceedingly small real velocity.

§ 174. The motion represented by (106) is not a wave
travelling into the denser medium : it is a clinging wave travelling
along the interface with velocity a"*'. The direction of the displace-
ment is approximately perpendicular to the interface. Its magni-
tude decreases inwards from the interface according to the law of
proportion represented by the real exponential factor; distance
inwards from the interface being — y. The period of "the wave is
27r/o) ; and the space travelled in this time with velocity q is 2irqjo>,
Hence the displacement at a distance from the interface equal to
this space, is 1/e-' or 1/535 of the displacement at the interface.

§ 175. Consideration of the largeness of any such distance as
2 . 10~* cm. (§ 80 above) between centres of neighbouring atoms,
and of the smallness of i) if real, will probably give good dynamical
reason for the assumption of ,v a pure imaginary. It is a very
important assumption, inasmuch as it implies that there is no
inward travelling condensational-rarefuctional wave carrying away
energy from the equivoluminal reflected and refracted rays.

§ 176. Let us now find the value of qju, the adamantinism
(§ 158); to give any observed amount of the adamantine property,
as represented by the tangent of the Principal Azimuth, which is
Jamin's k. In (59) of § 132 take for L the value given by (102),

and put iiP—Pl^N; ,v=cq; ^ = a (107),

T. L. 27

41S LECTURE XIX.

Molar, thus we find

(las)^,

where P^W'ii .pbf - {p,hf\ - (3V sin* %f\ ^

Q = 2u,pbXa sin* i i

Hence taking in § 128,/U) == c*^: and realising by taking half-som
of solutions for ± i, we find displacement

in incident wave= Gcoewf^ — j (11®X

in reflected wave =- G v(r* + J?*)co8 jw;'*--^ -^ |...(111X

= -^ a/ ,.^ — ^^^r^ — ;i^ — ^.^co8 »(' — )-♦ ...(111 )\

0 JF \

where ^ = tan~* ^ = tan~' ^ !

=? fowl— 1

= tan

2^' (a* — 1) cr COS i sin- i

....(112)l

/i- cos* 1 — sin* 1 — (/i* — I )* a* sin* i I

The developed form of Q P here given is found by putting

• • •
sin I

,pp = u?\ i/6 = coeji; M,6 = /xcos,i*: cos?,t = l -— ...<113X

and by dividing numerator and deiioniinator by {p?— 1) m'-

* According to Ihe notation of § ^' we lure Q;P=EIT, and
r=^r^r ... ^ and E= ^

u^{,pb^ p.bY -r {S ain« i g/tt>* «» (.p6 -r a*)* + l-V sin« i g/«)^ "

t This alternative form (111') comes firom (111) by molTing P*+^ inio tvo
factors according to the algebraic identity

(a* - 63 - c^-r 4aV=[(a + 6)*+c*] [(a - 4)« -r cT.

Bot it is foond directly by treating (108) as we treated a similar formola in (75).
following Green, on a plan which is simpler in respect to the resoltant magnitude,
bot lem simple in restpect to the pluue, than the plan of (HI).

ADAMANTINE REFLECTION. 419

§ 177. By (111) aud the notation of § 94', we have Molar.

\,pb + p,bj Ltan(t+,i)J

^ 4p,p6,6 (i^cr sinS^

{,pb + p,br [u^ (.pb^-pjbf + (NasinHf] •••^^^*^'

or approximately

■*"^ Ltan(i + ,i)J " u'i^pb-hpfiy -V^^^^

In (114) and (115) !r« + £« denotes the whole intensity of the
reflected light, due to incident light of unit intensity vibrating
in the plane of incidence, and the smallness of the right-hand
membei-8 of these equations shows how little this exceeds that
calculated according to Fresnel's formula

ftan (t — ,t)"|'
Ltan (i + ,t)J

Compare ^ 103 — 106 above.

A

Fig. 10.

§ 178. In (110) and (111), « denotes a length AEF (fig. 10)
in the line of the incident ray produced through E the point of
incidence, or AEF' an equal length in the path of the incident
and the reflected ray. Thus we see that the reflection causes
a phasal set-bivck <j) which increases from O"" through 90° to 180°
when the angle of incidence is increased from 0° to 90°, as shown by
(112); because (112) shows that tan <f> increases through positive
values from 0 to + oc when i increases from 0 to a value slightly
less than tan~^ /i : and, when i increases farther up to 90°, tan <j)
increases through negative values from — oo to 0. These variations
are illustrated by the accompanying diagrams (figs. 11, 12) drawn
accoifling to the following Table of values of tan (f> and <j) calculated

27-2

-If

"7 %. *«t « 7 f \f ^ ^ %.»-i x^ ^-^ft:

52

1

i ^

3 <«k.

DIAGRAM OF PHASAL LAG IN ADAMANTINE REFLECTION. 421

from (112) with cr = '00293 as for diamond accordiDg to §181 Molar,
below. Note how very slowly tan <j) and ^ increase for increasing

Diamond fi=2'iM ^= -00293

20

so ,40 so

/ tvciaLtn ce»

70

Fig. 12.

m

t

tan^
•0

0

•

t

tan^

±00

■

0^ 1

/

90°

V

•000008798

OP 03'

67°7

- 14-87770

93° 51'

10^

•0008890

0^3'

67°75

-6-95857

98° 11'

20'-' 1

•003678

QPIB'

6r»^8

1

-4-53967

102° 25'

' 30' '

•008835

(PS(y

: 67°-9

- 2-67607

110° 29'

40* 1

•01772

vr

68^0

-189599 '

117° 49'

50^ I

•03469

vw

68^1

-1-46724

124° 17'

60^' 1

•08784

5°r'

68°-2

-1 19615

129° 54'

' 05'

•25164

140 7/

1

68^3

-1-00921

134° 44'

67'

1^00386

45^7'

68^-4

- -872514

138° 54'

6r-i

1 18360

49° 48'

: 70^

-•27083

164° 51'

67' -2 •

1

144216

65^16'

75°

-•07694

175° 36'

67-3

1-85085

61** 37'

80°

-•01640

179° 4'

67-4

2 56306

68<>41'

89°

--00299

179° 60'

67-5

4-20197

76' 37'

90P

•00000

180° 0'

67-55

618014

80-49'

m

67' -6

11-70160

85'' 7'

tan "'a* -

0' -00883 = /

90' 0'

422 LECTCRE XIX.

Molmr. incideDce^, except thoee very slightly less than, and veiy slightly
greater than, the Principal Incidence, 67''65. Note also how very
suddenly tan ^ rises from a small positive valne to + x and from
— X to a very small negative value when the incidence increases
through the Principal Incidence. Note also how suddenly there-
fore (f>, the phasal retardation, increases fix>m a small positive
quantity up to W when the incidence reaches 67°'65 ; and, when
the incidence increases to slightly above this value, how suddenly
the phasal retardation rises to very nearly 180^ But carefully
bear this in mind that tan ^ is essentially zero for t = 0, and for
I = 90% as shown by (112).

§ 179. The sign minus before G in (111) signifies a phasal
advance or back-set of 180\ For incidence and reflection of ravs
having vibrations perpendicular to the plane of incidence, we have
also essentially the sign minus before C in (116) below, taken
from (20) and (23) of § 117 and (89) of § 123 as corresponding
to (110) and (111). Thus for amplitudes we have

incident wave C cos i»(t ] , |

reflected wave — C ' j — i cos \m{t — ]

Here there is neither retardation nor acceleration corresponding to
the ^ of (111). We infer that if plane polarized light &lls on the
reflecting sur&ce with its Wbrational plane inclined at any angle 2
to the plane of incidence, and if we resolve it ideally into two
components having their vibrational planes in and perpendicular
to that plane, the component of the reflected light due to the
former will be phasally behind that due to the latter by the
retardation^ given by (112). Hence for every incidence the
reflected light will be elliptically polarized ; or circulariy in case of
phasal difference 90', and equal amplitudes, of the two reflected
components. The values of i and of a which make the reflected
light circularly polarized, are called the " Principal Incidence "
and " Principal Azimuth " (see § 97 above).

§ 180. The Principal Incidence, /, is the value of t which
annuls the denominator of Ej-T as seen in (112). It is therefore
given by the equation

tan=/ = /A--(Ai«-l)»a»tan*/sin*/ (117).

(116).

PRINCIPAL INCIDENCE IN ADAMANTINE REFLECl'ION. 423

Notwithstanding the greatness of (/i'— 1)' for substance of high Molar,
refractivity (Example 126*7 for Sulphuret of Arsenic) the second
term of the second member of (117) is very small relatively to the
first because of the smallness of {qjuy for all transparent substances
for which we have the observational data from Jamin. (See § 182
below.) Hence in that second term we may take /a' for tan*/
and /*'/(/*' + 1) for sin'/. Thus instead of (117) to determine the
Principal Incidence / we have

^r^^^^^rf^ aw

§ 181. To find the Principal Azimuth a; let D be the vibra-
tional amplitude of plane polarized incident light ; and D cos a,
D sin a the vibrational amplitudes of the components in and
perpendicular to the plane of incidence. Thus with the notation
of g 176, 179 we may take

G=i)cosa, C=D8ina (119).

These must be such as to make the two components of the reflected
light equal when the angle of incidence is that given by (117)
(the Principal Incidence); that is to say, the value of i which
makes P= 0 ; or approximately according to (109), ,pb—p,b. This
makes the amplitude of (111) approximately equal to — OE, or

-Gj-r^c,, or -Go^^r"."ix^ (120).

4(u,p6)»' 2V(m" + 1)

Equating this to the second line of (116) and using (119), we see
that the condition for circular polarization is

^"^«w(/?+ir=^^'^«,6+6 ^^^^>-

Now the angle of incidence is approximately tan~^/ii; for which
we have

£^ = A*'_--l. (122)

and therefore for a, the principal azimuth, we have

tana = H/**-l)V(A*' + l)f (123).

4f4

^ Tuns :c

T

Jil^. Tiir siA^ziiat :c li

3Ii£lLlaiCi=il

*2mtic» L

SCiUEIlIli'

Finr

-■ *.

AJMianiat A.ksuxjjL

r=r I

*IL*~ h-i»?n "ir -»

-HP7W l--l«a?tt "TT-IJ

-«!2» I— llW« v

'W«.r4 r^tiW<i*f iff -11?

- A*H I- -M*!*:^^* 5? - IS

t i*5

-•iiifrr

. Ai

ilt-'T-ET^l

••z'.:^ -rr-.i .

1— ^t

m m

■=*'."iit:..— ;h. 2;cj^tj:c- liTrr inrf iiv-i ;c -z^r^i-viTef zz. & rr^^

HUYGHENS CONFIRMED EXPERIMENTALLY BY STOKES. 425

strain as the clastic solid theory has seemed to require. See MoUr.
Lecture I., pages 17, 18, particularly the words "//* the effect
" depends upon the return force in an elastic solid" In the same
lecture (page 19) I told of an explanation of this difficulty
suggested first by Rankine and afterwards by Stokes and by
Rayleigh, to the effect that the different propagational velocities
of light in different directions through a crystal are due, not to
aeolotropy of elastic action, but to aeolotropy of effective inertia.
But I had also to say that Stokes working on this idea, which
had occun-ed to himself independently of Rankine's suggestion,
had been compelled to abandon it for the reasons stated in the
following reproduction of a short paper of twenty-one lines which
appeared in the Philosophical Magazine for October, 1872 ; and
which is all that he published on the subject : —

"It is now some years since I carried out, in the case of Iceland
" spar, the method of examination of the law of refraction which
** I described in my report on Double Refiuction, published in the
"Report of the British Association for the year 1862, p. 272.
" A prism, approximately right-angled isosceles, was cut in such
a direction as to admit of scrutiny, across the two acute angles,
in directions of the wave-normal within the crystal comprising
" respectively inclinations of 90° and 45° to the axis. The directions
of the cut faces were referred by reflection to the cleavage-
planes, and thereby to the axis. The light observed was the
bright D of a soda-flame.

" The result obtained was, that Huyghens' construction gives
*' the true law of double refraction within the limits of errors of
" observation. The error, if any, could hardly exceed a unit in
" the fourth place of decimals of the index or reciprocal of the
" wave-velocity, the velocity in air being taken as unity. This
" result is sufficient absolutely to disprove the law resulting from
" the theory which makes double reftuction depend on a difference
"of inertia in different directions.

" I intend to present to the Royal Society a det^led account
" of the observations ; but in the meantime the publication of
" this preliminary notice of the result obtained may possibly be
" useful to those engaged in the theory of double refiuction."

It is well that the essence of the result of this very important
experimental investigation was published : it is sad that we have

(I

u

tt

tl

u

4itj LEcnrkE xix.

MsiMt. ittA ih'c Aa:h>.»^':^ itiieoilt^ <xrmmaiiicath>D Uf the BiAal Sicic^T
describing Lis work.

The ojrnEs^p>Dding experiiiHrDuJ t€s$t for a KaxiI Crystal.
rEs&aliiiig al3t> in a miDotelr acctxiate verincadoQ of FrEsake"^
ynkre-Stuiace vas carri€»i oat a few veais laier br GlaaebrociL*.
Most ve theref*jfe give op all idem of explaining the dxiKieflit
rehxities of light in different directions throogh a crystal bj
aecrlotropic inertia ? Yes, certainlr. if, a$ asimmted by Greetik aad
SixJaet, eiker in a crfUal is iwoompreagibU,

I 1S5. Bat Glazebrook has pointed oat as a caas&fiewice of
mj saggesti«jn of approximately zero Telocitj of coodenaadonal-
rare&cti'jnal waves in a transparent s»jlid or li-^oid, that with thi«
asBomption a<jlotropic inertia gives precisely Fresnel s shape of
wave-sariSMe aod Fresnel* (dependence of veUjcitr on direction
of vibration, irrespectivelv of the direction of the sUain-axiiSL
TTia5 after all we have a dvnamical exp!anati«>o k4 Freaiel's law«
of light in a crystal which we may accept as in all probability
abeolotety trae. To pn>ve this I»:t os nrst investigate the ooo-
ditions Pyr a plane wave in an is»>trop:c elastic solid with any
given values f*jr iu^ two m«jdalus^^ of elasticity : t balk-moduias
and n rigidity-modalos : and with aeolotP>pic inertia in respect lo
the motion *j{ any small part •A its sabstaoce. This s«>l«Xiopy «jf
effective inertia of ether thpjugh the sabstaDce of a transparent
crvstal follows natorallv. we mav almoe; sav inevitablv, from
the Molecalar Thet>ry of g 162 — loS. In Lecture XX- details
on which we need not at present enter will b^ carefally con-
sidered.

I 186. MeaDtime wtr may simply assume that Bp^r Bp^. Bp^
are the virtual masses, or inertia-e>:|aivaleuts. relatively to motiofKs
parallel to x. y. z. of ether within a ver^' small volume B containing
a large number of the ponderable atoms concerned : so that

^^^'% ^P'jt' ^f^§ ^'«»

are the forces which must act up*>n the ether to produce com-
ponent accelerations d( dt. etc. Hence the equations of molar

• ftt

An csperiiaaital drtermiittsioo of the Ta2aM of the TekxitiB of N
PrapBiBftSioB of Plftae WmTa in difieum dizcctioB* in m KaxsI CzritaL aad a
puiMO of the Rcsnhi vith tfaeotr.'' Br B. T. Glairtiook. CommnkaSed W
J. Clot MasveO. PkiL r-cw. Rmm- >*^ l^T*. Vcl. 170.

.*:OLOTROPIC INEIITIA FOR WAVE-MOTION IN A CRYSTAL. 427

motions of the ether will be (8) of § 113, modified by substituting MoUr.
for p in their first members p«, py, />, respectively.

§ 187. To express a regular train of plane waves, take

(125),

where X, fi, v denote the direction cosines of a perpendicular to the
wave-planes; and a, /8, 7 the direction cosines of the lines of
vibration. These, used in (7) of § 113, give

— cos^ d ^{. \x '\' im '\' vz\

__ — cos^ ^ ///

u--)

where

cos ^ = oX 4- fifi + yv.

(126).
(127).

Thus ^ is the inclination of the direction of the displacement, to
the wave-u*)rmal.

Using (124), (125), (126) in the etiuations of motion, and
removing from each side of each the factor

d' .,

dt

'./[t-^^T-"') <i2«).

we Knd

., , .Xcosft a

p,/^=(i + i«)^,-+«£; >

.(129).

§ 187'. From these equations we determine the direction-
cosines (a, /8,7) of the vibration; and the propagational velocity, u,
of the plane wave (X, /i, v) ; thu.s : First solving for o, /8, 7, and
putting

Px^nla\ py = nlb', Rz^ri/d" (129'),

428

LECTURE XIX.

Molmr. we find

a =

/8 =

7 =

k + i» Xa' cos ^

n

u^ — w

A? + Jn /ifc* cos Sr
i 4- in vc* cos ^

n

u»-c»

(130).

whence by a' + /8' + 7' = l,

Multiplying the first of (130) by X, the second by /*, the third by
V and adding ; and removing the common factor cos^ ; we find

^- n U'-a» + tt»-6» + u'-c»>' ^"^^

This is a cubic for determining u\ the square of the propagational
velocity^ The three roots are all obviously real ; one greater than
the greatest of a', 5*, c*, and the other two between the values of
these quantities. For each value of u* the corresponding direction
of vibration is given by (130).

§ 187". Calling m,», uJ two of the three roots of (132) we find,
by writing down the equation for each of these and subtracting
one equation from the other, the following

^• + |n r X»a» fi^b^

n \_{u^^ - a«) (t/,* - a^) "^ (Wi« - 6«) (a,« - fr*)

-^ r-i—^^^-^-:T^ («i'-««') (^32').

(W,» - C») (W2« - c*)J ^

And taking the corresponding notation in (130) we find

(Comparing these two equations, we see that «iO, + /8i/8j + 7i7«
cannot generally be zero : that is to say the vibrational lines
corresponding to parallel wave-planes travelling with different
velocities cannot generally be perpendicular to one another. This

CONDENSATIONAL-RAREFACTIONAL WAVE VERY SLOW. 429

result for the new theory, in which diflFerences of velocity are due Molar,
to inertial seolotropy, presents an interesting contrast to the
theorem of Lecture XII., page 136, that the vibrational lines in
any two of the three waves are mutually perpendicular; when there
is no inertial aeolotropy, and the differences of velocity are due to
seolotropy of elasticity.

§ 188. In each of the two extreme cases of A: = oo and A: = — |n
the cubic sinks to a quadratic, as we most readily* see by (in
virtue of X» + /i* + i/» = 1) writing (132) thus

1^"^^ "^ u' - 6* "^ w' - c* " n U"-a'"^u«-6»"^w»-c»y

(133).

§ 189. From this we see that t/ A: + fn is very greats one root
of the cubic for u^ is very great, being given approximately by

ti» = ^-i^(X«a» + /ii*« + i;*c») (134);

while the other two roots are approximately the roots of the
quadratic,

-^ i+ -rj;2+-i i = 0 (135).

This agrees with the propagational velocities for a given wave-
plane (X, fly I/), found by Stokes and by Rayleigh from Rankine s
hypothesis of aeolotropic inertia, and Green's assumption of a
virtually infinite resistance against compression. It implies a
wave-surface proved observationally by Stokes for Iceland spar
(uniaxal), and by Glazebrook for arragonite (biaxal), to differ from
the truth by far greater differences than could be accounted for
by errors of observation.

§ 190. But ifk^-^nia very small, one of the three values of m'
given by (133) is very small positive, being given approximately

&:-^S»--'in*-" w^

* Another way of managing this detail will be found in the inveHtigation of
chiral waves in § 20B of Lee. XX.

430 LECTURK XIX.

Molar, while the other two are approximately the roots of the quadratic

J?^ + ^+^-0 <137).

This is Fresnel's equation [see (94) of Lecture XV.] for the
two propagational velocities for a given direction of wave-plane
(X, /A, 1/). It implies precisely Fresnel's celebrated wave>8urfisu»

fri^^;^6»^i^^:^"^ ^^^^

or r* (rt-a:* + fcy + c»^) - a» (6» + c») ar* - 6* (c* + a*) y* - c»(a« + 6»)r«

+ a*6V=0...(138'X

This equation is got, not now *' after a very troublesome algebraic

process," as said by Air}"* in 1831, but by a very short and easy

symmetrical method given by Archibald Sfnith in 1835+ ; the

problem being to find the envelope of all the planes given by the

equation

Xx + /Ay + y^ = ti \

with X« + m' + »''=1 (138")-

and (137) for u )

§ 191. The theorems of Giazebrook and Basset, stated in
§ 195 below, are readily proved by using (139) in connection with
Archibald Smith's now well-known investigation of Fresnels wave-
surface. But the theor}' on which it is founded implies essen-
tially condensation and rarefaction and therefore a direction of
vibration not in the wave-plane : and not agreeing with Fresnel's
' which is exactly in the ware-plane (corresponding as it does to
strictly equivoluminal waves). For our present case of X: -h |n = 0,
the vibrational direction (a, /8, 7) as given by (130) is

(i*Xcos& _ b-Ltcos"^ c^vcos^ ,^-,,

w — a- w — (r »' — c**

and <131) for ^. the angle lx*t\veen the vibrational direction and
the wave-normal, becomes

soc--^ = h"M+ ( fV '+ ( -«V <1^)-

• .\iTy'8 Matknmatical Tract*, p. S53.

f TraMi. Camhridgt PkiL S,^. Vol. n. p. 85; also Phil. Mag. Vol. xn. 1838 (1st
hftlf-jMr), p. 335.

VIBRATIONAL LINES NOT EXACTLY IN WAVE-PLANE. 431

§ 192. For an example, take arragonite with, as in Lee. XY. Molar.
§ 37, (and n = 1 for simplicity)

1/a = 1-5301 ; a = -65355 ; a« = -42713'

1/6 = 16816; 6 = -59467; 6* =35363}^ ... (141);

1/c = 1-6859 ; c = '59315 ; c' = 35183

and let us find the two propagational velocities (Ui, li,) and the
inclinations {%, %) of the vibrational lines to the wave-normal, for
the case in which the wave-normal is equally inclined to the three

principal axes (X = /Lt = i; = l/JS). By the solution of the quadratic
(137) and by using the roots in (139) we find

tii» = -40234; wr' = l'57653; ai = 8r37'-68 ,

' V...(142).

ii,«= -35272; i/r' = 168377 ; ^a = 89°49'-24 ^

y..A

Remarking that i^"' and t/,""^ are the indices of refraction for the
two waves of which the normals are each equally inclined to the
three principal axes (x, y, z) it is interesting to notice how nearly
the greater of them is equal to ^(1 -6816 +1*6859), the mean of ^le
two greater of the three principal refractive indicea This, and the
nearness of ^2 to 90°, are due to the smallness of the diflFerence
between the two greater principal indices: that is to say, the
smallness of the difference between arragonite and a uniaxal
crystal. In fact u^^, the second of our solutions of the quadratic,
corresponds to what would be the ordinary ray if the two greater
principal indices were equal.

§ 193. To help in thoroughly undei-standing the condensations
and rarefactions which the theory gives us in any plane wave
through a biaxal crystal, take as wave-plane any plane parallel to
one of three principal axes, 0 Y for instance. It is clear without
algebra that the directions of vibration in the two waves for
every such direction of wave-plane are respectively parallel and
perpendicular to OY, The former corresponds to the ordinary
ray: its vibrational direction is parallel to OY: it has propa-
gational velocity '^(n/py), or 6, according to (129). It is a strictly
cquivoluminal wave. All this we verify readily in our algebra by
putting fi = 0; which gives for one root of the quadratic (135)
?/2= 6-, and gives by (139) soc*^ = 00 and therefore & = 90".

485 LECTURE XII.

Molar. ; 194. For the other root of the quadmtic (135> we hare

U*^€^ If* — C*

whence ii' = r*X* + ciV : )

M«-a> = (c«-a*)X-; -^ Cl-131

Using these in (139) we find

^ = ^ a«i

Patting now jf = 0 in ilSS'), the equation of Fiesners ware-
snrfiuTe, we find for its intersection with the plane XOZ

(r»-fc*>(aV + c'r-aVl = 0 a4oi

This expresses, for the ordinanr rav, a circle r = 6 : and for the
ry ray an ellipse.

•

2 + -.= l <1461

Tne waTe-plane touches this ellipse at the point {x, r); hence

X a*x _ -^^

- = — (l^il

w cz

Taking this for X f in <144l we see that the Tibrational line l<
perpendicniar to the ray-diiectioD. which is the radios rector of
the ellipse through the point {x, zy This is only a particular case
of the general theorem given by Glazebrook, PhiL Mag, Dec ISSS
page 52S, that the direction of vibration is perpendicular to the
ray in the new theory of double refraction founded on aeolotropic
inertia.

§ 195. Moreover in this theory* Basset has given a ma?t
interesting theorem •"• to the effect that the direction of the vibration
is a line drawn perpendicular to the radius vector from the foot
of the perpendicular to any plane touching the wave-sur&ce (the
perpendicular to the radius vector being drawn from the centre of

X'% Pk^meal Opiies (Cambridge, 1^921. j d&S.
f Given alio for m TeiydilEerait dynamiaJ iheorx inTohring zero Tdocitr of ooo-
(ipwtinnil-rmrHarrinnal vmTe, by Sama in his Second Pmper on the Piropagmtkn
and the Polariiatinn of Light in Crjital* ; LiomriiWs JomrmaL Tol. xin. ISfiS. p. d&.

OLAZEBROOK AND BASSET'S DRAWING FOR VIBRATIONAL LINE. 433

the wave-surface to the tangent-plane and to its point of contact). Molar.
This includes Glazebrook's theorem and appends to it a simpler
completion of the problem of drawing the vibrational line than
that given by Qlazebrook in pages 529, 530 of the volume of the
Philosophical Magazine already referred to. The construction is
illustrated in figure 13, drawn exactly to scale for the principal
section through greatest and least principal diameters of wave-
sur&ce for arragonite. P is the point of contact of the tangent-
plane KM. OP is the radius vector (optically the ray). F is the
foot of the perpendicular from the centre of the wave-surface.
FNf perpendicular to OP, not shown in the diagram, is the direc-
tion of the vibration. It would be interesting to construct on
a tenfold scale a portion of figure 13 around ^P; but it is perhaps
more instructive to calculate the angle NFP (which is equal to
FOP) and the radius of curvature, at P, of the ellipse. Figure 13
illustrates the construction for any wave-plane whatever touching
the wave-surface in the point P ; though it is drawn exactly to
scale only for the extraordinary ray in the principal section of
arragonite, through the greatest and least principal diameters.

b s 5-947

e s e*630

X

>

Fig. 13.

T. L.

28

434 LBCTURE XIX.

MoUr. I iQQ, Going back now to § 194 we see that the intersectious
of Fresnel's wave-surface with the three planes YOZ, ZOX, XOY,
are expressed respectively by the following equations

(r» - 6*) (c»z« + a»aj» - <?a«) = 0 ^ (148),

which prove that each intersection consists of an ellipse and a
circle ; the ellipse corresponding to the extraordinary ray, and the
circle to the ordinary.

§ 197. Lastly consider wave-planes perpendicular to one or
other of OX, OY, OZ. Take for example OX] we see that the
two propagational velocities are b and c, with vibrational lines
respectively parallel to OF and OZ. The physical explanation
is much more easily understood than anything which we thought
of in Lecture I. pages 17 to 20, when we were believing that
differences of velocity were due to aeolotropy of elasticity. We
now see that the two waves travelling a;-wards have different
velocities because of greater or less effective inertias of the moving
ether in its vibrations parallel respectively to 0 F and OZ.

§ 198. The fundamental view given by Fresnel for the
determination of his wave-surface by considering an infinite
number of wave-planes in all directions through one point, and
waves starting from them all at the same instant, is most im-
portant and interesting; truly an admirable work of genius!
It loaves something very definite to be desired in respect to the
geometry and the dynamics of a real source of light travelling
in all directions from a small portion of space in which the source
does its work. It therefore naturally occurs to consider what may
be the very simplest ideal element of a source of light.

§ 199. Preparation was made for this in the "molar" divisions
of Lectures III.... VI. and Vin....XIV. and particularly in pages
190 to 219 of the addition to Lecture XIV. What we now want
is an investigation of the motion of ether in a crjrstal (Jue to an
ideal molecular vibrator moving to and fro in a straight line in
any direction. The problem is simplified by supposing the direc-
tion to be one of the three lines OX, OY, OZ, of minimum, and of
minimax, and of maximum effective inertia of ether in the crystal

THEORY FOR VIBRATIONAL AMPLITUDE IN WAVE-SURFACE. 435

It seems to me that this should be found to be a practicable Molar,
problem. Towards its solution we have Fresnel's wave-surface
as the isophasal surface of the outward travelling disturbance or
wave-motion : and we have the direction of the vibration in each
pai-t of the surface by the theorems of Glazebrook and Basset
(§ 195 above). What remains to be found in our present problem
is the amplitude of the vibration at any point of the wave-surfisuie.
One thing we see without calculation is, that at distances from
the origin great in comparison with the greatest diameter of the
source, the vibrational amplitude is zero at the four points in
which the wave-surface is cut by the vibrational line produced
in both directions from the source ; when this line coincides with
one of the three axes of symmetry OX^ OY, OZ, The general
solution of the problem is to be had by mere superposition
of motions from the solutions for vibrations of the source in the
three principal directions. For the present I must regretfully
leave the problem hoping to be able to return to it later.

28—2

LECTURE XX.

Friday, October 17, 5 p.m., 1884. Written afresh, 1903.

Molecular. § 200. CoNSiDERiNQ how well Rankine's old idea of seolotropic
inertia has served us for the theory of double refraction, it natur-
ally occurs to try if we can found on it also a thorough dynamical
explanation of the rotation of the plane of polarization of light in
a transparent liquid, or crystal, possessing the chiral property.
I prepared the way for working out this idea in a short paper
communicated to the Royal Society of Edinburgh in Session
1870—71 under the title "On the Motion of Free Solids through
a Liquid " which was re-published in the Philosophical Magazine
for November 1871 as part of an article entitled " Hydrokinetic
Solutions and Observations," and which constitutes the greater
part of Appendix G of the present volume. The extreme diffi-
culty of seeing how atoms or molecules embedded in (ether), an
elastic solid could experience resistance to change of motion
practically analogous to the quasi-inertia conferred on a solid
moving through an incompressible liquid has, until a few weeks
ago, prevented me from attempting to explain chiral polarization
of light by sBolotropic inertia. Now, the explanation is rendered
easy and natural by the hypothesis explained in ^ 162 — 164 above
and in §§ 204, 205 below and in Appendix A.

§ 201. To explain aeolotropic inertia, whether chiral or not, of
molecules in ether, from the rudimentary statements in Appendix A,
take first the very simplest case; a diatomic molecule (^i, A^ con-
sisting of two equal and similar atoms held together by powerful
attraction ; so as, with a single electrion in each, to constitute a
rigid system when, as we shall at first suppose, the forces and
motions with which we are concerned are so small that the

ELECTRO-ETHERIAL EXPLANATION OF iKOLOTROPIC INERTIA. 437

electrions have only negligible motion relatively to the atoms. Molecular.
This supposition will be definitely modified when we come, in
§§ 232 — 242, to explain chromatic dispersion on the new theory.

§ 202. In fig. 14 the circles represent the bounding spherical
surfaces of the two atoms. According to the details suggested for
the sake of definiteness in Appendix A and illustrated by its
diagram of stream-lines (fig. 5), the two atoms must overlap as
indicated in our present diagram fig. 14, if the stream-lines of
ether through each atom are disturbed by the presence of the
other. Without attempting any definite solution of the extremely
difficult problem of determining stream-lines of ether through our
double atom we may be at present contented to know that the
quasi-inertias of the disturbed motion of ether within the molecule,
in the two cases in which the motion of the ether outside is parallel
to AiA^, and is perpendicular to AiA^, must be different. It seems

Fig. 14.

to mo probable that the former must be less than the latter: I shall
only assume however that they are different, which is certainly
true : and I denote the former by a and the latter by )8. This
means that if the molecule is at rest, and the ether outside it is
moving uniformly according to velocity-components (|, ^) respec-
tively parallel to AiA^, and perpendicular to it in the plane of
the diagram, the kinetic energy of the whole etherial motion will
be greater by i(^' + ^^0 ^^^^ ^^ ^^^ ether had the uniform
motion (f, rj) everywhere. Thus if B denote any volume of space

438 LECTURE XX.

Moleoular. completely surrounding one molecule but not including another,
and if p be the undisturbed density of ether, the kinetic energy of
otherial motion within B is

HBp(i' + n') + <4'+^v'] 049)-

Throughout §§ 202 — 205 we are supposing the different constituent
molecules of the assemblage to occupy separate spaces. In § 225
we shall find ourselves obliged to assume overlapping complex
molecules of silica in a quartz crystal.

§ 203. This gives a clear and definite explanation of the
seolotropy of inertia (§ 184 above) suggested by Rankine for
explaining double refraction. First for a uniaxal crystal, consider
any homogeneous assemblage (Appendix H, §§ 3, 6, 15, 16, 17, 18,
19) of our diatomic molecules and let B be the volume of space
allotted to each of them. The homogeneousness of the assemblage
implies that the lines AiA^ in all the molecules are parallel Take
this direction for OX; and for OY, OZ any two lines perpendicular
to it and to one another. In respect to double refraction it is of
no consequence what the character of the homogeneous assemblage
may be : though it may be expected to be symmetrical relatively
to the direction oi A^A^ because the equilibrium of the assemblage,
and the forces of elasticity called into play by deformation, depend
on mutual forces between the molecules not directly concerned
with the elasticity and motions of the ether called into play in
luminous waves. In short, the essence of our assumption is that
the molecules are unmoved, while the ether is moved, by waves of
light. Write (149) as follows

5[i(/'+j)i'+i(/>+D^'

= -B.iO>«p + /)y^)...(loO),

where

a. 8
px-^P + ^; />y=/> + | (151).

This shows clearly the meaning, and the physical explanation,
of the px, py, pt assumed in § 186 as virtual densities of ether
relatively to motions in the three rectangular directions. With
diatomic molecules having their axes parallel to OZ, we have
essentially py = p^. This gives the optical properties of a uniaxal
crystal, which essentially present no difference between different
directions perpendicular to the axis; though the homogeneous

DYNAMICAL EXPLANATION OF A GHIRAL MOLECULE. 439

assemblage of molecules, constituting the crystal, presents es- Moleoola
sentially the differences in different directions corresponding to
tactics of square order, or of equilateral-triangle order.

§ 204. When instead of the mere diatomic molecule of § 201 we
have in each molecule a molecular structure which is isotropically
symmetrical (square order or equilateral-triangle order) in planes
perpendicular to one line OZ, we still have py^p^g.^ Jfiving the
axially isotropic optical properties of a uniaxal crystal : and this
quite independently of any symmetry of the homogeneous assem-
blage of molecules constituting the crystal, if the virtual inertia
contributed to the ether by each molecule is independent of its
neighbours. If the structure of each molecule has no chirality
(Appendix H, § 22, footnote) the homogeneous assemblage has no
chirality. And if each molecule is geometrically and dynamically
symmetrical with reference to three rectangular axes (OX, OF, OZ),
but not isotropic in respect to these axes, we have generally three
different values for px, py, pz\ and (§§ 187, 190 above) exactly
Fresnel's wave-surface. This is quite independent of any sym-
metry of the homogeneous assemblage constituting the crystal : it
is merely because the axes of symmetry of all the molecules are
parallel in virtue of the assemblage being homogeneous. For
brevity I now call a molecule which has chirality, a chiroid.

§ 205. When each molecule is a chiroid it may (Appendix G,
Part 1) contribute a chiral property to the inertia of ether oscil-
lating to and fro in the space occupied by the assemblage. To
understand this chiral inertia, consider a volume B of ether, very
small in all its diameters in comparison with a wave-length of
light, and exactly equal in volume to the volume of space allotted
(§ 20 1 ) to each molecule. Let (f , ly, 5) and (^, 17, (;) be the components
of displacements and velocity of ether within B but not within
the part or parts of B occupied by the atoms of the molecule.
The components round x, y, z, of rotational velocity (commonly,
but perhaps less conveniently, called angular velocity) of ether in
space in the neighbourhood of B and not within any atom, are

1/4 dii\, l/dl 4V Ifdv dt\ ,3.
2\d^'Tz)' 2\dz''dx)' 2\dx dy) ^^^^^'

I'Ct Xx> Xy^ Xi ^^ coefficients which we may call the inertial
chiralities of the molecule relative to x, y, z respectively. The

440 LECTURE XX.

kfolecular. chiral inertia with which we are concerned may be defined by
asserting that acceleration of any one of the three components
(152) of angular velocity of the ether, implies a mutual force
between the molecule and the ether, in the direction of the axis
of the rotational acceleration, which may be expressed by the
following equations*

^dt'\dy dzj' ^~ ^df[dz dx)' ^ dtAdx dy)

(153);

where P, Q, -B, denote components of force per unit of volume,
exerted by the molecules on the moving ether. Hence the
^> y> ^-components of the elastic force on ether per unit of its
volume acting against inertial reaction are equal to

'■§-^' "S-ft "-g-* <'»*>•

The equations of motion of ether occupying the same space as
our homogeneous assemblage of molecules will be found by substi-
tuting (154) for the first members of (8) in § 113. Thus with (153)
we find

f''l^^^WATy'dz)=^^^^''^d.[d^^

(155),

and the symmetricals in relation to y and z ; three equations to
determine the three unknowns f, 17, f.

§ 206. The method of treatment by an arbitrary function, as
in §§ 115, 120, 121, 127, 130, would be interesting ; but because of
the triple differentiations in the chiral terms it is not convenient.
All that it can give is in reality, in virtue of Fourier's theorems,

* The negative sign is prefixed to x i° these equations in order to make x
positive for a medium in which right-handed circularly polarized light travels faster
than left-handed (see § 215 below). Such a medium is by all writers on the subject
called a right-handed medium, because the vibrational line of plane polarized light
travelling through it turns clockwise as seen by a person testing it with a NicoPs
prism next his eye. The molecule which, according to our inertial theory, produces
this result is analogous to a left-handed screw-propeller in water and is therefore
properly to be called left-handed. Thus left-handed molecules produce an optically
right-handed medium.

FORMULAS EXPRESSING CHIRAL INERTIA IN WAVE-MOTION. 441

included in the more convenient method of periodic functions ; of Molar,
which the most convenient form for our present problem is given
by the use of imaginaries with the assumption f, 17, f equal to
constants multiplied into

. ' ^-> (156).

where X, /a, v denote the direction-cosines of a perpendicular to
the wave-plane, and u the propagational velocity. This gives

V« = - -- ; -~ = - fl)»

d £a>X

— •

d ia>fi^

d Aft>i/

dx~ u '

dy~ tt '

dz u

w

which reduces (155) and its symmetricals to

d^
.(157),

ft)

fl>»X

ft)

,5 \

/>««'? + *X* -(I'l? -M?) = (* + i^) ■;ir(^ + /*^ + ^?) + ^ :nif

u

U"

w

ft>' r^ C «.V •! * \ ft>'/A X^ «. CA ft>'

/>yft)>i7 -h tXy - (X?- i/f) = (& + in)-j^ (\f + /^ + I/O -h n - 17

"...(159),

/>^'?+%jV?-Xi7) = (fc + in)^(Xf-h/ii7-hi/0 + n~?

(158)*

Multiplying each member of these equations by u^g)~* and arrang-
ing in order of f , 17, f, we find

(il - iV) f -h (tx^t^ft)!' - l(^fi) V + (- iXtUtofi - Ar'i/X) ? = 0 ,

(— tXi/^®^ "" ^'^m) ? + {B'^-k'fj?) 17 + (+ t;^Uft)X— Ar'fw/) f = 0

(+ iXiim^i, — A;'i/X) f + (— tx«t^ft)X — A;'/ai/) 17 + ((7 — Ar'i/') f = 0^

where

il=/>x^' — 7i; B = pyU^^n\ C^pzU^ — n; k' = {k + ^)...{160).

Forming the determinant for elimination of the ratios f, 17, f ; and
simplifying as much as possible, with reductions involving

X' + fi^ + v^r^l (161);

and putting

= ^' [X« (Py - P^) + Xv (P« - Pa:) + Xr (p» - Pi/)] . . ..(162),

* These formulas, implying as they do that the chirality expressed by them is
due, not to chirality of elasticity but to chirality of virtual inertia, are given by
Boussinesq on p. 456 of Vol. 11. (1903) of his TMorie Analytique de la Chaleur.

442 LECTURE XX.

Molar, we find

^i.a[i-i-g^SVg)]

+ mi'j&wX/ii/ = 0 (163).

In (163) we have an equation of the sixth degree for the
determination of m, the propagational velocity of waves whose
wave-normal has X, /a, v for its direction-cosines.

This equation is greatly simplified by our assumption of
practically zero propagational velocity of condensational-rare-
factional waves; which makes i' = — w, and therefore by (160),
and (129')

A u^'-a? u^ — a?
This, mth corresponding formulas for ^^jB and i/'/O, gives

By k* = — n, we also have

A—K = pxU^ ; B — k' = pyii^ ; C — k' = p^uK
Thus (163) is reduced to

(XyTCzRx^' + TCzXxRyf^'' + X^XyP'^) " f'UnEtoktiv = 0 . . . (163').

u

- 6)2/^2

§ 207. The imaginary term in (163), unless it vanishes, makes
every one of the six values of u imaginary. The realisation of the
corresponding result according to the principles of §§ 150, 151,
above will be very interesting. It essentially demands an ex-
tension of the dynamical theory to include the conversion of the
energy of wave-motion into thermal energy, — energy of irregular
intermolecular motions; that is to say the dynamics of the
absorption of luminous waves travelling through an assemblage
of atoms or of groups of atoms.

§ 208. Remark first that when E does not vanish, the
imaginary term in (163) vanishes if, and only if, one of the three

(164).

SPEED OF CIRCULARLY POLARIZED LIGHT IN CUIRAL MEDIUM. 443

direction-cosines \, /i, i/, vanishes ; that is to say if the wave-plane Molar,
is parallel to one of the three principal axes. It is certainly a
curious result that plane waves Ciin be propagated without
absorption if their plane is parallel to any one of the principal
axes; while there is absorption for all waves not fulfilling this
condition: in other words that the crystal should be perfectly
transparent for all rays perpendicular to a principal axis, and
somewhat absorptive for all ra3rs not perpendicular to a principal
axis. No such crystal is known in Nature or in chemical art.
Time forbids us to go farther at present into this most interesting
subject.

§ 209. Considering now cases in which E vanishes and there-
fore (163) has all its terms rea), we see that in all such cases it
becomes a cubic in u\ A first and simplest case of this condition
is indicated by the third member of (162), which shows that E
vanishes when px = py=^pz. For this case (163) becomes

(pit'- n - k') [{pu^ - nf - uWixyXz^'-^ X^XxM'+ XxXv i^)] = 0'
where p=zp^ = py = p^

The three roots of this cubic are given, one of them by equating
the first factor to zero, and the two others by the quadratic in u'
obtained by equating the second factor to zero. No crystals are
known to present the optical properties thus indicated for unequal
values of Xx> Xm %«• But it is conceivable that these properties
may be found in some crystals of the cubic class, which might
conceivably, while isotropic in respect to ordinary refraction, give
different degrees of rotation of the plane of polarization of polarized
rays travelling in different directions relatively to the three rect-
angular lines of geometrical symmetry.

§ 210. For the case of Xx^Xy'^X^ ^^^ quadratic of (164)
becomes {pu- — nf ^ ta^ii? whence

pu^'-n = i (oxif' (165).

In all known cases, I think we may safely say in all conceivable
cases, whether of chiral crystals, or of chiral liquids, g);^m is very
small in comparison with n\ in other words (§213 below) the
difference between the propagational velocities of right-handed
and left-handed circularly polarized light is exceedingly small
in comparison with the mean of the two velocities. Hence we
lose practically nothing of accuracy by taking u = ^(n/p) in the

444 LECTURE XX.

Molar, second member of (165). Thus if we denote by tii, ti, the two
positive values of u given by (165) and by u a mean of the two,
we find by (165)

«jL-«,'j^2«2et ^d«ir!^^«X?*^?LX (166).

m' * 71 u n pu

And, in accordance with (182), putting

^ = 9 (1660.

we have

ui-ti, = 5r (166"X

§ 211. To interpret this result go back to (158) simplified by
making p^^p^^ pg and j^ = j^ =a j^. The medium being isotropic
in respect to all directions of the wave-plane, we lose nothing of
generality by putting X = l, m = 0> 1^ = 0. With these simplifi-
cations, the first of the three equations (158) gives { = 0 and the
second and third multiplied by u^/col* become

(ptt* — n) 17 = — ix^^^ » (P^* "" ^) ?= f'X^^ (167).

Equating the product of the first members to the product of the
second members of these equations we find (pu^ ^ ny ^ j^eaW,
which verifias the determinantal quadratic as given at the
beginning of § 210.

Using (165) to eliminate (pu^^n) from (167) we find

C=±ti7 (168).

Hence as an imaginary solution according to (156) of § 206 we
have for f, 17, (f respectively

0; Ce"^"^^; ±»(7€*"^''s) (169).

Changing the sign of t gives another imaginary solution; and
taking the half-sum of the two imaginary solutions, we find as
a real solution

f = 0; fj = Ccoaa>(t--y, C= + Csinco («-^y...(l70);

§ 212. The interpretation of this is, sinusoidal vibrations of
equal amplitudes, parallel respectively to 0 F and OZ ; the former

COEXISTENT WAVES OF CONTRARY CIRCULAR MOTIONS. 445

being a quarter of a period behind or before the latter, according Molar,
as we choose the upper or the lower of the two signs in each
formula. The resultant (y, z) motion of the ether*, in the case

O R

Fig. 15.

. * %

Fig. 16.

* This (y, i) component is irrofeational. The (z, y) component of the etherial
motion is rotational haviqg OZ as axis ; and the (x, z) component is rotational having
or as axis. We are not concerned with this view at present. Bat we meet it
necessarily when we think out the geometry of § 205 (152) ; and it brings to us a
very simple synthetic investigation of the velocity of circularly polarized light in a
chiral liquid or in an isotropic chiral solid.

44G LECTURE XX.

Molar, represented by the lower signs, is in circular orbits in the direction
shown as anti-clockwise in the annexed diagram (fig. 15). This
motion is the same in phase for all points of the ether in every
plane perpendicular to OX, that is to say every wave-plane ; and
it varies from wave-plane to wave-plane so as to constitute a wave
of circularly polarized light travelling a?-wise with velocity u. The
motion corresponding to the upper signs is another circularly
polarized wave with opposite orbital motion, that is to say clock-
wise direction. The radius of the orbit is the same, C, in the two
cases. If* when looking to fig. 15, we take the positive OX as
towards the eye, a line of particles of the ether which is parallel to
OX when undisturbed, becomes, in the wave of clockwise orbits,
a right-handed spiral ; and in the wave of anti-clockwise orbits a
left-handed spiral. Thus the two waves are of opposite chiralities :
the former are called right-handed, the latter left-handed. The
steps of the two spirals (screws) are slightly different, being the
spaces travelled by the two waves in their common period ; that
is to say the wave-lengths of the two waves. To understand this
look at fig. 16 showing a right-handed spiral RRR of step 6 cm.
and a left-handed spiral LLL of step 5 cm. ; having a common
axis OXy and both wound on a cylinder of radius IJ cm. repre-
senting C of (170). If the radius of this cylinder were reduced to
a thousandth or a millionth of the step of either screw, and if the
step were reduced to about 4.10"' cm., either spiral might represent
the line in which particles of ether lying in OX when undisturbed
are displaced by homogeneous yellow or yellow-green circularly
polarized light, travelling in the direction of OX positive, through
a transparent liquid or solid of refractive index of about 1*5. With
this understanding as to the scale of the diagram consider the
resultant of the two equal coexisting displacements of wave-planes
represented at any instant by the two spirals. At points of inter-

* This is a oonyention which I have aniformly followed for sixty years in
respect to the positive and negative directions in three mutually perpendicular lines
OXf OYf OZ. It makes the positive direction for angular velocities be from OX to
OY, from OY to OZ^ and from OZ to OX. This agrees with the ordinary conven-
tions of English as well as foreign books on trigonometry, and geometry of two
dimensions, which make anti-clockwise rotation be positive. It is convenient for
inhabitants of the Northern Hemisphere as it makes positive the anti-clockwise
orbital and rotational motions of the sun and planets as viewed from space above
our North Pole, which we may call the nortii side of the mean plane of the
planetary motions.

fresnel's kinematics of coexistent waves. 447

section of the two spirals, for example D, ZX, iy\ the resultant Molar,
displacement of each of the corresponding wave-planes is 2(7, the
sum of the two equal components. The resultant displacement
of the wave-plane through AB '\% twice the distance from OX of
the middle point of AB, that is to say 2 ^((7' — \^^\

§ 213. Suppose now the two spirals to rotate in opposite
directions, with equal angular velocity cd, round the axis 0X\
the right-handed spiral clockwise and the left-handed anti-clock-
wise when viewed from X towards 0. This may be realised in an
instructive model having one spiral wound on a brass or wooden
cylinder, and the other on a glass tube fitting easily around it.
The two moving spirals will represent the motions of the ether in
two circularly polarized waves travelling a?- wards with velocities

ft)\, ft)X, /I "TON

Xi, Xg denoting the steps of the two screws. The motion of the
ether in any fixed plane perpendicular to OX, the plane through
AB for instance in the diagram, will be found by the geometrical
construction of § 212. Thus we see that a very short time after
the time of the configuration shown in the diagram, the point D
will come to the plane through AB, and the points A, B will
come together at 2): immediately after this they will separate,
A leftwards in the diagram, B rightwards. Considering the whole
movement we see that in any one fixed plane through OX, a
wave-plane of the ether moves to and fro in a fixed straight line.
The point D will travel ar-wards with a velocity equal to

ft)/27r ^

l/^^ 1\ 1/1 \\
2 U \) 2 Ui ■*" vj

(173),

being the harmonic mean of the velocities of the two compounded
circular waves; and will revolve slowly clockwise round the
cylinder of radius (7 at a rate, in radians per unit of distance
travelled a?-wise, equal to

7r

(i-a=fS-^) <"*>■

Compare with (179) and (180) below.

448 LECTURE XX.

Molar. § 214. The short algebraic expression and proof of all this is
found by taking the two solutions represented by (170), (171) and
writing down their sum with modification as follows : —

^=COSG)

(<-j) + cosa,(<-|)

= 2cosa,(«-2--^jco92(---J (175);

g=-sino,(«-^) + 9ina,(«-^)

For magnitude and direction of the resultant of these we find

V(,'+n = 20cosa,(<-^-|-) (177).

and

? = _fcan|(^-^) (178).

These equations express rectilinear vibration through a total

27r
range 40 in period — , in the line whose azimuth is

-ie-3 <"">■

This shows that the vibrational line in plane polarized light
revolves clockwise in the wave-plane at a rate r, in turns per unit
of space travelled, expressed by

== ^ fl _i) = .«- H>.-if?= J !fL=3 (180).

47r \a^ UiJ 4nr v^u^ ra^

In the last member r denotes the period of the vibrations, and a

denotes sfu^u^^ or the "geometric mean" of the propagaiional
velocities, of the two waves.

§ 215. Returning now to figure 16 we see that, as virtually
said in §§212, 213;

I. The orbital motions in right-handed circularly polarized
light coming towards the eye are clockwise, and in left-handed
an ti- clockwise.

r

LEFT-HANDED MOLECULES MAKE A RIGHT-HANDED MEDIUM. 449

n. The vibrational line of plane polarized light traversing Molar,
a chiral medium towards the eye, turns clockwise when the
velocity of right-handed circularly polarized light is greater, and
anti-clockwise when it is less, than that of lefb-handed circularly
polarized light.

III. A chiral medium is called optically right-handed or
left-handed according as the propagational velocity of right-handed
or of left-handed circularly polarized light travelling through it
is the greater.

All this (§§ 212 — 215) is Fresnel, pure and simple. It is his
kinematics of the optical right-handed and left-handed chirality
discovered by Arago and Biot in quartz, and by Biot in turpentine
and in a vast number of other liquids.

§ 216. For the dynamical explanation take the third member
of (180); and look back to (152) and (153) which show that,
according to our notation, twice the acceleration of angular velocity
of moving ether multiplied by ^ is, when x ^ negative, a force
per unit bulk of the ether in the positive direction in the axis of
the angular velocity, exerted on the ether by the fixed chiral
molecules. If x ^^ positive, the chiral molecule is, as said in the
footnote on § 205, analogous to a left-handed screw and its chirality
is properly to be called left-handed : because (§ 205 above) if the
ether concerned is viewed in the direction of the axis of the
angular velocity, the force of left-handed molecule on ether is
toward the eye (or positive) when the acceleration of the angular
velocity of the ether is clockwise (or negative according to my
convention regarding direction of rotation, stated in the footnote
on § 212). With this statement the simple synthetic investigation
of the velocity of circularly polarized light in a chiral medium,
indicated in the first footnote of § 212, is almost completed; and
by completing it we easily arrive, by a short cut, at the solution
expressed in the third member of (180), for an isotropic medium ;
without the more comprehensive analytical investigation of §§ 205
to 214.

By (166) we saw that the left-handed molecule (x positive)
gives the greater propagational velocity (wi) for right-handed
circularly polarized light, and therefore (§ 215, III.) the medium
is optically right-handed. Thus our conventions necessarily result
in left-handed molecules making a right-handed medium and
right-handed molecules a left-handed medium.

T. L. 29

450 LECTUKE XX.

Molar. § 217. Going back now to §§ 206, 207, 208, for light traversing
a chiral medium with three rectangular axes of symmetry cor-
responding to maximum, minimax and minimum wave velocities,
let us work out the solution for wave-plane perpendicular to one
of the principal axes, OZ for example. This makes X = 0, /a = 0,
i/=l ; and, with (160), reduces the determinantal equation (162)

to

(0 - k') [(p,ti« - n) (pyu^ - n) - u'to^xy] = 0 . . . (181 ).

I

Hence, removing the first factor (which, when k^ = -n, gives w = 0,
for the condensational-rarefactional wave) and putting

-?!: = a«; - = 6«; ^^2fe2&f=^» (182),

Px py pxpy

we find

(n« - a») (u« - 6») = grV (183);

a quadratic of which the two roots are the squares of the velocities
of the -er-ward waves. The third of equations (158) makes ^ = 0
for each of these waves; and therefore they are both exactly
equivoluminal. When XtXy ^ positive the two roots of the quad-
ratic are both positive ; one of them > a', the other < 6' if a* > 6*.

§ 218. In these formulas a and 6 denote the velocities of
light having its vibrational lines parallel to OX and OY respec-
tively; and 5^ is a comparatively very small velocity measuring
the chiral quality of the crystal. Judging by all that has been
hitherto discovered from observation of chiro-optic properties of
gases, liquids and solids, we may feel sure that g is in every case
exceedingly small in comparison with either a or 6 : it is about
one twenty- thousandth in quartz : and in cinnabar, which has the
greatest optic chirality hitherto recorded for any liquid or solid so
far as I know, g seems to be about fifteen times as great as in
quartz. Suppose now that g'^ is very small in comparison with
the difference between a} and 6*, we see, by the form of (183), that
the two values of u^ given by the quadratic must be to a first
approximation equal to a" and 6" respectively : hence to a second
approximation we have (taking forms convenient for the case of
a« > 6«),

§ 219. To find the character of the two waves corresponding
to the two roots of (183), put X = 0, /a = 0, i/ = l in the first and

OPTIC CHIRALITY IMPERCEPTIBLE IN A BIAXAL CRYSTAL. 451

second of (1 58), each multiplied by t*'©-* ; which, with (182), Molar,
reduces them to

px py

Equating the product of the first members to product of the
second members of these equations we verify the determinantal
quadratic (183). With either value of u^ given by (183), either
one or other of equations (185) may be used to complete the
solution. The first of (185), however, is the more convenient for
the root approximately equal to 6* ; and the second for the root
approximately equal to a*. Taking them accordingly and realising
in the usual manner, we find as follows for two completed in-
dependent solutions with arbitrary constants (7i, C^:

(186).

§ 220. The exceeding smallness of g/a* has rendered fruitless
all attempts hitherto made, so far as I know, to discover optic
chirality in a "biaxal crystal," that is to say a crystal having
three principal axes at right angles to one another of minimum,
minimax and maximum wave-velocities. If it were discoverable
at all, it certainly would be perceptible in light travelling along
one of these three principal axes.

[Dec. 19, 1903. I have only to-day seen in Phil Mag., Oct.
1901 that Dr H. C. Pocklington has found rotations, per cm.,
22° anti-clockwise, and 64° clockwise, of the vibrational line in
polarized light travelling through crystallized sugar along two
" axes," of which the first is nearly perpendicular to the cleavage
plane : a most interesting and important discovery.]

§ 221. For light travelling along the axis, OZ, of a uniaxal

crystal, take a = 6 in (183>--<186), in this case, and we have

by (183)

u^-'a^=±gu (187).

♦ For sodium-light in quartz it is -4606 . 10"* ; and it may be expected to be
correspondingly smaU in biaxal crystals.

29—2

452 LECTTRE

Ifobr. And, taking p^ = />,. and x» == X»- ^^ bave, by <1S2 1,

a = ^^ assi

With this notation and with « 187 1. \ 1S6> becomes
Vi>a: f= Ccosiw t ■: » = — Csin^lf '

iit<a; f=— CiSin»(# : if= C^cosm t i

)

Thus ve see that for light travelling along the axis of a nnaiTal
ciTStal the Telocities of right-handed and of left-handed circalariT
polarized light, and the rotation of the Tibrati*3nal tine of plane
polarized light, are in every detail the same as we found in §§ 211,
212 for a medium wholly isotropic Bat we shall see presently
(§ 226 below) that two or three degrees of deviation frxn the axis
prodaces a great change from the phenomena of a chiral is* > tropic
medinm ; and that for rays inclined 34)' or more ap to 90\ the
chirality is almost wholly swamped by the solotropy. even when
the aeolotropy is as small as it is for quartz i§ 223 below t As
remarked in § 220, there is no direction of light in a crystaL
having three unequal values f jr a, b, c. in which the chiro-optic
effect is not masked by the aeolotropy: and therefore it is nol
so interesting to work out for a biaxal crystal the realised
details of the solution 4159), < 162). (163). But it is exceedingly
interesting to work them out for a uniaxal crystal, because of the
great exaltation of the chiral phenomena when the ray is nearly in
the direction of the axis, and because of the beautiful phenomena of
AirVs spirals due to this exaltation ; and because of the admirable
experimental investigation of the wave-sur£Eu:e in quartz crystal
by McConnel referred to in § 228 below.

§ 222. For waves transmitted in any direction through a
uniaxal crystal ; choose OZ as the axis, and therefore let Pm = p,,
and x»=Xy» i» (158>— (163) of § 206. This reduces (163) to
E = 0 and therefore makes the waves in all directions real : that is
to say transmissible with no change of motional con^guration.
Corresponding to (182) we may n«>w put

n n

- = c»; '^ = ^: ^' = A (1891.

P9 Pjf fh Rm px

— =— =a»

OPTIC CHIRALITY IN QUARTZ CRYSTAL. 453

Without loss of generality, we may simplify (158) — (162) by Molar.

taking

;x> = 0; X*=sin«d; i;» = cos*d (190);

with these, (163') of § 206 gives

(w« - a*) [u^ - a« + (a« - c«) sin' 0] = g[g-(g''h) sin' ^ u\ . .(191).

This is a quadratic equation for the determination of the
squares of the velocities of the two plane waves whosl^ wave-
normals are inclined at the same angle 0 to OZ. For 6 = 90° we
fall back on the case of §§ 217, 218, 219, but with c instead of b;
and for ^ = 0*" we fall back directly on § 221. The exceedingly
interesting transition from the subject of § 221 to our present
subject is fully represented by the solution of the quadratic
equation for u' : and is best explained by tables of values of the
two roots from tf = 0*" to tf = 90° for some particular case or cases ;
or graphic representation by curves as in figs. 17, 18, 19.
I have chosen the case of quartz crystal traversed by sodium-
light ; for which observation shows the rotation of the vibrational
line to be 217° per centimetre of space travelled along the axis.
This makes g= 4'605 . 10~'. a; as we find by putting in the first
member of (180), r = 217/360 : and in the last member Ui — th = g\
and a = '647593, the reciprocal of 1*54418, the smallest refractive
index of quartz at 18°, according to Rudberg; and r = '58932, the
period in decimal of a michron, of the mean of sodium-lights
DiDi] the michron being the unit of time which makes the
velocity of light unity when the unit of space is the michron (or
millionth of a metre). (See footnote on p. 150 above.)

As two sub-cases I have chosen A = 0 and A = — ^ ; because
for all that our theory tells us, (§ 223), h/g mi^ht be negative ; or
might be zero or might have any positive value less than, or equal
to, or greater than, unity. McConnel's experimental investigation
seems (§ 228 below) to make it certain that h/g for quartz is less
than unity and probable that it is negative, and as small as — 1,
or perhaps smaller. As for ^r*, (189) shows that its value is
inversely proportional to the square of the period of the light,
if Xx is the same for light of all periods ; g is positive or negative
according as the crystal is optically right-handed or left-handed.

§ 223. c/a is the ratio of the smallest to the greatest refrac-
tive index of quartz; that* is 1*54418/1*55328, according to the

454 LECTURE XX.

MoUr. figures used by McConnel ; being (as I see in Landolt and
Borastein's Tables) Rudbergs results for temperature 18° and
sodium-light. From this we have (c/a)« = -98832 and (191)
becomes

S-')S-)-'r

W* ., .V U'

or -T — (1 + w' + a) -^ = — ti;*

(192),

where 1 -t£;»=:01168sin«^ = ^^^^(l -cos2d) ...(193),

Sd

and g = a"^5r[5r — (^ — A)sin'tf] (194?).

If i/i*, u} denote the greater and less roots of (191), we have

a"^(t^* + W2') = l + ^ + ? (195);

a-«(t^« - u}) = V[(l - 1(;») + 2 (1 + 1£;») g + ?»] ...(196).

Remark that when ^ = 90°, 1 -t£;» = -001168, q^a^gh. Hence
-^ar^gh might be positive, and as great as \ . 10"* (1-168)*, or a
little greater, without making the radical imaginary.

§ 224. In fig. 17, Curves 1 and 2 represent, from ^ = 0° to
0 = 30°, \ [a~* {u^ — u^) — (1 — vf)"] for the sub-cases A = gr, and
h = —g] and Curve 3 represents i (1 — tu^) on the same scale from
^ = 0"* to 6 = 5°'3. In fig. 18, Curves 1 and 2 represent, from

^ = 30° to ^ = 90°, i[«"*K'-^')-(l-'«^)] for the sub-cases
A = gr and A = — ^, on a scale of ordinates ten times, and abscissas
half, that of fig. 17.

Throughout the whole range of figs. 17 and 18, Curves 1 and
2 represent chiral differences from the squares of the aeolotropic
wave-velocities calculated according to the non-chiral constituent
of the aeolotropy of quartz crystal; Curve 3 of fig. 17, and
equations (193) — (196), show that through the range from ^ = 0°
to tf = 5°-3, the values of a'^Ui, a~^u^, and q differ from unity by
less than 51 . 10"". Hence, through this range, we have, very
approximately,

i[o-'(V-«,')-(l-«;')]=i^i^-(l-«;) ...(197).

TWO TRIAL ASSUMPTIONS FOR QUARTZ COMPARED. 455

II

1

*v

A

f

■/

=m!=j-

^

^

^

-^

' -

---

-~^

1

^

'S

456

LECTURK XX.

'To

o

o

o
10

00

0

o

«0

e

to

M

e

o

>
ft.
3

o

0

lO

>

ft.

3

o

to

o

o

1

to

e
lO

1

10

0

o

10

o
10

/

o

o

1

7

o
lO

/

/

CO

o

o

CO

00

i

10

I

MODIFICATION OF DOUBLE REFRACTION BYCHIRALITYIN QUARTZ. 457

§ 225. Fig. 19 illustrates the critical features of the crystal- MoUr.
line influence, and of the combined crystalline and chiral influence,
of quartz-crystal on light traversing it with wave-normal inclined
to the axis at any angle up to 14°. The crystalline influence
alone is represented in Curve 3 by downward ordinates equal to
the excess of the velocity of the ordinary ray above the velocity

-6 2* 4* 0«

8*

h

Qo

la*

14«

5.10

Cm

— a

•

—

===

6tt
10 •»

H

p^

\

^

^

16 n

20*1
26 H

\

L

\

\

\

^

\

30*1
35 tt

\

\

t

4

JO i

!• C

o

t

J-

1

00

1

2o

1

4«

Fig. 19.

of the extraordinary ray, divided by the former ; in an ideal crystal,
corresponding to the mean of a right-handed and a left-handed
quartz-crystal. Curve 1 represents the excess of the greater of
the two wave-velocities in a real quartz-crystal above the velocity
of the ordinary ray in the ideal mean crystal, divided by the
latter. Curve 2 represents the excess (negative) of the less of
the two wave- velocities in a real quartz-crystal above the velocity
of the extraordinary ray in the ideal mean crystal, divided by the
latter. According to our notation of § 223, the ordinates of Curves
3, 1, 2 are equal respectively to

w — 1; — — 1; — — t^.
a a

.(198).

458 LECTURE XX.

Molar. § 226. From (198) we see that Curve 3 continued through the
whole range from tf = 0"* to tf = 90°, represents the distance between
a tangent plane on a prolate ellipsoid of revolution of unit axial
semi-diameter, and the parallel tangent plane on the cii*cumscribed
spherical surface of unit radius (the ellipsoid and the sphere
constituting the wave-surface for the ideal uniaxal crystal cor-
responding to the mean of right-handed and left-handed quartz).
Curves 1 and 2 represent similarly the projections, outward from
the sphere and inwards from the ellipsoid, of the two sheets of
the wave-surface in a real quartz-crystal whether right-handed or
left-handed. In each case 0 is the inclination to the equatorial
plane, of the two tangent planes whose distance is represented by
ordinates in the diagram. It is interesting to see and judge by
Curve 1 how closely at tf = 14°, and thence up to ^ = 90°, one of
the sheets of the wave-surface in quartz agrees with the spherical
surface: and to see by Curves 2 and 3 how nearly, at and above 14°,
the other sheet agrees with the inscribed ellipsoid. On the other
hand, it is interesting to see by the three curves how prepon-
derating is chirality over aeolotropy, from ^ = 0° to ^ = 2°; and
to see how the preponderance gradually changes from chirality
to aeolotropy when d increases from 2° to 14°.

§ 227. The characters of the two plane waves, whose wave-
normals are inclined at angle 6 to the axis of a quartz crystal, are
to be discovered by commencing as in § 222 ; and, for the case
there defined, working out a realised solution from (169) of § 206.
We thus find that, for tf = 0 each wave is circularly polarized ;
and, for all values of d between 0° and 90°, each wave is
elliptically polarized ; the axes of the elliptic orbit being, one of
them perpendicular to, and the other in, the plane through the
wave-normal and the axis of the crystal. The former is the
greater of the two axes for the wave which has the greater
velocity (mi); the latter is the greater for the wave having the
less velocity {u^. For all values of d greater than six or seven
degrees the less axis of the elliptic orbit is very small in com-
parison with the greater: that is to say each wave consists of very
nearly rectilinear vibrations, or is very nearly " plane polarized *' ;
and one of the two waves approximates closely to the "ordinary ray,"
the other to the "extraordinary ray" in the ideal non-chiral crystal
corresponding to the mean of right-handed and left-handed quartz.

MATHEMATICAL THEORIES TESTED EXPERIMENTALLY. 459

§ 228. Looking at the two sub-cases represented by Curve 1 Molar,
and Curve 2 of figures 17 and 18, we see that the difference
between them is very small, probably quite imperceptible to the
most delicate observation practicable, when ^ < 6°. At tf = 10°
the difference of the ordinates for the two curves is about 1/17
of the ordinates for either ; and might be perceptible to observa-
tion, though it represents an exceedingly small proportion, about
1/500, of the whole difference of velocities between the two rays.
This difference, as shown in figure 19, is itself very small, being
only about 18.10""* of a, the mean velocity of the right-handed and
left-handed axial rays. How exceedingly searching McConnel's
experimental investigation was may be judged by the fact shown
in his figure 3, page 321*, that from ^ = 14° to ^ = 30° he found
definite systematic differences between " MacCuUagh's Theory"
and " Sarrau's Theory." The results of MacCullagh's Theory are
expressed by our Curve 1 ; and our Curve 2 shows results differing
from MacCullagh's in the same direction as Sarrau's but some-
what more. Thus McConnel's investigation was more than amply
sensitive to distinguish between our two sub-cases represented by
Curves 1 and 2 of figures 17 and 18. Looking at McConnel's
figure 3, and remarking that 003794 is the value of a — 6 (our
a — c), which he took as correct according to his statement of
Rudberg's results, we see that what he takes as Sarrau's Theory
under-corrects the error of MacCullagh's : and our sub-case 2,
which I chose on this account, must make the correction very
nearly perfect. But I see on looking to Sarrau's paperf that
his theory was not, as supposed by McConnel, confined to rela-

tively small deviations from J = ^ cos* 0 (McConnel's notation of

his page 314 translated into ours of § 226); and that proper
values of his constants (without the restriction of /i and gi
relatively small, stated by McConnel), may be found to give a
perfect agreement with McConnel's observations. The same. may
be said of Voigt's theory as pointed out by McConnel (p. 314).
Thus we may take it that Sarrau's and Voigt's theories lead
to results perfectly consistent with McConnel's observations.
Theories of MacCullagh, Clebsch, Lang, and Boussinesq, are re-
ferred to by McConnel as giving a constant for what we have

♦ Phil. Tram, Roy, Soc,, Part I., 1886.

t Sarraa, LiouvilU, b^. 2, tome xni. (1868), p. 101.

460

LECTURE XX.

Molar, denoted by 5, and as giving a fairly good approximation to the
results of his observations; but decidedly less good than Sairau
and Voigt, who allowed q to vary with 0. All these theories agree
in taking the mutual forces between different parts of the vibrating
medium as the origin of the chiral property, and differ essentially
from my theory which (g 162—166, 200—205) finds it in virtual
inertia of the ether as disturbed by chiral groups of atoms.

§ 229. The following Table shows, in degrees per centimetre,
the rotation of the vibrational line of polarized light travelling
through various substances; crystalline solid, and liquid; taken
from Landolt and Bomstein's Tables, Edition 1894.

Sabfltanoe

Quality
of Light

Direction

of
Botation*

Botation of
vibrational

line in
degrees per
centimetre

Observer

Solid crystals :
Cinnabar

Bed

f ^1
D ^ean)

D
D

D

D
D
D
D

D
D

B.
L.

2700 to 8000

217-27 )
21705
216*84 j

55-81

S8-85

19-8

21-7
81-04
81-6
2880

1-4147 xdt
16-155 xdt

DeBoloiseaox

Soret&Sarasin

Pape
Pape

H. Traabe

H. Traabe
Guye
Sohnoke
Groth

Landolt
Landolt

Quartz

Lead Hyposulphate + 4 aq

Potash Hyposulphate ...

Potassium Sulphate — \
Lithium Chromate
K2S04+Li2Cr04 j

Sodium Bromate

Sodium Chlorate

ij »» •

Sodium Periodate + 8 aq

Liquids :

Turpentine CinHm

Nicotine CiqHuNj

* Clockwise as seen by the observer is called right-handed (B.), anti-clockwise,
left-handed (L.).

t Where d denotes the specific gravity of the fluid. See p. 450 of Landolt and
B5msteiD. I do not see any good reason for the necessity of introducing d in the
manner indicated in my table in the text, and rendered necessary by the notation
adopted in the tables of Landolt and Bomstein. Some other embarrassing peculiari-
ties in their notation have prevented me from venturing to quote any one of the
numerous examples of the rotatory effect of "active" substances dissolved in
** non-active " liquids given in pp. 450 — 458 of these tables.

FARADAY'S MAGNETO-OPTIC ROTATION. 461

§ 230. Important and interesting information regarding chiro- Molar,
optic properties of liquids and solids is to be found in Mascart's
Traiti dVptiquey Volume il., Edition 1891, pages 247 to 369 ; and
regarding Faraday's magneto-optic rotation of the vibrational line
in pages 370 to 392. The first section of Appendix I in the
present volume contains an important statement, given to me by
the late Sir George Stokes, regarding chirality in crystals, and
in crystalline molecules. Appendix H (Molecular Tactics of a
Crystal, § 22 footnote, g 47 — 52) contains statements of funda-
mental principles in the pure geometry of chirality. Appendix G
contains a complete mathematical theory of the quasi-inertia of
a solid of any shape moving through a perfect liquid, with special
remarks on chirality of this quasi-inertia.

§ 231. Lecture XX. as originally given, and fully reported in
the papyrograph edition, and an appendix to it entitled ''Improved
Gyrostatic Molecule," contained unsatisfactory dynamical efforts to
illustrate or explain Faraday's magneto-optic rotation. These are
not reproduced in the present volume: but instead, an old paper
of date 1856 entitled "Dynamical Illustration of the Magnetic
and the Helicjoidal Rotatory Effects of Transparent Bodies on
Polarized Light " is reproduced as Appendix F. This paper con-
tains a statement of dynamical principles concerned in the two
kinds of rotation of the vibrational line of plane polarized light
travelling through transparent solids or fluids, which I believe
may even now be accepted as fundamentally correct. When we
have a true physical theory of the disturbance produced by a
magnet in pure ether, and in ether in the space occupied by
ponderable matter, fluid or solid, there will probably be no
difficulty in giving as thoroughly satisfactory explanation of the
magneto-optic rotation as we now have of the chiro-optic.

§ 232. In conclusion, let us consider what modification of the
original Maxwell-Sell meier dynamics of ordinary and anomalous
dispersion must be made when we adopt the atomic hypothesis of
Appendices A and E and of Lee. XIX., ^ 162 — 168.

In App. A it is temporarily assumed, for the sake of a definite
illustration, that the enormous variation of the etherial density
within an atom is due to a purely Boscovichian force acting on
the ether, in lines through the centre of the atom and varying

462 LECTURE XX.

foleoalar. as a function of the distance. This makes no provision for
vibrator or vibrators within an atom ; and, for the explanation of
molecular vibrators, it only grants such molecular groups of atoms,
as we have had for fifty years in the kinetic theory of gases,
according to Clausius' impregnable doctrine of specific heats with
regard to the partition of energy between translational and other
than translational movements of the molecules. Now, in App. E,
and in applications of it suggested in ^ 162 — 168 of Lea XIX.,
we have foundation for something towards a complete electro-
etherial theory, of the Stokes- KirchhofF vibrators* in the dynamics
of spectrum-analysis, and of the Maxwell-Sellmeier explanation of
dispersion.

§ 233. In our new theory, every single electrion within a
mono-electrionic atom, and every group of two, three, or more,
electrions, within a poly-electrionic atom, is a vibrator which, in
a source of light, takes energy from its collision with other atoms,
and radiates out energy in waves travelling through the sur-
rounding ether. But at present we are not concerned with the
source; and in bringing this last of our twenty lectures to an
end, I must limit myself to finding the effect of the presence of
electrionic vibrators in ether, on the velocity of light traversing it.

§ 234. The " fundamental modes" of which, in Lee. X., p. 120,
we have denoted the periods by #c, ac,, ac„, ... are now modes of
vibration of the electrions within a fixed atom, when the ether
around it and within it has no other motion than what is produced
by vibrations of the electrions. It is to be remarked however that
a steady motion of the atom through space occupied by the ether^
will not affect the vibrations of the electrions within it, relatively
to the atom.

§ 235. To illustrate, consider first the simple case of a mono-
electrionic atom having a single electrion within it. There is just
one mode of vibration, and its period is

* = 2-y^f = |'V^ (199).

where a denotes the radius of the atom, e the quantity of resinous
electricity in an electrion, and m its virtual mass ; and c denotes

* See Leo. IX. pp. 101, 102, IDS.

ELECTRO-ETHERIAL EXPLANATION OF UNIFORM RIGIDITY. 463

e^ar*. This we see because the atom, being mono-electrionic, has Molecul
the same quantity of vitreous electricity as an electrion has of
resinous ; and therefore (App. E, § 4) the force towards the centre,
experienced by an electrion held at a distance x from the centre,
is e^ar^x ; which is denoted in § 240 by ex,

§ 236. Consider next a group of i electrions in equilibrium,
or disturbed from equilibrium, within an i-electrionic atom. The
force exerted by the atom on any one of the electrions is ie^ar*D,
towards the centre, if D is its distance from the centre. Let now
the group be held in equilibrium with its constituents displaced
through equal parallel distances, x, from their positions of equi-
librium. Parallel forces each equal to ie^or^x, applied to the
electrions, will hold them in equilibrium*; and if let go, they will
vibrate to and fro in parallel lines, all in the same period

^.— VmS (200).

This therefore is one of the fundamental modes of vibration of
the group; and it is clearly the mode of longest period. Thus
we see that the periods of the gravest vibrational modes of different
electrionic vibrators are directly as the square roots of the cubes
of the radii of the atoms and inversely as the square roots of the
numbers of the electrions ; provided that in each case the atom
is electrically neutralised by an integral number of electrions.
Compare App. E, § 6.

§ 237. I now propose an assumption which, while greatly
simplifying the theory of the quasi inertia-loading of ether when
it moves through space occupied by ponderable matter as set forth
in App. A, perfectly explains the practical equality of the rigidity
of ether through all space, whether occupied also by, or void of,
ponderable matter. My proposal is that the radius of an electrion
is so extremely small that the quantity of ether within its sphere of
condeTisation (Lee. XIX., § 166) is exceedingly small in comparison
with the quantity of undisturbed ether in a volume equal to the
volume of the smallest atom.

This assumption, in connection with ^ 164, 166 of Lee. XIX.,
makes the density of the ether exceedingly nearly constant through

* Compare App. E, § 23.

464 LECTURE XX.

Moleoolar. all space outside the spheres of condensation of electrions. This
is true of space whether void of atoms, or occupied by closely
packed, or even overlapping, atoms; and the spheres of con-
densation occupy but a very small proportion of the whole space
even where most densely crowded with poly-electrionic atoms.
The highly condensed ether within the sphere of condensation
close around each electrion might have either greater or less
rigidity than ether of normal density, without perceptibly marring
the agreement between the normal rigidity of undisturbed ether,
and the working rigidity of the ether within the atom. This
seems to me in all probability the true explanation of what
everyone must have felt to be one of the greatest difficulties in
the dynamical theory of light; — the equality of the rigidity of
ether inside and outside a transparent body.

5 238. The smallness of the rarefaction of the ether within an
atom and outside the sphere or spheres of condensation around its
electrions, implies exceedingly small contribution to virtual inertia
of vibrating ether, by that rarefaction ; so small that I propose to
neglect it altogether. Thus if an atom is temporarily deprived of
its electrion or electrions (rendering it vitreously electrified to the
highest degree possible), ether vibrating to and fro through it will
experience the same inertial resistance as if undisturbed by the
atom. Its presence will not be felt in any way by the ether
existing in the same place. Thus the actual inertia-loading of
ether to which the refraction of light is due, is produced prac-
tically by the electrions, and but little if at all perceptibly by
the atoms, of the transparent body.

§ 239. For the present I assume an electrion to be massless,
that is to say devoid of intrinsic inertia, and to possess virtual
inertia only on account of the kinetic energy which accompanies
its steady motion through still ether. This is in reality an energy
of relative motion ; and does not exist when electrion and ether
are moving at the same speed. See App. A passim, and equation
(202) § 240 below.

§ 240. Come now to the wave- velocity problem and begin
with the simplest possible case, — only one electrion in each atom.
Consider waves of a?-vibration travelling y-wards according to the
formula (203) below. Take a sample atom in the wave-plane at
distance y from XOZ, The atom is unmoved by the ether- waves ;

ELECTRO-ETHERIAL THEORY OF REFRACTION. 465

while the electrion is set vibrating to and fro through its
centre.

At time t, let x be the displacement of the electrion, from the
centre of the atom (or its absolute displacement because at present
we assume the atom to be absolutely fixed) :

f the displacement of the ether around the atom :

p the mean density of the ether within and around the atom,
being, according to our assumptions, exactly the same as the
normal density of undisturbed ether :

n the rigidity of the ether within and around the atoms, being,
according to our assumptions, very approximately the same at
every point as the rigidity of undisturbed ether:

N the number of atoms per unit of volume :

ex the electric attraction towards the centre of its atom,
experienced by the electrion in virtue of its displacement, x :

m the virtual mass of an electrion :
^ E s, cube of ether equal to l/N of the unit of volume, having

the centre of one, and only one, atom within it.

The equation of motion of E, multiplied by N, is

I

I

I

and the equation of motion of the electrion within it, is

m^^^ = -cx (202).

§ 241. The solution of these two equations for the regular
regime of wave-motion is of the form

f = Osin 0) (f- 1) ; a:= 0' sin 0) (f - 1) (203),

where o) is given. Our present object is to find the two un-
knowns G/C (or f/a?), and v. By (203) we see that

cP . d' CO

H -.»«

dt—'^'-' df=-^ v-<2<^)'

This reduces (201) and (202) to

T, L. 30

466 LECTURE XX.

Molecular, from which we find

X mo)' ^^^

and l=e + ^fl_^) = P + ^»? ^ (207).

The last member is introduced with the notation

T

='J^ '-^V? <'"»>•

where t denotes the period of the waves, and k the period of an
electrion displaced from the centre of its atom, and left vibrating
inside, while the surrounding ether is all at rest except for the
outward travelling waves, by which its energy is carried away at
some very small proportionate rate per period ; perhaps not more
than lO"*. It is clear that the greater the wave-length of the
outgoing waves, in comparison with the radius of the sphere of
condensation of the vibrating electrion, the smaller is the pro-
portionate loss of energy per period. (Compare with the more
complex problem, in which there are outgoing waves of two
diflTerent velocities, worked out in the Addition to Lee. XIV.,
pp. 190 — 219. See particularly the examples in pp. 217, 218,
219.)

§ 242. Look back now to the diagram of Lee. XII., p. 145.
representing our complex molecular vibrator of Lee. I., pp. 12, 13,
reduced to a single free mass, m ; connected by springs with the
rigid sheath, the lining of an ideal spherical cavity in ether. In
respect to that old diagram, let x now denote what was denoted
on p. 145 by f — a;; that is to say the displacement of the ether,
relatively to m. Thus in the old illustrative ideal mechanism,
ex denotes a resultant force of springs acting on m : in the new
suggestion of an electro-etherial reality ex denotes simply the
electric attraction of the atom on its electrion m, when displaced
to a distance x from its centre. In the old mechanism it is the
pulls on ether by the springs, equal and opposite to their forces
on m, by which m acts on the ether (always admittedly an unreal
kind of agency, invoked only by way of dynamical illustration).
In the new electric design, m acts directly on the ether, in simple
proportion to acceleration of relative motion. It does so because,
in virtue of the ether s inertia when m is being relatively accelerated

END OF LECTURE XX. 467

the ether is less dense before than behind m, and therefore the Moleonlar.
resultant of m's attraction on it is backwards.

It is interesting to see that every one of the formulas of
^ 240, 241 (with the new notation of x, in the old dynamical
problem), are applicable to both the old and the new subjects :
and to know that the solution of the problem in terms of periods
is the same in the two cases, notwithstanding the vast difference
between the artificial and unreal details of the mechanism thought
of and illustrated by models in 1884, and the probably real details
of ether, electricity and ponderable matter, suggested in 1900 —
1903.

§ 243. The interesting question of energy referred to in
Lee. X., 11. 18 — 21 of p. Ill becomes more and more interesting
now when we seem to understand its real quadruple character in

(I) kinetic energy of pure ether,

(II) potential energy of elasticity of ether,

(III) electric potential energy of mutual repulsions of elec-

trions and of attractions between electrions and atoms,

(IV) potential energy of attraction of electrions on ether.

It is slightly and imperfectly treated in App. C. It must, when
fully worked out, include a dynamical theory of phosphorescence.
For the present I must leave it with much regret, to allow this
Volume to be prepared for publication.

30—2

APPENDIX A.

ON THE MOTION PRODUCED IN AN INFINITE ELASTIC SOLID
BY THE MOTION THROUGH THE SPACE OCCUPIED BY IT
OF A BODY ACTING ON IT ONLY BY ATTRACTION OR
REPULSION*

§ 1. The title of the present communication describes a
pure problem of abstract mathematical dynamics, without in-
dication of any idea of a physical application. For a merely
mathematical journal it might be suitable, because the dynamical
subject is certainly interesting both in itself and in its relation
to waves and vibrations. My reason for occupying myself with
it, and for offering it to the Royal Society of Edinburgh, is that
it suggests a conceivable explanation of the greatest diflSculty
hitherto presented by the undulatory theory of light; — the
motion of ponderable bodies through infinite space occupied by
an elastic solid -I*.

§ 2. In consideration of the confessed object, and for brevity,

I shall use the word atom to denote an ideal substance occupying

a given portion of solid space, and acting on the ether within it

and around it, according to the old-fashioned eighteenth century

idea of attraction and repulsion. That is to say, every infinitesimal

volume A of the atom acts on every infinitesimal volume B of the

ether with a force in the line PQ joining the centres of these two

volumes, equal to

Af(P,PQ)pB (1).

* Communicated to the Phil Mag, by the author, having been read before the
Royal Society of Edinburgh, July 16th, 1900; and before the **Congr^B'* of the
Paris Exhibition in August 1900.

t The so-called '* electro-magnetic theory of light'* does not out away this
foundation from the old undulatory theory of light. It adds to that primary theoiy
an enormous province of transcendent interest and importance ; it demands of us
not merely an explanation of all the phenomena of light and radiant heat by
transverse vibrations of an elastic solid called ether, but also the inclusion of
electric currents, of the permanent magnetism of steel and lodestone, of magnetic
force, and of electrostatic force, in a comprehensive etherial dynamics.

FORCE BETWEEN ETHER AND ATOM. 469

where p denotes the density of the ether at Q, and /(P, PQ)
denotes a quantity depending on the position of P and on the
distance PQ. The whole force exerted by the atom on the
portion pB of the ether at Q, is the resultant of all the forces
calculated according to (1), for all the infinitesimal portions A
into which we imagine the whole volume of the atom to be
divided.

§ 3. According to the doctrine of the potential in the well-
known mathematical theory of attraction, we find rectangular
components of this resultant as follows: —

^ = pB^<l>(^>y*^)'y Y=pB-j-4>{x,y,z)\

^=p^:ni>i^>y^^)

y ...(2),

dz

where x, y, z denote coordinates of Q referred to lines fixed with
reference to the atom, and ^ denotes a function (which we call
the potential at Q due to the atom) found by summation as
follows : —

*=//WpV^^^^'^> (^>'

where ^A denotes integration throughout the volume of the
atom.

§ 4. The notation of (1) has been introduced to signify that
no limitation as to admissible law of force is essential; but no
generality, that seems to me at present practically desirable, is
lost if we assume, henceforth, that it is the Newtonian law of the
inverse square of the distance. This makes

np,pQ)=-PQ, (4).

and therefore

ll^draP,r) = ^ (5).

where a is a coefficient specifying for the point, P, of the atom,
the intensity of its attractive quality for ether. Using (5) in (3)
we find

.=///

^PQ (6),

and the components of the resultant force are still expressed
by (2). We may suppose a to be either positive or negative

470 APPEXDIX A.

(positive for attractioo and negadTe for repulsion); and in &ct
in our first and simplest illustration of the problem we suppose
it to be positive in some parts and negative in other parts of the
atom, in such quantities as to fulfil the condition

ffJAa^O (7).

§ 5. As a first and very simple illustratiou, suppoese the atom
to be spherical, of radius unity, with concentric interior spherical
surCsures of equal density. This gives, for the direction of the
resultant force on any particle of the ether, whether inside or
outside the spherical boundary of the atom, a line through the
centre of the atom. We may now take A = 4ru r^dr. The further
assumption of (7) may thus be expressed by

f.

dri^^O (8);

and this, as we are now supposing the forces between every
particle of the atom and every particle of the ether to be subject
to the Nei't'tonian law, implies, that the resultant of its attractions
and repulsions is zero for ever}' particle of ether outside the
boundary of the atom. To simplify the case to the utmost, we
shall further suppose the distribution of positive and negative
density of the atom, and the law of compressibility of the ether,
to be such, that the average density of the ether within the atom
is equal to the undisturbed density of the ether outside. Thus
the attractions and repulsions of the atom in lines through its
centre produce, at different distances from its centre, condensa-
tions and rarefactions of the ether, with no change of the total
quantity of it withio the boundar}* of the atom ; and therefore
produce no disturbance of the ether outside. To fix the ideas,
and to illustrate the application of the suggested hypothesis to
explain the refractivity of ordinary isotropic transparent bodies
such as water or gla^^s, I have chosen a definite particular case in
which the distribution of the ether when at rest within the atom
is expressed by the following formula, and partially shown in
the accompanying diagram (fig. 1), and tables of calculated
numbers : —

r'»

Here, / denotes the undisturbed distance from the centre of the
atom, of a particle of the ether which is at distance r when at

ARBITRARY EXAMPLE OF DENSITY-DISTURBANCE.

471

rest under the influence of the attractive and repulsive forces.

49r
According to this notation ^ 8(t*) is the disturbed volume of a

o

spherical shell of ether whose undisturbed radius is r and thick-

ness 8r' and volume -^ 8 (/•). Hence, if we denote the disturbed

and undisturbed densities of the ether by p and unity respectively,
we have

pa(r») = a(r") (10);

whence, by (9),

Sjl + Kjl-'/yy

.(11).

This gives 1 + -K" for the density of the ether at the centre
of the atom. In order that the disturbance may suffice for
refractivities such as those of air, or other gases, or water, or
glass, or other transparent liquids or isotropic solids, according
to the dynamical theory explained in § 16 below, I find that K
may for some caises be about equal to 100, and for others must
be considerably greater. I have therefore taken K = 100, and
calculated and drawn the accompanying tables and diagram
accordingly.

Table I.

Col. 1

Col. 2

Col. 8

Col. 8'

Col. 4

Col. 5

/

J=1 + A: (!-/)«

r

t'-r

P

(/>-l)r»

000

101-0

0^000

0^000

1010

0-000

•05

91-25

•Oil

•039

88^1

•oil

•10

82 0

•023

•077

76-8

•039

•20

65^0

•049

•151

65 8

•132

•30

50 0

•082

•218

391

•266

•40

37^0

•120

•280

26-8

•367

•60

26 0

•169

•331

15-8

•428

•60

17 0

•233

-367

8^76

•423

•70

100

•326

•375

4-17

•338

•80

5 0

•468

•332

1-60

•131

•85

3-25

•678

•272

0-90

-0^038

•90

2 00

•715

•185

0^60

- -256

•95

1-25

•866

•085

•86

- -486

•96

116

•897

•063

•36

- 616

•97

1-09

•928

-042

•39

- 626

•98

104

•967

•023

•46

- ^495

•99

101

•982

•008

•61

- -876

100

1^00

1-000

•000

100

- •ooo

472

APPENDIX A.

Table II.

Col. 1

Col. 2

Col. 8

Col. 4

Col. 6

r

/

r-/

P

{p-l)f^

000

0000

0-000

101-00

0000

•02

•091

•071

78-6

•030

•04

•169

•129

644

•191

•06

•235

•176

49-6

•175

•08

•297

•217

89-5

•246

•10

•361

•261

318

•308

•20

•661

•361

11-8

•432

•30

•677

•377

6^00

•360

•40

•768

•368

246

•234

•50

•816

•316

1*34

•086

•60

•868

- ^258

0-82

- 0065

•70

•896

•196

053

- ^231

•80

•929

•129

0-38

- -397

•90

•961

•061

0-36

- ^518

100

1000

•000

100

•000

§ 6. The diagram (fig. 1) helps us to understand the dis-
placement of ether and the resulting distribution of density,
within the atom. The circular arc marked 100 indicates a
spherical portion of the boundary of the atom; the shorter of

the circular arcs marked 95, '90, '20, '10 indicate spherical

surfaces of undisturbed ether of radii equal to these numbers.
The positions of the spherical surfaces of the same portions of
ether under the influence of the atom, are indicated by the arc
marked 1*00, and the longer of the arcs marked 95, '90, ... '50,
and the complete circles marked "40, 30, 20, '10. It may be
remarked that the average density of the ether within any one
of the disturbed spherical surfaces, is equal to the cube of the
ratio of the undisturbed radius to the disturbed radius, and is
shown numerically in column 2 of Table I. Thus, for example,
looking at the table and diagram, we see that the cube of the
radius of the short arc marked '50 is 26 times the cube of the
radius of the long arc marked '50, and therefore the average
density of the ether within the spherical surface corresponding to
the latter is 26 times the density (unity) of the undisturbed ether
within the spherical surface corresponding to the former. The
densities shown in column 4 of each table are the densities of the
ether at (not the average density of the ether within) the con-
centric spherical surfaces of radius r in the atom. Column 5 in

DIAGRAM OF DENSITT-DISTURBANCE.

473

each table shows l/^ire of the excess (positive or negative) of the
quantity of ether in a shell of radius r and infinitely small
thickness e as disturbed by the atom above the quantity in a

1-00

95

-90

•80

70

60

•50

1*00

95

'90

SO

•70.

-60

50

•40

Fig. 1.

)*0d

•95

•M

-80

•70

474

APPENDIX A.

shell of the same dimensions of uDdisturbed ether. The formula
of col. 2 makes r = 1 when r' = 1, that ia to aay the total quantity
of the disturbed ether within the radius of the atom ia the same
as that of undisturbed ether in a sphere of the same radius.
Hence the sum of the quantities of ether calculated from col. 5
for consecutive values of r, with infinitely small di6Ferences from
r = 0 to r=l, must be zero. Without calculating for smaller
differences of r than those shown in either of the tables, we find
a close verification of this result by drawing, as in fig. 2, a curve
to represent (p — l)r^ through the points for which its value is
given in one or other of the tables, and measuring the areas on
the positive and negative sides of the line of abscissas. By
drawing on paper (four times the scale of the annexed diagram),
showing engraved squares of '5 inch and '1 inch, and counting
the smallest squares and parts of squares in the two areas, I have
verified that they are equal within less than 1 per cent, of either
sum, which is as close as can be expected from the numerical
approximations shown in the tables, and from the accuracy
attained in the drawing.

' I ^»

' J "s X X

, t \

\ -l-

1 \

I" ^

' ' ■ 5 " '

"^

^

s

: '^ J

^ t

\ t

§ 7. In Table I. (argument »■') all the quantities are shown
for chosen values of /, and in Table II, for chosen values of r.

ATOM MOVING THROUGH SPACE OCCUPIED BY ETHER. 475

The calculations for Table I. are purely algebraic, involving
merely cube roots beyond elementary arithmetic. To calculate
in terms of given values of r the results shown in Table IL
involves the solution of a cubic equation. They have been
actually found by aid of a curve drawn from the numbers of
col. 3 Table L, showing r in terms of r'. The numbers in col. 2
of Table II. showing, for chosen values of r, the corresponding
values of r', have been taken from the curve ; and we may verify
that they are approximately equal to the roots of the equation
shown at the head of col. 2 of Table I., regarded as a cubic for /
with any given values of r and K,

Thus, for example, taking / = *929 we calculate r = 'Sll,

„ r' = -816 „ r = -498,

„ r'=f-677 „ r = -301,

„ r'=091 „ r = -0208,

where we should have r = '8, '5, '3, and '02 respectively. These
approximations are good enough for our present purpose.

§ 8. The diagram of fig. 2 is interesting, as showing how,
with densities of ether varying through the wide range of from
'35 to 101, the whole mass within the atom is distributed among
the concentric spherical surfaces of equal density. We see by it,
interpreted in conjunction with col. 4 of the tables, that fi'om the
centre to '56 of the radius the density falls from 101 to 1. For
radii from *56 to 1, the values of (/o — l)r^ decrease to a negative
minimum of 025 at r = '93, and rise to zero at r = 1. The place
of minimum density is of course inside the radius at which
(/o — 1) r* is a minimum ; by cols. 4 and 3 of Table I., and cols. 4
and 1 of Table II., we see that the minimum density is about '35,
and at distance approximately 87 from the centre.

§ 9. Let us suppose now our atom to be set in motion through
space occupied by ether, and kept in motion with a uniform
velocity v, which we shall first suppose to be infinitely small in
comparison with the propagational velocity of equivoluminal*
waves through pure ether undisturbed by any other substance
than that of the atom. The velocity of the earth in its orbit

* That is to say, waves of transverse vibration, being the only kind of wave in
an isotropic solid in which every part of the solid keeps its volume unchanged
during the motion. See Phil. Mag.^ May, August, and October, 1899.

476 APPENDIX A.

round the sun being about 1/10,000 of the velocity of light, is
small enough to give results, kinematic and dynamic, in respect
to the relative motion of ether and the atoms constituting the
earth closely in agreement with this supposition. According to
it, the position of every particle of the ether at any instant is the
same as if the atom were at rest ; and to find the motion produced
in the ether by the motion of the atom, we have a purely
kinematic problem of which an easy graphic solution is found
by marking on a diagram the successive positions thus determined
for any particle of the ether, according to the positions of the
atom at successive times with short enough intervals between
them, to show clearly the path and the var)dng velocity of the
particle.

§ 10. Look, for example, at fig. 3, in which a semi-circum-
ference of the atom at the middle instant of the time we are
going to consider, is indicated by a semicircle C^ACq, with
diameter CqC^ equal to two units of length. Suppose the centre
of the atom to move from right to left in the straight line Cfi^
with velocity 1, taking for unit of time the time of travelling 1/10
of the radius. Thus, reckoning from the time when the centre
is at Co, the times when it is at Gg, (7b, Ck,, Gi^, G^ are 2, 5, 10,
18, 20. Let Q' be the undisturbed position of a particle of ether
before time 2 when the atom reaches it, and after time 18 when
the atom leaves it. This implies that QG^ = Q'Oig = 1, and
(/jCio = CioCia = '8, and therefore CwQ' = '6. The position of the
particle of ether, which when undisturbed is at Q', is found for
any instant t of the disturbance as follows : —

Take GoG = t/lO; draw Q'C, and calling this r' find r'-7' by
formula (9), or Table I. or II. : in Q'G take Q'Q = r'-r, Q is
the position at time t of the particle whose undisturbed position
is Q'. The drawing shows the construction for ^ = 2, and ^ = 5,
and t= 18. The positions at times 2, 3, 4, 5, ... 15, 16, 17, 18
are indicated by the dots marked 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4,
5, 6, 7, 8 on the closed curve with a corner at Q\ which has been
found by tracing a smooth curve through them. This curve,
which, for brevity, we shall call the orbit of the particle, is
clearly tangential to the lines Q'Oj and Q'Cig. By looking to the
formula (9), we see that the velocity of the particle is zero at
the instants of leaving Q and returning to it. Fig. 4 shows the

AN ABSOLUTE ORBIT AND ITS RELATIVE STREAM-LINE. 477

00

478 APPENDIX A.

particular orbit of fig. 3, and nine others drawn by the same
method ; in all ten orbits of ten particles whose undisturbed
positions are in one line at right angles to the line of motion of
the centre of the atom, and at distances 0, 1, '2, ... 9 from it.
All these particles are again in one straight line at time 10, being
what we may call the time of mid-orbit of each particle. The
numbers marked on the right-hand halves of the orbits are times
from the zero of our reckoning; the numbers 1, 2, 3 ... etc. on
the left correspond to times 11, 12, 13 ... of our reckoning as
hitherto, or to times 1, 2, 3 ... after mid-orbit passages. Lines
drawn across the orbits through 1, 2, 3 ... on the left, show
simultaneous positions of the ten particles at times 1,^2, 3 after
mid- orbit. The line drawn from 4 across seven of the curved
orbits, shows for time 4 after mid-orbit, simultaneous positions
of eight particles, whose undisturbed distances are 0, '1, ... '7.
Remark that the orbit for the first of these ten particles is a
straight line.

§ 11. We have thus in § 10 solved one of the two chief
kinematic questions presented by our problem: — to find the
orbit of a particle of ether as disturbed by the moving atom,
relatively to the surrounding ether supposed fixed. The other
question, to find the path traced through the atom supposed fixed
while, through all space outside the atom, the ether is supposed
to move uniformly in parallel lines, is easily solved, as follows : —
Going back to fig. 3, suppose now that instead of, as in § 10, the
atom moving from right to left with velocity '1 and the ether
outside it at rest, the atom is at rest and the ether outside it is
moving from left to right with velocity '1. Let '2, '3, '4, '5, '6, '7,
'8, '9, 0, 1, '2, % '4, '5, '6, '7, '8 be the path of a particle of ether
through the atom marked by seventeen points corresponding to
the same numbers unaccented showing the orbit of the same
particle of ether on the former supposition. On both suppositions,
the position of the particle of ether at time 10 from our original
era, (§ 10), is marked 0. For times 11, 12, 13, etc., the positions
of the particle on the former supposition are marked 1, 2, 3, 4, 6,
6, 7, 8 on the left half of the orbit. The positions of the same
particle on the present supposition are found by drawing from the
points 1, 2, 3, ... 7, 8 parallel lines to the right, 1 '1, 2 '2, 3 '3, ...
7 7, 8 '8, equal respectively to 1, % '3, ... "7, 8 of the radius of

ABSOLUTE OBBITS OF TEN PARTICLES OF ETHER.

479

the atom, being our unit of length. Thus we have the latter half
of the passage of the particle through the atom ; the first half is
equal and similar on the left-hand aide of the atom. Applying

i

i

K/J-^X^

f-

^^^^^!><\

i''j^><?tl4^

]

IZ^^^xT^^

/'■''''^^~~^^~—l~^ /_— i—

^_— -''''**^

3^^=*™)^^^^!^

- ~^^^^^^^^:_:=-^

'^^^— 1

■" ' "H!^

8(9

'5 5 4 J 2

C

9 t ' 7 6 S 43

Fig. 4.

the same process to every one of the ten orbits shown in
fig. 4, and to the nine orbits of particles whose undisturbed
distances from the central line on the other side are "l, '2, ... '9,
we find the set of stream-lines shown in tig. S (p. 480). The
dots on these lines show the positions of the particles at times

480

APPEN'DtX A.

0, 1, 2, ... 19, 20 of our origtoal reckoning (§ 10>. The nninbera
CD the stream-IiDe of the particle whose undisturbed distance
from the central line is "6 are marked for comparison with fig. 3.

Fig. fi.

The lines drawn across the stream-lines on the left-band side of
fig. 5 show simultaneous positions of rows of particles of ether
which, when undisturbed, are in straight lines perpendicular to

KINETIC ENERGY OF ETHER WITHIN A MOVING ATOM.. 481

the direction of motion. The quadrilaterals thus formed within
the left-hand semicircle show the figures to which the squares
of ether, seen entering from the left-hand end of the diagram,
become altered in passing through the atom. Thus we have
completed the solution of our second chief kinematic question.

§ 12. The first djmamic question that occurs to us, returning
to the supposition of moving atom and of ether outside it at rest,
is : — What is the total kinetic energy (k) of the portion of the
ether which at any instant is within the atom? To answer it,
think of an infinite circular cylinder of the ether in the space
traversed by the atom. The time-integral from any era t = 0 o(
the total kinetic energy of the ether in this cylinder is t/c] because
the ether outside the cylinder is undisturbed by the motion of the
atom according to our present assumptions. Consider any circular
disk of this cylinder of infinitely small thickness e. After the
atom has passed it, it has contributed to t/c, an amount equal to
the time-integral of the kinetic energies of all the orbits of small
parts into which we may suppose it divided, and it contributes
no more in subsequent time. Imagine the disk divided into
concentric rings of rectangular cross-section edr\ The mass of
one of these rings is 2irr'dr^e because its density is unity; and
all its parts move in equal and similar orbits. Thus we find that
the total contribution of the disk amounts to

lire

rdrVJd^/dt (12),

where jds^jdt denotes integration over one-half the orbit of a
particle of ether whose undisturbed distance from the central
line is r'; (because i^d^jd^ is the kinetic energy of an ideal
particle of unit mass moving in the orbit considered). Now the
time-integral kI is wholly made up by contributions of successive
disks of the cylinder. Hence (12) shows the contribution per
time ejq, q being the velocity of the atom ; and (k being the
contribution per unit of time) we therefore have

= 27rqCdrVJd^ldt (13).

§ 13. The double integral shown in (13) has been evaluated
with amply sufficient accuracy for our present purpose by seem-
ingly rough summations; firstly, the summations Jd^fdt for the

T. L. 31

482

APPENDIX A.

ten orbits shown in fig. 4, and secondly, summation of these sums
each multiplied by drr'. In the summations for each half-orbit,
da has been taken as the lengths of the curve between the con-
secutive points from which the curve has been traced. This
implies taking dt — \ throughout the three orbits corresponding
to undisturbed distances from the central line equal respectively
to 0, "6, '8 ; and throughout the other serai-orbits, except for the
portions next the comer, which correspond essentially to intervals
each < 1. The plan followed is suflRciently illustrated by the
accompanying Table III., which shows the whole process of
calculating and summing the parts for the orbit corresponding
to undisturbed distance 7.

Table IV. shows the sums for the ten orbits and the products
of each sum multiplied by the proper value of r', to prepare for'
the final integration, which has been performed by finding the
area of a representative curve drawn on conveniently squared
paper as described in § 6 above. The result thus found is '02115.
It is very satisfactory to see that, within •! per cent., this agrees
with the simple sum of the widely diflferent numbers shown in
col. 3 of Table IV.

Table III.

Orbit r' = -7.

Table IV.

d»

•006

•000036

dt
0-14

ds^ldt

•000257

•137

•018769

1-00

•018769

•112

•012544

1-00

•012544

•077

•0051)29

100

•005929

•or)0

•002500

1-00

•002500

•048

•002304

100

•002304

•060

•002600

100

•002500

•062

•002704

1-00
Sum

•002704

1

•047507

•

•0

jds^ldt
•0818

'l.r'.jdiflldt

•00000

•1

•0804

•00080

•2

•0781

•00156

•3

•0769

•00231

•4

•0722

•00289

•5

•0670

•00336 1

•6

•0567

•00310

•7

•0475

•00332

•8

•0310

•00248

•9

•0114
Sum

•00102

1
•02113

§ 14. Using in (13) the conclusion of § 13, and taking 5 = 1,
we find

/c = 27r. 002115 (14).

A convenient way of explaining this result is to remark that it is

EXTRA INERTIA OF ETHER MOVING THROUGH A FIXED ATOM. 483

/47r 1 \
•634 of the kinetic energy (-^- ai'^y) ^^ ^^ ideal globe of rigid

matter of the same bulk as our atom, moving with the same
velocity. Looking now at the definition of /c in the beginning
of § 12, we may put our conclusion in words, thus: — The dis-
tribution of etherial density within our ideal spherical atom
represented by (11) with K = 100, gives rise to kinetic energy
of the ether within it at any instant, when the atom is moving
slowly through space filled with ether, equal to 634 of the kinetic
energy of motion with the same velocity through ideal void space,
of an ideal rigid globe of the same bulk as the atom, and the
same density as the undisturbed density of the ether. Thus if
the atom, which we are supposing to be a constituent of real
'ponderable matter, has an inertia of its own equal to / per unit
of its volume, the effective inertia of its motion through space

occupied by ether will be ^«*(/ + '634); the diameter of the

atom being now denoted by s (instead of 2 as hitherto), and the
inertia of unit bulk of the ether being still (as hitherto) taken as
unit of inertia. In all that follows we shall suppose 7 to be very
great, much greater than 10'; perhaps greater than 10".

§ 15. Consider now, as in § 11 above, our atom at rest; and
the ether moving uniformly in the space around the atom, and
through the space occupied by the atom, according to the curved
stream-lines and the varying velocities shown in fig. 5. The
effective inertia of any portion of the ether containing the atom
will be greater than the simple inertia of an equal volume of the

ether by the amount ^ «* x 634. This follows from the well-known

dynamical theorem that the total kinetic energy of any moving
body or system of bodies is equal to the kinetic energy due to the
motion of its centre of inertia, plus the sum of the kinetic energies
of the motions of all its parts relative to the centre of inertia.

§ 16. Suppose now a transparent body — solid, liquid, or
gaseous — to consist of an assemblage of atoms all of the same
magnitude and quality as our ideal atom defined in § 2, and
with / enormously great as described in § 14. The atoms may
be all motionless as in an absolutely cold solid, or they may have
the thermal motions of the molecules of a solid, liquid, or gas at

31—2

484 APPENDIX A.

any temperature not so high but that the thermal velocities are
everywhere small in comparison with the velocity of light. The
eflfective inertia of the ether per unit volume of the assemblage
will be exceedingly nearly the same as if the atoms were all
absolutely fixed, and will therefore, by § 15, be equal to

l4-iV^^5'-634 (15),

0

where N denotes the number of atoms per cubic centimetre of the
assemblage, one centimetre being now our unit of length. Hence,
if we denote by V the velocity of light in undisturbed ether, its
velocity through the space occupied by the supposed assemblage
of atoms will be

F/(l+iV^^«»-634y (16).

§ 17. For example, let us take iV= 4 x 10** ; and, as I find
suits the cases of oxygen and argon, s = 1*42 x 10~®, which gives

i\r^ a* = '60 X 10"'. The assemblage thus defined would, if con-
densed one-thousand-fold, have '6 of its whole volume occupied by
the atoms and "4 by undisturbed ether ; which is somewhat denser
than the cubic arrangement of globes

(space unoccupied = 1 — ^ = •4764] ,

and less dense than the densest possible arrangement

(space unoccupied = 1 — ^-^ = •2595] .

Taking now N^^ = '60x 10~' in (16), we find for the refractive

u

index of our assemblage 1*00019, which is somewhat smaller than
the refractive index of oxygen (1*00027 3). By taking for K a larger
value than 100 in (11), we could readily fit the formula to give, in

* I am forced to take this very large number instead of Maxwell's 19 x 10^,
as I have found it otherwise impossible to reconcile the known viscosities and the
known condensations of hydrogen, oxygen, and nitrogen with MaxweU's theoretical
formulas. [In § 60 of Lect. XVII. of the present volume we saw that the smaUer
value 10^ is admissible and probably may be not far from the truth.] It must be
remembered that Avogadro's law makes N the same for all gases.

DIFFICULTY RAISED BY MICHELSON'S EXPERIMENT. 486

an assemblage in which '6 x 10~* of the whole space is occupied
by the atom, exactly the refractive index of oxygen, nitrogen, or
argon, or any other gas. It is remarkable that according to the
particular assumptions specified in § 5, a density of ether in the
centre of the atom considerably greater than 100 times the
density of undisturbed ether is required to make the refractivity
as great as that of oxygen. There is, however, no diflSculty in
admitting so great a condensation of ether by the atom, if we
are to regard our present problem as the basis of a physical
hypothesis worthy of consideration.

§ 18. There is, however, one serious, perhaps insuperable,
diflSculty to which I must refer in conclusion: the reconciliation
of our hypothesis with the result that ether in the earth's
atmosphere is motionless relatively to the earth, seemingly proved
by an admirable experiment designed by Michelson, and carried
out with most searching care to secure a trustworthy result, by
himself and Morley*. I cannot see any flaw either in the idea
or in, the execution of this experiment. But a possibility of
escaping from the conclusion which it seemed to prove may be
found in a brilliant suggestion made independently by Fitzgerald f,
and by LorentzJ of Leyden, to the effect that the motion of ether
through matter may slightly alter its linear dimensions; according
to which if the stone slab constituting the sole plate of Michelson
and Morley s apparatus has, in virtue of its motion through space
occupied by ether, its lineal dimensions shortened one one-hundred-
millionth§ in the direction of motion, the result of the experiment
would not disprove the free motion of ether through space occupied
by the earth.

* Phil Mag,, December, 18S7.

t Public Lectures in Trinity College, Dublin.

X Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten
K'&rpem, Leiden, 1896.

§ This being the square of the ratio of the earth's velocity round the son
(30 kilometres per sec.) to the velocity of light (300,000 kilometres per sec.).

APPENDIX B.

NINETEENTH CENTURY CLOUDS OVER THE DYNAMICAL

THEORY OF HEAT AND LIGHT*

(Friday evening Lecture, Royal Institution, April 27, 1900.)

[In the present article the substance of the lecture is repro-
duced— with large additions, in which work commenced at the
beginning of last year and continued after the lecture, during
thirteen months up to the present time, is described — ^with results
confirming the conclusions and largely extending the illustrations
which were given in the lecture. I desire to take this opportunity
of expressing my obligations to Mr William Anderson, my secretary
and assistant, for the mathematical tact and skill, the accuracy of
geometrical drawing, and the un&ilingly faithful perseverance in
the long-continued and varied series of drawings and algebraic
and arithmetical calculations, explained in the following pages.
The whole of this work, involving the determination of results
due to more than five thousand individual impacts, has been
performed by Mr Anderson. — K., Feb. 2, 1901.]

§ 1. The beauty and clearness of the dynamical theory, which
asserts heat and light to be modes of motion, is at present ob-
scured by two clouds. I. The first came into existence with the
imdulatory theor}' of light, and was dealt with by Fresnel and
Dr Thomas Young; it involved the question, How could the earth
move through an elastic solid, such as essentially is the lumini-
ferous ether ? II. The second is the Maxwell-Boltzmann doctrine
regarding the partition of energy.

§ 2, Cloud I. — Relative Motion of Ether and Ponder-
able Bodies; such as movable bodies at the earths surface,
stones, metals, liquids, gases; the atmosphere surrounding the
earth; the earth itself as a whole; meteorites, the moon, the sun,

* Journal of the Royal Institution, Also Phil, Mag. July, 1901.

MOTION OF ETHER THROUGH PONDERABLE MATTER. 487

and other celestial bodies. We might imagine the question satis-
factorily answered, by supposing ether to have practically perfect
elasticity for the exceedingly rapid vibrations, with exceedingly
small extent of distortion, which constitute light; while it behaves
almost like a fluid of very small viscosity, and yields with ex-
ceedingly small resistance, practically no resistance, to bodies
moving through it as slowly as even the most rapid of the
heavenly bodies. There are, however, many very serious objections
to this supposition; among them one which has been most noticed,
though perhaps not really the most serious, that it seems in-
compatible with the known phenomena of the aberration of light.
Referring to it, Fresnel, in his celebrated letter* to Arago, wrote
as follows:

" Mais il parait impossible d'expliquer Taberration des 6toiles
"dans cette hypoth^se; je n'ai pu jusqu'i present du moins con-
"cevoir nettement ce ph^nomfene qu'en supposant que Tdther
" passe librement au travers du globe, et que la vitesse communi-
" qu^e k ce fluide subtil n*est qu'une petite partie de celle de la
"terre; nen exc^de pas le centifeme, par exemple.

"Quelque extraordinaire que paraisse cette hypothfese au premier
" abord, elle n'est point en contradiction, ce me semble, avec Tid^e
" que les plus grands physiciens se sont faite de Textrfeme porosity
" des corps."

The same hypothesis was given by Thomas Young, in his
celebrated statement that ether passes through among the mole-
cules or atoms of material bodies like wind blowing through a
grove of trees. It is clear that neither Fresnel nor Young had
the idea that the ether of their undulatory theory of light, with
its transverse vibrations, is essentially an elastic solid, that is to
say, matter which resists change of shape with permanent or
sub-permanent force. If they had grasped this idea they must
have noticed the enormous difficulty presented by the laceration
which the ether must experience if it moves through pores or
interstices among the atoms of matter.

§3. It has occurred to me that, without contravening any-
thing we know from observation of nature, we may simply deny
the scholastic axiom that two portions of matter cannot jointly

* AnnaUs de Chimie, 1818 ; quoted in full by Larmor in his recent book, JEther
and Matter, pp. 320—322.

488 APPENDIX B.

occupy the same space, and may assert, as an admissible hypo-
thesis, that ether does occupy the same space as ponderable
matter, and that ether is not displaced by ponderable bodies
moving through space occupied by ether. But how then could
matter act on ether, and ether act on matter, to produce the
known phenomena of light (or radiant heat), generated by the
action of ponderable bodies on ether, and acting on ponderable
bodies to produce its visual, chemical, phosphorescent, thermal,
and photographic effects? There is no diflBculty in answering
this question if, as it probably is, ether is a compressible and
dilatable* solid. We have only to suppose that the atom exerts
force on the ether, by which condensation or rarefaction is pro-
duced within the space occupied by the atom. At present^ I
confine myself, for the sake of simplicity, to the suggestion of a
spherical atom producing condensation and rarefaction, with con-
centric spherical surfaces of equal density, but the same total
quantity of ether within its boundary as the quantity in an
equal volume of free undisturbed ether.

§4. Consider now such an atom given at rest anywhere in
space occupied by ether. Let force be applied to it to cause it to
move in any direction, first with gradually increasing speed, and
after that with uniform speed. If this speed is anything less
than the velocity of light, the force may be mathematically proved
to become zero at some short time after the instant when the
velocity of the atom becomes uniform, and to remain zero for
ever thereafter. What takes place is this :

§ 5. During all the time in which the velocity of the atom is
being augmented from zero, two sets of non-periodic waves, one
of them equi-voluminal, the other irrotational (which is therefore
condensational-rarefactional), are being sent out in all directions
through the surrounding ether. The rears of the last of these
waves leave the atom, at some time after its acceleration ceases.
This time, if the motion of the ether outside the atom, close

* To deny this property is to attribute to ether indefiDitely great resistance
against forces tending to condense it or to dilate it — which seems, in truth, an
infinitely difficult assumption.

t Further developments of the suggested idea have been contributed to the
Boyal Society of Edinburgh, and to the Congrds International de Physique held in
Paris in August. {Proc, R,8,E, July 1900 ; Vol. of reports, in French, of the Cong,
Inter,; and Phil. Mag,^ Aug., Sept., 1900.)

UNIFORM MOTION OF ATOM RESISTED IF QUICKER THAN LIGHT. 489

beside it, is infinitesimal, is equal to the time taken by the slower
wave (which is the equi-voluminal) to travel the diameter of the
atom, and is the short time referred to in §4. When the rears
of both waves have got clear of the atom, the ether within it and
in the space around it, left clear by both rears, has come to a
steady state of motion relatively to the atom. This steady motion
approximates more and more nearly to uniform motion in parallel
lines, at greater and greater distances, from the atom. At a
distance of twenty diameters it differs exceedingly little from
uniformity.

§6. But it is only when the velocity of the atom is very
small in comparison with the velocity of light, that the dis-
turbance of the ether in the space close round the atom is
infinitesimal. The propositions asserted in § 4 and the first sen-
tence of §5 are true, however little the final velocity of the atom
falls short of the velocity of light. If this uniform final velocity
of the atom exceeds the velocity of light, by ever so little, a
non-periodic conical wave of equi-voluminal motion is produced,
according to the same principle as that illustrated for sound by
Mach's beautiful photographs of illumination by electric spark,
showing, by changed refractivity, the condensational- rarefactional
disturbance produced in air by the motion through it of a rifle
bullet*. The semi-vertical angle of the cone, whether in air or
ether, is equal to the angle whose sine is the ratio of the wave
velocity to the velocity of the moving body*.

* On the same principle we see that a body moving steadily (and, with little
error, we may say also that a fish or water-fowl propelling itself by fins or web-feet)
through calm water, either floating on the surface or wholly submerged at some
moderate distance below the surface, produces no wave disturbance if its velocity is
less than the minimum wave velocity due to gravity and surface tension (being about
23 cms. per second, or '44 of a nautical mile per hour, whether for sea water or fresh
water) ; and if its velocity exceeds the minimum wave velocity, it produces a wave
disturbance bounded by two lines inclined on each side of its wake at angles each
equal to the angle whose sine is the minimum wave velocity divided by the velocity
of the moving body. It is easy for anyone to observe this by dipping vertically a
pencil or a walking-stick into still water in a pond (or even in a good-sized hand-
basin), and moving it horizontally, first with exceeding small speed, and afterwards
faster and faster. I first noticed it nineteen years ago, and described observations
for an experimental determination of the minimum velocity of waves, in a letter to
William Froude, published in Nature for Oct., in PhiL Mag. for Nov. 1871, and
in App. G below, from which the following is extracted. ''[Recently, in the
"schooner yacht LaUa Rookh], being becalmed in the Sound of Mull, I had an
" excellent opportunity, with the assistance of Professor Helmholtz, and my brother

490 APPENDIX a

§ 7. If, for a moment, we imagine the steady motion of the
atom to be at a higher speed than the wave velocity of the
condensational-rarefactional wave, two conical waves, of angles
corresponding to the two wave velocities, will be steadily pro-
duced; but we need not occupy ourselves at present with this
case because the velocitv of the condensational-rarefieustional wave
in ether is, we are compelled to believe, enormously great in
comparison with the velocity of light.

§ 8. Let now a periodic force be applied to the atom so as to
cause it to move to and fro continually, with simple harmonic
motion. By the first sentence of § 5 we see that two sets of
periodic waves, one equi-voluminal, the other irrotational, are
continually produced. Without mathematical investigation we
see that if^ as in ether, the condensational-rarefactional wave
velocity is very great in comparison with the equi-voluminal wave
velocity, the energy taken by the condensational-rarefSstctional
wave is exceedingly small in comparison with that taken by the
equi-voluminal wave; how small we can find easily enough by
regular mathematical investigation. Thus we see how it is that
the hj-pothesis of § 3 suffices for the answer suggested in that
section to the question. How could matter act on ether so as to
produce light ?

§ 9. But this, though of primary importance, is only a small
part of the very general question pointed out in § 3 as needing
answer. Another part, fundamental in the undulatory theory of

** from Belfast [the late Professor James Thomson], of determining by ohaerration
**the minimam wave-velocity with some approach to accuracy. The fishing-line
** was hong at a distance of two or three feet from the vessel's side, so as to cot the
** water at a point not sensibly disturbed by the motion of the vessel. The speed
** was determined by throwing into the sea pieces of paper previously wetted, and
** observing their times of transit across parallel planes, at a distance of 912 eenti-
** metres asunder, fixed relatively to the vessel by marks on the deck and gunwale.
**By watching carefully the pattern of ripples and waves which connected the
** ripples in front with the waves in rear, I had seen that it included a set of
** parallel waves 8lanting off obliquely on each side and presenting appearances
** which proved them to be waves of the critical length and corresponding minimum
** speed of propagation.'' When the speed of the yacht fell to but little above the
critical velocity, the front of the ripples was very nearly perpendicular to the line
of motion, and when it just fell below the critical velocity the ripples disappeared
altogether, and there was no perceptible disturbanoe on the surface of the water.
The sea was ** glassy"; though there was wind enough to propel the schooner at
speed varying between \ mile and 1 mile per hour.

MOTION OF THE EARTH THROUGH SPACE OCCUPIED BY ETHER. 491

optics, is, How is it that the velocity of light is smaller in trans-
parent ponderable matter than in pure ether? Attention was
called to this particular question in my address, to the Royal
Institution, of last April; and a slight explanation of my pro-
posal for answering it was given, and illustrated by a diagram.
The validity of this proposal is confirmed [in App. A] by a some-
what elaborate discussion and mathematical investigation of the
subject worked out since that time and communicated under the
title, ** On the Motion produced in an Infinite Elastic Solid by the
Motion through the Space occupied by it of a Body acting on it
only by Attraction or Repulsion," to the Royal Society of Edin-
burgh on July 17, and to the Congrfes International de Physique
for its meeting at Paris in the beginning of August.

§ 10. The other phenomena referred to in § 3 come naturally
under the general dynamics of the undulatory theory of light,
and the full explanation of them all is brought much nearer if
we have a satisfactory fundamental relation between ether and
matter, instead of the old intractable idea that atoms of matter
displace ether from the space before them, when they are in
motion relatively to the ether around them. May we then sup-
pose that the hypothesis which I have suggested clears away the
first of our two clouds? It certainly would explain the "aber-
ration of light" connected with the earth's motion through
ether in a thoroughly satisfactory manner. It would allow the
earth to move with perfect freedom through space occupied by
ether without displacing it. In passing through the earth the
ether, an elastic solid, would not be lacerated as it would be
according to Fresnel's idea of porosity and ether moving through
the pores as if it were a fluid. Ether would move relatively to
ponderables with the perfect freedom wanted for what we know
of aberration, instead of the imperfect freedom of air moving
through a grove of trees suggested by Thomas Young. According
to it, and for simplicity neglecting the comparatively very small
component due to the earth's rotation (only '46 of a kilometre per
second at the equator where it is a maximum), and neglecting
the imperfectly known motion of the solar system through space
towards the constellation Hercules, discovered by Herschel*, there

* The splendid spectroscopic method originated hy Hoggins thirty-three years
ago, for measuring the component in the line of vision of the relative motion of the

492 APPENDIX B.

would be at all points of the earth's surfiskce a flow of ether at the
rate of 30 kilometres per second in lines all parallel to the tangent
to the earth's orbit round the sun. There is nothing inconsistent
with this in all we know of the ordinary phenomena of terrestrial
optics; but, alas! there is inconsistency with a conclusion that
ether in the earth's atmosphere is motionless relatively to the
earth, seemingly proved by an admirable experiment designed by
Michelson, and carried qjit. with most searching care to secure a
trustworthy result, by himself and Morley*. I cannot see any
flaw either in the idea or in the execution of this experiment
But a possibility of escaping from the conclusion which it seemed
to prove may be found in a brilliant suggestion made indepen-
dently by Fitzgerald f and by LorentzJ of Leyden, to the effect
that the motion of ether through matter may slightly alter its
linear dimensions, according to which if the stone slab constituting
the sole plate of Michelson and Morley's apparatus has, in virtue
of its motion through space occupied by ether, its lineal dimensions
shortened one one-hundred-millionth§ in the direction of motion,
the result of the experiment would not disprove the free motion
of ether through space occupied by the earth.

§ 11. I am afraid we must still regard Cloud No. I. as very
dense.

eartli, and any visible star, has been carried on since that time with admirable
perseverance and skill by other observers, who have from their results made
estimates of the velocity and direction of the motion through space of the centre
of inertia of the solar system. My Glasgow colleague, Professor Becker, has kindlj
given me the following information on the subject of these researches :

** The early (1888) Potsdam photographs of the spectra of 51 stars brighter than
** 2^ magnitude have been employed for the determination of the apex and velocity
*' of the solar system. Kempf (Astronomische Nachrichteiu Vol. 132) finds for the
*• apex : right ascension, 206° ± 12°; declination, 46° ± 90°; velocity, 19 kilometres
** per second ; and Kisteen (Astronomical Journal, 1893) finds practicaUy the same
" quantities. The proper motions of the fixed stars assign to the apex a position
** which may be anywhere in a narrow zone parallel to the Milky-way, and extending
** 20° on both sides of a point of Bight Ascension 275° and Declination + 30°. The
** authentic mean of 13 values determined by the methods of Argelander or Airy
"gives 274° and +36° (Andr^, Traiti d' Astronomic Stellaire),"

• Phil. Mag., December 1887.

t Public Lectures in Trinity College, Dublin.

X Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegUn
KVrpem, Leiden, 1895.

§ This being the square of the ratio of the earth's velocity round the son
(30 kilometres per sec.) to the velocity of light (300,000 kilometres per sec).

WATERSTONIAN-MAXWELLIAN DISTRIBUTION OF ENERGIES. 493

§ 12. Cloud II. Waterston (in a communication to the Royal
Society, now famous/ which, after lying forty-five years buried
and almost forgotten in the archives, was rescued from oblivion
by Lord Rayleigh and published, with an introductory notice of
great interest and importance, in the Transactions of the Royal
Society for 1892), enunciated the following proposition : " In mixed
"media the mean square molecular velocity is inversely propor-
" tional to the specific weight of the molecule. This is the law
"of the equilibrium of vis viva." Of this proposition Lord
Rayleigh in a footnote* says, "This is the first statement of a
" very important theorem (see also Brit Assoc, Rep,j 1851). The
"demonstration, however, of §10 can hardly be defended. It
"bears some resemblance to an argument indicated and exposed
" by Professor Tait (Edinburgh Trans., Vol. 33, p. 79, 1886). There
"is reason to think that this law is intimately connected with
"the Maxwellian distribution of velocities of which Waterston
"had no knowledge."

§13. In Waterston's statement the "specific weight of a
molecule" means what we now call simply the mass of a mole-
cule; and "molecular velocity" means the translational velocity
of a molecule. Writing on the theory of sound in the Phil. Mag.
for 1858, and referring to the theory developed in his buried
paper f, Waterston said, "The theory... assumes... that if the
" impacts produce rotatory motion the vis viva thus invested bears
"a constant ratio to the rectilineal vis viva." This agrees with
the very important principle or truism given independently about
the same time by Clausius to the effect that the mean energy,
kinetic and potential, due to the relative motion of all the parts
of any molecule of a gas, bears a constant ratio to the mean
energy of the motion of its centre of inertia when the density and
pressure are constant.

§ 14. Without any knowledge of what was to be found in
Waterston's buried paper. Maxwell, at the meeting of the British
Association at Aberdeen, in 185 9 J, gave the following proposition

♦ Phil Trans. A, 1892, p. 16.

t ** On the PhysicB of Media that are Composed of Free and Perfectly Elastic
Molecules in a State of Motion." Phil. Trans. A, 1892, p. 13.

t ** Illustrations of the Dynamical Theory of Gases,*' Phil. Mag., January and
July 1860, and ColUcUd Papers, Vol i. p. 378.

494 APPEXDDL &

regarding the motioD and ccdliaons of perfectlr elastic sphere? :
"Two sTstems of particles more in the same veaoel: to ptove
" that the mean vis vi\*a of each particle will become the same
-in the two systems.* This is precisely Waterston's piopoatioB
regarding the law of partition of energy, quoted in § 12 above :
bot Maxwell's 1>60 pn:of was certainly not more soccessAil than
Waterston's. Maxwell's ISM prcof has always seemed to me
quite inconclusiTe, and many times I nrged my colleague, Fpo-
fessor Tait, to enter on the subject. This he did. and in 1S86
he communicated to the Royal Society of Edinbargfa a paper*
on the foundations of the kinetic theoTy of gases, whidi oootained
a critical examination of Maxwell's 1S60 paper, highly apprecianTe
c( the great origiDaliry and splendid value, for the kinetic theocy
of gasesv. of the ideas and principles set fonh in it : hot showing
that the demonstTati^*n of the :he«:*rem of tbe partiti<*n of eneq^r
in a mixed assemblage oi panicles of different masses was
inconclusive, and successfully substitutiijg for it a conchxaTe
demonstration.

§ l-"^. Watei^oiu MaxwelL and Tait, all assume that the
particles of the two systems are thorc^ughly mixed iTait. | ISt
and their theorem is of fundamental importance in respect to
the specific heats of mixed eases. But ther do noc in anv of
the i«i>ers alreaiv leferr^i to, irive anr indicaifc^i of a pf*x>f of
the cs>rresp>Dding tbe-Mem. regajding ifce pardiion of energy
between two sets of e*3-al panicles separaie^i bv a membia2>e
impermeable to :he mrlecules. while perminiiLg forces to act
ac7v>ss i: between ike mv!ec:i!es c^ iis rwo sides***, which is the
simplest illustration of :he mv'.ec:i*.ar dynamics of Avogadro's
law. It seems :o me, towevrr. that Tail's demonsrration of the
Waierstoc-Maxwell law may p.'«5^^:y be st:wn to Tirrcally iticlTide.
n-M ociIt this vitaliv imi»r:aE.: s-iib-rci. b-t also the verr interesi-
in^. tho^ieh c»?diaratiTelT Tmimt^rnant. case of an assemb-La^e of
parti'^lt^ of eq-al n:,asses '•-ith a sin^ile panicle of different mass
moriiiC aK^i am-MiZ them.

r-nw^ Mat «^ :»7*: CiC^ru^ f /yrrr. V.-L =_ rf. 71 i — 741.

VALID PRINCIPLE OF WATERSTON AND CLAUSIUS. 495

§ 16. In ^ 12, 14, 15, " particle " has been taken to mean what
is commonly, not correctly, called an elastic sphere, but what is in
reality a Boscovich atom acting on other atoms in lines exactly
through its centre of inertia (so that no rotation is in any case
produced by collisions), with, as law of action between two atoms,
no force at distance greater than the sum of their radii, infinite
force at exactly this distance. None of the demonstrations, unsuc-
cessfiil or successful, to which I have referred would be essentially
altered if, instead of this last condition, we substitute a repulsion
increasing with diminishing distance, according to any law for
distances less than the sum of the radii, subject only to the
condition that it would be infinite before the distance became
zero. In fact the impact, oblique or direct, between two Boscovich
atoms thus defined, has the same result after the collision is
completed (that is to say, when their spheres of action get outside
one another) as collision between two conventional elastic spheres,
imagined to have radii dependent on the lines and velocities of
approach before collision (the greater the relative velocity the
smaller the effective radii); and the only assumption essentially
involved in those demonstrations is, that the radius of each
sphere is very small in comparison with the average length of
free path.

§ 17. But if the particles are Boscovich atoms, having centre
of inertia not coinciding with centre of force; or quasi-Boscovich
atoms, of non-spherical figure ; or (a more acceptable supposition)
if each particle is a cluster of two or more Boscovich atoms;
rotations and changes of rotation would result from collisions.
Waterston's and Clausius' leading principle, quoted in § 13 above,
must now be taken into account, and Tait's demonstration is no
longer applicable. Waterston and Clausius, in respect to rotation,
both wisely abstained from saying more than that the average
kinetic energy of rotation bears a constant ratio to the average
kinetic energy of translation. With magnificent boldness Boltz-
mann and Maxwell declared that the ratio is equality ; Boltzmann
having found what seemed to him a demonstration of this re-
markable proposition, and Maxwell having accepted the supposed
demonstration as valid.

§ 18. Boltzmann went further*, and extended the theorem

* "Studien iiber das G'leiohgewicht der lebendigen Kraft zwischen bewegten
materiellen Pankten." Sittb. K, Akad. Wien, October 8, 1868.

496 APPENDIX a

of equality of mean kinetic energies to any system of a finite
number of material points (Boscovich atoms) acting on one
another, according to any law of force, and moving freely among
one another: and finally. Maxwell* gave a demonstration extend-
ing it to the generalised Lagrangian coordinates of any system
whatever, with a finite or infinitely great namber of degrees of
freedom. The words in which he enunciated his supposed theorem
are as follows :

** The only assumption which is necessary for the direct |xoof
" is that the svstem, if left to itself in its actual state of motion,
" will, sooner or later, pass [infinitely neariy+] through every phase
"which is consistent with the equation of energy" (p. 714X and
again (p. 716) :

''It appears from the theorem, that in the ultimate state of
"the system the average* kinetic energy of two portions of the
" system must be in the ratio of the number of degrees of freedom
" of those portions.

"This, therefore, must be the condition of the equality of
" temperature of the two portions of the system."

I have never seen validity in the demonstration § on which
Maxwell founds this statement, and it has always seemed to me
exceedingly improbable that it can be true. If true, it would
be ver}' wonderfiil, and most interesting in pure mathematical
dynamics. Having been published by Boltzmann and Maxwell
it would be worthy of most serious attention, even without con-

^ **0n Boltzmann^s Theorem on the Avenge Distribution of Energy in a
System of MAterial Points,'* MaxweU's ColUettd Papen, YoL n. pp. 713—741, And
Camb. Phil. Trans., May 6, 1878.

t I hATe inserted these two words as oertAinly belonging to HaxwcII's meaning.
— K.

X The ATerage here meant is a time-axerage through a sufficiently long time.

{ The mode of proof followed by Maxwell, and its connection with antecedent
considerations of his own and of Boltzmann, imply, as included in the general
theorem, that the average kinetic energy of any one of three rectangular com-
ponents of the motion of the centre of inertia of an isolated system, acted upon
only by matual forces between its parts is equal to the average kinetic eneigy of
each generalised component of motion relatively to the centre of inertia. Consider,
for example, as ** parts of the system" two particles of masses in and m* free to move
only in a fixed straight line, and connected to one another by a massless spring.
The Boltzmann-Maxwell doctrine asserts that the average kinetic energy of the
motion of the inertial centre is equal to the average kinetic energy of the motion
relative to the inertial centre. This is included in the wording of MaxweU^s state-
ment in the text if, but not unless, si = iii'. See footnote in g 7 of my paper, **0n
■ome Test-Cases for the BoltzmAun-MAxweU Doctrine regArding Distribution of
Energy," Proe. Boy. Soe., June 11, 189L

ATOMIC ACTION AT DISTANCE. 497

sideration of its bearing on thermo-dynamics. But, when we
consider its bearing on thermo-dynamics, and in its first and
most obvious application we find it destructive of the kinetic
theory of gases, of which Maxwell was one of the chief founders,
we cannot see it otherwise than as a cloud on the dynamical
theory of heat and light.

§ 19. For the kinetic theory of gases, let each molecule be
a cluster of Boscovich atoms. This includes every possibility
("dynamical," or "electrical," or "physical," or "chemical") re-
garding the nature and qualities of a molecule and of all its
parts. The mutual forces between the constituent atoms must
be such that the cluster is in stable equilibrium if given at rest ;
which means, that if started from equilibrium with its consti-
tuents in any state of relative motion, no atom will fly away
from it, provided the total kinetic energy of the given initial
motion does not exceed some definite limit. A gas is a vast
assemblage of molecules thus defined, each moving fi'eely through
space, except when in collision with another cluster, and each
retaining all its own constituents unaltered, or only altered by
interchange of similar atoms between two clusters in collision.

§ 20. For simplicity we may suppose that each atom, -4, has
a definite radius of activity, a, and that atoms of different kinds,
A, A', have different radii of activity, a, a'; such that A exercises
no force on any other atom, A\ A", when the distance between
their centres is greater than a + a' or a + a". We need not per-
plex our minds with the inconceivable idea of " virtue," whether
for force or for inertia, residing in a mathematical point* the
centre of the atom ; and without mental strain we can distinctly
believe that the substance (the " substratum " of qualities) resides
not in a point, nor vaguely through all space, but definitely in
the spherical volume of space bounded by the spherical sur&ce
whose radius is the radius of activity of the atom, and whose
centre is the centre of the atom. In our intermolecular forces
thus defined we have po violation of the old scholastic law,
"Matter cannot act where it is not," but we explicitly violate
the other scholastic law, "Two portions of matter cannot simul-
"taneously occupy the same space." We leave to gravitation,

* See Math, and Phys. Papers, Vol. in. Art. xcvn. ** Molecular Constitution of
Matter," § 14.

T. L. 32

498 APPENDIX B.

and possibly to electricity (probably not to magnetism), the at
present very unpopular idea of action at a distance.

§ 21. We need not now (as in § 16, when we wished to keep
as near as we could to the old idea of colliding elastic globes)
suppose the mutual force to become infinite repulsion before the
centres of two atoms, approaching one another, meet. Following
Boscovich, we may assume the force to vary according to any
law of alternate attraction and repulsion, but without supposing
any infinitely great force, whether of repulsion or attraction, at
any particular distance ; but we must assume the force to be zero
when the centres are coincident. We may even admit the idea
of the centres being absolutely coincident, in at all events some
cases of a chemical combination of two or more atoms ; although
we might consider it more probable that in most cases the
chemical combination is a cluster, in which the volumes of the
constituent atoms overlap without any two centres absolutely
coinciding.

§22. The word "collision" used without definition in §19
may now, in virtue of §§29, 21, be unambiguously defined thus:
Two atoms are said to be in collision during all the time their
volumes overlap after coming into contact. They necessarily in
virtue of inertia separate again, unless some third body intervenes
with action which causes them to remain overlapping ; that is to
say, causes combination to result from collision. Two clusters of
atoms are said to be in collision when, after being separate, some
atom or atoms of one cluster come to overlap some atom or atoms
of the other. In virtue of inertia the collision must be followed
either by the two clusters separating, as described in the last
sentence of § 19, or by some atom or atoms of one or both systems
being sent fl3dng away. This last supposition is a matter-of-fact
statement belonging to the magnificent theory of dissociation;
discovered and worked out by Sainte-Clair Deville without any
guidance from the kinetic theory of gases. In gases approxi-
mately fulfilling the gaseous laws (Boyle's and Charles'), two
clusters must in general fly asunder after collision. Two clusters
could not possibly remain permanently in combination without at
least one atom being sent flying away after collision between two
clusters with no third body intervening*.

* See Kelvin's Math, and Phys. Papers, Vol. iii. Art. xcvii. § 33, In this
reference, for "scarcely" substitute **not."

CLAUSIUS' THEOREM KEGARDINQ RATIO OF THERMAL CAPACITIES. 499

§ 23. Now for the application of the Boltzmann-Maxwell
doctrine to the kinetic theory of gases: consider first a homo-
geneous single gas, that is, a vast assemblage of similar clusters
of atoms moving and colliding as described in the last sentence of
§ 19 ; the assemblage being so sparse that the time during which
each cluster is in collision is very short in comparison with the
time during which it is unacted on by other clusters, and its centre
of inertia, therefore, moves uniformly in a straight line. If there
are i atoms in each clusTter, it has 3i freedoms to move, that is to
say, freedoms in three rectangular directions for each atom. The
Boltzmann-Maxwell doctrine asserts that the mean kinetic energies
of these 3i motions are all equal, whatever be the mutual forces
between the atoms. From this, when the durations of the col-
lisions are not included in the time-averages, it is easy to prove
algebraically (with exceptions noted below) that the time-average
of the kinetic energy of the component translational velocity of
the inertial centre*, in any direction, is equal to any one of the
3i mean kinetic energies asserted to be equal to one another in
the preceding statement. There are exceptions to the algebraic
proof corresponding to the particular exception referred to in
the last footnote to §18 above; but, nevertheless, the general
Boltzmann-Maxwell doctrine includes the proposition, even in
those cases in which it is not deducible algebraically from the
equality of the Si energies. Thus, without exception, the average
kinetic energy of any component of the motion of the inertial
centre is, according to the Boltzmann-Maxwell doctrine, equal to

. of the whole average kinetic energy of the system. This makes

the total average energy, potential and kinetic, of the whole
motion of the system, translational and relative, to be 3i(l +P)
times the mean kinetic energy of one component of the motion
of the inertial centre, where P denotes the ratio of the mean
potential energy of the relative displacements of the parts to
the mean kinetic energy of the whole system. Now, according
to Clausius' splendid and easily proved theorem regarding the
partition of energy in the kinetic theory of gases, the ratio of
the difference of the two thermal capacities to the constant-
volume thermal capacity is equal to the ratio of twice a single

* This expression T use for brevity to signify the kinetic energy of the whole
mass ideally collected at the centre of inertia.

32—2

500 APPENDIX B.

component of the translational energy to the total energy. Hence,
if according to our usual notation we denote the ratio of the
thermal capacity pressure-constant to the thermal capacity volume-
constant by ky we have,

2

i-l =

3i(l + P)-

§ 24. Example 1. For first and simplest example, consider a
monatomic gas. We have i = 1, and according to our supposition
(the supposition generally, perhaps universally, made) regarding
atoms, we have P = 0. Hence, ft — 1 = J.

This is merely a fundamental theorem in the kinetic theory of
gases for the case of no rotational or vibrational energy of the
molecule ; in which there is no scope either for Clausius' theorem
or for the Boltzmann-Maxwell doctrine. It is beautifully illus-
trated by mercury vapour, a monatomic gas according to chemists,
for which many years ago Kundt, in an admirably designed ex-
periment, found ft — 1 to be very approximately f : and by the
newly discovered gases argon, helium, and krypton, for which also
ft — 1 has been found to have approximately the same value, by
Kayleigh and Ramsay. But each of these four gases has a large
number of spectrum lines, and therefore a large number of vibra-
tional freedoms, and therefore, if the Boltzmann-Maxwell doctrine
were true, ft — 1 would have some exceedingly small value, such
as that show^n in the ideal example of § 26 below. On the other
hand, Clausius' theorem presents no difficulty; it merely asserts
that ft — 1 is necessarily less than | in each of these four cases,
as in every case in which there is any rotational or vibrational
energy whatever; and proves, from the values found experi-
mentally for ft — 1 in the four gases, that in each case the total
of rotational and vibrational energy is exceedingly small in
comparison with the translational energy. It justifies admirably
the chemical doctrine that mercury vapour is practically a mon-
atomic gas, and it proves that argon, helium, and krypton, are
also practically monatomic, though none of these gases has hitherto
shown any chemical affinity or action of any kind from which
chemists could draw any such conclusion.

But Clausius' theorem, taken in connection with Stokes' and
Kirchhoflf's dynamics of spectrum analysis, throws a new light on
what we are now calling a " practically monatomic gas." It shows

RATIOS OF CAPACITIES BY BOLTZMANN-MAXWELL DOCTRINE. 501

that, unless we admit that the atoms can be set into rotation or
vibration by mutual collisions (a most unacceptable hypothesis),
each atom must have satellites connected with it (or ether con-
densed into it or around it) and kept, by the collisions, in motion
relatively to it with total energy exceedingly small in comparison
with the translational energy of the whole system of atom and
satellites. The satellites must in all probability be of exceedingly
small mass in comparison with that of the chief atom. Can they
be the " ions " by which J. J. Thomson explains the electric con-
ductivity induced in air and other gases by ultra-violet light,
Rcintgen rays and Becquerel rays?

Filially, it is interesting to remark that all the values of A: — 1
found by Rayleigh and Ramsay are somewhat less than J ; argon
•64, -61 ; helium '652 ; krypton '666. If the deviation from '667
were accidental they would probably have been some in defect and
some in excess.

Example 2. As a next simplest example let i = 2, and as a
very simplest case let the two atoms be in stable equilibrium
when concentric, and be infinitely nearly concentric when the
clusters move about, constituting a homogeneous gas. This sup-
position makes P = i, because the average potential energy is
equal to the average kinetic energy in simple harmonic vibrations ;
and in our present case half the whole kinetic energy, according
to the Boltzmann-Maxwell doctrine, is vibrational, the other half
being translational. We find A; - 1 = f = '2222.

Exa7nple 3. Let i = 2; let there be stable equilibrium, with
the centres C, C of the two atoms at a finite distance a asunder,
and let the atoms be always very nearly at this distance asunder
when the clusters are not in collision. The relative motions of the
two atoms will be according to three freedoms, one vibrational,
consisting of very small shortenings and lengthenings of the
distance GG\ and two rotational, consisting of rotations round
one or other of two lines perpendicular to each other and perpen-
dicular to CO' through the inertial centre. With these conditions
and limitations, and with the supposition that half the average
kinetic energy of the rotation is comparable with the average
kinetic energy of the vibrations, or exactly equal to it as according
to the Boltzmann-Maxwell doctrine, it is easily proved that in
rotation the excess of CC above the equilibrium distance a, due
to centrifugal force, must be exceedingly small in comparison

502 APPENDIX &

with the maximum value of CC' — a due to the vibration. Hence
the average potential energy of the rotation is negligible iu
comparison with the potential eneigy of the vibration. Hence,
of the three freedoms for relative motion there is only one con-
tributory to P, and therefore we have P = i- Thus we find

fc-l=f = -2857.

The best way of experimentally determining the ratio of the
two thermal capacities for any gas is by comparison between the
observed and the Newtonian velocities of sound. It has thus
been ascertained that, at ordinary temperatures and pressures,
k—1 differs but little from '406 for common air, which is a mixture
of the two gases nitrogen and oxygen, each diatomic according
to modem chemical theory; and the greatest value that the
Boltzmann-Maxwell doctrine can give for a diatomic gas is the
'2857 of Ex. 3. This notable discrepance from obser^'ation suffices
to absolutely disprove the Boltzmann-Maxwell doctrine. What is
really established in respect to partition of energy is what Clausius'
theorem tells us (§ 23 above). We find, as a result of observation
and true theory, that the average kinetic energy of translation
of the molecules of common air is '609 of the total energy,
potential and kinetic, of the relative motion of the constituents
of the molecules.

§ 25. The method of treatment of Ex. 3 above, carried out for

a cluster of any number of atoms greater than two not in one line,

J + 2 atoms, let us say, shows us that there are three translational

freedoms; three rotational freedoms, relatively to axes through

the ineitial centre ; and 3j vibrational freedoms. Hence we have

1

P = . •^^^, and we find Xr— 1 = ^. ■- . . The values of fc— 1 thus
j + 2' 3(l+j)

calculated for a triatomic and tetratomic gas, and calculated as

above in Ex. 3 for a diatomic gas, are shown in the following

table, and compared with the results of obsen-ation for several

such gases.

It is interesting to see how the dynamics of Clausius' theorem

is verified by the results of observation shown in the table. The

values of k—1 for all the gases are less than |, as they must be

when there is any appreciable energj* of rotation or vibration in

the molecule. They are different for different diatomic gases;

ENORMOUSLY GREAT ERRORS PROVED BY SPECTRl/M-ANALYSIS. 603

ranging from '42 for oxygen to '32 for chlorine, which is quite as
might be expected, when we consider that the laws of force
between the two atoms may differ largely for the different kinds
of atoms. The values of A: — 1 are, on the whole, smaller for the

Values of ik-l

(ias

According to the

By

H.-M. doctrine.

Observat

Air

^ = •2867

• -406

H^

•40

0,

•41

CL

c6

» * WW

•32

•39

NO

•39

CO-
N.O
NH,

^=•1667

•30

»» f»

•331

i=llll

•311

tetratomic and triatomic than for the diatomic gases, as might be
expected from consideration of Clausius* principle. It is probable
that the differences of i — 1 for the different diatomic gases are
real, although there is considerable uncertainty with regard to
the observational results for all or some of the gases other than
air. It is certain that the discrepancies from the values,
calculated according to the Boltzmann-Maxwell doctrine, are
real and great; and that in each case, diatomic, triatomic, and
tetratomic, the doctrine gives a value for k—l much smaller
than the truth.

§ 26. But, in reality, the Boltzmann-Maxwell doctrine errs
enormously more than is shown in the preceding table. Spectrum
analysis showing vast numbers of lines for each gas makes it
certain that the number of freedoms of the constituents of each
molecule is enormously greater than those which we have been
counting, and therefore that unless we attribute vibratile quality
to each individual atom, the molecule of every one of the ordinary
gases must have a vastly greater number of atoms in its consti-
tution than those hitherto reckoned in regular chemical doctrine.
Suppose, for example, there are forty-one atoms in the molecule
of any particular gas ; if the doctrine were true we should have
^' = 39. Hence there are 117 vibrational freedoms, so that there
might be 117 visible lines in the spectrum of the gas; and we

504 APPENDIX B.

have & — 1 = = -0083. There is, in fact, no possibility of

reconciling the Boltzmann-Maxwell doctrine with the truth re-
garding the specific heats of gases.

§ 27. It is, however, not quite possible to rest contented with
the mathematical verdict not proven, and the experimental verdict
not true, in respect to the Boltzmann-Maxwell doctrine. I have
always felt that it should be mathematically tested by the con-
sideration of some particular case. Even if the theorem were
true, stated as it was somewhat vaguely, and in such • general
terms that great difficulty has been felt as to what it is really
meant to express, it would be very desirable to see even one other
simple case, besides that original one of Waterston's, clearly stated
and tested by pure mathematics. Ten years ago* I suggested
a number of test cases, some of which have been courteously
considered by Boitzmann; but no demonstration either of the
truth or untruth of the doctrine as applied to any one of them has
hitherto been given. A year later, I suggested what seemed to
mc a decisive test case disproving the doctrine ; but my statement
was quickly and justly criticised by Boitzmann and Poincare;
and more recently Lord Rayleighf has shown very clearly that
my simple test case was quite indecisive. This last article of
Rayleighs has led mc to resume the consideration of several
diflferent classes of dynamical problems, which had occupied me
more or less at various times during the last twenty years, each
presenting exceedingly interesting features in connection with the
double question : Is this a case which admits of the application
of the Boltzmann-Maxwell doctrine; and, if so, is the doctrine
true for it ?

§ 28. Premising that the mean kinetic energies with which
the Boltzmann-Maxwell doctrine is concerned are time-integrals
of energies divided by totals of the times, we may conveniently
divide the whole class of problems, with reference to which the
doctrine comes into question, into two classes.

Class I. Those in which the velocities considered are either

* *'0n some Test-Cases for the Mazwell-Boltzmann Doctrine regarding Distribn*
tion of Energj'," Proc. Roy. Soc, June 11, 1891.

t PhiL Mag., Vol. xxxin. 1892, p. 356, '^Kemarks on Maxwell's Investigation
respecting Boltzmann's Theorem.'*

CLASSIFIED CASES FOR THE B.-M. DOCTRINE. 505

coiiiitant or only vary suddenly — that is to say, in inBnitely small
times — or in times so short that they may be omitted from the
time-integration. To this class belong :

(a) The original Waterston-Maxwell case and the collisions
of ideal rigid bodies of any shape, according to the assumed law
that the translatory and rotatory motions lose no energy in the
collisions.

(b) The frictionless motion of one or more particles constrained
to remain on a surface of any shape, this surface being either closed
(commonly called finite though really endless), or being a finite area
of plane or curved surface, bounded like a billiard-table, by a wall
or walls, from which impinging particles ai-e reflected at angles
equal to the angles of incidence.

(c) A closed surface, with non-vibratory particles moving
within it freely, except during impacts of particles against one
another or against the bounding surface.

(d) Cases such as (a), (6), or (c), with impacts against
boundaries and mutual impacts between particles, softened by
the supposition of finite forces during the impacts, with only
the condition that the durations of the impacts are so short as
to be practically negligible, in comparison with the durations of
free paths.

Class 1 1. Cases in which the velocities of some of the particles
concerned sometimes vary gradually ; so gradually that the times
during which they vary must be included in the time-integration.
To this class belong examples such as (d) of Class I. with durations
of impacts not negligible in the time-integration.

§ 29. Consider first Class I. (6) with a finite closed surface as
the field of motion and a single particle moving on it. If a particle
is given, moving in any direction through any point / of the field,
it will go on for ever along one determinate geodetic line. The
question that first occurs is, Does the motion fulfil MaxwelFs con-
dition (see § 18 above) ; that is to say, for this case, if we go along
the geodetic line long enough, shall we pass infinitely nearly to
any point Q whatever, including /, of the surface an infinitely
great number of times in all directions ? This question cannot be
answered in the aflSrmative without reservation. For example,
if the surface be exactly an ellipsoid it must be answered in the
negative, as is proved in the following ^ 30, 31, 32.

504 APPENDIX B.

have A; — 1 = = -0083. There is, in fact, no possibility of

reconciling the Boltzmann-Maxwell doctrine with the truth re-
garding the specific heats of gases.

§ 27. It is, however, not quite possible to rest contented with
the mathematical verdict not proven, and the experimental verdict
not true, in respect to the Boltzmann-Maxwell doctrine. I have
always felt that it should be mathematically tested by the con-
sideration of some particular case. Even if the theorem were
true, stated as it was somewhat vaguely, and in such general
terms that great difficulty has been felt as to what it is really
meant to express, it would be very desirable to see even one other
simple case, besides that original one of Waterston s, clearly stated
and tested by pure mathematics. Ten years ago* I suggested
a number of test cases, some of which have been courteously
considered by Boltzmann; but no demonstration either of the
truth or untruth of the doctrine as applied to any one of them has
hitherto been given. A year later, I suggested what seemed to
me a decisive test case disproving the doctrine ; but my statement
was quickly and justly criticised by Boltzmann and Poincare;
and more recently Lord Rayleighf has shown very clearly that
my simple test case was quite indecisive. This last article of
Rayleigh's has led me to resume the consideration of several
different classes of dynamical problems, which had occupied me
more or less at various times during the last twenty years, each
presenting exceedingly interesting features in connection with the
double question : Is this a case which admits of the application
of the Boltzmann-Maxwell doctrine; and, if so, is the doctrine
true for it?

§ 28. Premising that the mean kinetic energies with which
the Boltzmann-Maxwell doctrine is concerned are time-integrals
of energies divided by totals of the times, we may conveniently
divide the whole class of problems, with reference to which the
doctrine comes into question, into two classes.

Class I. Those in which the velocities considered are either

* **0n some Test-Cases for the Mazwell-Boltzmann Doctrine regarding Difltribn^
tion of Energj'," Proc. Roij. Soc, June 11, 1891.

t Phil. Mag,, Vol. zzxin. 1892, p. 356, **Bemark8 on Maxwell's Investigation
respecting Boltzmann^s Theorem."

CLASSIFIED CASES FOB THE B.-M. DOCTBINE. 505

constant or only vary suddenly — that is to say, in inBnitely small
times — or in times so short that they may be omitted from the
time-integration. To this class belong :

(a) The original Waterston-Maxwell case and the collisions
of ideal rigid bodies of any shape, according to the assumed law
that the translatory and rotatory motions lose no energy in the
collisions.

(6) The frictionless motion of one or more particles constrained
to remain on a surface of any shape, this surface being either closed
(commonly called finite though really endless), or being a finite area
of plane or curved surface, bounded like a billiard-table, by a wall
or walls, from which impinging particles are reflected at angles
equal to the angles of incidence.

(c) A closed surface, with non-vibratory particles moving
within it freely, except during impacts of particles against one
another or against the bounding surface.

(d) Cases such as (a), (6), or (c), with impacts against
boundaries and mutual impacts between particles, softened by
the supposition of finite forces during the impacts, with only
the condition that the durations of the impacts are so short as
to be practically negligible, in comparison with the durations of
free paths.

Class I F. Cases in which the velocities of some of the particles
concerned sometimes vary gradually ; so gradually that the times
during which they vary must be included in the time-integration.
To this class belong examples such as {d) of Class I. with durations
of impacts not negligible in the time-integration.

§ 29. Consider first Class I. (6) with a finite closed surface as
the field of motion and a single particle moving on it. If a particle
is given, moving in any direction through any point / of the field,
it will go on for ever along one determinate geodetic line. The
question that first occurs is, Does the motion fulfil Maxwell's con-
dition (see § 18 above) ; that is to say, for this case, if we go along
the geodetic line long enough, shall we pass infinitely nearly to
any point Q whatever, including /, of the surface an infinitely
great number of times in all directions ? This question cannot be
answered in the aflSi'mative without reservation. For example,
if the sur&ce be exactly an ellipsoid it must be answered in the
negative, as is proved in the following ^ 30, 31, 32.

506

APPENDIX B.

§ 30. Let AA\ BR, CC he the ends of the greatest, mean,
and least diameters of an ellipsoid. Let Ui, I/,, Uj, U4 he the
umbilics in the arcs AC\ CA', A'C\ C'A. A known theorem in
the geometry of the ellipsoid tells us that every geodetic through
Ui passes through U^y and every geodetic through 17, passes through
U^, This statement regarding geodetic lines on an ellipsoid of
three unequal axes is illustrated by fig. 1, a diagram showing
for the extreme case in which the shortest axis is zero, the exact
construction of a geodetic through Ui which is a focus of the
ellipse shown in the diagram. U^, C\ U^ being infinitely near to
Uiy C, L\ respectively are indicated by double letters at the same

Fig. 1.

points. Starting from t", draw the geodetic UiQU:-; the two parts
of which UiQ and QU3 are straight lines. It is interesting to
remark that in whatever direction we start from l\ if we continue
the geodetic through ^^'3, and on through Ui again and so on
endlessly, as indicated in the diagram by the straight lines
^iQ» t^iO^f ^\Q\ ^zQ'\ and so on, we come very quickly to lines
approaching successively more and more nearly to coincidence with
the major axis. At every point where the path strikes the ellipse
it is reflected at equal angles to the tangent. The construction is
most easily made by making the angle between the reflected path
and a line to one focus, equal to the angle between the incident
path and a line to the other focus.

REFLECTIONS OF BILLIARD-BALL FROM ELLIPTIC BOUNDARY. 507

§ 31. Returning now to the ellipsoid: — From any point i,
between Uj and f/j, draw the geodetic /Q, and produce it through
Q on the ellipsoidal surface. It must cut the arc A'C'A at some
point between U^ and C/4, and, if continued on and on, it must cut
the ellipse ACA'C'A successively between I/, and l/g, or between
IT, and 11^] never between U^ and U^, or 11^ and Uy, This, for
the extreme case of the smallest axis zero, is illustmted by the
path IQQ'q'Q:''Q''Q^ in fig. 2.

§ 32. If now, on the other hand, we commence a geodetic
through any point J between U^ and f/4, or between IT, and IT,, it
will never cut the principal section containing the umbilics, either
between U^ and U^ or between U^ and U^. This for the extreme
case of CO' = 0 is illustrated in fig. 3.

§ 33. It seems not improbable that if the figure deviates by
ever so little from being exactly ellipsoidal, Maxwell's condition
might be fulfilled. It seems indeed quite probable that Maxwell's
condition (see ^ 13, 29, above) is fulfilled by a geodetic on a
closed surface of any shape in general, and that exceptional cases,

in which the question of § 29 is to be answered in the negative,
are merely particular surfaces of definite shapes, infinitesimal
deviations from which will allow the question to be answered in
the affirmativa

508

APPENDIX B.

§ 34. Now with an affirmative answer to the question — is
Maxwells condition fulfilled ? — what does the Boltzmann-Maxwell
doctrine assert in respect to a geodetic on a closed surfeu^ ? The
mere wording of Maxwells statement, quoted in § 13 above, is
not applicable to this case, but the meaning of the doctrine as
interpreted from previous writings both of Boltzmann and Maxwell,
and subsequent writings of Boltzmann, and of Rayleigh*, the most
recent supporter of the doctrine, is that a single geodetic drawn
lung enough w*ill not only fulfil Maxwell's condition of passing
infinitely near to every point of the surface in all directions, but
will pass with equal frequencies in all directions; and as many
times within a certain infinitesimal distance ± S of any one point
P as of any other point P' anywhere over the whole surface. This,
if true, would be an exceoilingly interesting theorem.

Fig. 3.

§ 35. I have made many efforts to test it for the case in which
the closed surface is reduced to a plane with other boundaries
than an" exact ellipse (for which as we have seen in §§ 30, 31, 32,
the investigation fails through the non-fulfilment of Maxwells
preliminary condition). Every such case gives, as we have seen,
straight lines drawn across the enclosed area turned on meeting
the boundary, according to the law of equal angles of incidence
and reflection, which corresponds also to the case of an ideal

♦ Phil, Mag,, January 1900.

REFLECTIONS OF BALL FROM BOUNDARY OF ANY SHAPE. 509

perfectly smooth non-rotating billiard-ball moving in straight lines
except when it strikes the boundary of the table ; the boundary
being of any shape whatever, instead of the ordinary rectangular
boundary of an ordinary billiard-table, and being perfectly elastic.
An interesting illustration, easily seen through a large lecture-
hall, is had by taking a thin wooden board, cut to any chosen
shape, with the corner edges of the boundary smoothly rounded,
and winding a stout black cord round and round it many times,
beginning with one end fixed to any point, /, of the board. If the
pressure of the cord on the edges were perfectly frictionless the
cord would, at every turn round the border, place itself so as to
fulfil the law of equal angles of incidence and reflection, modified
in virtue of the thickness of the board. For stability it would be
necessary to fix points of the cord to the board by staples pushed
in over it at sufficiently frequent intervals, care being taken
that at no point is the cord disturbed from its proper straight
line by the staple. [Boards of a considerable variety of shape
with cords thus wound on them were shown as illustrations of
the lecture.]

§ 36. A very easy way of drawing accurately the path of a
particle moving in a plane and reflected from a bounding wall of
any shape, provided only that it is not concave externally in any
part, is furnished by a somewhat interesting kinematicaJ method
illustrated by the accompanying diagram (fig. 4). It is easily

Fig, 4.

realised by using two equal and similar pieces of board, cut to any
desired figure, one of them being turned upside down relatively

510 APPENDIX B.

to the other, so that when the two are placed together with
corresponding points in contact, each is the image of the other
relative to the plane of contact regarded as a mirror. SufBcientlv
close corresponding points should be accurately marked on the
boundaries of the two figures, and this allows great accuracy Ui
be obtained in the drawing of the free path after each reflection.
The diagram shows consecutive free paths 74-6 — 32*9 given, and
32-9— 54-7, found by producing 74-6— 32-9 through the point of
contact The process involves the exact measurement of the
length (l) — ^say to three significant figures — and its inclination (0)
to a chosen line of reference XX\ The summations S/ cos 20
and ^ sin 20 give, as explained below, the difference of time-
integrals of kinetic energies of component motions pamllel and
perpendicular respectively to XX\ and parallel and perpendicular
respectively to KK\ inclined at 45' to XX'. From these differ-
ences we find (by a procedure equivalent to that of finding the
principal axes of an ellipse) two lines at right angles to one another,
such that the time-integrals of the components of velocity parallel
to them are respectively greater than and less than those of the
components parallel to any other line. [This process was illustrated
by models in the lecture.]

§ 37. Virtually the same process as this, applied in the case
of a scalene triangle ABC (in which BC= 20 centimetres and the
angles A = 97", 5 = 29*5', C= 53*5'), was worked out in the Royal

Fig. 5.

Institution during the fortnight after the lecture, by Mr Anderson,
with very interesting results. The length of each free path (/),
and its inclination to BC{0), reckoned acute or obtuse according
to the indications in the diagram (fig. 5), were measured to the

REFLECTIONS OF BALL SHOW ERRORS FROM B.-M. DOCTRINK. 511

nearest millimetre and the nearest integral degree. The first free
path was drawn at random, and the continuations, after 509 reflec-
tions (in all 600 paths), were drawn in a manner illustrated by
fig. 5, which shows, for example, a path PQ on one triangle
continued to QR on the other. The two when folded together
round the line AB shows a path PQ, continued on QR after
reflection. For each path I cos 20 and i sin 2^ were calculated
and entered in tables with the proper algebraic signs. Thus, for
the whole 600 paths, the following summations were found :

2Z = 3298; 2i cos 2^ = + 1288; 2Z sin 2^ = - 201*9.

Remark, now, if the mass of the moving particle is 2, and the

velocity one centimetre per second, 11 cos 20 is the excess of the

time-integral of kinetic energy of component motion parallel to

BC above that of component motion perpendicular to BC, and

XI sin 20 is the excess of the time-integral of kinetic energy of

component motion perpendicular to KK' above that of component

motion parallel to KK' ; KK' being inclined at 45"" to BC in the

direction shown in the diagram. Hence the positive value of

2Z cos 20 indicates a preponderance of kinetic energy due to

component motion parallel to BC above that of component motion

perpendicular to BC ; and the negative sign of 2Z sin 20 shows

preponderance of kinetic energy of component motion parallel

to KK\ above that of component motion perpendicular to KK\

Deducing a determination of two axes at right angles to each

other, corresponding respectively to maximum and minimum

kinetic energies, we find LL\ being inclined to KK^ in the

128*8
direction shown, at an angle = ^ tan~^ ^ ^ , is what we may

call the axis of maximum energgr, and a line perpendicular to LL'
the axis of minimum energy ; and the excess of the time-integral
of the energy of component velocity parallel to ZZ' exceeds that
of the component perpendicular to LL' by 239*4, being

-v/l28-8» + 201-9*.

This is 7*25 per cent, of the total of XI which is the time-integral
of the total energy. Thus, in our result, we find a very notable
deviation from the Boltzmann-Maxwell doctrine, which asserts for
the present case that the time-integrals of the component kinetic
energies are the same for all directions of the component. The

512

APPENDIX B.

percentage which we have found is not very large'; and, most
probably, summations for several successive 600 flights would
present considerable diffei-ences, both of the amount of the devia-
tion fix>m equality and the direction of the axes of maximum and
minimum energy. Still, I think there is a strong probability that
the disproof of the Boltzmann-Maxwell doctrine is genuine, and
the discrepance is somewhat approximately of the amount and
direction indicated. I am supported in this view by scrutinising
the thirty sums for successive sets of twenty flights : thus I find
S/ cos 20 to be positive for eighteen out of thirty, and S2 sin 20 to
be negative for nineteen out of the thirty.

§ 38. A very interesting test-case is represented in the ac-
companying diagram, fig. 6 — a circular boundary of semicircular
corrugations. In this case it is obvious from the symmetry that
the time-integral of kinetic energy of component motion parallel

Fig. C.

to any straight line must, in the long run, be equal to that parallel
to any other. But the Boltzmann-Maxwell doctrine asserts, that
the time-integrals of the kinetic energies of the two components,
radial and transversal, according to polar coordinates, would be

STATISTICS OF REFLECTIONS FROM CORRUGATED BOUNDARY. 513

equal. To* test this I have taken the case of an infinite number
of the semicircular corrugations, so that in the time-integral it
is not necessary to include the times between successive impacts
of the particle on any one of the semicircles. In this case the
geometrical construction would, of course, fail to show the precise
point Q at which the free path would cut the diameter AB of the
semicircular hollow to which it is approaching ; and I have evaded
the difficulty in a manner thoroughly suitable for thermodynamic
application, such as the kinetic theory of gases. I arranged to
draw lots for one out of the 199 points dividing AB into 200 equal
parts. This was done by taking 100 cards*, 0, 1...98, 99, to
represent distances from the middle point, and, by the toss of a
coin, determining on which side of the middle point it was to
be (plus or minus for head or tail, frequently changed to avoid
possibility of error by bias). The draw for one of the hundred
numbers (0...99) was taken- after very thorough shuffling of the
cards in each case. The point of entry having been found, a large
scale geometrical construction was used to determine the successive
points of impact and the inclination 6 of the emergent path to the
diameter AB, The inclination of the entering path to the diameter
of the semicircular hollow struck at the end of the flight has the
same value 0. If we call the diameter of the large circle unity
the length of each flight is sin 0. Hence, if the velocity is unity
and the mass of the particle 2, the time-integral of the whole
kinetic energy is sin 0 ; and it is easy to prove that the time-
integrals of the squares of the components of the velocity, per-
pendicular to and along the line from each point of the path
to the centre of the large circle, are respectively ^cos^ and
sin ^ — ^ cos 0, By summation for 143 flights we have found

2 sin ^ = 121-3 ; 2^ cos 0 = 6415 ;

whence 2 (sin ^ - ^ cos ^) = 2d cos ^ + 130.

This is a notable deviation from the Boltzmann-Maxwell doc-
trine, which makes 2 (sin d — d cos 0) equal to %0 cos ft We have

* I had tried nambered biUets (small squares of paper) drawn from a bowl, but
foand (his very onsatisfaotory. The best mixing we could make in the bowl seemed
to be quite insufficient to secure equal chances for all the billets. Full-sized cards
like ordinary playing-cards, well shuffled, seemed to give a very fairly equal chance
to every card. Even with the full-sized cards, electric attraction sometimes inter-
venes and causes two of them to stick together. In using one's fingers to mix dry
billets of card, or of paper, in a bowl, vexy considerable disturbance may be expected
from electrification.

T. L. 33

514

APPENDIX B.

found the former to exceed the latter by a diflference
which amounts to 10'7 of the whole 2 sin 0.

Out of fourteen sets of ten flights I find that the
time-integral of the transverse component is less than
half the whole in twelve sets, and greater in only two.
This seems to prove beyond doubt that the deviation
from the Boltzmann-Maxwell doctrine is genuine ; and
that the ultimate time-integral of the transverse com-
ponent is certainly smaller than the time-integral of
the radial component.

§ 39. It is interesting to remark that, on Brewster's
kaleidoscopic principle of successive images, our present
result is applicable (see § 38 above) to the motion of
a particle, flying about in an enclosed space, of the
same shape as the surface of a marlin-spike (fig. 7),
with its angle any exact submultiple of 360°, (360° -r-t)*.
Symmetry shows that the axes of maximum or minimum
kinetic energy must be in the direction of the middle
line of the length of the figure and perpendicular to it.
Our conclusion is that the time-integral of kinetic
energy is maximum for the longitudinal component
and minimum for the transverse. In the series of
flights, corresponding to the 143 of fig. 6 which we
have investigated, the number of flights is of course
many times 143 in fig. 7, because of the reflections
at the straight sides of the marlin-spike. It will be
understood, that we are considering merely motion in
one plane through the axis of the marlin-spike.

§ 40. The most difficult and seriously troublesome
statistical investigation in respect to the partition of
energy which I have hitherto attempted, has been to
find the proportions of translational and rotational
energies in various cases, in each of which a rotator
experiences multitudinous reflections at two fixed
parallel planes between which it moves, or at one
plane to which it is brought back by a constant force
through its centre of inertia, or by a force var3ring
directly as the distance from the plane. Two

* See my Electrostatict and Magnetism, § 20S, for same principle
as to electric images.

CHATTERING IMPACTS OF ROTATORS ON A FIXED PLANE. 515

(liflFerent rotators were considered, one of them consisting of
two equal balls, fixed at the ends of a rigid massless rod, and
each ball reflected on striking either of the planes; the other
consisting of two balls, 1 and 100, fixed at the ends of a rigid
massless rod, the smaller ball passing freely across the plane
without experiencing any force, while the greater is reflected
every time it strikes. The second rotator may be described,
in some respects more simply, as a hard massless ball having a
mass = 1 fixed anywhere eccentrically within it, and another
mass = 100 fixed at its centre. It may be called, for brevity,
a biassed ball.

§ 41. In every case of a rotator whose rotation is changed by
an impact, a transcendental problem of pure kinematics essentially
occurs to find the time and configuration of the first impact ; and
another such problem to find if there is a second impact, and, if so,
to determine it. Chattering collisions of one, two, three, four, five
or more impacts, are essentially liable to occur, even to the extreme
case of an infinite number of impacts and a collision consisting
virtually of a gradually varying finite pressure. Three is the
greatest number of impacts we have found in any of our calcula-
tions. The first of these transcendental problems, occurring
essentially in every case, consists in finding the smallest value
of 6 which satisfies the equation

where co is the angular velocity of the rotator before collision;
a is the length of a certain rotating arm ; % its inclination to the
reflecting plane at the instant when its centre of inertia crosses
a plane F, parallel to the reflecting plane and distant a from it ;
and V is the .velocity of the centre of inertia of the rotator. This
equation is, in general, very easily solved by calculation (trial and
error), but more quickly by an obvious kinematic method, the
simplest form of which is a rolling circle carrying an arm of
adjustable length. In our earliest work we performed the solution
arithmetically, after that kinematically. If the distance between
the two parallel planes is moderate in comparison with 2a (the
effective diameter of the rotator), i for the beginning of the
collision with one plane has to be calculated from the end of
the preceding collision against the other plane by a transcendental

33—2

516 AK»ENDIX B

equation, on the same principle as that which we have just been
considering. But I have supposed the distance between the two
planes to be very great, practically infinite, in comparison with 2a,
and we have therefore found i by lottery for each collision, using
180 cards corresponding to 180° of angle. In the case of the
biassed globe, different equally probable values of i through a
range of 360° was required, and we found them by drawing from
the pack of 180 cards and tossing a coin for plus or minus.

§ 42. Summation for 110 flights of the rotator, consisting of
two equal masses, gave as the time-integral of the whole energy
200*03, and an excess of rotatory above translatory 42*05. This
is just 21 per cent, of the whole; a large deviation from the
Boltzmann-Maxwell doctrine, which makes the time-integrals of
translatory and rotatory energies equal.

§ 43. In the solution for the biassed ball (masses 1 and 100)
we found great irregularities due to " runs of luck " in the toss for
plus or minus, especially when there was a succession of Ave or six
pluses or Ave or six minuses. We therefore, after calculating a
sequence of 200 flights with angles each determined by lottery,
calculated a second sequence of 200 flights with the equally
probable set of angles given by the same numbers with altered
signs. The summation for the whole 400 gave 555*55 as the
time-integral of the whole energy, and an excess, 82*5, of the
time-integral of the translatory, over the time-integral of the
rotatory energy. This is nearly 16 per cent. We cannot, how-
ever, feel great confidence in this result, because the first set of
200 made the translatory energy less than the rotatory energy by
a small percentage (2*3) of the whole, while the second 200 gave
an excess of translatory over rotatory amounting to 35*9 per cent,
of the whole.

§ 44. All our examples considered in detail or worked out,
hitherto, belong to Class I. of § 28. As a first example of Class 11.
consider a case merging into the geodetic line on a closed surface S.
Instead of the point being constrained to remain on the sur&ce,
let it be under the influence of a field of force, such that it is
attracted towards the surface with a finite force, if it is placed
anywhere very near the surface on either side of it, so that if the
particle be placed on S and projected perpendicularly to it, either

PARTICLE CONSTRAINED TO REMAIN NEAR A CLOSED SURFACE. 6l7

inwards or outwards, it will be brought back before it goes farther
from the surface than a distance h, small in comparison with the
shortest radius of curvature of any part of the surface. The Boltz-
mann-Maxwell doctrine asserts that the time-integral of kinetic
energy of component motion normal to the surface would be equal
to one-third of the kinetic energy of component motion at right
angles to the normal; by normal being meant, a straight line drawn
frx>m the actual position of the point at any time perpendicular to
the nearest part of the surfece S. This, if true, would be a very
remarkable proposition. If h is infinitely small we have simply
the mathematical condition of constraint to remain on the surface,
and the path of the particle is exactly a geodetic line. If the
force towards S is zero, when the distance on either side of 8 is
± h, we have the case of a particle placed between two guiding
surfaces with a very small distance 2h between them. If 8 and
therefore each of the guiding surfaces, is in every normal section
convex outwards, and if the particle is placed on the outer guide-
surface and projected in any direction in it with any velocity,
great or small, it will remain on that guide-surface for ever, and
travel along a geodetic line. If now it be deflected very slightly
from motion in that surface, so that it will strike against the
inner guide-surface, we may be quite ready to learn that the
energy of knocking about between the two surfaces will grow up
from something very small in the beginning till, in the long run,
its time-integral is comparable with the time-integral of twice the
energy of component motion parallel to the tangent plane of
either surface. But will its ultimate value be exactly one-third
that of the tangential energy, as the doctrine tells us it would be ?
We are, however, now back to Class I. ; we should have kept to
Class II. by making the normal force on the particle always finite,
however great.

§ 45. Very interesting cases of Class II. § 28 occur to us
readily in connection with the cases of Class I. worked out in
§§ 38, 41, 42, 43.

§ 46. Let the radius of the large circle in § 38 become in-
finitely great : we have now a plane F (floor) with semicircular
cylindric hollows, or semicircular hollows as we shall say for
brevity; the motion being confined to one plane perpendicular
to jP, and to the edges of the hollows. For definiteness we shall

518

APPENDIX B.

take for F the plane of the edges of the hollows. Instead now of
a particle after collision flying along the chord of the circle of § 38
it would go on for ever in a straight line. To bring it back to the
plane F, let it be acted on either (a) by a force towards the plane
in simple proportion to the distance, or (yS) by a constant force.
This latter supposition (^) presents to us the very interesting case
of an elastic ball bouncing from a corrugated floor, and describing
gravitational parabolas in its successive flights, the durations of

BALL BOUNCING FROM CORRUGATED FLOOR. 519

the different flights being in simple proportion to the component
of velocity perpendicular to the plane F. The supposition (a)
is purely ideal; but it is interesting because it gives a half
curve of sines for each flight, and makes the times of flight
from F after a collision and back again to F the same for all the
flights, whatever be the inclination on leaving the floor and return-
ing to it. The supposition (yS) is illustrated in fig. 8, with only
the variation that the corrugations are convex instead of concave,
and that two vertical planes are fixed to reflect back the particle
instead of allowing it to travel indefinitely, either to right or
to left.

§ 47. Let the rotator of §§ 41 to 43, instead of bouncing to
and fro between two parallel planes, impinge only on one plane F^
and let it be brought back by a force through its centre of inertia,
either (a) varying in simple proportion to the distance of the centre
of inertia from jP, or (yS) constant. Here, as in § 46, the times of
flight in case (a) are all the same, and in (fi) they are in simple
proportion to the velocity of its centre of inertia when it leaves F
or returns to it.

§ 48. In the cases of §§ 46, 47 we have to consider the time-
integral for each flight of the kinetic energy of the component
velocity of the particle perpendicular to Fy and of the whole
velocity of the centre of inertia of the rotator, which is itself
perpendicular to F, If q denotes the velocity perpendicular to F
of the particle, or of the centre of inertia of the rotator, at the
instants of crossing F at the beginning and end of the flight, and
if 2 denotes the mass of the particle or of the rotator so that the
kinetic energy is the same as the square of the velocity, the time-
integral is in case (a) ig*r, and in case ()8) ^q^T, the time of the
flight being denoted in each case by T. In both (a) and (yS), § 46,
if we call 1 the velocity of the particle, which is always the same,
we have g* = sin' 6, and the other component of the energy is cos' 0.
In § 47 it is convenient to call the total energy 1 ; and thus 1 — j'
is the total rotational energy which is constant throughout the
flight. Hence, remembering that the times of flight are all the
same in case (a) and are proportional to the value of g in case (/3) ;
in case (a), whether of § 46 or § 47, the time-integrals of the kinetic
energies to be compared are as ^Sg* to 2 (1 — 3'), and in case iff)
they are as J2j* to 2g (1 — (f).

520 APPENDIX B,

§ 49. Hence with the following notation : —

T R ift (time-integral of kinetic energy perpendicular to JF\ = K
^^^*^t „ „ parallel to F.^U

translatory energy = V
rotatory „ = iJ

In § 47 {
we have

( _ 2 gg*-!)

v-r\ i{i-w

>*

»

(iS)

(«)

V + R

^ 2J|3» -q)

2 (? - W

»

(fi)

§ 49'. By the processes described above q was calculated for
the single particle and corrugated floor (§ 46), and for the rotator
of two equal masses each impinging on a fixed plane (§§ 41, 42),
and for the biassed ball (central and eccentric masses 100 and 1
respectively, ^ 41, 43). Taking these values of g, summing q, g«
and ^ for all the flights, and using the results in § 48, we find the
following six results :

Single particle bounding from corrugated floor (semicircular
hollows), 143 flights: —

F — t/' f = + '197 for isochronous sinusoidal flights.
F + 1/' ( = + '136 for gravitational parabolic „

Rotator of two equal masses, bounding from plane floor, 110
flights : —

V — jR I = — '179 for isochronous sinusoidal flights.
V+R\ = — '150 for gravitational parabolic „

Biassed ball, bounding from plane floor, 400 flights : —

F — jR J = + '025 for isochronous sinusoidal flights.

V + ^ I = — '014 for gravitational parabolic „

The smallness of the deviation of the last two results, from what
the Boltzmann- Maxwell doctrine makes them, is very remarkable
when we compare it with the 15 per cent, which we have found
(§ 43 above) for the biassed ball bounding free from force, to and
fro between two parallel planes.

CAGED ATOM STKUCK BY OTHERS OUTSIDE. 521

§ 50. The last case of partition of energy which we have
worked out statistically relates to an impactual problem, belong-
ing partly to Class I. § 28, and partly to Class II. It was designed
as a nearer approach to practical application in thermodynamics
than any of those hitherto described. It is, in fact, a one-dimen-
sional illustration of the kinetic theory of gases. Suppose a row
of a vast number of atoms, of equal masses, to be allowed freedom
to move only in a straight line between fixed bounding planes L
and K, Let P the atom next K be caged between it and a parallel
plane C, at a distance from it very small in comparison with the
average of the free paths of the other particles; and let Q, the
atom next to P, be perfectly free to cross the cage-front G, without
experiencing force from it. Thus, while Q gets freely into the cage
to strike P, P cannot follow it out beyond the cage-front. The
atoms being all equal, every simple impact would produce merely
an interchange of velocities between the colliding atoms, and no
new velocity could be introduced, if the atoms were perfectly hard
(§ 16 above), because this implies that no three can be in collision
at the same time. I do not, however, limit the present investigation
to perfectly hard atoms. But, to sim'plify our calculations, we shall
suppose P and Q to be infinitely hard. All the other atoms we
shall suppose to have the property defined in § 21 above. They
may pass through one another in a simple collision, and go asunder
each with its previous velocity unaltered, if the differential velocity
be sufficiently great ; they must recoil from one another with inter-
changed velocities if the initial differential velocity was not great
enough to cause them to go through one another. Fresh velocities
will generally be introduced by three atoms being in collision at
the same time, so that even if the velocities were all equal, to
begin with, inequalities would supervene in virtue of three or
more atoms being in collision at the same time; whether the
initial differential velocities be small enough to result in two
recoils, or whether one or both the mutual approaches lead to a
passage or passages through one another. Whether the distribu-
tion of velocities, which must ultimately supervene, is or is not
according to the Maxwellian law, we need not decide in our minds ;
but, as a first example, I have supposed the whole multitude to be
given with velocities distributed among them according to that
law (which, if they were infinitely hard, they would keep for ever
after); and we shall further suppose equal average spacing in

522 APPENDIX &

different parts of the row, so that we need nd be troubled with
the consideration of waves, as it were of soiuid, mimiDg to aod firo
along the row because of inequalities of density.

§ 51. For our present problem we require two lotteries to find
the influential conditions at each instant, when Q enters l^s cage
— lottery I. for the velocity (r) of Q at impact; lottery IL for the
phaese of P's motion. For lottery I. (after trying 837 small squares
of paper with velocities written on them and mixed in a bowl, and
finding the plan unsatisfactory) we took nine stiff cards, numb^ed
1, 2... 9, of the size of ordinary playing-cards with rounded aNmerB,
with one hundred numbers written on each in ten lines ot ten
numbers. The velocities on each card are shown cm the follow-
ing table.

Table showing the XrnBER of the Different VELocmES

*

ON THE DlFFEREN-T CaRDS.

W

1

Veiodcy

1
100

-*

•4 -4

-i

•i -: -* -9 1-0 1-1

1-2 l-a 1-4 1-4

1-:

i-ft

1

If

*«

»

«

Cardl

., 2

7

03

1

\

„ 3

10

90

1

„ 4

9 91

i

>. 5

1

f^

15

1

,. 6

60 40

■

„ 7

26 57 17

.. «

31 40 29

1

.. 9

107

3

103 99 92 iH 75 66 57 id 40 32

i

•26 19 15 11
26 19 15 11

9
9

6
6

4

4

3
3

2

1

li

1

1 It

1 900

Sams of 1
vekxiiieb i

t

1 '

The number of times each velocity occurs was chosen to fulfil

as nearly as may be the Maxwellian law, which is Cdve * » the
number of velocities between v-^^dv and v~^dv. We took
k = l, which, if dv were infinitely small, would make the mean
of the squares of the velocities equal exactly to '5; we took
dv=^'l and Cdv =108 to give, as nearly as circumstanoeB would
allow, the Maxwellian law, and to make the total number of

CAGED ATOM KEEPS LESS ENERGY THAN FREE ASSAILANT. 623

different velocities 900. The sum of the squares of all these 900
velocities is 468*4, which divided by 900 is '52. In the practice
of this lotteiy the numbered cards were well shuffled and then
one was drawn ; the particular one of the hundred velocities on
this card to be chosen was found by drawing one card from a pack
of one hundred numbered 1, 2.. .99, 100. In lottery II. a pack of
one hundred cards is used to draw one of one hundred decimal
numbers from '01 to 1*00. The decimal drawn, called a, shows
the proportion of the whole period of P from the cage-front (7,
to K, and back to (7, still unperformed at the instant when Q
crosses C, Now remark, that if Q overtakes P in the first half of
its period it gives its velocity v to P, and follows it inwards ; and
therefore there must be a second impact when P meets it after
reflection from Ky and gives it back the velocity v which it had on
entering. If Q meets P in the second half of its period, Q will,
by the first impact, get P's original velocity, and may with this
velocity escape from the cage. But it may be overtaken by P
before it gets out of the cage, in which case it will go away from
the cage with its own original velocity v unchanged. This occurs
always if, and never unless, u is less than va\ Ps velocity being
denoted by ti, and 0*8 by v. This case of Q overtaken by P can
only occur if the entering velocity of Q is greater than the speed
of P before collision. Except in this case P*8 speed is unchanged
by the collision. Hence we see that it is only when P's speed is
greater than Q*8 before collision that there can be interchange, and
this interchange leaves P with less speed than Q. If every collision
involved interchange, the average velocity of P would be equalised
by the collisions to the average velocity of Q, and the average
distribution of different velocities would be identical for Q and P.
Non-fulfilment of this equalising interchange can, as we have seen,
only occur when Q*s speed is less than P*s, and therefore the average
speed and the average kinetic energy of P must be less than the
average kinetic energy of Q.

§ 52. We might be satisfied with this, as directly negativing
the Boltzmann-Maxwell doctrine for this case. It is however,
interesting to know, not only that the average kinetic energy of
Q is greater than that of the caged atom, but, further, to know
how much greater it is. We have therefore worked out sum-
mations for 300 collisions between P and Q, beginning with

524 APPENDIX B.

u« = '5 (w = '71), being approximately the mean of v* as given by
the lottery. It would have made no appreciable difference in the
result if we had begun with any value of u, large or small, other
than zero. Thus, for example, if we had taken 100 as the first
value of u, this speed would have been taken by Q at the first
impact, and sent away along the practically infinite row, never to
be heard of again; and the next value of u would have been the
first value drawn by lottery for v. Immediately before each of
the subsequent impacts, the velocity of P is that which it had
from Q by the preceding impact. In our work, the speeds which
P actually had at the first sixteen times of Q*s entering the cage
were 71, 5, '3, 2, % 1, 1, 2, -2, 5, 7, 2, 3, 6, 1-6, o— from
which we see how little effect the choice of '71 for the first speed
of P had on those that follow. The summations were taken in
successive groups of ten; in every one of these St;* exceeded 2u'.
For the 300 we found Xt^ = 14853 and 2u* = 61*62, of which the
former is 2*41 times the latter. The two ought to be equal
according to the Boltzmann-Maxwell doctrine. Dividing 2i^ by
300 we find '495 ; which chances to be more nearly equal to the
intended *5 than the '52 which is on the cards (§ 51 above). A
still greater deviation from the B.-M. equality (2'7l instead of
2'41) was found by taking Xv^ and Xu'^v to allow for greater
probability of impact with greater than with smaller values of
V ; w' being the velocity of P after collision with Q,

§ 53. We have seen in § 52 that Xu* must be less than St;*,
but it seemed interesting to find how much less it would be with
some other than the Maxwellian law of distribution of velocities.
We therefore arranged cards for a lottery, with an arbitrarily
chosen distribution, quite different from the Maxwellian. Eleven
cards, each with one of the eleven numbers 1, 3... 19, 21, to cor-
respond to the different velocities '1, •3... 1*9, 2'1, were prepared
and used instead of the nine cards in the process described in § 51
above. In all except one of the eleven tens, Xi^ was greater than
2w*, and for the whole 110 impacts we found 2t;* = 179*90, and
2w*= 97'66; the former of these is 1*84 times the latter. In this
case we found the ratio of Xv* to Xu'h) to be 1*87.

§ 54. In conclusion, I wish to refer, in connection with Class II.
§ 28, to a very interesting and important application of the doctrine,
made by Maxwell himself, to the equilibrium of a tall column of

GAS UNDER INFLUENCE OF GRAVITY IN VERTICAL TUBE. 525

gas under the influence of gravity. Take, first, our one-dimen-
sional gas of § 50, consisting of a straight row of a vast number
of equal and similar atoms. Let now the line of the row be
vertical, and let the atoms be under the influence of terrestrial
gravity, and suppose, first, the atoms to resist mutual approach,
sufiiciently to prevent any one from passing through another with
the greatest relative velocity of approach that the total energy
given to the assemblage can allow. The Boltzmann-Maxwell
doctrine (§ 18 above) asserting as it does that the time-integral
of the kinetic energy is the same for all the atoms, makes the
time-average of the kinetic energy the same for the highest as
for the lowest in the row. This, if true, would be an exceedingly
interesting theorem. But now, suppose two approaching atoms
not to repel one another with infinite force at any distance
between their centres, and suppose energy to be given to the
multitude sufficient to cause frequent instances of two atoms
passing through one another. Still the doctrine can assert nothing
but that the time-integral of the kinetic energy of any one atom
is equal to that of any other atom, which is now a self-evident
proposition, because the atoms are of equal masses, and each one
of them in turn will be in every position of the column, high or
low. (If in the row there are atoms of different masses, the
Waterston-Maxwell doctrine of equal average energies would, of
course, be important and interesting.)

§ 55. But now, instead of our ideal one-dimensional gas, con-
sider a real homogeneous gas, in an infinitely hard vertical tube,
with an infinitely hard floor and roof, so that the gas is under
no influence from without, except gravity. First, let there be
only two or three atoms, each given with sufiicient velocity to
fly against gravity from floor to roo£ They will strike one an-
other occasionally, and they will strike the sides and floor and
roof of the tube much more frequently than one another. The
time-averages of their kinetic energies will be equal So will
they be if there are twenty atoms, or a thousand atoms, or a
million, million, million, million, million atoms. Now each atom
will strike another atom much more frequently than the sides
or floor or roof of the tube. In the long run each atom will be
in every part of the tube as often as is every other atom. The
time-integral of the kinetic energy of any one atom will be equal

526 APPENDIX B.

to the time-integral of the kinetic energy of any other atom.
This truism is simply and solely all that the Boltzmann-Maxwell
doctrine asserts for a vertical column of a homogeneous monatomic
gas. It is, I believe, a general impression that the Boltzmann-
Maxwell doctrine, asserting a law of partition of the kinetic part
of the whole energy, includes obviously a theorem that the average
kinetic energies of the atoms in the upper parts of a vertical column
of gas, are equal to those of the atoms in the lower parts of the
column. Indeed, with the wording of Maxwell's statement, § 18,
before us, we might suppose it to assert that two parts of our
vertical column of gas, if they contain the same number of atoms,
must have the same kinetic energy, though they be situated, one
of them near the bottom of the column, and the other near the
top. Maxwell himself, in his 1866 paper ("The Dynamical Theory
of Gases'*)*, gave an independent synthetical demonstration of
this proposition, and did not subsequently, so far as I know, regard
it as immediately deducible from the partitional doctrine general-
ised by Boltzmann and himself several years after the date of
that paper.

§ 56. Both Boltzmann and Maxwell recognised the experi-
mental contradiction of their doctrine presented by the kinetic
theory of gases ; and felt that an explanation of this incompatibility
was imperatively called for. For instance. Maxwell, in a lecture
on the dynamical evidence of the molecular constitution of bodies,
given to the Chemical Society, Feb. 18, 1875, said: "I have put
" before you what I consider to be the greatest diflSculty yet en-
" countered by the molecular theory. Boltzmann has suggested
"that we are to look for the explanation in the mutual action
" between the molecules and the ethereal medium which surrounds
" them. I am afraid, however, that if we call in the help of thLs
"medium we shall only increase the calculated specific heat,
"which is already too great." Rayleigh, who has for the last
twenty years been an unwavering supporter of the Boltzmann-
Maxwell doctrine, concludes a paper " On the Law of Partition of
Energy,'* published a year ago in the Phil. Mag., Jan. 1900, with
the following words: "The diflRculties connected with the appli-
" cation of the law of equal partition of energy to actual gases
"have long been felt. In the case of argon and helium and

* AdditioD, of date December 17, 1866. Collected Papers, Vol. ix. p. 76.

ABANDON THE DOCTRINE CONSTITUTING CLOUD II. 527

"mercury vapour, the ratio of specific heats (1*67) limits the
"degrees of freedom of each molecule to the three required for
" translatory motion. The value (1'4) applicable to the principal
" diatomic gases, gives room for the three kinds of translation and
"for two kinds of rotation. Nothing is lefb for rotation round
** the line joining the atoms, nor for relative motion of the atoms
" in this line. Even if we regard the atoms as mere points, whose
"rotation means nothing, there must still exist energy of the
"last-mentioned kind, and its amount (according to law) should
" not be inferior.

" We are here brought face to face with a fundamental difficulty,
" relating not to the theory of gases merely, but rather to general
"dynamics. In most questions of dynamics, a condition whose
" violation involves a large amount of potential energy may be
"treated as a constraint. It is on this principle that solids are
" regarded as rigid, strings as inextensible, and so on. And it is
" upon the recognition of such constraints that Lagrange's method
" is founded. But the law of equal partition disregards potential
"energy. However great may be the energy required to alter
" the distance of the two atoms in a diatomic molecule, practical
" rigidity is never secured, and the kinetic energy of the relative
" motion in the line of junction is the same as if the tie were of
" the feeblest. The two atoms, however related, remain two atoms,
" and the degrees of freedom remain six in number.

"What would appear to be wanted is some escape from the
" destructive simplicity of the general conclusion."

The simplest way of arriving at this desired result is to deny
the conclusion; and so, in the beginning of the twentieth century,
to lose sight of a cloud which has obscured the brilliance of the
molecular theory of heat and light during the last quarter of the
nineteenth century.

APPENDIX a

ON THE DISTURBANCE PRODUCED BY TWO PARTICULAR
FORMS OF INITIAL DISPLACEMENT IN AN INFINITELY
LONG MATERIAL SYSTEM FOR WHICH THE VELOCITY
OF PERIODIC WAVES DEPENDS ON THE WAVE-LENGTH.

§ 1. One of the chosen forms of initial displacement is

C€-**/«* (1),

where x is distance from 0, the centre of the origin of the
disturbance ; a is a length-parameter of the initial disturbance ;
and c is a length very small (regarded in the mathematics as
infinitely small) in comparison with a.

§ 2. Let the displacement at {x^ t) in an infinite train of
periodic waves be expressed by

DcoBq{x — a — vt) (2).

Here i'frlq is the wave-length; and v is the propagational
velocity which we may regard as a function of q ; and a is the
distance from 0 of a point where the displacement is a positive
maximum at time, ^ = 0. Any number of solutions such as (2)
with different values of D, q, and a, or with — v for v, super-
imposed, give a complex solution, found by simple summation
because the displacements and their sums are infinitely small.
Hence a solution, the solution for standing vibrations of period
2'ir/qVf is found by taking the half-sum of (2) for ± v ; being

this : —

i) cos g (a? — a) cos v^ (3).

And a summation of (3) for our present purpose gives a solution,
1;=-/ dql da€"**^*'cosg(a? — a)cost;^ (4).*

When ^ = 0 this gives d/dt 17 = 0; and, by "Fourier's theorem,"
the second member becomes C€"^'*' which agi*ees with (1). Hence
(4) expresses the solution 0/ our problem.

* See Lee. X. Part i., top of p. 110.

TRA.N3VERSE WA.VES IN ELAJSTIC ROD: FINITE SOLUTION. 529

§3. If in (4) we expand cos q(x^a), the factor of sin ga?
disappears in the summation I da ; and the factor of cos qx, being

da€-«W cos gra, is equal to yjirae"^'^'^ according to an evaluation

J -00

given by Laplace in 1810*. Thus we reduce (4) to the following
simpler form

ca r*

i7 = -T-| dqe"^^'^ aos qx QOB qyt (5).

The verification that this becomes C€~**'** when ^ = 0, is most
interesting : it is done by a second application of the same evalua-
tion of Laplace's with a curiously inverted notation.

§ 4. Whatever function v may be of the wave-length, iirjq,
we have in (5) a thoroughly convenient solution of our problem ;
calculable by highly convergent quadratures whether the definite
integral is reducible to finite terms or not. For one very interest-
ing case, an infinitely long elastic rod, taken as an example by
Fourier f, the definite integral is reducible to finite terms.

In this case the velocity of periodic waves is inversely as the
wave-length, and, by choosing our units conveniently, we may
put t; = gr, in (5). Thus, and by taking a= 1, and by substituting

- I dq tov \ dq, we find

2 j -00 Jo

^— _ j dq€~^f* cos qH COS qx (6).

' 2v^J-oo

Denoting by R the half-sum for ± i, we may write this thus ;

V = o^R r d?€-i<i-*««*-*«*l (7).

Putting i'-tt — h\ we reduce the index of the exponential to
- (hq - ixl2hy - (x/2hy ; and putting hq - tx/2h = z, we find for
the definite integral,

J -00

d^€-^.€-<''>*)'=^.€-<*'»*>' (8).

• MSmoires de VInstitut, 1810. See Gregory's Examples, p. 480.
t Chaleur, Chap. xix. Article 406.

T. L. 34

530 APPENDIX C.

Hence (7) becomes

rf = ^cRh-'€-^f^^* = ^cRh-'€ !+'«• (9).

Now 1 /A = (^ + ^«)-i (cos J <^ + t sin i<^),

if <^ = tan-i« (10).

Hence, if we put fZTr^"^ ^^^^'

we find by (9), i; = c(l + 16<»)~*ei'''^cos(i<^-tf) (12).

§ 5. This is a very interesting solution in "finite terms."
To illustrate it let first a? = 0; we have, ^=0, and

17 = c(H- 16i»)-*cos i<^ = a/^2^» where t = V(l + 16^),.. (13).

Hence the middle point of the originally displaced part of the
rod subsides to its undisturbed position not vibrationcUly, but
continuously, and ultimately in proportion to 1r^ when t is very
great. And when t is very great we may neglect <f> in (12), and
we have 0 =? x/r ; so that we find

17 = CT-* €-<«'^>" cos - (14).

This shows that, in going from the origin we first find a zero
of displacement at a? = V(w^/2) ; and beyond this there is an
infinite number of zeros, and vibratory subsidence to repose.
Interesting graphic illustrations will require but little labour.

§ 6. As another even more interesting case, take the two-
dimensional case of the old deep-water wave problem of Poisson
and Cauchy. In it the velocity of periodic waves is proportional
to »/K and we may therefore conveniently take v = q~^. This in
(5), with a = l, gives

V = ~7~ I dq€~^'^co^qxco^s/qt (15),

an irreducible definite integral which expresses the displacement
of the water at time t and distance x from the origin of the
disturbance; by a convergent formula thoroughly convenient for
calculation by quadratures for all values large or small of the
two variables. Analytical expedients may no doubt be found to
diminish the labour: but it is satisfactory to know that the
method of quadratures, with moderate labour, gives very interest-
ing illustrations; though great labour would be needed for fiiU
graphical illustrations by time-curves and space-curves.

TWO-DIMENSIONAL DEEP-WATER WAVES: FINITE SOLUTION. 531

§ 7. My assistant Mr Witherington has evaluated i; by (15)
with cjs/ir = 2 ; for four cases as follows :

Case 1. a? = 0, i = 0

„ 2. a: = 0, ^ = 1

„ 3. a? = 0, i = 2

„ 4. a?=2, ^ = 2

17= 1-763
17= -911
17 = - -451
17 = -559.

Vtt = 1-77245,

Case 1 was worked as a test for accuracy in a first trial of
quadrature. The result is about 2/3 per cent, too small; the
correct result being V^- The accuracy of the quadratures is
sufficient for merely illustrative purposes.

§ 8. Interesting as is the solution by quadratures for water-
waves resulting from the initial disturbance (1), it is less important
than the following solution in finite terms for another fomi of
initial disturbance, which was given in my short paper " On the
Front and Rear of a Procession of free Waves in Deep Water"*: —

^ d<t> d<f> ^ 2c f, ., gt^x . , . gt^x] 1^

^^£^ ^=dj' <^=^-{(^ + y)*cos^ + (r-y)^sin^}e4^

(16).

Here (^, rj) denote the displacement of the particle of water
whose equilibrium-position is (x, y); r denotes (aJ* + y')*; and
g, gravity. According to a beautiful discovery of Cauchy and
Poisson, for waves in infinitely deep water, the pressure is con-
stant at all the particles (ar + fo» y + Vo) for any constant value, y©*
of y; and therefore the water above every such surface may be
removed without disturbing the motion of the water below it.
Hence 170 is the vertical displacement of any point in a free surfece
of particles whose undisturbed position is y = yo- The value of 170
for e = 0, from (16), is

c(ro»-royo-2yo»)/ro»(ro-hyo)* (17).

This, with any value we please to assign to yo> is tihe initiating
free-surface for (16) ; r^ denotes {af^ + yo*)*

I hope to reproduce, with extensions, for early publication,
curves of the wave-motion expressed by (16), shown in Section A
of the British Association, Sep. 1886, and in the Royal Society of
Edinburgh, Dec. 20, 1886.

* Proc. R.S.E, Jan. 7, 1887; Phil. Mag, Feb. 1887.

34—2

APPENDIX D.

ON THE CLUSTERING OF GRAVITATIONAL MATTER IN ANY

PART OF THE UNIVERSE*.

Gravitational matter, according to our ideas of universal
gravitation, would be all matter. Now is there any matter which
is not subject to the law of gravitation ? I think I may say with
absolute decision that there is. We are all convinced, with our
President, that ether is matter, but we are forced to say thalt we
must not expect to find in ether the ordinary properties of molar
matter which are generally known to us by action resulting
from force between atoms and matter, ether and ether, and
atoms of matter and ether. Here I am illogical when I say
between matter and ether, as if ether were not matter. It is
to avoid an illogical phraseology that I use the title "gravi-
tational matter." Many years ago I gave strong reason to feel
certain that ether was outside the law of gravitation. We
need not absolutely exclude, as an idea, the possibility of there
being a portion of space occupied by ether beyond which there
is absolute vacuum — no ether and no matter. We admit that
that is something that one could think of; but I do not believe
any living scientific man considers it in the slightest degree
probable that there is a boundary around our universe beyond
which there is no ether and no matter. Well, if ether extends
through all space, then it is certain that ether cannot be subject
to the law of mutual gravitation between its parts, because if
it were subject to mutual attraction between its parts its equi-
librium would be unstable, unless it were infinitely incom-
pressible. But here, again, I am reminded of the critical
character of the ground on which we stand in speaking of pro-
perties of matter beyond what we see or feel by experiment. I
am afraid I must here express a view different from that which

* Ck)mmnnicated by the Author to the Phil, Mag,, having been read before the
British Association at the Olasgow meeting.

ETHER IS GRAVITATIONLESS MATTER FILLING ALL SPACE. 533

Professor Rlicker announced in his Address, when he said that
continuity of matter implied absolute resistance to condensation.
We have no right to bar condensation as a property of ether.
While admitting ether not to have any atomic structure, it is
postulated as a material which performs functions of which we
know something, and which may have properties allowing it to
perform other functions of which we are not yet cognisant. If
we consider ether to be matter, we postulate that it has rigidity
enough for the vibrations of light, but we have no right to say
that it is absolutely incompressible. We must admit that suf-
ficiently great pressure all round could condense the ether in a
given space, allowing the ether in surrounding space to come in
towards the ideal shrinking surface. When I say that ether
must be outside the law of gravitation, I assume that it is not
infinitely incompressible. I admit that if it were infinitely in-
compressible, it might be subject to the law of mutual gravitation
between its parts ; but to my mind it seems infinitely improbable
that ether is infinitely incompressible, and it appears more con-
sistent with the analogies of the known properties of molar
matter, which should be our guides, to suppose that ether has
not the quality of exerting an infinitely great force against
compressing action of gravitation. Hence, if we assume that it
extends through all space, ether must be outside the law of
gravitation — that is to say, truly imponderable. I remember the
contempt and self-complacent compassion with which sixty years
ago I myself — I am afraid — ^and most of the teachers of that
time looked upon the ideas of the elderly people who went before
us, and who spoke of " the imponderables." I fear that in this, as
in a great many other things in science, we have to hark back to
the dark ages of fifly, sixty, or a hundred years ago,. and that we
must admit there is something which we cannot refuse to call
matter, but which is not subject to the Newtonian law of gravitation.
That the sun, stars, planets, and meteoric stones are all of them
ponderable matter is true, but the title of my paper implies that
there is something else. Ether is not any part of the subject of
this paper; what we are concerned with is gravitational matter,
ponderable matter. Ether we relegate, not to a limbo of impon-
derables, but to distinct species of matter which have inertia,
rigidity, elasticity, compressibility, but not heaviness. In a paper
I have already published I gave strong reasons for limiting to a

534 APPENDIX D.

definite amount the quantity of matter in space known to astro-
nomers. I can scarcely avoid using the word "universe," but I
mean our universe, which may be a very small affair after all,
occupying a very small portion of all the space in which there is
ponderable matter.

Supposing a sphere of radius 3"09.10^' kilometres (being the
distance at which a star must be to have parallax O'^'OOl) to have
within it, uniformly distributed through it, a quantity of matter
equal to one thousand million times the sun's mass ; the velocity
acquired by a body placed originally at rest at the surface would,
in five million years, be about 20 kilometres per second, and in
twenty -five million years would be 108 kilometres per second (if
the acceleration remained sensibly constant for so long a time).
Hence, if the thousand million suns had been given at rest
twenty-five million years ago, uniformly distributed throughout
the supposed sphere, many of them would now have velocities of
20 or 30 kilometres per second, while some would have less and
some probably greater velocities than 108 kilometres per second ;
or, if they had been given thousands of million years ago at rest
so distributed that now they were equally spaced throughout the
supposed sphere, their mean velocity would now be about 60 kilo-
metres per second*. This is not unlike the measured velocities
of stars, and hence it seems probable that there might be as
much matter as one thousand million suns within the distance
309. 10^' kilometres. The same reasoning shows that ten thousand
million suns in the same sphere would produce, in twenty-
five million years, velocities far greater than the known star
velocities: and hence there is probably much less than ten
thousand million times the sun's mass in the sphere considered.
A general theorem discovered by Green seventy-three years ago
regarding force at a surface of any shape, due to matter (gravi-
tational, or ideal electric, or ideal magnetic) acting according to
the Newtonian law of the inverse square of the distance, shows
that a non-uniform distribution of the same total quantity of
matter would give greater velocities than would the uniform dis-
tribution. Hence we cannot, by any non-uniform distribution of
matter within the supposed sphere of 309 . 10" kilometres radius,
escape from the conclusion limiting the total amount of the

♦ Phil Mag, August 1901, pp. 169, 170.

TOTAL BRIGHT OR DARE STAR-DISC APPARENT AREA. 635

matter within it to something like one thousand million times
the sun's mass.

If we compare the sunlight with the light from the thousand
million stars, each being supposed to be of the same size and
brightness as our sun, we find that the ratio of the apparent
brightness of the star-lit sky to the brightness of our sun's disc
would be 3'87.10~". This ratio* varies directly with the radius
of the containing sphere, the number of equal globes per equal
volume being supposed constant ; and hence to make the sum of
the apparent areas of discs 3'87 per cent, of the whole sky, the
radius must be 309.10*' kilometres. With this radius light would
take 3^.10" years to travel from the outlying stars to the centre.
Irrefragable d3mamics proves that the life of our sun as a luminary
may probably be between twenty-five and one hundred million
years ; but to be liberal, suppose each of our stars to have a life of
one hundred million years as a luminary : and it is found that the
time taken by light to travel from the outlying stars to the centre
of the sphere is three and a quarter million times the life of a
star. Hence it follows that to make the whole sky aglow with
the light of all the stars at the same time the commencements
of the stars must be timed earlier and earlier for the more and
more distant ones, so that the time of the arrival of the light of
every one of them at the earth may fall within the durations of
the lights of all the others at the earth. My supposition as to
uniform density is quite arbitrary; but nevertheless I think it
highly improbable that there can be enough of stars (bright or
dark) to make a total of star- disc area more than 10~" or 10~" of
the whole sky.

To help to understand the density of the supposed distribution
of one thousand million suns in a sphere of 309.10" kilometres
radius, imagine them arranged exactly in cubic order, and the
volume per sun is found to be 123*5. 10*® cubic kilometres, and
the distance from one star to any one of its six nearest neighbours
would be 4*98.10" kilometres. The sun seen at this distance
would probably be seen as a star of between the first and second
magnitudes ; but supposing our thousand million suns to be all of
such brightness as to be stars of the first magnitude at distance
corresponding to parallax 1"'0, the brightness at distance 309. 10"
kilometres would be one one-millionth of this; and the most

* Phil, Mag, Aagast 1901, p. 175.

636 APPENDIX D.

distant of our stars would be seen through powerful telescopes
as stars of the sixteenth magnitude. Newcomb estimated from
thirty to fifty million as the number of stars visible in modem
telescopes. Young estimated at one hundred million the number
visible through the Lick telescope. This larger estimate is only
one-tenth of our assumed one thousand million masses equal to
the sun, of which, however, nine hundred million might be either
non-luminous, or, though luminous, too distant to be seen by us
at their actual distances from the earth. Remark, also, that it is
only for facility of counting that we have reckoned our universe
as one thousand million suns; and that the meaning of our
reckoning is that the total amount of matter within a sphere of
3*09.10^^ kilometres radius is one thousand million times the
sun's mass. The sun's mass is 1*99.10*' metric tons, or 1*99. 10*
grammes. Hence our reckoning of our supposed spherical uni-
verse is that the ponderable part of it amounts to 199. 10**
grammes, or that its average density is 1 '61.10"" of the density
of water.

Let us now return to the question of sum of apparent areas.
The ratio of this sum to 47r, the total apparent area of the sky
viewed in all directions, is given by the formula*:

3JV/a'

"4 \r) '

provided its amount is so small a fraction of unity that its dimi-
nution by eclipses, total or partial, may be neglected. In this
formula, iVis a number of globes of radius a uniformly distributed
within a spherical surface of radius r. For the same quantity of
matter in N' globes of the same density, uniformly distributed
through the same sphere of radius r, we have

N ~ Uv

and therefore — = - .

a a

With N=^\0\ r = 3-09.10" kilometres; and a (the sun's radius)
= 7.10' kilometres; we had a = 3'87.10~". Hence a' = 7 kilo-
metres gives a'= 3'87.10"®; and a"= 1 centimetre gives a"= 1/36*9.
Hence if the whole mass of our supposed universe were reduced

• PhiU Mag, August 1901, p. 175.

VELOCITIES DUE TO DISTANT ATTRACTIONS BETWEEN ATOlfS. 637

to globules of density 1*4 (being the sun's mean density), and of
2 centimetres diameter, distributed uniformly through a sphere
of 309. 10" kilometres radius, an eye at the centre of this sphere
would lose only 1/369 of the light of a luminary outside it!
The smallness of this loss is easily understood when we consider
that there is only one globule of 2 centimetres diameter per
364,000,000 cubic kilometres of space, in our supposed universe
reduced to globules of 2 centimetres diameter. Contrast with
the total eclipse of the sun by a natural cloud of water spherules,
or by the cloud of smoke from the funnel of a steamer.

Let now all the matter in our supposed universe be reduced
to atoms (literally brought back to its probable earliest condition).
Through a sphere of radius r let atoms be distributed uniformly
in respect to gravitational quality. It is to be understood that
the condition "uniformly** is fulfilled if equivoluminal globular
or cubic portions, small in comparison with the whole sphere,
but large enough to contain large numbers of the atoms, contain
equal total masses, reckoned gravitationally, whether the atoms
themselves are of equal or unequal masses, or of similar or dis-
similar chemical qualities. As long as this condition is fulfilled,
each atom experiences very approximately the same force as if
the whole matter were infinitely fine-grained, that is to say,
utterly homogeneous.

Let us therefore begin with a uniform sphere of matter of
density /?, gravitational reckoning; with no mutual forces except
gravitation between its parts, given with every part at rest at
the initial instant: and let it be required to find the subsequent
motion. Imagining the whole divided into infinitely thin con-
centric spherical shells, we see that every one of them falls
inwards, as if attracted by the whole mass within it collected
at the centre. Hence our problem is reduced to the well known
students* exercise of finding the rectilinear motion of a particle
attracted according to the inverse square of the distance from a

fixed point. Let x^ be the initial distance, -^oc^ the attracting

mass, V and x velocity and distance from the centre at time t
The solution of the problem for the time during which the particle
is falling towards the centre is

*-^'-'e-3

538 APPENDIX D.

where 0 denotes the acute angle whose sine is a / — . This shows

that the time of falling through any proportion of the initial
distance is the same whatever be the initial distance; and that
the time (which we shall denote by T) of falling to the centre is

^TT^ /— . Hence in our problem of homogeneous gravitational

matter given at rest within a spherical surface and left to fall
inwards, the augmenting density remains homogeneous, and the
time of shrinkage to any stated proportion of the initial radius
is inversely as the square root of the density.

To apply this result to the supposed spherical universe of
radius 309.1()*' kilometres, and mass equal to a thousand million
times the mass of our sun, we find the gi*avitational attraction on
a body at its surface gives acceleration of 1 '37.10"" kilometres

per second per second. This therefore is the value of —J^x^,

with one second as the unit of time and one kilometre as the
unit of distance; and we find T=52'8.10" seconds = 16*8 million
years. Thus our formulas become

Jt;»=r37.10-"a?o(J'-l)

givmg

and

t; = 5-23.19-'A/a?o(^-l)

whence, when sin 0 is very small,

« = 52-8. 10"

(-S-

Let now, for example, a?o = 3*09.10" kilometres, and —=10';

and, therefore, sin tf=tf = 316.10-*; whence, t; = 291,000 kilo-
metres per second, and t^T — 7080 seconds = T — 2 hours ap-
proximately.

By these results it is most interesting to know that our sup-
posed sphere of perfectly compressible fluid, beginning at rest
with density 1*61.10-^ of that of water, and of any magnitude

CONDENSATION FROM UNIFORM ASSEMBLAGE GIVEN AT REST. 539

large or small, and left unclogged by ether to shrink under the
influence of mutual gravitation of its parts, would take nearly
seventeen million years to reach '0161 of the density of water,
and about two hours longer to shrink to infinite density at its
centre. It is interesting also to know that if the initial radius is
3*09. 10^' kilometres, the inward velocity of the surface is 291,000
kilometres per second at the instant when its radius is 3*09.10®
and its density OlGl of that of water. If now, instead of an
ideal compressible fluid, we go back to atoms of ordinary matter
of all kinds as the primitive occupants of our sphere of 3*09. 10^'
kilometres radius, all these conclusions (provided all the velocities
are less than the velocity of light) would still hold, notwith-
standing the ether occupying the space through which the atoms
move. This would, I believe*, exercise no resistance whatever
to uniform motion of an atom through it ; but it would certainly
add quasi-inertia to the intrinsic Newtonian inertia of the atom
itself moving through ideal space void of ether ; which, according
to the Newtonian law, would be exactly in proportion to the
amount of its gravitational quality. The additional quasi-inertia
must be exceedingly small in comparison with the Newtonian in-
ertia, as is demonstrated by the Newtonian proofe, including that
founded on Kepler s laws for the groups of atoms constituting the
planets, and movable bodies experimented on at the earth s surface.
In one thousand seconds of time after the density 0161 of the
density of water is reached, the inward surface velocity would be
305,000 kilometres per second, or greater than the velocity of
light ; and the whole surface of our condensing globe of gas or
vapour or crowd of atoms would begin to glow, shedding light
inwards And outwards. All this is absolutely realistic, except
the assumption of uniform distribution through a sphere of the
enormous radius of 309. 10*' kilometres, which we adopted tem-
porarily for illustrational purpose. The enormously great velocity
(291,000 kilometres per second) and rate of acceleration (13*7
kilometres per second per second) of the boundary inwards, which
we found at the instant of density 0161 of that of water, are due
to greatness of the primitive radius, and the uniformity of density
in the primitive distribution.

* See App. A, *'0n the Motion produced in an Infinite Elastic Solid by the
Motion through the Space occupied by it of a Body acting on it only by Attraction
and Bepulsion."

540 APPENDIX D.

To come to reality, according to the most probable judgment
present knowledge allows us to form, suppose at many millions,
or thousands of millions, or millions of millions of years ago, all
the matter in the universe to have been atoms very nearly at
rest* or quite at rest; more densely distributed in some places
than in others; of infinitely small average density through the
whole of infinite space. In regions where the density was then
greater than in neighbouring regions, the density would become
greater still ; in places of less density, the density would become
less ; and large regions would quickly become void or nearly void
of atoms. These large void regions would extend so as to com-
pletely surround regions of greater density. In some part or
parts of each cluster of atoms thus isolated, condensation would
go on by motions in all directions not generally convergent to
points, and w4th no perceptible mutual influence between the
atoms until the density becomes something like 10~* of our
ordinary atmospheric density, when mutual influence by collisions
would begin to become practically effective. E^h collision would
give rise to a train of waves in ether. These waves would carry
away energy, spreading it out through the void ether of infinite
space. The loss of energy, thus taken away from the atoms,
would reduce large condensing clusters to the condition of gas
in equilibrium t under the influence of its own gravity only, or
rotating like our sun or moving at moderate speeds as in spiral
nebulas, &c. Gravitational condensation would at first produce
rise of temperature, followed later by cooling and ultimately
freezing, giving solid bodies; collisions between which would
produce meteoric stones such as we see them. We cannot regard
as probable that these lumps of broken-looking solid matter
(something like the broken stones used on our macadamised
roads) are primitive forms in which matter was created. Hence
we are forced, in this twentieth century, to views regarding the
atomic origin of all things closely resembling those presented by
Democritus, Epicurus, and their majestic Roman poetic expositor,
Lucretius.

* *' On Mechanical Antecedents of Motion, Heat, and Light," Brit. Assoc, Rep.,
Part 2, 1854; Edin, New PhiLJoum. Vol. i. 1865; Comptes Reiidus, Vol. xl. 1856;
Kelvin's Collected Math, and Phys. Papers^ Vol. xi. Art. Izix.

t Homer Lane, American Journal of Science^ 1870, p. 67 ; Sir W. Thomson,
Phil, Mag, March 1867, p. 287.

APPENDIX E.

AEPINUS ATOMIZED*

§ 1. According to the well-known doctrine of Aepinus,
commonly referred to as the one-fluid theory of electricity, positive
and negative electrifications consist in excess above, and deficiency
below, a natural quantum of a fluid, called the electric fluid,
permeating among the atoms of ponderable matter. Portions of
matter void of the electric fluid repel one another ; portions of the
electric fluid repel one another ; portions of the electric fluid and
of void matter attract one another.

§ 2. My suggestion is that the Aepinus' fluid consists of
exceedingly minute equal and similar atoms, which I call elec-
trionsf , much smaller than the atoms of ponderable matter ; and

* From the Jubilee Volame presented to Prof. Bosoha in November, 1901.
t I ventured to suggest this name ii? a short article published in Nature,
May 27, 1S97, in which, after a slight reference to an old idea of a ** one-fluid
theory of electricity " with resinous electricity as the electric fluids the foUowing
expression of my views at that time occurs : '* I prefer to consider an atomic theory
'* of electricity foreseen as worthy of thought by Faraday and Clerk Maxwell, very
" definitely proposed by Helmholtz in his last lecture to the Boyal Institution, and
"largely accepted by present-day workers and teachers. Indeed Faraday's law of
" electro-chemical equivalence seems to necessitate something atomic in electricity,
''and to justify the very modern name electron [given I believe originaUy by
Johnstone Stoney, and now largely used to denote an atom of either vitreous or
resinous electricity]. The older, and at present even more popular, name ion
''given Bixty years ago by Faraday, suggests a convenient modification of it; elec-
" trion, to denote an atom of resinous electricity. And now, adopting the essentials
" of Aepinus' theory, and dealing with it according to the doctrine of Father Bos-
"covich, each atom of ponderable matter is an electron of vitreous electricity;
" wLich, with a neutralizing electron of resinous electricity close to it, produces a
" resulting force on every distant electron and electrion which varies inversely as the
cube of the distance, and is in the direction determined by the well-known re-
quisite application of the parallelogram of forces.'' It will be seen that I had
not then thought of the hypothesis suggested in the present communication, that
while electrions permeate freely through all space, whether occupied only by ether
or occupied also by the volumes of finite spheres constituting the atoms of pon-
derable matter, each electrion in the interior of an atom of ponderable matter ex-
periences electric force towards the centre of the atom, just as if the atom contained
within it, fixed relatively to itself, a uniform distribution of ideal electric matter.

542 APPENDIX E.

that they permeate freely through the spaces occupied by these
greater atoms and also freely through space not occupied by them.
As in Aepinus* theory we must have repulsions between the
electrions ; and repulsions between the atoms independently of the
electrions ; and attractions between electrions and atoms without
electrions. For brevity, in ftiture by atom I shall mean an atom of
ponderable matter, whether it has any electrions within it or not.

§ 3. In virtue of the discovery and experimental proof by
Cavendish and Coulomb of the law of inverse square of distance
for both electric attractions and repulsions, we may now suppose
that the atoms, which I assume to be all of them spherical, repel
other atoms outside them with forces inversely as the squares of
distances between centres ; and that the same is true of electrions,
which no doubt occupy finite spaces, although at present we are
dealing with them as if they were mere mathematical points,
endowed with the property of electric attraction and repulsion.
We must now also assume that every atom attracts every electrion
outside it with a force inversely as the square of the distance
between centres.

§ 4. My assumption that the electrions freely permeate the
space occupied by the atoms requires a knowledge of the law of
the force experienced by an electrion within an atom. As a tenta-
tive hypothesis, I assume for simplicity that the attraction ex-
perienced by an electrion approaching an atom varies exactly
according to the inverse square of the distance from the centre,
as long as the electrion is outside ; has no abrupt change when
the electrion enters the atom ; but decreases to zero simply as the
distance from the centre when the electrion, approaching the
centre, is within the spherical boundary of the atom. This is
just as it would be if the electric virtue of the atom were due
to uniform distribution through the atom of an ideal electric
substance of which each infinitely small part repels infinitely
small portions of the ideal substance in other atoms, and attracts
electrions, according to the inverse square of the distance. But
we cannot make the corresponding supposition for the mutual
force between two overlapping atoms ; because we must keep our-
selves free to add a repulsion or attraction according to any law of
force, that we may find convenient for the explanation of electric,
elastic, and chemical properties of matter.

VITREOUS RESINOUS INSTEAD OF POSITIVE NEGATIVE. 543

§ 6. The neutralizing quantum of electrions for any atom or
group of atoms has exactly the same quantity of electricity of one
kind as the atom or group of atoms has of electricity of the
opposite kind. The quantum for any single atom may be one or
two or three or any integral number, and need not be the same
for all atoms. The designations mono-electrionic, dielectrionic,
trielectrionic, tetraelectrionic, polyelectrionic, etc., will accordingly
be convenient. It is possible that the differences of quality of
the atoms of diflferent substances may be partially due to the
quantum-numbers of -their electrions being different ; but it is
possible that the diflferences of quality are to be wholly explained
in merely Boscovichian fashion by diflFerences in the laws of force
between the atoms, and may not imply any differences in the
numbers of electrions constituting their quantums.

§ 6. Another possibility to be kept in view is that the
neutralizing quantum for an atom may not be any integral number
of electrions. Thus for example the molecule of a diatomic gas,
oxygen, or nitrogen, or hydrogen, or chlorine, might conceivably
have three electrions or some odd number of electrions for its
quantum so that the single atoms, O, N, H, CI, if they could exist
separately, must be either vitreously or resinously electrified and
cannot be neutral.

§ 7. The present usage of the designations, positive and
negative, for the two modes of electrification originated no doubt
with the use of glass globes or cylinders in ordinary electric
machines giving vitreous electricity to the insulated prime con-
ductor, and resinous electricity to the not always insulated rubber.
Thus Aepinus and his followers regarded the prime conductors of
their machines as* giving the true electric fluid, and leaving a
deficiency of it in the rubbers to be supplied from the earth. It
is curious, in Beccaria's account of his observations made about
1760 at Garzegna in Piedmont on atmospheric electricity, to read
of " The mild excessive electricity of the air in fair weather."
This in more modem usage would be called mild positive
electricity. The meaning of either expression, stated in non-
hypothetical language, is, the mild vitreous electricity of the air
in fair weather.

§ 8. In the mathematical theory of electricity in equilibrium,
it is a matter of perfect indifference which of the opposite electric

544 APPENDIX E.

manifestations we call positive and which negative. But the
gi'eat diflFerences in the disruptive and luminous effects, when the
forces are too strong for electric equilibrium, presented by the two
modes of electrification, which have been known from the earliest
times of electric science, show physical properties not touched by
the mathematical theory. And Varley's comparatively recent
discovery* of the molecular torrent of resinously electrified
particles from the "kathode" or resinous electrode in apparatus
for the transmission 'of electricity through vacuum or highly
rarefied air, gives strong reason for believing that the mobile
electricity of Aepinus* theory is resinous, and not vitreous as he
accidentally made it. I shall therefore assume that our electrions
act as extremely minute particles of resinously electrified matter ;
that a void atom acts simply as a little globe of atomic substance,
possessing as an essential quality vitreous electricity uniformly
distributed through it or through a smaller concentric globe ; and
that ordinary ponderable matter, not electrified, consists of a vast
assemblage of atoms, not void, but having within the portions of
space which they occupy, just enough of electrions to annul electric
force for all places of which the distance from the nearest atom is
large in comparison with the diameter of an atom, or molecular
cluster of atoms.

§ 9. This condition respecting distance would, because of the

Fig. 1. Fig. 2.

Badii 8 and 1
C'C= 2-7 C'E'=1458 0^=0462

inverse square of the distance law for the forces, be unnecessary
and the electric force would be rigorously null throughout all

♦ Proc, Roy. Soc. Vol. xix. 1871, pp. 289, 240.

OVERLAPPING MONO-ELECTRIONIC ATOMS. 545

space outside the atoms, if every atom had only a single electrion
at its centre, provided that the electric quantities of the opposite
electricities (reckoned according to the old definition of mathe-
matical electrostatics) are equal in the atom and in the electrion.
But even if every neutralized separate atom contains just one
electrion in stable equilibrium at its centre, it is obvious that,
when two atoms overlap so far that the centre of one of them is
within the spherical boundary of the other, the previous equi-
librium of the two electrions is upset, and they must find positions
of equilibrium elsewhere than at the centres. Thus in fig. 1 each
electrion is at the centre of its atom, and is attracted and repelled
with equal forces by the neighbouring atom and electrion at its
centre. In fig. 2, if E and E' were at the centres C, C\ of the
two atoms, E would be repelled by E' more than it would be
attracted by the atom A\ Hence both electrions being supposed
free, E will move to the right; and because of its diminished
repulsion on E\ E' will follow it in the same direction. The
equations of equilibrium of the two are easily written down, not
so easily solved without some slight arithmetical artifice. The
solution is correctly shown in fig. 2, for the case in which one
radius is three times the other, and the distance between the
centres is 2*7 times the smaller radius*. The investigation in

* Calling e the quantity of electricity, vitreous or resinous, in each atom or
electrion : ^ the distance between the centres of the atoms ; a, a' the radii of the
two atoms ; Xy x' the displacements of the electrions from the centres ; X, X' the
forces experienced by the electrions; we have

r-..r-- + ^^ - 1

Each of these being equated to zero for equilibrium gives us two equations which
are not easily dealt with by frontal attack for the determination of two unknown
quantities x, x'\ but which may be solved by a method of successive approximations,
as follows:— Let x^, Xj,... x<, x©', x/, ... x/, be successive approximations to the
values of x and x\ and take

where I><' = (f+x<-x/)*. As an example, take a=l, a'=3. To find solutions for
gradual approach between centres, take successively ^=2*9, 2*8, 2-7, 2*6. Begin
with Xo=0, x<,'=0, we find X4= -01243, X4'=0297, and the same values for Xj,
aud x„'. Take next f=2-8, Xo = -01243, Xo' = -0297; we find X4=X5 = -0269,
j-^' = .r5' = '()702. Thus we have the solution for the second distance between

T. L. 35

646

APPENDIX E.

the footnote shows that if the atoms are brought a little nearer,
the equilibrium becomes unstable; and we may infer that both
electrions jump to the right, E' to settle at a point within the
atom A on the left-hand side of its centre ; and E outside A\ to

settle at a point still within A. If,
lastly, we bring the centres closer
and closer together till they coincide,
E comes again within A'y and the
two electrions settle, as shown in
fig. 3 at distances on the two sides of
the common centre, each equal to

1 y 2
2V T

Fig. 3.
£C'=C£ = -622

which for the case a = 3a is

1 72-27 _
2"V 28 "'

622«.

§ 10. Mutual action of this kind might probably be presented
in such binary combinations as Oj, Nj, Hj, Cl^, CO, SO, NaCl (dry
common salt) if each single atom, O, N, H, CI, C*, S, Naf, had
just one electrion for its neutralizing quantum. If the combina-
tion is so close that the centres coincide, the two electrions will
rest stably at equal distances on the two sides of the common
centre as at the end of § 9. I see at present no reason for con-
sidering it excessively improbable that this may be the case for
SO, or for any other binary combinations of two atoms of different
quality for neither of which there is reason to believe that its
neutralizing quantum is not exactly one electrion. But for the
binary combinations of two atoms of identical quality which the

centres. Next take f=2-7, Xo = -0269, Xo' = *0702; we find x, = Xy = -0462,
x^=x^'=*\4l^* Working similarly for ^=2*6, we do not find convergence, and
we infer that a position of unstable equilibrium is reached by the electrions for
some value of ^ between 2*7 and 2'6.

* The complexity of the hydrocarbons and the Van H Hofl and Le Bel doctrine
of the asymmetric results (chirality) produced by the quadrivalence of carbon
makes it probable that the carbon atom takes at least four electrions to neotralize
it electrically.

t The fact that sodium, solid or liquid, is a metallic conductor of electricity
makes it probable that the sodium atom, as all other metallic elements, takes
a large number of electrions to neutralize it. (See below § SO.)

FRICTION AL AND SEPARATION AL ELECTRICITY. 547

chemists have discovered in diatomic gases (Og, Nj, etc.) there
mast, over and above the electric repulsion of the two similar
electric globes, be a strong atomic repulsion preventing stable
equilibrium with coincident centres, however strongly the atoms
may be drawn together by the attractions of a pair of mutually
repellent electrions within them ; because without such a repulsion
the two similar atoms would become one, which no possible action
in nature could split into two.

§ 11. Returning to § 9, let us pull the two atoms gradually
asunder from the concentric position to which we had brought
them. It is easily seen that they will both remain within the
smaller atom A, slightly disturbed from equality of distance on
the two sides of its centre by attractions towards the centre of -4';
and that when A' is infinitely distant they will settle at distances
each equal to Ja \/2 = •62996a on the two sides of (7, the centre
of A. If, instead of two mono-electrionic atoms, we deal as in
§ 9 with two polyelectrionic atoms, we find after separation the
number of electrions in the smaller atom increased and in the
larger decreased ; and this with much smaller difference of magni-
tude than the three to one of diameters which we had for our
mono-electrionic atoms of § 9. This is a very remarkable conclusion,
pointing to what is probably the true explanation of the first
known of the electric properties of matter; attractions and
repulsions produced by rubbed amber. Two ideal solids consisting
of assemblages of mono-electrionic atoms of largely different sizes
would certainly, when pressed and rubbed together and separated,
show the properties of oppositely electrified bodies; and the
preponderance of the electrionic quality would be in the assem-
blage of which the atoms are the smaller. Assuming as we do
that the electricity of the electrions is of the resinous kind, we
say that after pressing and rubbing together and separating the
two assemblages, the assemblage of the smaller atoms is resinously
electrified and the assemblage of the larger atoms is vitreously
electrified. This is probably the true explanation of the old-
known fact that ground glass is resinous relatively to polished
glass. The process of polishing might be expected to smooth down
the smaller atoms, and to leave the larger atoms more effective in
the surface.

§ 12. It probably contains also the principle of the explana-

35—^

646

APPENDIX E.

the footnote shows that if the atoms are brought a little nearer,
the equilibrium becomes unstable; and we may infer that both
electrions jump to the right, E' to settle at a point within the
atom A on the left-hand side of its centre ; and E outside A\ to

settle at a point still within A, If,
lastly, we bring the centres closer
and closer together till they coincide,
E comes again within A\ and the
two electrions settle, as shown in
fig. 3 at distances on the two sides of
the common centre, each equal to

1 y 2

2V T

Fig. 8.
£C=C£=-622

which for the case a' = 3a is

1 y 2-27
2*V 28 ~*

622a.

§ 10. Mutual action of this kind might probably be presented
in such binary combinations as O,, Ng, Ha, CI,, CO, SO, NaCl (dry
common salt) if each single atom, O, N, H, CI, C*, S, Naf, had
just one electrion for its neutralizing quantum. If the combina-
tion is so close that the centres coincide, the two electrions will
rest stably at equal distances on the two sides of the common
centre as at the end of § 9. I see at present no reason for con-
sidering it excessively improbable that this may be the case for
SO, or for any other binary combinations of two atoms of different
quality for neither of which there is reason to believe that its
neutralizing quantum is not exactly one electrion. But for the
binary combinations of two atoms of identical quality which the

centres. Next take f=2-7, iro = -0269, Xo' = '0702; we find x, = Xy = -0462,
x^'=Xj='li5S, Working similarly for ^=2*6, we do not find convergence, and
we infer that a position of unstable equilibrium is reached by the electrions for
some value of ^ between 2*7 and 2*6.

* The complexity of the hydrocarbons and the Van 't Hofl and Le Bel doctrine
of the asymmetric results (chirality) produced by the quadrivalence of carbon
makes it probable that the carbon atom takes at least four electrions to neutralize
it electrically.

t The fact that sodium, solid or liquid, is a metallic conductor of electricity
makes it probable that the sodium atom, as all other metallic elements, takes
a large number of electrions to neutralize it. (See below § SO.)

FRICTIONAL AND SEPARATIONAL ELECTRICITY. 547

chemists have discovered in diatomic gases (0„ N„ etc.) there
mast, over and above the electric repulsion of the two similar
electric globes, be a strong atomic repulsion preventing stable
equilibrium with coincident centres, however strongly the atoms
may be drawn together by the attractions of a pair of mutually
repellent electrions within them ; because without such a repulsion
the two similar atoms would become one, which no possible action
in nature could split into two.

§ 11. Returning to § 9, let us pull the two atoms gradually
asunder from the concentric position to which we had brought
them. It is easily seen that they will both remain within the
smaller atom A, slightly disturbed from equality of distance on
the two sides of its centre by attractions towards the centre of A';
and that when A' is infinitely distant they will settle at distances
each equal to Ja \/2 = '629960 on the two sides of C, the centre
of A, If, instead of two mono-electrionic atoms, we deal as in
§ 9 with two polyelectrionic atoms, we find after separation the
number of electrions in the smaller atom increased and in the
larger decreased ; and this with much smaller difference of magni-
tude than the three to one of diameters which we had for our
mono-electrionic atoms of § 9. This is a very remarkable conclusion,
pointing to what is probably the true explanation of the first
known of the electric properties of matter; attractions and
repulsions produced by rubbed amber. Two ideal solids consisting
of assemblages of mono-electrionic atoms of largely different sizes
would certainly, when pressed and rubbed together and separated,
show the properties of oppositely electrified bodies; and the
preponderance of the electrionic quality would be in the assem-
blage of which the atoms are the smaller. Assuming as we do
that the electricity of the electrions is of the resinous kind, we
say that after pressing and rubbing together and separating the
two assemblages, the assemblage of the smaller atoms is resinously
electrified and the assemblage of the larger atoms is vitreously
electrified. This is probably the true explanation of the old-
known fact that ground glass is resinous relatively to polished
glass. The process of polishing might be expected to smooth down
the smaller atoms, and to leave the larger atoms more eflfective in
the surface.

§ 12. It probably contains also the principle of the explana-

35—2

548

APPENDIX E.

tion of Erskine Murray's* experimental discovery that surfaces of
metals, well cleaned by rubbing with glass-paper or emery-paper,
become more positive or less negative in the Volta contact
electricity scale by being burnished with a smooth round hard
steel burnisher. Thus a zinc plate brightened by rubbing on
glass-paper rose by '23 volt by repeated burnishing with a hard
steel burnisher, and fell again by the same difference when rubbed
again with glass-paper. Copper plates showed differences of
about the same amount and in the same direction when similarly
treated. Between highly burnished zinc and emeiy-cleaned
copper, Murray found a Volta-diflFerence of 1'13 volts, which is,
I believe, considerably greater than the greatest previously found
Yolta-difference between pure metallic surfaces of zinc and
copper.

§ 13. To further illustrate the tendency (§ 9) of the smaller
atom to take electrions from the larger, consider two atoms ; A\
o{ radius a\ the greater, having an electrion in it to begin with ;
and A, radius a, the smaller, void.

By ideal forces applied to the atoms while the electrion is free
let them approach gradually from a very great distance apart.
The attraction of A draws the electrion from the centre of A'; at
first very slightly, but farther and farther as the distance between
the atoms is diminished. What will be the position of the

Fig. 4.

= 1 C'C=2 C'JE;=-38

I

Fig. 5.
CC= 1-89 C'JE;=0-6S

electrion when the distance between the centres is, as in fig. 4,
2a'? Without calculation we see that the electrion would be in
ec^uilibrium if placed at the point in which the surface of A' is
cut by the line of centres ; but the equilibrium would be obviously

♦ ♦• On Contact Electricity of Metals,** Proc. Roy, Soc, Vol. Lxra. 1898, p. 118.
See alao Lord Kelvin, '* Contact Electricity of Metals," Phil. Mag. Vol. xlvi. 1898,
pp.'96— 98.

SMALLER ATOM TAKES ELECTRION FROM LARGER. 549

unstable, and a simple calculation* shows that the stable position
actually taken by the electrion is SSoi from C\ when the distance
between the centres is 2a' (fig. 4). If the distance between the
centres is now diminished from 2a' to l*89a' (a being now
supposed to be anything less than 'SQa') the electrion comes
gradually to distance 'GSa' from C" (fig. 5) ; its equilibrium there
becomes unstable ; and it jumps out of A' towards A (like a cork
jumping out of a bottle). It will shoot through A (A' and A
being held fixed); and after several oscillations to and fro, perhaps'f
ten or twenty [or perhaps a million (K., Aug. 9, 1903)], if it has
only quasi-inertia due to condensation or rarefaction^ produced by
it in ether ; or perhaps many times more if it has intrinsic inertia
of its own ; it will settle, with decreasing range of excursions,
sensibly to rest within A, attra<:ted somewhat from the centre
by A\ If, lastly, A' and A be drawn asunder to their original
great distance, the electrion will not regain its original position
in A\ but will come to the centre of A and rest there. Here
then we have another illustration of the tendency found in § 9,
of the smaller atom to take electrions from the larger.

§ 14. In preventing the two atoms from rushing together
by holding them against the attractive force of the electrion, we
shall have gained more work during the approach than we after-
wards spent on the separation ; and we have now left the system

* Denoting by ^ the distance between the centres, and by X the force on E when
its distance from C is x\ we have

^~^lW^)*~ a-*}'

Hence for eqailibrium jr-— ,:^ = -^ . This is a cubic for x' of which the proper

root (the smallest root) for the case ^=2a is 'd8a\ The formula for X has
a minimum value when ^-x'=a'4^2, which makes

-SG^'-g-

Hence the value of x' for equilibrium coincides with the value of X, a minimum,

and the equilibrium becomes unstable, when ^ is diminished to — ^ a' = l*890a'.

V2
For this, the value of x' is ^ a' = -63a'.

t ** On the Production of Wave Motion in an Elastic Solid," Phil, Mag. Oct.
isyg, § 44.

:;: " On the Motion of Ponderable Matter through Space occupied by Ether,'*
Phil. Mag. Aug. 1900, §§ 15, 17. [Reproduced in the present Volume as App. A.]

550 APPENDIX E.

deprived of the farther amount of energy carried away by etherial
waves into space.

§ 15. The system in its final state with the electrion at the
centre of the smaller atom has less potential energy in it than it
had at the beginning (when the electrion was at the centre of 4'),
by a difference equal to the excess of the work which we gained
during the approach above that which we spent on the final
separation of A' and A^ plus the amount carried away by the
etherial waves. All these items except the last are easily calcu-
lated from the algebra of the footnote on § 13 ; and thus we find
how much is our loss of energy by the etherial waves.

§ 16. Very interesting statical problems are presented to us
by consideration of the equilibrium of two or more electrions
within one atom, whether a polyelectrionic atom with its saturat-
ing number, or an atom of any electric strength with any number
of electrions up to the greatest number that it can hold. To help
to clear our ideas, first remark that if the number of electrions is
infinite, that is to say if we go back to Aepinus* electric fluid, but
assume it to permeate freely through an atom of any shape
whatever and having any arbitrarily given distribution of elec-
tricity of the opposite kind fixed within it, the gi-eatest quantity
of fluid which it can take is exactly equal to its own, and lodges
with density equal to its own in every part. Hence if the atom
is spherical, and of equal electric density throughout as we have
supposed it, and if its neutralizing quantum of electrions is a very
large number, their configuration of equilibrium will be an
assemblage of more and more nearly uniform density from surface
to centre, the greater the number. Any Bravais homogeneous
assemblage whatever would be very neai-ly in equilibrium if all
the electrions in a surface-layer of thickness a hundred times the
shortest distance from electrion to electrion were held fixed ; but
the equilibrium would be unstable except in certain cases. It
may seem probable that it is stable if the homogeneous assemblage
is of the species which I have called* equilateral, being that in
which each electrion with any two of its twelve next neighbours
forms an equilateral triangle. If now all the electrions in the

* Afolecular Tactics of a Crystal, § 4, being the second Robert Boyle Lecture,
delivered before the Oxford University Junior Scientific Club, May 16, 1893
(Clarendon Press, Oxford). [Reproduced in the present Volume as App. H.]

STABLE EQUILIBRIUM OF SEVERAL ELECTRIONS IN AN ATOM. 551

surface layer are left perfectly free, a slight rearrangement among
themselves and still slighter among the neighbouring electrions
in the interior will bring the whole multitude (of thousands or
millions) to equilibrium. The subject is of extreme interest,
geometrical, dynamical and physical, but cannot be pursued
further at present.

§ 17. To guide our ideas respecting the stable equilibrium of
moderate numbers of electrions within an atom, remark first that
for any number of electrions there may be equilibrium with all
the electrions on one spherical surface concentric with the atom.
To prove this, discard for a moment the atom and imagine the
electrions, whatever their number, to be attached to ends of equal
inextensible strings of which the other ends are fixed to one
point C. Every string will be stretched in virtue of the mutual
repulsions of the electrions ; and there will be a configuration or
configurations of equilibrium with the electrions on a spherical
surface. Whatever their number there is essentially at least one
configuration of stable equilibrium. Remark also that there is
always a configuration of equilibrium in which all the strings are
in one plane, and the electrions are equally spaced round one
great circle of the sphere. This is the sole configuration for two
electrions or for three electrions : but for any number exceeding
three it is easily proved to be unstable, and is therefore not the
sole configuration of equilibrium. For four electrions it is easily
seen that, besides the unstable equilibrium in one plane, there is
only one stable configuration, and in this the four electrions are at
the four comers of an equilateral tetrahedron.

§ 18. For five electrions we have clearly stable equilibrium
with three of them in one plane through (7, and the other two at
equal distances in the line through C perpendicular to this plane.
There is also at least one other configuration of equilibrium : this
we see by imagining four of the electrions constrained to remain
in a freely movable plane, which gives stable equilibrium
with this plane at some distance from the centre and the fifth
electrion in the diameter perpendicular to it. And similarly
for any greater number of electrions, we find a configuration of
equilibrium by imagining all but one of them to be constrained
to remain in a freely movable plane. But it is not easy, without
calculation, to see, at all events for the case of only five electrions.

552 APPENDIX K.

whether that equilibrium would be stable if the constraint of all
of them but one, to one plane is annulled. For numbers greater
than five it seems certain that that equilibrium is unstable.

§ 19. For six we have a configuration of stable equilibrium
with the electrions at the six comers of a regular octahedron ;
for eight at the comers of a cube. For ten, as for any even
number, we should have two configurations of equilibrium (both
certainly unstable for large numbers) with two halves of the
number in two planes at equal distances on the two sides of the
centre. For twelve we have a configuration of stable equilibrium
with the electrions at positions of the twelve nearest neighbours
to (7 in an equilateral homogeneous assemblage of points* : for
twenty at the twenty comers of a pentagonal dodecahedron. All
these configurations of § 19 except those for ten electrions are
stable if, as we are now supposing, the electrions are constrained
to a spherical surface on which they are free to move.

§ 20. Except the cases of § 18, the forces with which the
strings are stretched are the same for all the electrions of each
case. Hence if we now discard the strings and place the electrions .
in an atom on a spherical surface concentric with it, its attraction
on the electrions towards the centre takes the place of the tension
of the string, provided it is of the proper amount. But it does
not secure, as did the strings, against instability relatively to
radial displacements, different for the different electrions. To
secure the proper amount of the radial force the condition is

— T- = T; where i denotes the number of electrions; e the electric

Or

quantity on each (and therefore, § 8, ie the electric quantity of
vitreous electricity in the atom); r denotes the radius of the
spherical surface on which the electrions lie : a the radius of the
atom; and T the tension of the string in the arrangement of

§ 17. We have generally 7= g— where g is a numeric depending

on the number and configuration of the electrions found in each

case by geometry. Hence we have ~ = a/ • f'or the ratio of the

radius of the smaller sphere on which the electrions lie to the
radius of the atom. For example, take the case of eight electrions

♦ App. H, " Molecular Tactics of a Crystal," § 4.

EXHAUSTION OF ENERGY IN STABLE GROUPS OF ELECTRIONS. 553

at the eight coraers of a cube. T is the resultant of seven
repulsions, and we easily find j = } ( V3 + Vf + i) * ^J^d finally

T

- = '6756. Dealing similarly with the cases of two, three, four,
and six electrions, we have the following table of values of
( - j and - ; to which is added a last column showing values of

i* -^"^ ^ 7) » ^^^°S 2 ^^ *^® work required to remove the

electrions to infinite distance. D is the distance between any two
of the electrions.

^xwork

Number

of
Electriona

Conflguration

(i)'

r

required to r»-
more the elec-
triona to infi-
nite dietanoe

1

At the centre

0

0

1-600

2

At quarter points from the
ends of a diameter

1

8

•5000

4-600

3

At the comers of an equi-
lateral triangle

1
3^8

•6774

9-000

4

At the comers of a square

V2 1
8 ^16

•6208

14-760

4

At the comers of an equi-
lateral tetrahedron

3 /3
16 V 2

•6124

15 000

6

At the comers of an equi-
lateral octahedron

1 + 4^2
24

•6522

33-336

8

At the comers of a cube

3l(^/«VI

*1)

•6766

62180

§ 21. In the configurations thus expressed the equilibrium is
certainly stable for the cases of two electrions, three electrions,
and four electrions at the comers of a tetrahedron. It seems to me,
without calculation, also probably stable for the case of six, and
possibly even for the case of eight. For the case of twenty at the
comers of a pentagonal dodecahedron the equilibrium is probably
not stable ; and even for the cases of twelve electrions and ten
electrions, the equilibrium in the configurations described in §§ 18,
19 may probably be unstable, when, as now, we have the attraction
of the atom towards the centre instead of the inextensible strings.

§ 22. In fact when the number of electrions exceeds four,

5S4 Aprcn>ix kl

ve most think of the tendency to be aowded oot of ooe spbericml
WKohee, vhich with renr harge inunben gives m tgndencr to
umfarm distribotion throogfaoat the Tcdome of the atom as
described in § 16 abore. Thus, in the case of fire dectnons.
§ IS shows a oonfigniatioD of eqnilitHinm in which the two
electrioDs lying in ooe diameter are, bj the motnal repolsioos;
pushed veiT sligfatlj fai%beT from the centre than are the three
in the equatorial plane In this case the eqmlibrnim is dearlj
stableL Another obvioos oonfigoration, also stable, of fire elec-
trioDS within an atom is one at the centre, and four on a ooooentric
qiherical sor&oe at the comers, of a tetrahednML From any case
of any number of electricms aD on ooe spherical saAce, we may
pass to another omfiguration with one more electrion placed at
the centre and the proper proportionate increase in the electric
strength of the atom. Thus from the cases described in § 19, we
may pass to configurations of equilibrium for seven, nine, eleTen,
thirteen, and twenty-one electrions. All these cases, with questions
of stability or instabilitr and of the different amounts of work
required to pluck all the electrions out of the atom and remove
them to infinite distances, present most interesting subjects for
not diflicult mathematical wcnk: ; and I regret not being able to
pursue them at present

I 23. Consider now the electric properties of a real body,
gaseous, liquid, or solid, constituted by an assemblage of atoms
with their electrions. It follows immediately from our hypothesis,
that in a monatomic gas or in any sufficiently sparse assemblage
of single atoms, fixed or moving, Faraday s " conducting power /or
lines of electric force," or what is now commonly called the specific
dectro-inductive capacity, or the electro-inductive permeability,
exceeds unity by three times the ratio of the sum of the volumes of
the atoms to the whole volume of spcce occupied by the assemblage,
whether the atoms be mono-electrionic or polyelectrionic, and how-
ever much the electrion, or group of electrions, within each atom
is set to vibrate or rotate with each collision, according to the
kinetic theory of gases. To prove this, consider, in a unifinm
field of electrostatic force of intensity F, a single atop of radius a;
and, at rest within it, a group of t electrions in stable equi-
librium. The action of F produces simply displacements of the
electrions relatively to the atom, equal and in parallel lines, with
therefore no change of shape and no rotation ; and, x denoting the

ELECTRO-INDUCTIVE CAPACITY. 556

amount of this displacement, the equation for the equilibrium of

each electrion is — j- = F, This gives iex = a^F for the electric

moment of the electrostatic polarization induced in the atom by F,
In passing, remark that o^F \& also equal to the electric moment
of the polarization produced in an insulated unelectrified metal
globe of radius a, when brought into an electrostatic field of
intensity F: and conclude that the electric inductive capacity of
a uniformly dense assemblage of fixed metallic globules, so sparse
that their mutual influence is negligible, is the same as that of
an equal and similar assemblage of our hypothetical atoms, what-
ever be the number of electrions in each, not necessarily the same
in all. Hence our hypothetical atom realizes perfectly for sparse
assemblages Faraday's suggestion of '' small globular conductors,
as shot" to explain the electro-polarization which he discovered in
solid and liquid insulators. {Experimental Researches^ § 1679.)

§ 24. Denoting now by N the number of atoms per unit
volume we find NVa^F as the electric moment of any sparse
enough assemblage of uniform density occupying volume V in
a uniform electric field of intensity F, Hence No^ is what
(following the analogy of electro-magnetic nomenclature) we may
call the electro-inductive susceptibility* of the assemblage; being
the electric moment per unit bulk induced by an electric field of
unit intensity. Denoting this by /a, and the electro-inductive
permeability by cw we have [Electrostatics and Magnetism, § 629,
(14)],

Q) = 1 +47r/i= 1 + 3 (iV*^"\

which proves the proposition statcii at the commencement of § 23.

§ 25. To include vibrating and rotating groups of electrions
in the demonstration, it is only necessary to remark that the time-
average of any component of the displacement of the centre of
inertia of the group relatively to the centre of the atom will, under
the influence of Fy be the same as if the assemblage were at rest
in stable equilibrium.

«

§ 26. The consideration of liquids consisting of closely packed
mobile assemblages of atoms or groups of atoms with their elec-

* Suggested in my Electrostatics and MagnetUm, §§ 62S, 629.

556 APPENDIX E.

trions, forming compjund molecules, as in liquid argon or helium
(monatomic), nitrogen, oxygen, etc. (diatomic), or pure water, or
water with salts or other chemical substances dissolved in it,
or liquids of various complex chemical constitutions, cannot be
entered on in the present communication, further than to remark
that the suppositions we have made regarding forces, electric and
other, between electrions and atoms, seem to open the way to a
very definite detailed dynamics of electrolysis, of chemical aflSnity,
and of heat of chemical combination. Estimates of the actual
magnitudes concerned (the number of molecules per cubic centi-
metre of a gas, the mass in grammes of an atom of any substance,
the diameters of the atoms, the absolute value of the electric
quantity in an electrion, the eflFective mass or inertia of an elec-
trion) seem to show that the intermolecular electric forces are
more than amply great enough to account for heat of chemical
combination, and every mechanical action manifested in chemical
interactions of all kinds. We might be tempted to assume that
all chemical action is electric, and that all varieties of chemical
substance are to be explained by the numbers of the electrions
required to neutralize an atom or a set of atoms (§ 6 above) ; but
we can feel no satisfaction in this idea when we consider the great
and wild variety of quality and affinities manifested by the
different substances or the different "chemical elements"; and as
we are assuming the electrions to be all alike, we must fall back
on Father Boscovich, and require him to explain the difference of
quality of different chemical substances, by different laws of force
between the different atoma

§ 27. Consider lastly a solid ; that is to say, an assemblage in
which the atoms have no relative motions, except through ranges
small in comparison with the shortest distances between their
centres*. The first thing that we remark is that every solid
would, at zero of absolute temperature (that is to say all its atoms
and electrions at rest), be a perfect insulator of electricity under
the influence of electric forces, moderate enough not to pluck
electrions out of the atoms in which they rest stably when there
is no disturbing force. The limiting value of F here indicated

* I need scarcely say that it is only for simplicity in the text that we con-
veniently ignore Roberts- Austen's admirable discovery of the interdiffosion of solid
gold and solid lead, found after a piece of one metal is allowed to rest on a piece of
the other for several weeks, months, or years.

RADIO-ACTIVITY EXPLAINED BY ELECTRIONS SHOT OUT. 567

for perfect insulation, I shall for brevity call the disruptional
force or disruptional intensity. It is clear that this disruptional
force is smaller the greater the number of electrions within an
atom.

§ 28. The electro-inductive permeability of a solid at zero
temperature is calculable by the static dynamics of § 24, modified
by taking into account forces on the electrions of one atom due to
the attractions of neighbouring atoms and the repulsions of their
electrions. Without much calculation it is easy to see that
generally the excess of the electro-inductive permeability above
unity will be much greater than three times the sum of the
volumes of the electric atoms per unit volume of space, which we
found in § 24 for the electro-inductive permeability of an assem-
blage of single atoms, sparse enough to produce no disturbance
by mutual actions. Also without much calculation, it is easy to
see that now the induced electric moment will not be in simple
proportion to F, the intensity of the electric field, as it was
rigorously for a single atom through the whole range up to the
disruptional value of F\ but will tend to increase more than in
simple proportion to the value of F\ though for small practical
values of F the law of simple proportion is still very nearly
fulfilled.

§ 29. Raise the temperature now to anything under that at
which the solid would melt. This sets the electrions to per-
forming wildly irregular vibrations, so that some of them will
occasionally be shot out of their atoms. Each electrion thus shot
out will quickly either fall back into the atom from which it
has been ejected, or will find its way into another atom. If the
body be in an electric field F, a considerable proportion of the
electrions which are shot out will find their way into other atoms
in the direction in which they are pulled by F\ that is to say,
the body which was an infinitely perfect insulator at zero absolute
temperature has now some degree of electric conductivity, which
is greater the higher the temperature. There can be no doubt
that this is a matter-of-fact explanation of the electric con-
ductivity, which so nearly perfect an insulator as the flint glass
of my quadrant electrometer at atmospheric temperatures shows,
when heated to far below its melting point (according to Professor
T. Gray*, 98 . 10-»* at 60'' Cent.; 49 . lO"** at 100°; 8300 . lO""^ at

* Proc. R, S, Vol. xzxiY. Jan. 12, 1S82. The figares of the text are in o. o. a.

558 APPENDIX E.

200** Cent.). It explains also the enormous increase of electric
conductivity of rare earths at rising temperatures above 800° C,
80 admirably taken advantage of by Professor Nemst in his now
celebrated electric lamp.

§ 30. If the hypotheses suggested in the present communica-
tion are true, the electric conductivity of metals must be explained
in the same way as that of glass, gutta-percha, vulcanite, Nemst
filament, etc., with only this difference, that the metallic atom
must be so crowded with electrions that some of them are always
being spilt out of each atom by the intermolecular and electrionic
thermal motions, not only at ordinary atmospheric temperatures,
and higher, but even at temperatures of less than 16° Centigi-ade
above the absolute zero of temperature. I say 16° because in
Dewar's Bakerian Lecture to the Royal Society of London, June
13, 1901, The Nadir of Temperature, we find that platinum, gold,
silver, copper and iron have exceedingly high electric conductivity
at the temperature of liquid hydrogen boiling under 30 mms. of
mercury, which must be something between 20°'5, the boiling
point of hydrogen at 760 mms. pressure, and 16°, the temperature
of melting solid hydrogen, both determined by Dewar with his
helium thermometer. There is no difficulty in believing that the
electrions in each of the metallic atoms are so numerous that
though they rest in stable equilibrium within the atoms, closely
packed to constitute the solid metal at 0° absolute, and may move
about within the atom with their wildly irregular thermal motions
at 1° of absolute temperature, they may between 1° and 2° begin
to spill from atom to atom. Thus, like glass or a Nemst filament
below 300° absolute, a metal may be an almost perfect insulator
of electricity below 1° absolute: may, like glass at 333° absolute,
show very notable conductivity at 2° absolute : and, like glass at
473° absolute as compared with glass at 333° absolute, may show
8000 times as much electric conductivity at 2°'8 as at 2°. And,
like the Nemst filament at 1800° or 2000° absolute, our hypo-
thetical metal may at 6° absolute show a high conductivity, com-
parable with that of lead or copper at ordinary temperatures.
The electric conductivity in the Nemst filament goes on increasing
as the temperature rises till the filament melts or evaporates.
Nevertheless it is quite conceivable that in our hypothetical metal
with rising temperature fi'om 2° to 16° absolute the electric con-
ductivity may come to a maximum and decrease with further rise

ELECTRIONIC EXPLANATION OF ELECTRIC CONDUCTIVITY. 559

of temperature up to and beyond ordinary atmospheric tempera-
tures. In fact, while some extent of thermal motions is necessary
for electric conductivity (because there can be no such thing as
" lability '' in electrostatic equilibrium), too much of these motions
must mar the freedom with which an electrion can thread its
way through the crowd of atoms to perform the Ainction of electric
conduction. It seems certain that this is the matter-of-fact
explanation of the diminution of electric conductivity in metals
with rise of temperature.

§ 31. Regretting much not to be able (for want of time) to
include estimates of absolute magnitudes in the present com-
munication, I end it with applications of our hjrpothesis to the
P3nro-electricity and piezo-electricity of crystals. A crystal is a
homogeneous assemblage of bodies. Conversely, a homogeneous
assemblage of bodies is not a crystal if the distance between
centres of nearest neighbours is a centimetre or more ; it is a
crystal if the distance between nearest neighbours is 10~* of a cm.
or less. Pyro-electricity and piezo-electricity are developments
of vitreous and resinous electric forces such as would result from
vitreous and resinous electrification on different parts of the
surface of a crystal, produced respectively by change of tem-
perature and by stress due to balancing forces applied to the
sur&ces.

§ 32. To see how such properties can or must exist in crystals
composed of our hypothetical atoms with electrions, consider first

Fig. 6.

merely a row of equal tetra-electrionic atoms in a straight line,
each having its quantum of four electrions within it. Fig. 6
shows a configuration of stable equilibrium of the electrions not,
however, truly to scale. The sets of three dots indicate trios of
electrions at the corners of equilateral triangles, the middle dot
in each row being alternately on the far side and the near side of
the plane of the paper, which contains the centres of the atoms
and the remaining electrion of each four. Let d. C, C\ he the

560 APPENDIX E.

centres of the atoms Ai, A, A\ Au easy calculation shows that
the quartet of electrious within A, regarded for the moment as a
group of four material points rigidly connected, is attracted to the
left with a less force by A^ than to the right by A' (in making the
calculation remember that A^ attracts all the electrions within A
as if it were a quantity e of vitreous electricity collected at C, ,
and similarly in respect to A'). There are corresponding smaller
differences between the opposite attractions of the more and more
remote atoms on the two sides of A, Let S denote the excess of
the sum of the rightwards of these attractions above the leftwards.
The geometrical centre of the electrions within A is displaced

rightwards from (7 to a distance, Z, equal to ^^ •

§ 33. Imagine now a crystal or a solid of any shape built up
of parallel rows of atoms such as those of § 32. The amount of
the displacing force on each quartet of electrions will be some-
what altered by mutual action between the rows, but the general
character of the result will be the same; and we see that
throughout the solid, except in a thin superficial layer of perhaps
five or tens atoms deep, the whole interior is in a state of homo-
geneous electric polarization, of which the electric moment per
unit of volume is ^eNl ; where N is the number of atoms per
unit volume, and I is the displacement of the geometrical centre
of each quartet from the centre of its atom. This is the
interior molecular condition of a di-polar pyro-electric crystal,
which I described in 1860* as probably accounting for their
known pyro-electric quality, and as in accordance with the free
electro-polarities of fractured surfaces of tourmaline discovered
by Canton f. If a crystal, which we may imagine as given with
the electrions wholly undisturbed from their positions according

* Collected Mathematical and Physical Papers^ Vol. i. p. 315.

t Wiedemann {Die Lehre von der Elektricitat, Second Edition, 1894, Vol. u.
§ 878) mentions an experiment without fully desoribing it by which a noU resalt,
seemingly at variance with Canton's experimental discovery and condemnatory of
my suggested theory, was found. Interesting experiments might be made by
pressing together and reseparating fractured surfaces of tourmaline, or by pressing
and rubbing polished surfaces together and separating them. It would be very
difficult to get trustworthy results by breakages, becaufle it would be almost
impossible to avoid irregular electrifications by the appliances used for making
th6 breakage. The mode of electric measurement followed in the experiment
referred to by Wiedemann is not described.

AEPINUS* DISCOVERY OP PYRO-ELECTRICITY EXPLAINED. 561

to § 32, is dipped in water and then allowed to dry, electrions
would by this process be removed from one part of its surface and
distributed over the remainder so as to wholly annul its external
manifestation of electric quality. If now either by change of
temperature or by mechanical stress the distances between the
atoms are altered, the interior electro-polarization becomes neces-
sarily altered ; and the masking superficial electrification previously
got by the dipping in water and drying, will now not exactly
annul the electrostatic force in the air around the solid. If at
the altered temperature or under the supposed stress the solid
is again dipped in water and dried, the external electric force
will be again annulled. Thus is explained the pyro-electricity
of tourmaline discovered by Aepinus.

§ 34. But a merely di-polar electric crystal with its single
axis presents to us only a small, and the very simplest, part of the
whole subject of electro-crystallography. In boracite, a crystal of
the cubic class, Hauy found in the four diagonals of the cube, or
the perpendiculars to the pairs of faces of the regular octahedron,
four di-polar axes: the crystal on being irregularly heated or
cooled showed as it were opposite electricities on the surfaces in
the neighbourhood of opposite pairs of corners of the cube, or
around the centres of the opposite pairs of triangular faces of the
octahedron. His discoveries allow us to conclude that in general
the electric aeolotropy of crystals is octo-polar with four axes, not
merely di-polar as in the old-known electricity of the tourmaline.
The intensities of the electric virtue are generally diflFerent for
the four axes, and the directions of the axes are in general un-
symmetrically oriented for crystals of the unsymmetrical classes.
For crystals of the optically uniaxal class, one of the electro-polar
axes must generally coincide with the optic axis, and the other
three may be perpendicular to it. The intensities of the electro-
polar virtue are essentially equal for these three axes : it may be
null for each of them : it may be null or of any value for the so-
called optic axis. Hauy found geometrical differences in respect
to crystalline facets at the two ends of a tourmaline ; and between
the opposite ^omers of cubes, as leucite, which possess electro-
polarity. There are no such diflFerences between the two ends of
a quartz crystal (hexagonal prism with hexagonal pyramids at the
two ends) but there are structural differences (visible or invisible)

T. L. a^

560 APPENDIX E.

centres of the atoms Ai, A, A'. An easy calculation shows that
the quartet of electrious within A, regarded for the moment as a
group of four material points rigidly connected, is attracted to the
left with a less force by Ai than to the right by ^'(in making the
calculation remember that A^ attracts all the electrions within A
as if it were a quantity e of vitreous electricity collected at C,,
and similarly in respect to A'). There are corresponding smaller
differences between the opposite attractions of the more and more
remote atoms on the two sides of A. Let S denote the excess of
the sum of the rightwards of these attractions above the leftwards.
The geometrical centre of the electrions within A is displaced

rightwards from (7 to a distance, l, equal to ^^ •

§ 33. Imagine now a crystal or a solid of any shape built up
of parallel rows of atoms such as those of § 32. The amount of
the displacing force on each quartet of electrions will be some-
what altered by mutual action between the rows, but the general
character of the result will be the same; and we see that
throughout the solid, except in a thin superficial layer of perhaps
five or tens atoms deep, the whole interior is in a state of homo-
geneous electric polarization, of which the electric moment per
unit of volume is ieNl ; where N is the number of atoms per
unit volume, and I is the displacement of the geometrical centre
of each quartet from the centre of its atom. This is the
interior molecular condition of a di-polar pyro-electric crystal,
which I described in 1860* as probably accounting for their
known pyro-electric quality, and as in accordance with the free
electro-polarities of fractured surfaces of tourmaline discovered
by Canton f. If a crystal, which we may imagine as given with
the electrions wholly undisturbed from their positions according

* Collected Mathematical and Physical Papers^ Vol. i. p. 315.

t Wiedemann {Die Lehre von der Elektricitdt, Second Edition, 1894, Vol. n.
§ 878) mentions an experiment without fully describing it by which a nail resalt,
seemingly at variance with Canton's experimental discovery and condemnatory of
my suggested theory, was found. Interesting experiments might be made by
pressing together and reseparating fractured surfaces of tourmaline, or by pressing
and rubbing polished surfaces together and separating them. It would be very
difficult to get trustworthy results by breakages, because it would be almost
impossible to avoid irregular electrifications by the appliances used for making
the breakage. The mode of electric measurement foUowed in the experiment
referred to by Wiedemann is not described.

AEPINUS' DISCOVERY OP PYRO-ELECTRICITY EXPLAINED. 561

to § 32, is dipped in water and then allowed to dry, electrions
would by this process be removed from one part of its surface and
distributed over the remainder so as to wholly annul its external
manifestation of electric quality. If now either by change of
temperature or by mechanical stress the distances between the
atoms are altered, the interior electro-polarization becomes neces-
sarily altered ; and the masking superficial electrification previously
got by the dipping in water and drying, will now not exactly
annul the electrostatic force in the air around the solid. If at
the altered temperature or under the supposed stress the solid
is again dipped in water and dried, the external electric force
will be again annulled. Thus is explained the pyro-electricity
of tourmaline discovered by Aepinus.

§ 34. But a merely di-polar electric crystal with its single
axis presents to us only a small, and the very simplest, part of the
whole subject of electro-crystallography. In boracite, a crystal of
the cubic class, Haiiy found in the four diagonals of the cube, or
the perpendiculars to the pairs of faces of the regular octahedron,
four di-polar axes: the crystal on being irregularly heated or
cooled showed as it were opposite electricities on the surfaces in
the neighbourhood of opposite pairs of corners of the cube, or
around the centres of the opposite pairs of triangular faces of the
octahedron. His discoveries allow us to conclude that in general
the electric leolotropy of crystals is octo-polar with four axes, not
merely di-polar as in the old-known electricity of the tourmaline.
The intensities of the electric virtue are generally different for
the four axes, and the directions of the axes are in general un-
symmetrically oriented for crystals of the unsjmnmetrical classes.
For crystals of the optically uniaxal class, one of the electro-polar
axes must generally coincide with the optic axis, and the other
three may be perpendicular to it. The intensities of the electro-
polar virtue are essentially equal for these three axes : it may be
null for each of them : it may be null or of any value for the so-
called optic axis. HaUy found geometrical differences in respect
to crystalline facets at the two ends of a tourmaline ; and between
the opposite ^omers of cubes, as leucite, which possess electro-
polarity. There are no such diflFerences between the two ends of
a quartz crystal (hexagonal prism with hexagonal p3nramids at the
two ends) but there are structural diflFerences (visible or invisible)

T. L. '^^

562 APPENDIX E.

between the opposite edges of the hexagonal prism. The electro-
polar virtue is null for the axis of the prism, and is proved to
exist between the opposite edges by the beautiful piezo-electric
discovery of the brothers Curie, according to which a thin flat
bar, cut with its faces and its length perpendicular to two parallel
faces of the hexagonal prism and its breadth parallel to the edges
of the prism, shows opposite electricities on its two faces, when
stretched by forces pulling its ends. This proves the three electro-
polar axes to bisect the 120'' angles between the consecutive plane
faces of the prism.

§ 35. For the present let us think only of the octo-polar
electric aeolotropy discovered by Hatiy in the cubic class of
crystals. The quartet of electrions at the four corners of a
tetrahedron presents itself readily as possessing intrinsically the
symmetrical octo-polar quality which is realized in the natural
crystal. If we imagine an assemblage of atoms in simple cubic order
each containing an equilateral quartet of electrions, all similarly
oriented with their four faces perpendicular to the four diagonals
of each structural cube, we have exactly the required SBolotropy ;
but the equilibrium of the electrions all similarly oriented would
probably be unstable ; and we must look to a less simple assem-
blage in order to have stability with similar orientation of all the
electrionic quartets.

§ 36. This, I believe, we have in the doubled equilateral
homogeneous assemblage of points described in § 69 of my paper
on Molecular Constitution of Matter republished from the Trans-
actions of the Royal Society of Edinburgh for 1889 in Volume in.
of my Collected Mathematical and Physical Papers (p. 426);
which may be described as follows for an assemblage of equal and
similar globes. Beginning with an equilateral homogeneous
assemblage of points. A, make another similar assemblage of
points, B, by placing a J3 in the centre of each of the similarly
oriented quartets of the assemblage of ^'s. It will be found that
every A is at the centre of an oppositely oriented quartet of
the jB's. To understand this, let -4j, A^, -4,, A^, be an equilateral
quartet of the A's ; and imagine A^, A^, A4, placed on a horizontal
glass plate* with A^ above it. Let jBj be Q.t the centre of

* ParaUel glass plates canylng little white or black or coloured paper-oiroles are
nsefol aaxiliaries for graphic consiraction and illostratiye models in the moleoolar
theory of crystals.

EQUILIBRIUM OF ELECTRIONS IN DOUBLED ASSEMBLAGE. 563

Ai, Aif A^, A^, and let Bi, B2, B^, B4, be a quartet of the B*a
similarly oriented to Ai^ A2, At, A^, We see that B2, B^, -B4, lie
below the glass plate, and that the quartet Bi, B^, B^, B^, has
none of the A's at its centre. But the vertically opposite quartet
jBj, B2', Bt\ B4, contains ^li within it; and it is oppositely oriented
to the quartet Ai, A2, A^, A4. Thus we see that, while the half
of all the quartet^ of -4*8 which are oriented oppositely to
ill, -4j, 4s, A4, are void of B% the half of the quartets of J^s
oppositely oriented to Ai, ila, A^, A^, have each an A within it,
while the other half of the quartets of the jffs are all void

of A'8,

§ 37. Now let all the points A and all the points -B, of § 36,
be centres of equal and similar spherical atoms, each containing a
quartet of electrions. The electrions will be in stable equilibrium
under the influence of their own mutual repulsions, and the at-
tractions of the atoms, if they are placed as equilateral quartets
of proper magnitude, concentric with the atoms, and oriented all
as any one quartet of the A*8 or J^s. To see that this is true,
confine attention first to the five atoms A^ ^„ As, A4, Bi. If the
electrions within A^, A2, -4,, ^4 are all held similarly oriented to
the quartet of the centres of these atoms, the quartet of electrions
within Bi must obviously be similarly oriented to the other
quartets of electrions. If again, these be held oriented oppositely
to the quartet of the atoms, the stable configuration of the
electrions within jBi, will still be similar to the orientation of the
quartets within Ai, A2, -4j, A^, though opposite to the orientation
of the centres of these atoms. If, when the quartets of electrions
are all thus similarly oriented either way, the quartet within B^
is turned to reverse orientation, this will cause all the others to
turn and settle in stable equilibrium according to this reversed
orientation. Applying the same consideration to every atom of
the assemblage and its four nearest neighbours, we have proof of
the proposition asserted at the commencement of the present
section. It is most interesting to remark that if, in a vast homo-
geneous assemblage of the kind with which we are dealing, the
orientation of any one of the quartets of electrions be reversed
and held reversed, all the others will follow and settle in stable
equilibrium in the reversed orientation.

§ 38. This double homogeneous assemblage of tetra-electrionic

562 APPENDIX E.

between the opposite edges of the hexagonal prism. The electro-
polar virtue is null for the axis of the prism, and is proved to
exist between the opposite edges by the beautiful piezo-electric
discovery of the brothers Curie, according to which a thin flat
bar, cut with its faces and its length perpendicular to two parallel
faces of the hexagonal prism and its breadth parallel to the edges
of the prism, shows opposite electricities on its two faces, when
stretched by forces pulling its ends. This proves the three electro-
polar axes to bisect the 120'^ angles between the consecutive plane
faces of the prism.

§ 35. For the present let us think only of the octo-polar
electric aeolotropy discovered by Haiiy in the cubic class of
crystals. The quartet of electrions at the four corners of a
tetrahedron presents itself readily as possessing intrinsically the
symmetrical octo-polar quality which is realized in the natural
crystal. If we imagine an assemblage of atoms in simple cubic order
each containing an equilateral quartet of electrions, all similarly
oriented with their four faces perpendicular to the four diagonals
of each structural cube, we have exactly the required seolotropy ;
but the equilibrium of the electrions all similarly oriented would
probably be unstable ; and we must look to a less simple assem-
blage in order to have stability with similar orientation of all the
electrionic quartets.

§ 36. This, I believe, we have in the doubled equilateral
homogeneous assemblage of points described in § 69 of my paper
on Molecular Constitidion of Matter republished from the Trans-
actions of the Royal Society of Edinburgh for 1889 in Volume ill.
of my Collected Maihematical and Physical Papers (p. 426);
which may be described as follows for an assemblage of equal and
similar globes. Beginning with an equilateral homogeneous
assemblage of points. A, make another similar assemblage of
points, B, by placing a J5 in the centre of each of the similarly
oriented quartets of the assemblage of -4's. It will be found that
every A is at the centre of an oppositely oriented quartet of
the -B's. To understand this, let A^y 4,, -4,, A^, be an equilateral
quartet of the il*s ; and imagine A^, -4,, A^y placed on a horizontal
glass plate* with A^ above it. Let B^ be Q.t the centre of

* Parallel glass plates canying little white or black or coloured paper-circles are
nsefol auxiliaries for graphic oonsiraotion and illostratiye models in the Tn5>)ftffniftT
theory of crystals.

EQUILIBRIUM OF ELECTRIONS IN DOUBLED ASSEMBLAGE. 663

Ai, Ai, A^, A^, and let -Bj, B^, B^, B4, be a quartet of the Rs
similarly oriented to Ai, 2I9, -4,, A4. We see that B2, 5j, ^4, lie
below the glass plate, and that the quartet Bi^ B^, B^, B^^ has
none of the -4*s at its centre. But the vertically opposite quartet
Bi, B^t B^, 5/, contains Ai within it; and it is oppositely oriented
to the quartet A^, A^, -4,, A4. Thus we see that, while the half
of all the quartet^ of A*s which are oriented oppositely to
Ai, ^2, Ai, J.4, are void of 5's, the half of the quartets of J^s
oppositely oriented to Aiy A^y -4,, A^, have each an A within it,
while the other half of the quartets of the ^'s are all void
of A'b.

§ 37. Now let all the points A and all the points -B, of § 36,
be centres of equal and similar spherical atoms, each containing a
quartet of electrions. The electrions will be in stable equilibrium
under the influence of their own mutual repulsions, and the at-
tractions of the atoms, if they are placed as equilateral quartets
of proper magnitude, concentric with the atoms, and oriented all
as any one quartet of the A*& or J^s. To see that this is true,
confine attention first to the five atoms Ai, A^, -4,, A4, Bi, If the
electrions within Ai, A2, A^, A^ are all held similarly oriented to
the quartet of the centres of these atoms, the quartet of electrions
within Bi must obviously be similarly oriented to the other
quartets of electrions. If again, these be held oriented oppositely
to the quartet of the atoms, the stable configuration of the
electrions within Bi, will still be similar to the orientation of the
quartets within Ai, A^, -4j, A^, though opposite to the orientation
of the centres of these atoms. If, when the quartets of electrions
are all thus similarly oriented either way, the quartet within B^
is turned to reverse orientation, this will cause all the others to
turn and settle in stable equilibrium according to this reversed
orientation. Applying the same consideration to every atom of
the assemblage and its four nearest neighbours, we have proof of
the proposition asserted at the commencement of the present
section. It is most interesting to remark that if, in a vast homo-
geneous assemblage of the kind with which we are dealing, the
orientation of any one of the quartets of electrions be reversed
and held reversed, all the others will follow and settle in stable
equilibrium in the reversed orientation.

§ 38. This double homogeneous assemblage of tetra-electrionic

36—2

564

APPENDIX E.

atoms seems to be absolutely the simplest* molecular stnicture in
which Haiiy's octo-polar electric quality can exist. To see that it
has octo-polar electric quality, consider an octahedron built up
according to it. The faces of this octahedron, taken in proper
order, will have, next to them, alternately points and triangular
faces of the electrionic quartets within the atoms. This itself is
the kind of electric aeolotropy which constitutes octo-polar quality.
Time prevents entering fully at present on any djmamical investi-
gation of static or kinetic resulta

§ 39. [Added Oct, 23, 1901.] Since what precedes was written,
I have seen the explanation of a difficulty which had prevented
me from finding what was wanted for octo-polar electric aeolotropy
in a homogeneous assemblage of single atoms. I now find (§ 40
below) that quartets of electrions will rest stably in equilibrium,
under the influence of the
mutual repulsion between
electrion and electrion and
attraction between atom
and electrion, in an equi-
lateral homogeneous as-
semblage in the configura-
tion indicated in fig. 7.
The quartets of electrions
are supposed to have their
edges parallel to the six
lines of symmetry of the
assemblage. The plane of
the paper is supposed to
be that of the centres of
the seven atoms. The central point in each circle represents a
simple electrion which is at distance r, according to the notation
of § 20 above, from the plane of the paper on the near side ; and
therefore the other three at the comers of an equilateral triangle
at distance ^r on the far side to make the electric centre of gravity
of the quartet coincide with the centre of its atom. The radius of

2 /2
the circle on which these three lie is —^ r or 94. The diagram

o

is drawn correctly to scale according to the value '612a given

Fig. 7.

* Not ihe simplest See $ 40 below.

QUARTETS IN EQUILIBRIUM ARE ALL ORIENTED ONE WAY. 565

for r in the table of § 20, on the supposition that the circles
shown in the diagram represent the electric spheres of the atoms
in contact.

§ 40. Imagine now the electrions of each quartet to be rigidly
connected with one another and given freedom only to rotate
about an axis perpendicular to the plane of the paper. To all of
them apply torques ; turning the central quartet of the diagram
slowly and keeping all the others at rest. It is clear that the
first 60° of turning brings the central quartet to a position of
unstable equilibrium, and 60° more to a position of stable equi-
librium corresponding to the first position, which we now see is
stable when the others are all held fixed. We are now judging
simply from the mutual actions between our central quartet and
the six shown around it in the diagram; but it may be easily
proved that our judgment is not vitiated by the mutual action
between the central quartet and all around it, including the six
in the diagram. Similarly we see that any one quartet of the
assemblage, free to turn round an axis perpendicular to the plane
of the paper while all the others are fixed, is in stable equilibrium
when oriented as are those shown in the diagram. And similarly
again we see the same conclusion in respect to three other
diagrams in the three other planes parallel to the faces of the
tetrahedrons or corresponding octahedrons of the assemblage.
Hence we conclude that if the axial constraints are all removed,
and the quartets left perfectly free, every one of them rests in
stable equilibrium when oriented either as one set or as the other
set of equilateral tetrahedronal quartets of the assemblage. It is
interesting to remark that if, after we turned the central quartet
through 60°, we had held it in that position and left all the others
free to rotate, rotational vibrations would have spread out among
them from the centre; and, after losing in waves spreading
through ether outside the assemblage the energy which we gave
them by our torque acting on the central quartet, they would come
to stable equilibrium with every one of them turned 60° in one
direction or the other from its primitive position, and oriented
as the central quartet in the position in which we held it.

§ 41. We have thus found that an equilateral homogeneous
assemblage of atoms each having four electrions within it,
arranges these electrions in equilateral quartets all oriented in one

566 APPENDIX E.

or other of two ways. The assemblage of atoms and electrions
thus produced is essentially octo-polar. Of the two elementary
structural tetrahedrons, of the two orientations, one will have
every one of its elect rionic quartets pointing towards, the other
from, its faces. The elementary structural octahedron has four of
its faces next to corners, and four next to triangles, of its electrionic
quartets. This is essentially a dynamically octo-polar* assemblage ;
and it supplies us with a perfect explauation of the piezo-electric
quality to be inferred from the brothers Curie's experimental
discovery, and Voigt's mathematical theory.

§ 42. Look at the diagram in § 39; and remember that it
indicates a vast homogeneous assemblage consisting of a vast
number of parallel plane layers of atoms on each side of the
plane of the paper, in which seven atoms are shown. The
quartets of electrions were described as all similarly oriented, and
each of them equilateral, and having its geometrical centre at the
centre of its atom ; conditions all necessary for equilibrium.

§ 43. Let now the assemblage of atoms be homogeneously
stretched from the plane on both sides to any extent, small or
great, without any component motions of the centres of the atoms
parallel to the planes of the layers. First let the stretch be very
great; great enough to leave undisturbed by the other layers
the layer for which the centres of atoms arc, and the geometrical
centres of the quartets were, in the plane of the paper. The
geometrical centres of the quartets are not now in the plane of
the paper. The single electrions on the near side seen in the
diagram over the centres of the circles are drawn towards the
plane of the paper ; the equilateral triangles on the far side are
also drawn nearer to the paper ; and the equilateral triangles are
enlarged in each atom by the attractions of the surrounding
atoms. The contrary inward movements of the single atoms on
one side of the plane, and of the triplets on the other side, cannot

* The octo-polar pyro-eUctricity, which, is supposed to have been proved by
Haiiy's experiment, must have been due to something SBolotropic in the heating.
Uniform heating throughout a regular cube or octahedron could not give opposite
electric manifestations in the four pairs of alternate comers of the cube, or
alternate faces of the octahedron. Nevertheless the irregular finding of electric
octo-polarity by Haiiy is a splendid discovery ; of which we only now know the
true and full significance, through the experimental and mathematical labours of
the brothers Curie, of Friedel, and of Voigt.

VOIOT*S OCTO-POLAft PlEZO-ELECTRlCItY OP CttBIC CRYSTAL. 567

in general be in the proportion of three to one. Hence the
geometrical centres of gravity of the quartets are now displaced
perpendicularly to the plane of the paper to far side or near side ;
I cannot tell which without calculation. The calculation is easy
but essentially requires much labour; involving as it does the
determination of three unknowns, the length of each side of the
equilateral triangle seen in the diagram, the distance of each of
its comers from the electrion on the near side of the paper, and
the displacement of the geometrical centre of gravity of the four
to one side or other of the plane. Each one of the three equations
involves summations of infinite convergent series, expressing force
components due to all the atoms surrounding any chosen one in
the plane. A method of approximation on the same general plan
as that of the footnote to § 9 above would give a practicable
method of calculation.

§ 44. Return to § 42 ; and consider the diagram as represent-
ing a crystal in its natural unstressed condition, consisting of a
vast train of assemblages of atoms with centres in the plane of
the paper, and in parallel planes on each side of it. We now see
that the forces experienced by the electrions of one quartet from
all the surrounding atoms in the plane of the paper would, if
uncompensated, displace the geometrical centre of gravity of the
quartet to one side or other of the plane of the paper, and we
infer that the forces experienced from all the atoms on the two
isides of this plane give this compensation to keep the centre of
gravity of the quartet in the plane. Stretch now the assemblage
to any degree equally in all directions. The quartets remain
equilateral with their centres of gravity in the plane of the paper
and parallel planes. Lastly stretch it farther equally in all direc-
tions parallel to the plane of the paper, with no component motion
perpendicular to this plane. This last stretching diminishes
the influence of all the atoms whose centres are in the plane of
the paper tending to displace the centres of gravity of their
electrions in one direction from this plane ; and therefore leaves
all the atoms out of this plane to predominate, and to cause a
definite calculable displacement of the centres of gravity of all
the quartets in the contrary direction to the former.

§ 45. To realize the operations of § 44, cut a thin hexagonal
plate from the middle between two opposite comers of a cubic

568 APPENDIX K.

crystal, or parallel faces of an octahedron. Fix clamps to the six
edges of this plate, and apply forces pulling their pairs equally in
contrary directions. The whole material of the plate becomes
electro-polar with electric moment per unit bulk equal to AiNex\
of which the measurable result is uniform electrostatical potentials*
in vacuous ether close to the two sides of the plate, differing by
47r . 4s Next ; where t denotes the thickness of the plate, x the
calculated displacement of the centre of gravity of each quartet
from the centres of the atoms parallel to the two faces of the
plate, e the electric mass of an electrion, and N the number of
atoms per cubic centimetre of the substance. This crystal of the
cubic class is, in Voigt's mathematical theory, the analogue to
the electric effect discovered in quartz by the brothers Curie, and
measured by aid of thin metal foils attached to the two faces of
the plate and metrically connected to the two principal electrodes
of an electrometer.

* See my Electrostatics and Magnetism, § 512, Cor. 3.

APPENDIX F.

(Having been in tyiHj more than twelve years as Articles XCIIT. — XCVII.
for projected Vol. iv. of Mathematical and Physical Papers,)

Art. XCIII. Dynamical Illustrations of the Magnetic
and the heligoidal rotatory effects of transparent
Bodies on Polarized Light.

[From the Proc, Roy. Soc.y VoL viiL, June 1856 ; PhiL Mag.y March 1857.]

The elastic reaction of a homogeneously strained solid has a
character essentially devoid of all heli9oidal and of all dipolar
asymmetry. Hence the rotation of the plane of polarization of
light passing through bodies which either intrinsically possess the
heH9oidal property (syrup, oil of turpentine, quartz crystals, &c.),
or have the magnetic property induced in them, must be due to
elastic reactions dependent on the heterogeneousness of the strain
through the space of a wave, or to some heterogeneousness of the
luminous motions* dependent on a heterogeneousness of parts of
the matter of lineal dimensions not infinitely small in comparison
with the wave length. An infinitely homogeneous solid could
not possess either of those properties if the stress at any point of
it was influenced only by parts of the body touching it; but if
the stress at one point is directly influenced by the strain in
parts at distances from it finite in comparison with the wave
length, the heli<;oidal property might exist, and the rotation of
the plane of polarization, such as is observed in many liquids and
in quartz crystals, could be explained as a direct dynamical
consequence of the statical elastic reaction called into play by
such a strain as exists in a wave of polarized light. It may,
however, be considered more probable that the matter of trans-
parent bodies is reaUy heterogeneous from one part to another of
lineal dimensions not infinitely small in comparison with a wave

* As would be were there different sets of vibrating particles, or were Rankine's
important hypothesis true, that the vibrations of luminiferous particles are directly
affected by pressure of a surrounding medium in virtue of its inertia. [In Lectures
XIX. and XX. we have seen reason to believe that this is true.]

570 APPENDIX P.

length, than that it is infinitely homogeneous and has the property
of exerting finite direct " molecular " force at distances comparable
with the wave length : and it is certain that any spiral hetero-
geneousness of a vibrating medium must, if either right-handed
or left-handed spirals predominate, cause a finite rotation of the
plane of polarization of all waves of which lengths are not
infinitely great multiples of the steps of the structural spirals.
Thus a liquid filled homogeneously with spiral fibres, or a solid
with spiral passages through it of steps not less than the forty-
millionth of an inch, or a crystal with a right-handed or a left-
handed geometrical arrangement of parts of some such lineal
dimensions as the forty-millionth of an inch, might be certainly
expected to cause either a right-handed or a left-handed rotation

of ordinary light (the wave length being ^^^th of an inch for
homogeneous yellow).

But the magnetic influence on light discovered by Faraday
depends on the direction of motion of moving particles. For
instance, in a medium possessing it, particles in a straight line
parallel to the lines of magnetic force, displaced to a helix round
this line as axis, and then projected tangentially with such veloci-
ties as to describe circles, will have diflferent velocities according
as their motions are round in one direction (the same as the
nominal direction of the galvanic current in the magnetizing
coil), or in the contrary direction. But the elastic reaction of the
medium must be the same for the same displacements, whatever
be the velocities and directions of the particles; that is to say,
the forces which are balanced by centrifugal force of the circular
motions are equal, while the luminiferous motions are unequal.
The absolute circular motions being therefore either equal or such
as to transmit equal centrifugal forces to the particles initially
considered, it follows that the luminiferous motions are only com-
ponents of the whole motion ; and that a less luminiferous com-
ponent in one direction, compounded with a motion existing in
the medium when transmitting no light, gives an equal resultant
to that of a greater luminiferous motion in the contrary direction
compounded with the same non-luminous motion. I think it is
not only impossible to conceive any other than this dynamical
explanation of the fact that circularly polarized light transmitted
through magnetized glass parallel to the lines of magnetizing force,
with the same quality, right-handed always, or left-handed alwaysj

MOUNTING OF TWO-PERIOD PENDULUM ROTATED. 571

is propagated at different rates according as its course is in the
direction or is contrary to the direction in which a north magnetic
pole is drawn ; but I believe it can be demonstrated that no
other explanation of that fact is possible. Hence it appears that
Faraday's optical discovery affords a demonstration of the reality
of Ampfere*s explanation of the ultimate nature of magnetism;
and gives a definition of magnetization in the dynamical theory
of heat. The introduction of the principle of moments of momenta
(" the conservation of areas ") into the mechanical treatment of
Mr Raokine's hypothesis of " molecular vortices," appears to in-
dicate a line perpendicular to the plane of resultant rotatory
momentum ("the invariable plane") of the thermal motions as
the magnetic axis of a magnetized body, and suggests the resultant
moment of momenta of these motions as the definite measure of
the " magnetic moment." The explanation of all phenomena of
electro-magnetic attraction or repulsion, and of electro-magnetic
induction, is to be looked for simply in the inertia and pressure
of the matter of which the motions constitute heat. Whether
this matter is or is not electricity, whether it is a continuous, fluid
interpermeating the spaces between molecular nuclei, or is itself
molecularly grouped; or whether all matter is continuous, and
molecular heterogeneousness consists in finite vortical or other
relative motions of contiguous parts of a body; it is impossible
to decide, and perhaps in vain to speculate, in the present state
of science.

I append the solution of a dynamical problem for the sake of
the illustrations it suggests for the two kinds of effect on the
plane of polarization referred to above.

Let the two ends of a cord of amy length he attached to two
points at the ends of a horizontal arm made to rotate round a
vertical axis through its middle point at a constant angular velocity^
Q), and let a second cord beamng a weight he attached to the middle
of the first cord. The two cords being each perfectly light and
fieocihle, and the weight a material point, it is required to determine
its motion wlien infinitely little disturbed from its position of equi-
librium*.

Let I be the length of the second cord, and m the distance from

* By means of this arrangement, but without the rotation of the bearing arm,
a very beautiful experiment, due to Professor Blackburn, may be made by attach-
ing to the weight a bag of sand discharging its contents through a tiue aperture.

572 APPEXDIX F.

the weight to the middle point of the arm bearing the first Let
X and y be, at any time t, the rectangular coordinates of the
position of the weight, referred to the position of equilibrium 0,
and two rectangular lines OX, OT, revolving uniform! v in a hori-
zontal plane in the same direction, and with the same angular
velocity as the bearing arm ; then, if we choose OX parallel to
this arm, and if the rotation be in the direction with 0 T preceding
OX, we have, for the equations of motion,

df ^ dt m^

If for brevity we assume

we find, by the usual methods, the following solution : —

X = ^ cos {[»' + 7i» + (X* + 4/1 V)*]*< + a)

+ 5 cos {[«• + n' - (X* H- 4nV;*]*e + /3},

2«*-X*+;X* + 4n*«V . . ^
V = -r-i ^ sm 6

2«*-X*-(X* + 4iiV}* ^ . ,
2« [«' + w* - (X* + 4/1 V)*3 * ^

where A, n^B, ^ are arbitrary constants, and ^ and ^ are used for
brevity to denote the arguments of the cosines appearing in the
expression for x.

The interpretation of this solution, when e» is taken equal to
the component of the earth's angular velocity round a vertical at
the locality, affords a full explanation of curious phenomena which
have been observed by many in billing to repeat Foucault s admir-
able pendulum experiment. When the mode of suspension is
perfect, we have X = 0 ; but in many attempts to obtain Foucault s
result, there has been an asymmetry in the mode of attachment
of the head of the cord or wire used, or there has been a slight
lateral unsteadiness in the bearings of the point of suspension,
which has made the observed motion be the same as that expressed
by the preceding sjlution, where X has some small value either

PROBLEM OF THE MOTION SOLVED. 573

greater than or less than o), and n has the value a/^- The only

case, however, that need be considered as illustrative of the subject
of the present communication is that in which co is very great in
comparison with n. To obtain a form of solution readily inter-
preted in this case, let

[a)« + w' + (X* + 4wV)*]* =? Q) + p, [co' + n« - (X* + 4wV)*]* = oi - cr,

2q,« - x« + (X* + imWf

2a) [o)' + 7i« - (X* + 4:vW)^] * "" •^*
The preceding solution becomes

x= Acos{(€o + p)t+ a] + B cos {{co — <r)t + ^}
y = — A sin {{co -\- p) t -hoi} — B sin {(o) — cr) i + /8}

- e^ sin {(o) -\-p)t+a] -j-fB sin {(w - cr) ^ + /8}.

To express the result in terms of coordinates f, rjy with reference

to fixed axes, instead of the revolving axes OX, OY, we may

assume

^ = x cos cot — y sin cd^,

t; = a? sin ft)^ + y cos cd^.
Then we have

f = ^ cos (pt + a) + £ cos {at - /8)

+ {eA sin ((cd + /3) ^ + a} -/£sin {{co — a)t-\- /8}) sin cot

rj=^ — A sin {pt-\- a) -\-B sin (<r< — 13)

+ (— eil sin {{co -\- p)t'^ a] +/B sin {(o) - cr) < + /8}) cos cot

When o) is veiy large, e and / are both very small, and the last
two terms of each of these equations become very small periodic
terms, of very rapidly recurring periods, indicating a slight tremor
in the resultant motion. Neglecting this, and taking a = 0 and
/3 = 0, as we may do without loss of generality, by properly choosing
the axes of reference, and the era of reckoning for the time, we
have finally, for an approximate solution of a suitable kind,

^= Acqs pt + B cos crt^
rf = — A sin /:>f + -B sin (T^.

574 APPENDIX F.

The terms 5, in this expression, represent a circular motion of

27r .
period , in the positive direction (that is, from the positive axis

of f to the positive axis of tf), or in the same direction as that of

the rotation oi; and the terms A represent a circular motion, of

27r
period — , in the contrary direction. Now, a> being very great,

p and a are very nearly equal to one another ; but p is rather less
than a, as the following approximate expressions derived from
their exact values expressed above, show : —

1 X* 1 X* 1 X* 1 X*

'^ 8 0)71 8 CD 8 arn 8 w

Hence the form of solution simply expresses that circular vibrations
of the pendulum in the contrary directions have slightly different

periods, the shorter, — , when the motion of the pendulum follows

27r
that of the arm supporting it, and the longer, — , when it is in the

contrary direction. The equivalent statement, that if the pendulum
be simply drawn aside from its position of equilibrium, and let go
vrithout initial velocity, the vertical plane of its motion will rotate
slowly at the angular rate ^ (o- — p), is expressed most shortly by
taking A = B, and reducing the preceding solution to the form

f = 2^ cos mt cos n't,

rf = 2 A sin mt cos n't,

where n' = ^ (<r -f p), or, approximately, n =n + - ,- ,

^ 8 o) n

and ^ = o (^ "" P)» ^^> approximately, ©•=- — .

Z 8 o)

It is a curious part of the conclusion thus expressed, that the

faster the bearing arm is carried round, the slower does the plane

of a simple vibration of the pendulum follow it. When the bearing

arm is carried round infinitely fast, the plane of a vibration of the

pendulum will remain steady, and the period will be n ; in other

words, the motion of the pendulum will be the same as that of

2
a simple pendulum whose length is = — , or a harmonic mean

- + -
e m

SOLUTION EXAMINED. 676

between the eflfective lengths in the two principal planes of the
actual pendulum.

It is easy to prove from this, that if a long straight rod, or
a stretched cord possessing some rigidity, unequally elastic or of
unequal dimensions, in diflferent transverse directions, be made to
rotate very rapidly round its axis, and if vibrations be maintained
in a line at right angles to it through any point, there will result,
running along the rod or cord, waves of sensibly rectilineal trans-
verse vibrations, in a plane which in the forward progress of the
wave, turns at a uniform rate in the same direction as the rotation

27r
of the substance : and that if — be the period of rotation of the

substance, and I and m the lengths of simple pendulums respec-
tively isochronous with the vibrations of two plane waves of the
same length, a, in the planes of maximum and of minimum
elasticity of the substance, when destitute of rotation, the period
of vibration iu a wave of the same length in the substance when

made to rotate will be

27r

and the angle through which the plane of vibration turns, in the
propagation through a wave length, will be

or the number of wave lengths through which the wave is pro-
pagated before its plane turns once round, will be

where, as before,

and 0) denotes the angular velocity with which the substance is
made to rotate.

If next we suppose the rod or cord to be slightly twisted about
its axis, so that its directions of maximum and minimum elasticity
shall lie on two rectangular helifoidal surfaces {helifoides gauches),
and if, while regular rectilineal vibrations are maintained at one

576 APPENDIX F.

point of it with a period to which the wave length corresponding
is a very large multiple of the step of the screw, the substance be
made to rotate so rapi^lly as to make the velocity of a point carried
along one of the screw surfaces in a line parallel to the axis be
equal to the velocity of propagation of a wave, it is clear that a
series of sensibly plane waves will run along the rod or cord with
no rotation of the plane of vibration. The period of vibration
of a particle will be, approximately, the same as before, that is,

approximately, equal to — . Its velocity of propagation will

It

therefore be ^ , and, if s be the step of the screw, the period of

rotation of the substance, to fulfil the stated condition, must be

— , or its angular velocity — . Now it is easily seen that the

eflFects of the rapid rotation, and the effects of the slight twist,
may be considered as independently superimposed ; and therefore
the effect of the twist, with no rotation of the substance, must be
to give a rotation to the plane of vibration equal and contrary to
that which the rotation of the substance would give if there were
no twist. But the effect on the plane of vibration, due to an
angular velocity Wr of rotation of the substance, is, as we have

seen, one turn in -—y- wave lengths ; and therefore it is one turn
in 8 -j -J wave lengths when the angular velocity is — . Hence

A, 8 8

the effect of a twist amounting to one turn in a length, 8, a small

fraction of the wave length, is to cause the plane of vibration of a

wave to turn round with the forward propagation of the wave, at

n* a'
the rate of one turn in 8 -^ -, wave lengths, in the same direction

A. S

as that of a point kept on one of the screw surfaces.

From these illustrations it is easy to see in an infinite variety
of ways how to make structures, homogeneous when considered on
a large enough scale, which (1) with certain rotatory motions of
component parts having, in portions large enough to be sensibly
homogeneous, resultant axes of momenta arranged like lines of
magnetic force, shall have the dynamical property by which the
optical phenomena of transparent bodies in the magnetic field are
explained; (2) with spiral arrangements of component parts, having
axes all ranged parallel to a fixed line, shall have the axial rotaiory

INTERIOR MELTING OF ICE. 577

jyroperty corresponding to that of quartz crystal) and (3) with
spiral arrangements of component groups, having axes totally un-
arranged, shall have the isotropic rotatory property possessed by
solutions of sugar and tartaric add, by oil of turpentine, and many
other liquids.

Art. XCIV. Sui fenomeni magnetocristalline.

[iVtiovo CifMnto, IV., 1856.]
[Elkctrostatics and Magnetism, Art. xxx.]

Art. XCV. On the Alteration of Temperature accompany-
ing Changes of Pressure in Fluids.

[Proc, Roy, Soc, June, 1857 ; Phil. Mag.^ June Suppl., 1858.]

The subject of this paper is given in Mathematical and Physical
Papers, Art. XLViiL (Vol. I.).

Art. XCVI. Remarks on the interior Melting of Ice.

[Part of a letter to Prof. Stokes ; Proc. Roy. Soc. ix., Feb. 1858.]

In the Number of the Proceedings just published, which I
received yesterday, I see some very interesting experiments de-
scribed in a communication by Dr Tyndall, " On some Physical
Properties of Ice." I write to you to point out that they afford
direct ocular evidence of my brother's theory of the plasticity of
ice, published in the Proceedings of the 7th of May last ; and to
add, on my own part, a physical explanation of the blue veins
in glaciers, and of the lamellar structure which Dr Tyndall has
shown to be induced in ice by pressure, as described in the sixth
section of his paper.

Thus, my brother, in his paper of last May, says, "If we
" commence with the consideration of a mass of ice perfectly free
" from porosity, and free from liquid particles diffused through its
" substance, and if we suppose it to be kept in an atmosphere at
" or above 0° Cent., then, as soon as pressure is applied to it, pores
" occupied by liquid water must instantly be formed in the com-
" pressed parts, in accordance with the fundamental principle of
" the explanation I have propounded — the lowering, namely, of the

T. L. 37

578 APPENDIX F.

" freezing-point or melting-point, by pressure, and the fact that ice
" cannot exist at 0° Cent, under a pressure exceeding that of the
"atmosphere." Dr Tyndall finds that when a cylinder of ice is
placed between two slabs of box -wood, and subjected to gradually
increasing pressure, a dim cloudy appearance is observed, which
he finds is due to the melting of small portions of the ice in the
interior of the mass. The permeation into portions of the ice,
for a time clear, "by the water squeezed against it from such
parts as may be directly subjected to the pressure," theoretically
demonstrated by my brother, is beautifully illustrated by Dr Tyn-
dall's statement, that "the hazy surfaces produced by the com-
"pression of the mass were observed to be in a state of intense
" commotion, which followed closely upon the edge of the surface
" as it advanced through the solid. It is finally shown that these
" surfaces are due to the liquefaction of the ice in planes perpen-
" dicular to the pressure."

There can be no doubt but that the "oscillations" in the
melting-point of ice, and the distinction between strong and weak
pieces in this respect, described by Dr Tyndall in the second
section of his paper, are consequences of the varying pressures
which different portions of a mass of ice must experience when
portions within it become liquefied.

The elevation of the melting tempei^ature which my brother's
theory shows must be produced by diminishing the pressure of ice
below the atmospheric pressure, and to which I alluded as a
subject for experimental illustration, in the article describing my
experimental demonstration of the low^ering effect of pressure
(Proceedings, Roy. Soc. Edin. Feb. 1850), demonstrates that a
vesicle of water cannot form in the interior of a solid of ice
except at a temperature higher than 0' Cent. This is a conclusion
which Dr Tyndall expresses as a result of mechanical considera-
tions : thus, " Regarding heat as a mode of motion," ** liberty of
" liquidity is attained by the molecules at the surfeu^ of a mass of
" ice before the molecules at the centre of the mass can attain this
" liberty."

The physical theory shows that a removal of the atmospheric
pressure would raise the melting-point of ice by ^Jx^ths of a degree
Centigrade. Hence it is certain that the interior of a solid of
ice, heated by the condensation of solar rays by a lens, will rise
to at least that excess of temperature above the superficial parts.

STRATIFICATION OF VESICULAR ICE BY PRESSURE. 579

It appears very nearly certain that cohesion will prevent the
evolution of a bubble of vapour of water in a vesicle of water
forming by this process in the interior of a mass of ice, until a
high " negative pressure " has been reached, that is to say, until
cohesion has been called largely into operation, especially if the
water and ice contain little or no air by absorption (just as water
freed from air may be raised considerably above its boiling-point
under any non-evanescent hydrostatic pressure). Hence it appears
nearly certain that the interior of a block of ice originally
clear, and made to possess vesicles of water by the concentration
of radiant heat, as in the beautiful experiments described by
Dr Tyndall in the commencement of his paper, will rise very
considerably in temperature, while the vesicles enlarge under the
continued influence of the heat received by radiation through the
cooler enveloping ice and through the fluid medium (air and a
watery film, or water) touching it all round, which is necessarily
at 0° Cent, where it touches the solid.

I find I have not time to execute my intention of sending you
to-day a physical explanation of the blue veins of glaciers which
occurred to me last May, but I hope to be able to send it in
a short time.

Art. XCVII. On the Stratification of Vesicular Ice

BY Pressure.

[Part of a letter to Prof. Stokes ; Proc, Roy, Soc, ix., April, 1858.]

In my last letter to you I pointed out that my brother's
theory of the effect of pressure in lowering the freezing-point of
water, affords a perfect explanation of various remarkable pheno-
mena involving the internal melting of ice, described by Professor
Tyndall in the Number of the Proceedings which has just been
published. I wish now to show that the stratification of vesicular
ice by pressure observed on a large scale in glaciers, and the
lamination of clear ice described by Dr Tyndall as produced in
hand specimens by a Bramah's press, are also demonstrable as
conclusions from the same theory.

Conceive a continuous mass of ice, with vesicles containing
either air or water distributed through it; and let this mass be
pressed together by opposing forces on two opposite sides of it.

87—2

578 APPENDIX F.

" freezing-point or melting-point, by pressure, and the fact that ice
" cannot exist at 0° Cent, under a pressure exceeding that of the
"atmosphere." Dr Tyndall finds that when a cylinder of ice is
placed between two slabs of box -wood, and subjected to gradually
increasing pressure, a dim cloudy appearance is observed, which
he finds is due to the melting of small portions of the ice in the
interior of the mass. The permeation into portions of the ice,
for a time clear, "by the water squeezed against it from such
parts as may be directly subjected to the pressure," theoretically
demonstrated by my brother, is beautifully illustrated by Dr Tyn-
dall's statement, that "the hazy surfaces produced by the com-
"pression of the mass were observed to be in a state of intense
" commotion, which followed closely upon the edge of the surfEu^
" as it advanced through the solid. It is finally shown that these
" surfaces are due to the liquefaction of the ice in planes perpen-
" dicular to the pressure."

There can be no doubt but that the "oscillations" in the
melting-point of ice, and the distinction between strong and weak
pieces in this respect, described by Dr Tyndall in the second
section of his paper, are consequences of the varying pressures
which different portions of a mass of ice must experience when
portions within it become liquefied.

The elevation of the melting temperature which my brother's
theory shows must be produced by diminishing the pressure of ice
below the atmospheric pressure, and to which I alluded as a
subject for experimental illustration, in the article describing my
experimental demonstration of the lowering effect of pressure
(Proceedings, Roy, Soc, Edin, Feb. 1850), demonstrates that a
vesicle of water cannot form in the interior of a solid of ice
except at a temperature higher than 0"^ Cent. This is a conclusion
which Dr Tyndall expresses as a result of mechanical considera-
tions : thus, " Regarding heat as a mode of motion," " liberty of
" liquidity is attained by the molecules at the surface of a mass of
'* ice before the molecules at the centre of the mass can attain this
" liberty."

The physical theory shows that a removal of the atmospheric
pressure would raise the melting-point of ice by ^^tt*^^*^ ^f ^ degree
Centigrade. Hence it is certain that the interior of a solid of
ice, heated by the condensation of solar rays by a lens, will rise
to at least that excess of temperature above the superficial parts.

STRATIFICATION OF VESICULAR ICE BY PRESSURE. 579

It appears very nearly certain that cohesion will prevent the
evolution of a bubble of vapour of water in a vesicle of water
forming by this process in the interior of a mass of ice, until a
high " negative pressure " has been reached, that is to say, until
cohesion has been caUed largely into operation, especially if the
water and ice contain little or no air by absorption (just as water
freed from air may be raised considerably above its boiling-point
under any non-evanescent hydrostatic pressure). Hence it appears
nearly certain that the interior of a block of ice originally
clear, and made to possess vesicles of water by the concentration
of radiant heat, as in the beautiful experiments described by
Dr Tyndall in the commencement of his paper, will rise very
considerably in temperature, while the vesicles enlarge under the
continued influence of the heat received by radiation through the
cooler enveloping ice and through the fluid medium (air and a
watery film, or water) touching it all round, which is necessarily
at 0° Cent, where it touches the solid.

I find I have not time to execute my intention of sending you
to-day a physical explanation of the blue veins of glaciers which
occurred to me last May, but I hope to be able to send it in
a short time.

Art. XCVII. On the Stratification of Vesicular Ice

BT Pressure.

[Part of a letter to Prof. Stokes ; Proc, Roy, Soc, ix., April, 1858.]

In my last letter to you I pointed out that my brother's
theory of the effect of pressure in lowering the freezing-point of
water, affords a perfect explanation of various remarkable pheno-
mena involving the internal melting of ice, described by Professor
Tyndall in the Number of the Proceedings which has just been
published. I wish now to show that the stratification of vesicular
ice by pressure observed on a large scale in glaciers, and the
lamination of clear ice described by Dr Tyndall as produced in
hand specimens by a Bramah's press, are also demonstrable as
conclusions from the same theory.

Conceive a continuous mass of ice, with vesicles containing
either air or water distributed through it ; and let this mass be
pressed together by opposing forces on two opposite sides of it.

87—2

580 APPENDIX r.

The vesicles will gradually become arranged in strata perpen-
dicular to the lines of pressure, becaiLse of the melting of ice in tfie
localities of greatest pressure and the regelation of the water in the
localities of least pressure, in the neighbourhood of groups of these
cavities. For, any two vesicles nearly in the direction of the
condensation will afford to the ice between them a relief from
pressure, and will occasion an aggravated pressure in the ice
round each of them in the places farthest out from the line join-
ing their centres; while the pressure in the ice on the far sides
of the two vesicles will be somewhat diminished from what it
would be were their cavities filled up with the solid, although
not nearly as much diminished as it is in the ice between the
two. Hence, as demonstrated by my brother's theory and my
own experiment, the melting temperature of the ice round each
vesicle will be highest on its side nearest to the other vesicle,
and lowest in the localities on the whole farthest from the line
joining the centres. . Therefore, ice will melt from these last-
mentioned localities, and, if each vesicle have water in it, the
partition between the two will thicken by freezing on each side
of it. Any two vesicles, on the other hand, which are nearly in
a line perpendicular to the direction of pressure will agree in
leaving an aggravated pressure to be borne by the solid between
them, and will each direct away some of the pressure from the
portions of the solid next itself on the two sides farthest from
the plane through the centres, perpendicular to the line of pres-
sure. This will give rise to an increase of pressure on the whole
in the solid all round the two cavities, and nearly in the plane
perpendicular to the pressure, although nowhere else so much
as in the part between them. Hence these two vesicles will
gradually extend towards one another by the melting of the
intervening ice, and each will become flattened in towards the
plane through the centres perpendicular to the direction of pres-
sure, by the freezing of water on the parts of the bounding surface
farthest from this plane. It may be similarly shown that two
vesicles in a line oblique to that of condensation will give rise to
such variations of pressure in the solid in their neighbourhood, as
to make them, by melting and freezing, to extend, each obliquely
towards the other and/?'07/i the parts of its boundary most remote
from a plane midway between them, perpendicular to the direction
of pressure.

FORBES EXPLAINS BLUE VEINS IN GLACIERS BY PRESSURE. 581

The general tendency clearly is for the vesicles to become
flattened and arranged in layers, in planes perpendicular to the
direction of the pressure from without.

It is clear that the same general tendency must be experienced
even when there are bubbles of air in the vesicles, although no
doubt the resultant eflfect would be to some extent influenced by
the ninning down of water to the lowest part of each cavity.

I believe it will be found that these principles afford a satis-
factory physical explanation of the origin of that beautiful veined
structure which Professor Forbes has shown to be an essential
organic property of glaciers. Thus the first effect of pressure not
equal in all directions, on a mass of snow, ought to be, according
to the theory, to convert it into a stratified mass of layers of
alternately clear and vesicular ice, perpendicular to the direction
of maximum pressure. In his remarks " On the Conversion of the
Nev^ into Ice*," Professor Forbes says, " that the conversion into ice
is simultaneoiLS " (and in a particular case referred to " identical ")
'' luith the formation of the blue hands;.,, and that these bands are
" formed where the pressure is most intense, and where the dif-
'' ferential motion of the parts is a maximum, that is, near the walls
" of a glacier." He farther states, that, after long doubt, he feels
satisfied that the conversion of snow into ice is due to the effects
of pressure on the loose and porous structure of the former ; and
he formally abandons the notion that the blue veins are due to
the freezing of infiltrated water, or to any other cause than the
kneading action of pressure. All the observations he describes
seem to be in most complete accordance with the theory indicated
above. Thus, in the thirteenth letter, he says, "the blue veins
" are formed where the pressure is most intense and the differential
" motion of the parts a maximum."

Now the theory not only requires pressure, but requires differ-
ence of pressure in different directions to explain the stratification
of the vesicles. Difference of pressure in different directions pro-
duces the "differential motion" referred to by Professor Forbes.
Further, the difference of pressure in different directions must be
continued until a very considerable amount of this differential
motion, or distortion, has taken place, to produce any sensible
degree of stratification in the vesicles. The absolute amount of
distortion experienced by any portion of the viscous mass is

* Thii-tceuth Letter ou Glaciers, section (2), dated Dec. 184G.

5>2 APPEXDIX F.

iherefon: an iz>dex c^f the persistence of the diffieienttfti preKoie.
bv the vvntinued ftcncA of vhich the blue x&ns are iwhifwl
HeiKv als^^ vr s^^ vfav blue veins aie not fonned in any mas,
ever <o deefv of sno« fyatimo in a hcdlow or com^.

As TO the dii^cQon in which the Une reins mppear to lie, tibej
must, akWMuing t-c- the theonr. be something intennediate betweeo
the surfaces perf^ndicTiiar to the greatest preasnre, and the scr-
&ces of s!idin&: : >inc^ ibev wiU commence being fonned exacdr
perf^'ndiciilar to ;he dineciion of gneatest preasore, and will, bt
the didPex^nti*! motion acoompanving their formatMXi, become
gradualiT laid v^nt more and moie nearlj parallel to the sides of
the channel Umnigh which the glacier is foioed. This circnm-
stance. alvHig with the cv^mpaiativelv weak mechanical condition
of the white straxa vesicular iayeR between the blue straSa^i
must. I think, make ihese white strata become ultimatelj. in
Kvtliiy, the surfaces c^f "sliding'' or of " tearing,' or of chief
diffei>^ntia! motion, as acx»niin£: to Professor Forbes's obeerrations
they seem to K\ His nist statement on the subject, made as
early as 1S42, thai "the blue veins seem to be perpendicolar ic
the lines v>i maxiuium pressunE^.' is. however, more in aocordanciL
with :hcir nit^rhanioal oii^iu, according to the theonr I nov
suggt^si. than :he supposition ihat they are oaugtd by the tearing
aciion which :s found lo take place along them when formed. It
appears :o r^c. ihcrvtV-r^, that Dr Tyiidaii's conclusion, that tht
xesioular s;i:a tinea: ion is prcouoM by piressuie in surfaces per-
pendicular :o the dirtv^Tiv-ns of max^mam pressure, is correct as
recMvls the mechanical vriiTJi ^i the veiced structure; while there
seems everv reason. tv:h ttva\ vtiservaiion and from mechanical
the\^". to acxxpt the >'iew given by Prc-fea&iM" Forbes <rf their
function in c-aoial moiion.

The mechanical tbeoiy I have indicaied as the explanadoL
of tht veined structure c f glacial ice is especially applicable t^
account for the snatincation oi ihe vesicles observed in ice
originally clear, and subject€ti to dinerential pressure, by Dr
Tyndall : the fomiadoii oi the vesicles themselves being, as re-
marked in my last letter*, anticipsiteu by my brother s theor}*
published in the Pr.K^i rjt for May. ISoT.

I believe the the\rv I have ci'^^en ab-jve contains the true
explanation of one remarkable fact observed hv Dr Tyndall in

tyndall's vesicles in clear ice at focus of kadiant heat. 583

connexion with the beautiful set of phenomena which he dis-
covered to be produced by radiant heat, concentrated on an
internal portion of a mass of clear ice by a lens; the fact, namely,
that the planes in which the vesicles extend are generally parallel
to the sides when the mass of ice operated on is a flat slab ; for
the solid will yield to the "negative** internal pressure due to
the contractility of the melting ice, most easily in the direction
perpendicular to the sides. The so-called negative pressure is
therefore least, or which is the same thing, the positive pressure
is greatest in this direction. Hence the vesicles of melted ice,
or of vapour caused by the contraction of melted ice, must, as
I have shown, tend to place themselves parallel to the sides of
the slab.

The division of the vesicular layers into leaves like six-petaled
flowers is a phenomenon which does not seem to me as yet so
easily explained; but I cannot see that any of the phenomena
described by Dr Tyndall can be considered as having been proved
to be due to ice having mechanical properties of a uniaxal crystal.

[It now seems to me most probable Tyndall was right in
attributing the six-rayed structure to the moleculai* mechanics of
a uniaxal crystal. K., Dec, 13, 1903.]

APPENDIX G.

HYDROKINETIC SOLUTIONS AND OBSERVATIONS.

Part I. Or tke Motion of Free Solids tkrmgh a LiqukL

This paiper oommences with the following extract from the
aathor*s prirmte jounuJ. of date JaDoarr 6. 1858: —

" Let S, 1^. 2;, l. i&y ^ be rectangular compoo^its of an
impukive foroe and an impulsive coaple applied to a solid of
invariable shape, with or without inertia of its own, in a perfect
liqoid, and let m, r. ir, v. p, o- be the components of linear and
angular velocity generated. Then, if the rif rica^ (twice the
mechanical value) of the whole motic«i be, as it cannot bat be.
given by the expression

Q = [m, ii]M- + [r, r]f^-|-...-i-2[r, ii]r(f + 2[ir, ii]vic+...

where =[m. y], [r, r]. AlC, denote 21 constant coefficients deter-
minable bv trandcendeDtal analvsis h\>m the form of the surfiw^
of the solid, probably involving only elliptic transeendentals when
the surOEu^e is ellipsoidal : involving, of coarse, the moments of
inertia of the solid itself: we must have

[u, w] u -r [r, u]r-^ [w. u] ir + [w, u] «^ + [p, «] p + [a, m] «• = 3E, ic.

[M,v]u + [r,v]r + [ir.ir]if + >,ir]v+[p.v]p + [a.v]ir = U,^-.

If now a continuous force X, 1^ Z, and a continuous coaple
L, M, X, referred to axes fixed in the body, are applied, and if
S, ..., ^c denote the impulsive force and couple capable of

* Puu I. &nd Q. from the PruCftdimg* of the Roy^l Soeietff of Edimkmrfk,
1670-71. Partff III. and FV. from letters to Profenor Tait of AogiKt l^^TL
Pmn T. from Phil. Mas- Norrmber. 1871.

In Pmn n.. T, inste«d of 4 Q. is naed to denote the ** mechanicftl raloe,** or, as it
IS oov called, the " kinetic eueigr " of the motion.

MOTION OF A FREE SOLID THROUGH LIQUID.

585

generating from rest the motion w, v, w, «r, p, <7, which exists in
reality at any time t ; or, merely mathematically, if X &c. denote
for brevity the preceding linear functions of the components of
motion, the equations of motion are as follow: —

dW^

dt

= &c., \

dt

- Xv +|?w- %p +i»t!r=iV

.(1).

Three first integrals, when

X = 0, F=0, ^ = 0, Z = 0, if=0, iV^ = 0...(2).

must of course be, and obviously are,

3£* + ^» + ®' = const (3),

resultant momentum constant ;

UX + i«|? + iaSS = const (4),

resultant of moment of momentum constant ; and

uX + 1;|^ + ^'Z + ^% + p^Vl + a£i = Q = const. . . .(5)."

These equations were communicated in a letter to Professor
Stokes, of date (probably January) 1858, and they were referred
to by Professor Rankiue, in his first paper on Stream-lines,
communicated to the Royal Society of London*, July 1863.

They are now communicated to the Royal Society of Edin-
burgh, and the following proof is added: —

Let P be any point fixed relatively to the body ; and at time t,
let its coordinates relatively to axes OX, OY, OZ, fixed in space,
be x, y, z. Let PA, PB, PC be three rectangular axes fixed
relatively to the body, and (A, X), {A, F), ... the cosines of the
nine inclinations of these axes to the fixed axe3 OX, OF, OZ,

* These equations will be very conveniently called the Eulerian equations of the
motion. They correspond precisely to Euler's equations for the rotation of a rigid
body, and include them as a particular case. As Euler seems to have been the first
to give equations of motion in terms of coordinate components of velocity and force
referred to lines fixed relatively to the moving body, it will be not only convenient,
but just, to designate as *' Eulerian equations *' any equations of motion in which the
lines of reference, whether for position, or velocity, or moment of momentum, or
force, or couple, move with the body, or the bodies, whose motion is the subject.

586

APPENDIX G.

Let the components of the "impulse*" or generalized mo-
mentum parallel to the fixed axes be f, 17, f, and its moments
round the same axes \, fiy v\ so that if X, Y, Z he components
of force acting on the solid, in line through P, and X, M, N
• components of couple, we have

df_,, ^^v ^-y
dt'^' dt"^' dt' '

Let 3E, ^, 'Z and %, ifSt, JS be the components and moments
of the impulse relatively to the axes PA, PB, PC moving with
the body. We have

f=X(4,Z)-h|9(5,Z) + S£(0,Z), \

\ = 1L(AZ) + iW(5,Z) + ia(C,X) + Z.y-|^^,

>...(7).

Now let the fixed axes OX, OY, OZ be chosen coincident
with the position at time t of the moving axes PA, PB, PC: we
shall consequently have

x = Qy y = 0, ^ = 0,

dz

dx _ ^y ^ _

dt ^ dt ' dt
{A,X)^{B,Y) = {C,Z) = l,
{A, Y) = {A,Z) = (B, X) = {B, Z) = (C,X) = (C. Y) = 0,

(8).

= «".

diA,Y)
dt

diA.Z)^_
di

d jB, X)
dt

d{Bj_Z)_
dt

d{G,X)_

-„ d{C,Y)_
-*'■ —dt "^

'...(9).

Using (7), (8), and (9) in (6), we find (1).

One chief object of this investigation was to illustrate dyna-
mical effects of helitj'oidal property (that is, right or left-handed
asytnnietry). The case of complete isotropy, with helifoidal
quality, is that in which the coefficients in the quadratic ex-
pression Q fulfil the following conditions: —

* See '■ Vortex Motion," § 6, Traiu. Roy. Hoc. Edin. (ItMW).

AXES FIXED RELATIVELY TO BODY OK FIXED ABSOLUTELY. 587

»

}}

if

i>

a

»

...(10);

\u, u\ = [v, v\ = [Wy v)\ (let m be their common value)/

[cr,tar] = [/>, /»] = [<7, <7] „ 7i

[«,t!r] = [y, p\=^[w,<t] „ h

[v, ti;] = [u;, m] = [w, i;] = 0; [/>, o"] = [<7, tar] = [tar, /»] = 0,

and

[w., />] = [a, <7] = [v, <r\ = [v, tsr] = [w, tsr] = [t(;, /»] = 0

so that the formula for Q is

Q = 7/i (i^^ + ir» + 1^) + n (tsr'^ + /^'^ + a^) + 2A (utsr + v/> + wcr). . .(11).
For this case, therefore, the Eulerian equations (1) become
d{mu-\'hvi)

and

dt

d (nw + hu)
Jt

- VI (va- — wp) = Xy &c.,

= Z, &c.

>-...(ir).

[Memorandum : — Lines of reference fixed relatively
to the body.] r

But inasmuch as (11") remains unchanged when the lines of
reference are altered to any other three lines at right angles to
one another through P, it is easily shown directly from (6), (7),
and (9) that if, altering the notation, we take u, v, w to denote
the components of the velocity of P parallel to three fixed
rectangular lines, and ^, p, a the components of the body's
angular velocity round these lines, we have

\

\ ...(12),

d(mw + Atj) Y Q d{mv-\-hp) ^
~dt =^'^^- —dt ^^'

, d (ntj + hu) , , V r 0

and — ^ — -7- + h {<ro - pw) = L, &c.

[Memorandum : — Lines of reference fixed in space],,
which are more convenient than the Eulerian equations.

The integration of these equations, when neither force nor
couple acts on the body (X = 0 &c., i = 0 &c.), presents no
difficulty ; but its result is readily seen from § 21 (" Vortex
Motion") to be that, when the impulse is both translatory and
rotational, the point P, round which the body is isotropic, moves
uniformly in a circle or spiral so as to keep at a constant
distance from the " axis of the impulse," and th<at the components

588 APPENDIX G.

of angiilur velocity round the three fixed rectangular axes are
constant.

An isotropic heli^oid [chiroid as I now call it ; Lee. XX.,
§ 204] may be made by attaching projecting vanes to the surface
of a globe in proper positions ; for instance, cutting at 45*^ each,
at the middles of the twelve quadrants of any three great circles
dividing the globe into eight quadrantal triangles. By making
the globe and the vanes of light paper, a body might probably be
obtained rigid enough and light enough to illustrate by its motions
through air the motions of an isotropic heli^oid through an incom-
pressible liquid. But curious phenomena, not deducible from the
present investigation, will, no doubt, on account of viscosity, be
observed.

Part II.

Still considering only one movable rigid body, infinitely
remote from disturb«ince of other rigid bodies, fixed or movable,
let there be an aperture or apertures through it, and let there be
irrotational circulation or circulations (§ 60, " Vortex Motion ")
through them. Let f , 17, f be the components of the " impulse "
at time t, parallel to three fixed axes, and \, /t, 1/ its moments
round these axes, as above, with all notation the same, we still
have (§ 26, " Vortex Motion '*)

(6) (repeated).

^- = Z + Zi/-F^, &c.

But, instead of for ^Q a quadratic function of the components of
velocity as before, we now have

r=i; + i{[M, a] ?**+... + 2 [m, v]hv-\' ...] (13),

where E is the kinetic energy of the fluid motion when the solid
is at rest, and ^ {[u, u] n^ -^ , , ,} is the same quadratic as before.
The coefficients [u, u], [a, v], &c. are determinable by a tran-
scendental analysis, of which the character is not at all influenced
by the circumstance of there being apertures in the solid. And

instead of f = -7- , &c., as above, we now have

du

LIQUID CIRCULATING THROUGH HOLE IN MOVABLE BODY. 589

dT •"^^*^'

X. = ;^ — h / {ny — mz) -¥01, fi = &c., p = &c

where / denotes the resultant "impulse" of the cyclic motion
when the solid is at rest, I, m, n its direction-cosines, G its
" rotational moment " (§ 6, " Vortex Motion "), and x, y, z the
coordinates of any point in its "resultant axis.'* These, (14)
with (13), used in (6) give the equations of the solid's motion
referred to fixed rectangular axes. They have the inconvenience
of the coefficients [i/, w], [m, v], &c. being functions of the angular
coordinates of the solid. The Eulerian equations (free from this
inconvenience) are readily found on precisely the same plan as
that adopted above for the old case of no cyclic motion in the
fluid.

The formulae for the case in which the ring is circular, has no
rotation round its axis, and is not acted on by applied forces,
though, of course, easily deduced from the general equations
(14), (13), (6), are more readily got by direct application of first
principles. Let P be such a point in the axis of the ring, and
(ST, -4, B such constants that J (CTcd' -h Au^ + B^f^) is the kinetic
energy due to rotational velocity cd round D, any diameter
through P, and translational velocities u along the axis and v
perpendicular to it. The impulse of this motion, together with
the supposed cyclic motion, is therefore compounded of

,. ^, , y. (ilw + / alone the axis,
momentum m hues throucfh F <^. i. , ,

° {Bv perpendicular to axLs,

and moment of momentum CTcd round the diameter D,

Hence if OX be the axis of resultant momentum, (a?, y) the
coordinates of P relatively to fixed axes OX, 0Y\ 6 the inclina-
tion of the axis of the ring to 0 ; and f the constant value of the
resultant momentum, we have

^cosO^Au + I; -fsin^ = 5t;; ^ = CP
and x = u cos O — v sin 0\ y = usin0 -{-v cos 0

. L.(15).

; 0^(0}

Hence for 0 we have the differential equation

/sin e + :4_J?f sin 2^1=0 (16\

Mr^d?0 ^

590 APPENDIX G.

which shows that the ring oscillates rotationally according to the
law of a horizontal magnetic needle canying a bar of soft iron
rigidly attached to it parallel to it^ magnetic axi&

When 0 is and remains infinitely small, 6, y, and y are each
infinitely small, x remains infinitely nearly constant, and the
ring experiences an oscillatory motion in period

"^ y [I ^^(A ^ B)x:\{i ^^ AxY

compounded of translation along OY and rotation round the
diameter 2). This result is curiously comparable with the well-
known gyroscopic vibrations.

Part IIL The Influence of firictionless Wind on Waves in water
supposed JridionUss. {Letter to Professor Tait, of date
August 16, 1871.)

Taking OX vertically downwards and OY horizontal, let

jr = A sin n(y — erf) (1)

be the equation of the section of the water by a plane perpen-
dicular to the wave-ridges; and let A (the half ¥rave-height) be

infinitely small in comparison with " (the wave-length). The

j>component of the velocity of the water at the surfiice is then

— nah cos N (y — erf) (iV

dA
This (because h is infinitesimah must be the value of — ^ for

the point (0, y\ if ^ denote the veKvity-potential at any point
(jp, y) of the water. Now because*

djT * rfy* ~

aixi ^ is a periodic function of y, and a function of x which (as
we suppose the water infinitely deep^ becomes zero when x = x .
it must bo of the form

Pcostiiy~^l€~*'.
where P and e arv independent ot* x and y. Hence, taking -j- .
putting ^^ = 0 in it, and e^^uaiing it to \t), we have

— P*i Q\y^ \ f^ » — f > = — waA c«"*s • wy — fiaf^ :

FRICTIONLESS INFLUENCE OF WIND ON WAVES IN WATER. 591

and therefore P = a//, and e = nat ; so that we have

<^ = aA€"*»* cos n (y — flrf) (3).

This, it is to be remarked, results simply from the assumptions
that the water is frictionless, that it has been at rest, and that its
surface is moving in the manner specified by (1).

If the air were a frictionless liquid moving irrotationally, with
a constant velocity V at heights above the water (that is to say,
values of — x) considerably exceeding the wave-length, its velocity-
potential 1^, found on the same principle, would be

-i^=(F-a)/i€"*co8n(y-aO+ Vy (4).

Let now q denote the resultant velocity at any point (a?, y) of the
air. Neglecting infinitesimals of the order (nA)*, we have

Jgr3^^F--F(F-a)nA€"*sin7i(y-Qrf) (5).

Now, if p denote the pressure at any point (a?, y) in the air, and
a the density of the air, we have by the general equation for
pressure in an irrotationally moving fluid,

C-2) = <r(^-hi9»-5r^) (6).

Using (4) and (5) in this and putting C= JcrF*, we find

|} = <r {nA ( F - a)»€*^ sin n (y — flrf) -h 5ra:} (7 ).

Similarly if p' denote the pressure at any point (a?, y) of the
water, since in this case c^ is infinitesimal, we have

— p' = -^—5ric = nAa*€~^ sin n(y'-at)—gx (8),

the density of the water being taken as unity.

Now let T be the cohesive tension of the separating surfiEu^e of

air and water. The curvature of this surface being -v-^ derived

from (1), is equal to

— nVt sin ?i(j? — at) (9).

Hence at any point in the water-surfiwe,

p-p'=rn«Asinn(«-flrf) (10);

and by (7) and (8), with for x its value by (1) (which, as A is

infinitesimal, may be taken as zero except in their last terms), we

have

p-p =h{n[(r{V- ay + a!']-g{l''a)}8mn{x-at)... (11).

This, compared with (10), gives

n[cr(F-a)' + a«]-^(l-<7)=7Vi« (12).

592 APPENDIX G.

Let ,£;= /^A^ — ' nS),

which (being the value of a for F=0) is the velocity of propa-

gation of waves with no wind, when the wave-length Ls — .
Then (12) becomes

\\a '' whence a» = (l + o-)w»- o-(F- «)»... (14).

This determines a, the velocity of the waves when there is wind,
of velocity F, in the «lirection of their advance. It shows that,
for given wave-length, ^tt/w, the greatest wave-velocity is

which is reached when this is the velocity of the wind. It is
interesting to see that with wind of any other speed than that of
the waves, and in the direction of the waves, their speed is less.
For instance, the wave-speed with no wind, which is w, is less by
approximately | <r of w, (or about 1/1650,) than the speed when the
wind is with the waves and of their speed. The explanation
clearly is that when the air is motionless relatively to the wave
crests and hollows its inertia is not called into play. Solving (14)
for a, as a quadratic, we have

This result leads to the following conclusions : —

(1) When ^/«' = /v^^2«-7x(l + ^^^).

one of the values of a is zero, that is to say, static corrugations
of wave-length 2ir/w, would be equilibrated by wind of velocity

w V[(l H- cr)/o-].

But the equilibrium would be unstable.

(2) When >^/«' = '7/-287x(l+gL).
the two values of a are equal.

(3) When 17f(;>?-t5,

both values of a are imaginary, and therefore the wind would blow
into spin-drift, waves of length 27r/w or shorter.

WAVES AND RIPPLES. 593

Looking back to (13), we see that it gives a minimum value
for w equal to _ _ _

y/'-^'^^ <■«)■

Hence the water with a plane level surface would be unstable,
even if air were frictionless, when the velocity of the wind exceeds

/2V^f^-a') (jg,^

W. T.

Part IV. (Letter to Professor Tait, of daU August 23, 1871.)

Defining a ripple as any wave on water whose length

/2"
<2ir^Y*, where

, 1-<T

s'=^i-t;

and r =

l+a

(17);

(<r= 00121), you always see an exquisite pattern of ripples in
front of any solid cutting the surface of water and moving
horizontally at any speed, fast or slow [if not less than about
23 cm. per sec.]. The ripple-length is the smaller root of the
equation

\''^'^h^-'^ <!«)'

where w is the velocity of the solid. The latter may be a sailing-
vessel or a row-boat, a pole held vertically and moved horizontally,
an ivory pencil-case, a penknife-blade either edge or flat side
foremost, or (best) a fishing-line kept approximately vertical by
a lead weight hanging down below water, while carried along at
about half a mile per hour by a becalmed vessel. The fishing-line
shows both roots admirably ; ripples in front, and waves of same
velocity (X the greater root of same equation) in rear. If so
fortunate as to be becalmed again, I shall try to get a drawing of
the whole pattern, showing the transition at the sides from ripples

* Whioh for pure waters 1*7 centim. (see Part V.).
T. L. 38

594 APPEXDIX G.

to waves. When the speed with which the fishing-liDe is dragged
is diminished towards the critical velocity

s/2 s^^T\

which is the minimum velocity of a wave, being [see Pan V.
below] for pure water 23 centim& per second <or ^^^ of a

nautical mile per hour), the ripples in front elongate and become
leas carved, and the waves in rear become shorter, till at the
critical velocity waves and ripples seem neariy eqoal, and with
ridges neariy in straight lines perpendicular to the line of motioD.
(This is observation.) It seems that the critical velocity may
be determined with some accuracy by experiment thus [see
Pkrt V. below]:—

Remark that the shorter the ripple-length the greater is the
velocity of propagation: and that the moving f<»ce of the ripple-
motion is pa^ly gravity, but chiefly cohesion; and with veiy
short ripple-length it is almost altogether cohesicm, i^ the same
force as that which nudges a dew-drop tremble^ The least
velocity of frictionless air that can raise a ripple on rigorously
quiescent frictionless water is [(16'> above]

660 centimetres per second

1 being x minimum wave- velocity |

= 12'8 nautical miles per. hour.

Observation shows the sea to be ruffled bv wind of a much

smaller velocity than this. Such ruffling, therefore, is due to

\iscositv of the air.

W. T.

Postsenpi to Part IV. {October 17, 1871)l

The influence of viscosity gives rise to a greater pressure on
the anterior than on the posterior side of a solid moving uniformlv
relatively to a fluid. A symmetrical solid, as for example a
globe, moving uniformly through a frictionless fluid, experiences
augmentation of pressure in front and behind equally: and
diminished pressure over an intervening zone. Observation (as

STOKES'S WORK AGAINST VISCOSITY TO MAINTAIN WAVE. 595

for instance in Mr J. R. Napier's experiments on his "pressure
log," for measuring the speed of vessels, and experiments by
Joule and myself*, on the pressure at diflferent points of a solid
globe exposed to wind) shows that, instead of being increased,
the pressure is sometimes actually diminished on the posterior
side of a solid moving through a real fluid such as air or water.
Wind blowing across ridges and hollows of a fixed solid (such as
the furrows of a field) must, because of the viscosity of the air,
press with greater force on the slopes facing it than on the
sheltered slopes. Hence if a regular series of waves at sea
consisted of a solid* body moving with the actual velocity of the
waves, the wind would do work upon it, or it would do work
upon the air, according as the velocity of the wind were greater
or less than the velocity of the waves. This case does not afford
an exact parallel to the influence of wind on waves, because the
surface particles of water do not move forward with the velocity
of the waves as those of the furrowed solid do. Still it may be
expected that when the velocity of the wind exceeds the velocity
of propagation of the waves, there will be a greater pressure on
the posterior slopes than on the anterior slopes of the waves;
and vice versdy that when the velocity of the waves exceeds the
velocity of the wind, or is in the direction opposite to that of
the wind, there will be a greater pressure on the anterior than
on the posterior slopes of the waves. In the first case the
tendency will be to augment the wave, in the second case to
diminish it. The question whether a series of waves of a certain
height gradually augment with a certain force of wind or gradually
subside through the wind not being strong enough to sustain
them, cannot be decided off'hand. Towards answering it Stokes's
investigation of the work against viscosity of water required to
maintain a wavef, gives a most important and suggestive instal-
ment. But no theoretical solution, and very little of experimental
investigation, can be referred to with respect to the eddyings of
the air blowing across the tops of the waves, to which, by its
giving rise to greater pressure on the posterior than on the

♦ "Thermal Effects of Fluids in Motion," Trans. Roy, Soc, 1860; Phil. Mag.
1860, Vol. XX. p. 662.

t Trann. Camb. Phil. Soc. 1861 (•• Effect of Internal Friction of Fluids on the
Motion of Pendulums," § V.).

38—2

596 APPENDIX Q.

anterior slopes, the iDfluence of the wind in sustaining and
maintaining waves is chiefly if not altogether due.

My attention having been called three days ago, by Mr Froude,
to Scott Russell's Report on Waves (British Association, York,
1844), I find in it a remarkable illustration or indication of the
leading idea of the theory of the influence of wind on waves,
that the velocity of the wind must exceed that of the waves, in
the following statement: — ^'^Let him [an observer studying the
surface of a sea or large lake, during the successive stages of an-
increasing wind, from a calm to a storm] begin his observations
'' in a perfect calm, when the surface of the water is smooth and
"reflects like a mirror the images of surrounding objects. This
" appearance will not be affected by even a slight motion of the
" air, and a velocity of less than half a mile an hour (8^ in. per
" sec.) does not sensibly disturb the smoothness of the reflecting
"surface. A gentle zephyr flitting along the surface from point
"to point, may be observed to destroy the perfection of the
"mirror for a moment, and on departing, the surface remains
" polished as before ; if the air have a velocity of about a mile an
" hour, the surface of the water becomes less capable of distinct
"reflexion, and on observing it in such a condition, it is to be
"noticed that the diminution of this reflecting power is owing
" to the presence of those minute corrugations of the superficial
"film which form waves of the third order. These corrugations
" produce on the surface of the water an effect very similar to the
" effect of those panes of glass which we see corrugated for the
" purpose of destroying their transparency, and these corrugations
" at once prevent the eye from distinguishing forms at a consider-
"able depth, and diminish the perfection of forms reflected in
"the water. To fly-fishers this appearance is well known as
"diminishing the facility with which the fish see their captors.
"This first stage of disturbance has this distinguishing circum-
" stance, that the phenomena on the surface cease almost
"simultaneously with the intermission of the disturbing cause,
"so that a spot which is sheltered from the direct action of the
"wind remains smooth, the waves of the third order being in-
" capable of travelling spontaneously to any considerable distance,
" except when under the continued action of the original disturb-
"ing force. This condition is the indication of present force,
" not of that which is past. While it remains it gives that deep

SCOTT RUSSELL'S CAPILLARY WAVES. 597

'' blackness to the water which the sailor is accustomed to regard
" as an index of the presence of wind, and often as the forerunner
" of more.

" The second condition of wave motion is to be observed when
"the velocity of the wind acting on the smooth water, has in-
" creased to two miles an hour. Small waves then begin to rise
" uniformly over the whole surface of the water ; these are waves
"of the second order, and cover the water with considerable
" regularity. Capillary waves disappear from the ridges of these
"waves, but are to be found sheltered in the hollows between
" them, and on the anterior slopes of these waves. The regularity
" of the distribution of these secondary waves over the surface is
" remarkable ; they begin with about an inch of amplitude, and
" a couple of inches long ; they enlarge as the velocity or duration
" of the wave increases ; by and by conterminal waves unite ; the
"ridges increase, and if the wind increase the waves become
" cusped, and are regular waves of the second order. They con-
" tinue enlarging their dimensions ; and the depth to which they
"produce the agitation increasing simultaneously with their
" magnitude, the surface becomes extensively covered with waves
" of nearly uniform magnitude."

The " Capillary waves " or " waves of the third order " referred
to by Russell are what I, in ignorance of his observations on this
branch of his subject, had called "ripples." The velocity of
8^ inches (21 J centimetres) per second is precisely the velocity
he had chosen (as indicated by his observations) for the velocity
of propagation of the straight-ridged waves streaming obliquely
from the two sides of the path of a small body moving at speeds
of from 12 to 36 inches per second ; and it agrees remarkably
with my theoretical and experimental determination of the
absolute minimum wave- velocity (23 centimetres per second;
see Part Y.). Russell has not explicitly pointed out that his
critical velocity of 8| inches per second was an absolute minimum
velocity of propagation. But the idea of a minimum velocity of
waves can scarcely have been far from his mind when he fixed
upon 8^ inches per second as the minimum of wind that can
sustain ripples. In an article to appear in Nature on the
26th of this month, I have given extracts frx)m Russell's Report
(including part of a quotation which he gives from Poncelet
and Lesbros in the memoirs of the French Institute for 1829),

598 APPENDIX G.

showiDg how far my observations on ripples had been anticipated.
I need say no more here than that these anticipations do not
include any indication of the dynamical theory which I have
given, and that the subject was new to me when Parts IIL, IV.,
and V. of the present communication were written.

Part V. Waves under motive power of Gravity and
Cohesion jointly, without wind.

Leaving the question of wind, consider (13), and introduce
notation of (16), (17) in it. It becomes

vj'^^-^Tn (19).

n ^ ^

This has a minimum value,

when

" = \/r J

.(20).

In applying these formulae to the case of air and water, we
may neglect the difference between g and g\ as the value of a is
about j^\ and between T and T\ although it is to be remarked
that it is T rather than T that is ordinarily calculated from
experiments on capillary attraction. From experiments of Gay-
Lussac's it appears that the value of T' is about '074 of a
gramme weight per centimetre; that is to say, in terms of the
kinetic unit of force founded on the gramme as unit of mass,

T = gx 073.

To make the density of water unity (as that of the lower liquid
has been assumed), we must take one centimetre as unit of length.
Lastly, with one secoud as unit of time, we have

5r = 982;
and (18) gives

= a/982 (i + -074 X n\

for the wave- velocity in centimetres per second, corresponding

'V 1 /

to wave-length . When - = v073 = -27 (that is, when the

w

n n

FRINGE OF RIPPLES IN FRONT, SHIP-WAVES ABAFT, 599

wave-length is 1*7 centimetre), the velocity has a minimum value
of 23 centimetres per second.

The part of the preceding theory which relates to the eflfect of
cohesion on waves of liquids occurred to me in consequence of
having recently observed a set of very short waves advancing
steadily, directly in front of a body moving slowly through water,
and another set of waves considerably longer following steadily
in its wake. The two sets of waves advanced each at the same
rate as the moving body; and thus I perceived that there were
two different wave-lengths which gave the same velocity of pro-
pagation. When the speed of the bod/s motion through the
water was increased, the waves preceding it became shorter, and
those in its wake became longer. Close before the cut-water of
a vessel moving at a speed of not more than two or three knots*
through very smooth water, the surface of the water is marked
with an exquisitely fine and regular fringe of ripples, in which
several scores of ridges and hollows may be distinguished (and
probably counted, with a little practice) in a space extending 20
or 30 centimetres in advance of the solid. Right astern of either
a steamer or sailing vessel moving at any speed above four or
five knots, waves may generally be seen following the vessel at.
exactly its own speed, and appearing of such lengths as to verify
as nearly as can be judged the ordinary formula

,_2^
9

for the length of waves advancing with velocity w, in deep
water. In the well-known theory of such waves, gravity is
assumed as the sole origin of the motive forces. When cohesion
was thought of at all (as, for instance, by Mr Froude in his
important nautical experiments on models towed through water,
or set to oscillate to test qualities with respect to the rolling of
ships at sea), it was justly judged to be not sensibly influential
in waves exceeding 5 or 10 centimetres in length. Now it
becomes apparent that, for waves of any length less than 5 or
10 centimetres, cohesion contributes sensibly to the motive
system ; and that, when the length is a small fraction of a

* The speed *od6 knot* is a velocity of one nautical mile per hoar, or 61*6 centi-
metres per second.

600 APPENDIX G.

centimetre, cohesion is much more influential than gravity as
"motive" for the vibrations.

The following extract from part of a letter to Mr Froude,
forming part of a communication to Nature (to appear on the
26th of this month, October 1871), describes observations for an
experimental determination of the minimum velocity of waves in
sea- water : —

"About three weeks later, being becalmed in the Sound of
"Mull, I had an excellent opportunity, with the assistance of
"Professor Helmholtz, and my brother fix)m Belfast, of deter-
"mining by observation the minimum wave-velocity with some
" approach to accuracy. The fishing-line was hung at a distance
" of two or three feet from the vessels side, so as to cut the water
"at a point not sensibly disturbed by the motion of the vessel
"The speed was determined by throwing into the sea pieces of
"paper previously wetted, and observing their times of transit
"across parallel planes, at a distance of 912 centimetres asunder,
" fixed relatively to the vessel by marks on the deck and gunwale.
"By watching carefully the pattern of ripples and waves which
" connected the ripples in front with the waves in rear, 1 had seen
" that it included a set of parallel waves slanting off obliquely on
" each side, and presenting appearances which proved them to be
"waves of the critical length and corresponding minimum speed
" of propagation. Hence the component velocity of the fishing-
"line perpendicular to the fronts of these waves was the true
"minimum velocity. To measure it, therefore, all that was
"necessary was to measure the angle between the two sets of
"parallel lines of ridges and hollows sloping away on the two
" sides of the wake, and at the same time to measure the velocity
"with which the fishing-line was dragged through the water.
"The angle was measured by holding a jointed two-foot rule,
" with its two branches, as nearly as could be judged by the eye,
" parallel to the set of lines of wave ridges. The angle to which
"the ruler had to be opened in this adjustment was the angle
"sought. By laying it down on paper, drawing two straight
"lines by its two edges, and completing a simple geometrical
"construction with a length properly introduced to represent
" the measured velocity of the moving solid, the required
" minimum wave-velocity was readily obtained. Six observa-
"tions of this kind were made, of which two were rejected as

MINIMUM VELOCITY OF SOLID TO PRODUCE RIPPLES OR WAVES. 601

"not satisfactory. The following are the results of the other
" four : —

Velocity of moTing
solid.

51 centimetres per second.

38

26

24

Mean.

f>

n

*i

>»

i>

Deduced minimam wave-
velocity.

23'0 centimetres per second.

23-8

23-2

22-9

»

ti

>*

t»

»>

.23-22

"The extreme closeness of this result to the theoretical
"estimate (23 centimetres per second) was, of course, merely a
" coincidence ; but it proved that the cohesive force of sea- water
"at the temperature (not noted) of the observation cannot be
" very diflferent from that which I had estimated fix)m Qay-
"Lussac's observations for pure water."

APPENDIX H.

ON THE MOLECULAR TACTICS OF A CRYSTAL*

§ 1. My subject this evening is not the physical properties
of crystals, not even their dynamics ; it is merely the geometry of
the structure — the arrangement of the molecules in the constitu-
tion of a crystal. Every crystal is a homogeneous assemblage of
small bodies or molecules. The converse proposition is scarcely
true, unless in a very extended sense of the term crystal (§ 20
below). I can best explain a homogeneous assemblage of molecules
by asking you to think of a homogeneous assemblage of people.
To be homogeneous every person of the assemblage must be equal
and siipilar to every other : they must be seated in rows or stand-
ing in rows in a perfectly similar manner. Each person, except
those on the borders of the assemblage, must have a neighbour
on one side and an equi-distant neighbour on the other : a neigh-
bour on the left fi'ont and an equi-distant neighbour behind on
the right, a neighbour on the right front and an equi-distant
neighbour behind on the left. His two neighbours in front and his
two neighbours behind are members of two rows equal and similar
to the rows consisting of himself and his right-hand and left*hand
neighbours, and their neighbours* neighbours indefinitely to right
and left. In particular cases the nearest of the front and rear
neighbours may be right in front and right in rear; but we must
not confine our attention to the rectangularly grouped assemblages
thus constituted. Now let there be equal and similar assemblages
on floors above and below that which we have been considering,
and let there be any indefinitely great number of floors at equal
distances from one another above and below. Think of any one
person on any intermediate floor and of his nearest neighbours on
the floors above and below. These three persons must be exactly

* The Robert Boyle Lecture, delivered before the Oxford UniTersity Junior
Scientific Club, May 16, 1893.

HOMOGENEOUS ASSEMBLAGE OF BODIES. 603

in one line ; this, in virtue of the homogeneousness of the assem-
blages on the three floors, will secure that every person on the
intermediate floor is exactly in line with his nearest neighbours
above and below. The same condition of alignment must be
fulfilled by every three consecutive -floors, and we thus have a
homogeneous assemblage of people in three dimensions of space.
In particular cases every person's nearest neighbour in the floor
above may be vertically over him, but we must not confine our
attention to assemblages thus rectangularly grouped in vertical
lines.

§ 2. Consider now any particular person C (fig. 1) on any
intermediate floor, D and U his nearest
neighbours, E and E' his next nearest
neighbours all on his own floor. His
next next nearest neighbours on that
floor will be in the positions F and F'
in the diagram. Thus we see that
each person C is surrounded by six
persons, DU, EE\ and FF\ being his
nearest, his next nearest, and his next
next nearest neighbours on his own
floor. Excluding for simplicity the Fig. 1.

special cases of rectangular grouping

we see that the angles of the six equal and similar triangles CDE,
CEFy &c., are all acute : and because the six triangles are equal
and similar we see that the three pairs of mutually remote sides
of the hexagon DEFD'E'F' are equal and parallel.

§ 3. Let now A, A\ A'\ &c., denote places of persons of the
homogeneous assemblage on the floor immediately above, and
B, B\ E\ &c. on the floor immediately below, the floor of C. In
the diagram let a, a\ a" be points in which the floor of CDE is cut
by perpendiculars to it through Ay A\ A'' of the floor above, and
6, 6', 6" by perpendiculars from 5, Ey £" of the floor below. Of all
the perpendiculars from the floors immediately above and below,
just two, one from each, cut the area of the parallelogram CDEF:
and they cut it in points similarly situated in respect to the
oppositely oriented triangles into which it is divided by either of
its diagonals. Hence if a lies in the triangle CDEy the other five
triangles of the hexagon must be cut in the corresponding points,

604 APPENDIX H.

as shown in the diagram. Thus, if we think only of the floor of C
and of the floor immediately above it, we have points A^ A\ A"
vertically above a, a\ o!\ Imagine now a triangular pyramid, or
tetrahedron, standing on the base GBE^ and having A for vertex :
we see that each of its sides ACB^ ADEy A EC is an acute-angled
triangle because, as we have already seen, CDE is an acute-angled
triangle, and because the shortest of the three distances CA^ DA,
EA is (§ 2) greater than CE (though it may be either greater than
or less than DE), Hence the tetrahedron CDEA has all its angles
acute ; not only the angles of its triangular faces, but the six angles
between the planes of its four face& This important theorem
regarding homogeneous assemblages was given by Bravais, to
whom we owe the whole doctrine of homogeneous assemblages in
its most perfect simplicity and complete generality. Similarly
we see that we have equal and similar tetrahedrons on the bases
UCF, EF'C ; and three other tetrahedrons below the floor of C,
having the oppositely oriented triangles GUE'y &c. for their bases
and B, E, 5" for their vertices. These three tetrahedrons are
equal and heterochirally* similar to the first three. The con-
sideration of these acute-angled tetrahedrons is of fundamental
importance in respect to the engineering of an elastic solid, or
crystal, according to Boscovich. So also is the consideration of
the cluster of thirteen points C, and the six neighbours DEFUE'F
in the plane of the diagram, and the three neighbours A A' A" on
the floor above, and BRR' on the floor below.

§ 4. The case in which each of the four faces of each of
the tetrahedrons of § 3 is an equilateral triangle is particularly
interesting. An assemblage fulfilling this condition may con-
veniently be called an * equilateral homogeneous assemblage,' or,
for brevity, an * equilateral assemblage/ In an equilateral assem-
blage (7*s twelve neighbours are all equi-distant from it. I hold
in my hand a cluster of thirteen little black balls, made up by
taking one of them and placing the twelve others in contact with
it (and therefore packed in the closest possible order), and fixing
them all together by fish-glue. Yon see it looks, in size, colour,
and shape, quite like a mulberry. The accompanying diagram
shows a stereoscopic view of a similar cluster of balls painted
white for the photograph.

* See footnote on § 22 below.

' THIRTEEN NEAREST NEIGHBOURS IN B0H0GENE0U3 ASSEMBLAGE. 605

§ 5. By adding ball after ball to such a cluster of thirteen,
and always taking care to place each additional ball in some

Fig. 2.

position in which it is properly in line with others, so as to make
the whole a^emblage homogeneous, we can exercise oumelves in a
very interesting manner in the building up of any possible form of
crystal of the class called ' cubic ' by some writers and ' octahedral '
by others. You see before you several examples. I adviae any of
you who wish to study crystallography to contract with a wood-
turner, or a maker of beads for furniture tassels or for rosaries, for
a thousand wooden balls of about half an inch diameter each.
Holes through them will do no harm, and may even be useful ;
but make sure that the balls are as nearly equal to one another,
and each as nearly spherical as possible.

606

APPENDIX H.

§ 6. You see here before you a large model which I have
made to illustrate a homogeneous assemblage of points, on a plan
first given, I believe, by Mr William Barlow (Nature, December 20
and 27, 1883). The roof of the model is a lattice-frame (fig. 3)
consisting of two sets of eight parallel wooden bars crossing one
another, and kept together by pins through the middles of the
crossings. As you see, I can alter it to make parallelograms of all
degrees of obliquity till the bars touch, and again you see I can
make them all squares.

§ 7. The joint pivots are (for cheapness of construction) of
copper wire, each bent to make a hook below the lattice ^firame.
On these sixty-four hooks are hung sixty-four
fine cords, firmly stretched by little lead weights.
Each of these cords (fig. 4) bears eight short
perforated wooden cylinders, which may be
slipped up and down to any desired position*.
They are at present actually placed at distances
consecutively each equal to the distance from
joint to joint of the lattice frame.

§ 8. The roof of the model is hung by four
cords, nearly vertical, of independently variable
lengths, passing over hooks from fixed points
above, and kept stretched by weigh t»s, each
equal to one-quarter of the weight of roof and
pendants. You see now by altering the angles
of the lattice- work and placing it horizon t^il or
in any inclined plane, as I am allowed to do
readily by the manner in which it is hung,
I have three independent variables, by varying
which I can show you a\\ varieties of homo-
geneous assemblages, in which three of the
neighbours of every point are at equal distances
from it. You see here, for example, we have the equilateral
assemblage. I have adjusted the lattice roof to the proper angle,
and its plane to the proper inclination to the vertical, to make
a wholly equilateral assemblage of the little cylinders of wood on

* The holes in the cylinders are bored obliquely, as shown in fig. 4, which
eanses them to remain at any desired po5(ition on the cord, and allows them to be
fteed to move up and down by slackening the cord for a moment

Fig. 4.

HOMOGENEOUS ASSEMBLAGE HAS HOMOGENEOUS ORIENTATIONS. 607

the vertical cords, a case, as we have seen, of special importance.
If I vary also the distances between the little pieces of wood on
the cords; and the distances between the joints of the lattice
work (variations easily understood, though not conveniently pro-
ducible in one model without more of mechanical construction
than would be worth making), I have three other independent
variables. By properly varying these six independent variables,
three angles and three lengths, we may give any assigned value to
each edge of one of the fundamental tetrahedrons of § 3.

§ 9. Our assemblage of people would not be homogeneous
unless its members were all equal and similar and in precisely
similar attitudes, and were all looking the same way. You under-
stand what a number of people seated or standing on a floor or
plain and looking the same way means. But the expression
* looking' is not conveniently applicable to things that have no
eyes, and we want a more comprehensive mode of expression.
We have it in the words * orientation,' * oriented,' and (verb) * to
orient,' suggested by an extension of the idea involved in the
word ' orientation,' first used to signify positions relatively to east
and west, of ancient Greek and Egyptian temples and Christian
churches. But for the orientation of a house or temple we have
only one angle, and that angle is called 'azimuth' (the name
given to an angle in a horizontal plane). For orientation in three
dimensions of space we must extend our ideas and consider position
with reference to east and west and up and down. A man lying
on his side, with his head to the north and looking east, would not
be similarly oriented to a man standing upright and looking east.
To provide for the complete specification of how a body is oriented
in space we must have in the body a plane of reference, and a line
of reference in this plane, belonging to the body and moving with
it. We must also have a fixed plane and a fixed line of reference
in it, relatively to which the orientation of the moveable body is
to be specified ; as, for example, a horizontal plane, and the east
and west horizontal line in it. The position of a body is completely
specified when the angle between the plane of reference belonging
to it and the fixed plane is given ; and when the angles between
the line of intersection of the two planes and the lines of reference
in them are also given. Thus we see that three angles are neces-
sary and sufficient to specify the orientation of a moveable body,

608 APPENDIX H.

aod we see how the specification is coDveniently given in terms of
three angles.

§ 10. To illustrate this take a book lying on the table before
you with its side next the title-page up, and its back to the north.
I now lift the east edge (the top of the hook), keeping the bottom
edge north and south on the table till the book ia inclined, let us
say, 20° to the table. Nezt, without altering this angle of 20°,
between the side of the book and the table, I turn the book round
a vertical axis through 45° till the bottom edge lies north-east
and south-west. Lastly, keeping the book in the phine to which
it has been thus brought, I turn it round in this phtne throng 35°.
These three angles of 20°, 45°, and 35° specify, with reference to
the horizontal plane of the table and the east and west line in it,
the orientation of the book in the position to which you have seen
me bring it, and in which I hold it before you.

§ 11. In figs. 5 and 6 you see two assemblages, each of
twelve equal and similar molecules in a plana Fig. 5, in which
the molecules are all same-ways oriented, is one homogeneous

HOMOGENEOUS ASSEMBLAGE OF SYMMETRICAL DOUBLETS. 609

assembl^e of twenty-four molecules. Fig. 6, in which in one set
of rows the molecules are alternately oriented two different ways,
may either be regarded as two homogeneous assemblagefi, each of
twelve single molecules ; or one homogeneous assemblage of twelve
pairs of those single molecules.

§ 12. I must now call your attention to a purely geometrical
question • of vital interest with respect to homogeneous asBeroblages
in general, and particularly the homogeneous assemblage of mole-
cules constituting a crystal; — what can we take as ' the' boundary
or 'a' boundary enclosing each molecule with whatever portion of
space around it we are at liberty to choose for it, and separating ti
from, neighbours avd their portions of space given to them m homO'
geneous fairness ?

§ 13. If we had only mathematical points to consider we
should he at liberty to choose the simple obvious partitioning by
three sets of parallel planes. Even this may be done in an infinite

* "On the HomogeDeons Diviaion of Spaoe," bj Loid Eelvm, Royal Socittg
Proettdittgt, Vol. lt., Jan. IB, 169i.

610 APPENDIX H.

number of ways, thus: — Beginning with any point P of the
assemblage, choose any other three points A, B, C, far or near,
provided only that they are not in one plane with P, and that
there is no other point of the assemblage in the lin^s PA, PB, PC^
or within the volume of the parallelepiped of which these lines are
conterminous edges, or within the areas of any of the faces of this
parallelepiped. There will be points of the assemblage at each of
the comers of this -parallelepiped and at all the comers of the
parallelepipeds equal and similar to it which we find by drawing
sets of equi-distant planes parallel to its three pairs of faces.
(A diagram is unnecessary.) Every point of the assemblage is
thus at the intersection of three planes, which is also the point
of meeting of eight neighbouring parallelepipeds. Shift now any
one of the points of the assemblage to a position within the volume
of any one of the eight parallelepipeds, and give equal parallel
motions to all the other points of the assemblage. Thus we have
every point in a parallelepipedal cell of its own, and all the points
of the assemblage are similarly placed in their cells, which are
themselves equal and similar.

§ 14. But now if, instead of a single point for each member
of the assemblage, we have a group of points, or a globe or cube
or other geometrical figure, or an individual of a homogeneous
assemblage of equal, similar, similarly dressed, and similarly
oriented ladies, sitting in rows, or a homogeneous assemblage of
trees closely planted in regular geometrical order on a plane with
equal and similar distributions of molecules, and parallel planes
above and below, we may find that the best conditioned plane-
faced parallelepipedal partitioning which we can choose would cut
oflF portions properly belonging to one molecule of the assemblage
and give them to the cells of neighbours. To find a cell enclosing
all that belongs to each individual, for example, every part of each
lady's dress, however complexly it may be folded among portions
of the equal and similar dresses of neighbours ; or, every twig, leaf,
and rootlet of each one of the homogeneous assemblage of trees ;
we must alter the boundary by give-and-take across the plane
faces of the primitive parallelepipedal cells, so that each cell shall
enclose all that belongs to one molecule, and therefore (because
of the homogeneousness of the partitioning) nothing belonging to
any other molecule. The geometrical problem thus presented,

HOW TO DRAW PARTITIONAL BOUNDARY IN THREE DIMENSIONS. 611

wonderfully complex as it may be in cases such as some of those
which I have suggested, is easily performed for any possible case
if we begin with any particular parallelepipedal partitioning deter-
mined for corresponding points of the assemblage, as explained in
§ 13, for any homogeneous assemblage of single points. We may
prescribe to ourselves that the corners are to remain unchanged,
but if 80 they must to begin with be either in interfaces of contact
between the individual molecules, or in vacant space among the
molecules. If this condition is fulfilled for one corner it is fulfilled
for all, as the corners are essentially corresponding points relatively
to the assemblage.

§ 15. Begin now with any one of the twelve straight lines
between corners which constitute the twelve edges of the parallel-
epiped, and alter it arbitrarily to any curved or crooked line between
the same pair of comers, subject only to the conditions (1) that it
does not penetrate the substance of any member of the assemblage,
and (2) that it is not cut by equal and similar parallel curves*
between other pairs of comers.

Considering now the three fours of parallel edges of the
parallelepiped, let the straight lines of one set of four be altered
to equal and similar parallel curves in the manner which I have
described ; and proceed by the same rule for the other two sets of
four edges. We thus have three fours of parallel curved edges
instead of the three fours of parallel straight edges of our primitive
parallelepiped with comers (each a point of intersection of three
edges) unchanged. Take now the quadrilateral of four curves
substituted for the four straight edges of one face of the parallel-
epiped. We may call this quadrilateral a curvilineal parallelogram,
because it is a circuit composed of two pairs of equal parallel
curves. Draw now a curved surface (an infinitely thin sheet of
perfectly extensible india-rubber if you please to think of it so)
bordered by the four edges of our curvilineal parallelogram, and
so shaped as not to cut any of the substance of any molecule of
the assemblage. Do the same thing with an exactly similar and
parallel sheet relatively to the opposite face of the parallelepiped ;
and again the same for each of the two other pairs of parallel
faces. We thus have a curved-faced parallelepiped enclosing the

* Similar curves are said to be parallel when the tangents to them at correspond-
ing points are parallel.

39—2

612 APPENDIX H.

whole of one molecule and no part of any other ; and by similar
procedure we find a similar boundary for every other molecule
of the assemblage. Each wall of each of these cells is common
to two neighbouring molecules, and there is no vacaut space any-
where between them or at comers. Fig. 7 illustrates this kind of

partitioning by showing a plane section parallel to one pair of
plane faces of the primitive parallelepiped for an ideal case. The
plane diagram is in fact a realization of the two-dimensional
problem of partitioning the pine pattern of a Persian carpet by
parallelograms about us nearly rectilinear as we can make them.
In the diagram faint straight lines are drawn to show the primitive
parallelogrammatic partitioning. It will be seen that of all the
crossings (marked with dots in the diagram) every one is sinailarly
situated to every other in respect to the homogeneously repeated
pattern figures : A, B, G, D are four of them at the comers of one
cell.

§ 16. Confining our attention for a short time to the homo-
geneous division of a plane, remark that the division into
parallelograms by two sets of crossing parallels is singular in
this respect — each cell is contiguous with three neighbours at
every corner. Any shifting, large or small, of the parallelograms

PAKTITIONAL BOUNDARY FOR HOMOGENEOUS PLANE FIGURES. 613

by relative sliding in one direction or another violates this con-
dition, brings us to a configuration like that of the faces of
regularly hewn stones in ordinary bonded masonry, and gives
a partitioning which fulfils the condition that at each comer
each cell has only two neighbours. Each cell is now virtually
a hexagon, as will be seen by the letters A, B, C, Z), E, F in the
diagram fig. 8. A and D are to be reckoned as comers, each

Fig. 8.

witli an interior angle of 180°. In this diagram the continuous
heavy lines and the continuous faint lines crossing them show
a primitive parallelogrammatic partition by two sets of continuous
parallel intersecting lines. The interrupted crossing lines (heavy)
show, for the same homogeneous distribution of single points or
molecules, the virtually hexagonal partitioning which we get by
shifting the boundary from each portion of one of the light
lines to the heavy line next it between the same continuous
parallels.

Fig. 8 bis represents a further modification of the boundary
by which the 180° angles -4, D become angles of less than 180°.
The continuous parallel lines (light) and the short light portions
of the crossing lines show the configuration according to fig. 8,
from which this diagram is derived.

§ 17. In these diagrams (figures 8 and 8 bis) the object
enclosed is small enough to be enclosable by a primitive parallelo-
grammatic partitioning of two sets of continuous crossing parallel
straight lines, and by the partitioning of * bonded ' parallelograms
both represented in fig. 8, and by the derived hexagonal partition-

sasv r ikssteyva^ vcssioiezsc "**• ~*n res-
ism i ^ * -fiiuT"

3rt»n*m ■• arsf ia:&ur:'. urates-, -r »ti:;= ,r. 3. ' J »

^ -

HOMOGENEOUS ASSEMBLAGE OF TETRAKAIDEKAHEDRONAL CELLa615

parallel straight lines in fig. 9 show the same primitive parallelo-
grammatic partitioning as in fig. 7, and the same slightly shifted
to suit points chosen for well-conditionedness of hexagonal par-
titioning.

§ 18. For the division of continuous three-dimensional space*
into equal, similar, and similarly oriented cells, quite a correspond-
ing transformation from partitioning by three sets of continuous
mutually intersecting parallel planes to any possible mode of
homogeneous partitioning, may be investigated by working out
the three-dimensional analogue of § 16 — 17. Thus we find that
the most general possible homogeneous partitioning of space with
plane interfiices between the cells gives us fourteen walls to each
cell, of which six are three pairs of equal and parallel parallelo-
grams, and the other eight are four pairs of equal and parallel
hexagons, each hexagon being bounded by three pairs of equal
and parallel straight lines. This figure, being bounded by four-
teen plane faces, is called a tetrakaidekahedron. It has thirty-six
edges of intersection between faces; and twenty-four corners, in
each of which three faces intersect. A particular case of it, which
I call an orthic tetrakaidekahedron, being that in which the six
parallelograms are equal squares, the eight hexagonal faces are
equal equilateral and equiangular hexagons, and the lines joining
corresponding points in the seven pairs of parallel faces are
perpendicular to the planes of the faces, is represented by a
stereoscopic picture in fig. 10. The thirty-six edges and the
twenty-four corners, which are easily counted in this diagram,
occur in the same relative order in the most general possible
partitioning, whether by plane-faced tetrakaidekahedrons or by
the generalized tetrakaidekahedron described in § 19.

§ 19. The most general homogeneous division of space is not
limited to plane-faced cells; but it still consists essentially of
tetrakaidekahedronal cells, each bounded by three pairs of equal
and parallel quadrilateral faces, and four pairs of equal and
parallel hexagonal faces, neither the quadrilaterals nor the
hexagons being necessarily plane. Each of the thirty-six edges
may be straight or crooked or curved ; the pairs of opposite edges,
whether of the quadrilaterals or hexagons, need not be equal and

* See footnote to § 12 above.

61<

AFTKSWX H.

panDel ; oeither the fbor comen of eadi quadrihleial nor tke st
coraers of eacfa hexagon need be Id one plant. Bat ercfj piv of
Mcnspoading edges of evefT pair of paimllel camBpoa^ag haa.

EQUIVOLUMINAL CELLS WITH MINIBfUM PARTITION AL AREA. 617

whether quadrilateral or hexagonal, must be equal and parallel.*
I have described an interesting case of partitioning by tetrakai-
dekahedrons of curved faces with curved edges in a paper*
published about seven years ago. In this case each of the
quadrilateral faces is plane. Each hexagonal face is a slightly
curved surface having three rectilineal diagonals through its
centre in one plane. The six sectors of the face between these
diagonals lie alternately on opposite sides of their plane, and are
bordered by six arcs of plane curves lying on three pairs of parallel
planes. This tetrakaidekahedronal partitioning fulfils the con-
dition that the angles between three faces meeting in an edge are
everywhere each 120°; a condition that cannot be fulfilled in any
plane-faced tetrakaidekahedron. Each hexagonal wall is an anti-
clastic surface of equal opposite curvatures at every point, being
the surface of minimum area bordered by six curved edges. It
is shown easily and beautifully, and with a fair approach to
accuracy, by choosing six little circular arcs of wire, and soldering
them together by their ends in proper planes for the six edges
of the hexagon; and dipping it in soap solution and taking
it out.

§ 20. Returning now to the tactics of a homogeneous assem-
blage remark that the qualities of the assemblage as a whole
depend both upon the character and orientation of each molecule,
and on the character of the homogeneous assemblage formed by
corresponding points of the molecules. After learning the simple
mathematics of crystallography, with its indicial system f for
defining the faces and edges of a crystal according to the Bravais
rows and nets and tetrahedrons of molecules in which we think
only of a homogeneous assemblage of points, we are apt to forget
that the true crystalline molecule, whatever its nature may be,
has sides, and that generally two opposite sides of each molecule
may be expected to be very different in quality, and we are almost
surprised when mineralogists tell us that two parallel faces on two
sides of a crystal have very different qualities in many natural
crystals. We might almost as well be surprised to find that an
army in battle array, which is a kind of large-grained crystal,

* ** On the DivisioD of Space with Minimum Partitional Area," Philosophieat
Magazine^ Vol. xxiv., 18S7, p. 502, and Acta Mathematica of the same year.

t A. Levy, Edinburgh Philosophical Journal, April, 1822 ; Whewell, Phil. TranSm.
Royal Society, 1825 ; Miller, TreatUe on Crystallography,

618 APPENDIX H.

presents very different appearance to anyone looking at it from
outside, according as every man in the ranks with his rifle and
bayonet faces to the front or to the rear or to one flank or to the
other.

§ 21. Consider, for example, the ideal case of a crystal consist-
ing of hard, equal and similar tetrahedronal solids all same-ways
oriented. A thin plate of crystal cut parallel to any one set of the
faces of the constituent tetrahedrons would have very different
properties on its two sides ; as the constituent molecules would all
present points outwards on one side and flat surfaces on the other.
We might expect that the two sides of such a plate of crystal
would become oppositely electrified when rubbed by one and the
same rubber ; and, remembering that a piece of glass with part of
its surface finely ground but not polished and other parts polished
becomes, when rubbed with white silk, positively electrified over
the polished parts and negatively electrified over the non-polished
parts, we might almost expect that the side of our supposed
crystalline plate towards which flat faces of the constituent
molecules are turned would become positively electrified, and
the opposite side, showing free molecular comers, would become
negatively electrified, when both are rubbed by a rubber of
intermediate electric quality. We might also from elementary
knowledge of the fact of piezo-electricity, that is to say, the
development of opposite electricities on the two sides of a crystal
by pressure, expect that our supposed crystalline plate, if pressed
perpendicularly on its two sides, would become positively electrified
on one of them and negatively on the other.

§ 22. Intimately connected with the subject of enclosing cells
for molecules of given shape, assembled homogeneously, is the
homogeneous packing together of equal and similar molecules
of any given shape. In every possible case of any infinitely great
number of similar bodies the solution is a homogeneous assem-
blage. But it may be a homogeneous assemblage of single solids
all oriented the same way, or it may be a homogeneous assemblage
of clusters of two or more of them placed together in different
orientations. For example, let the given bodies be halves (oblique
or not oblique) of any parallelepiped on the two sides of a dividing
plane through a pair of parallel edges. The two halves are homo-

CASE OF SINGLE CONTACT BETWEEN NEIOHBOUBS. 619

chirally* similar; and, being equal, we may make a liomt^eDeous
assemblage of them by orieoting them all the same way and
placing them properly in rows. But the closest packing of this
assemblage would necessarily leave vacant spaces between the
bodies : and we get in reality the closest possible packing of the
given bodies by taking them in pairs oppositely oriented and
placed together to form parallelepipeds. These clusters may be
packed together so as to leave no unoccupied space.

Whatever the number of pieces in a cluster in the closest
possible packing of solids may be for any particular shape we
may consider each cluster as itself a given single body, and
thus reduce the problem to the packing closely together of
assemblages of individuals all same-ways oriented; and to this
problem therefore it is convenient that we should now confine our
attention.

§ 23. To avoid complexities, such as those which we find in
the familiar problem of homogeneous packing of
forks or spoons or tea-cupa or bowls of any ordinary
shape, we shall suppose the given body to be of such
shape that no two of them similarly oriented can
touch one another in more than one point. Wholly
convex bodies essentially fulfil this condition ; but it
may also be fulfilled by bodies not wholly convex, as
is illustrated in fig. 11.

§ 24. To find close and closest packing of any
number of our solids S,, S,, S, ... of shape fulfilling
the condition of § 23 proceed thus : —

(1) Bring S^ to touch Sj at any chosen point p
of its surface (fig. 12).

(2) Bring «, to touch S, and S, at r and q ^'« "■
respectively.

(3) Bring 84 (not shown in the diagram) to touch £,, S„
and S,.

* I coll an; geonetricaJ figure, or groap of poiDla, chirat, and say that it has
obiralit? if its imag? in a plene miiror, ideally realized, cannot be broaght to
coincide with itself. Two equal and aimilar tight hands are bomoohirally BimUar.
Equal and eimilar right and left hands are he terochi rally limilar or ' olloohirally '
similar (bat heterochirally is better). TheBe are also called ' enantiomorphs,' after
a usage introdaced, I believe, by German vriterB. Any ehiral object and ita image
in a plane mirror are beteroohiraUy similar.

620 APPENDIX H.

(4) Place any number of the bodies together in three rows
continuing the lines of SiS^, SiS^, S^S^, and in three sets of
equi-distant rows parallel to these. This makes a homogeneous
assemblage. In the assemblage so formed the molecules are
necessarily found to be in three sets of rows parallel respectively
to the three pairs 8^^^ 8^^, SA, The whole space occupied by
an assemblage of n of our solids thus arranged has clearly 6n times
the volume of a tetrahedron of corresponding points of /S^, j8^,
St, 84. Hence the closest of the close packings obtained by
the operations (1)...(4) is found if we perform the operations
(1), (2), and (3) as to make the volume of this tetrahedron least
possible.

§ 25. It is to be remarked that operations (1) and (2) leave
for (3) no liberty of choice for the place of 84, except between two
determinate positions on opposite sides of the group Siy S^, iS,.

Fig. 12.

The volume of the tetrahedron will generally be diflFerent for these
two positions of 8^, and, even if the volume chance to be equal in
any case, we have differently shaped assemblages according as we
choose one or other of the two places for 84,

This will be understood by looking at fig. 12, showing Si and
neighbours on each side of it in the rows of 8182, SiS^, and in a
row parallel to that of 8^^, The plane of the diagram is parallel
to the planes of corresponding points of these seven bodies, and
the diagram is a projection of these bodies by lines parallel to the
intersections of the tangent planes through p and r. If the three
tangent planes through p, g, and r, intersected in parallel lines,
q would be seen like p and r as a point of contact between the
outlines of two of the bodies ; but this is only a particular case,

SCALENE CONVEX SOLIDS HOMOGENEOUSLY ASSEMBLED. 621

and in general q must, as indicated in the diagram, be concealed
by one or other of the two bodies of which it is the point of
contact. Now imagining, to fix our ideas and facilitate brevity
of expression, that the planes of corresponding points of the seven
bodies are horizontal, we see clearly that S^ may be brought into
proper position to touch Si, jS, and S, either from above or from
below ; and that there is one determinate place for it if we bring
it into position from above, and another determinate place for it
if we bring it from below.

§ 26. If we look from above at the solids of which fig. 12
shows the outline we. see essentially a hollow leading down to
a perforation between Si, jS',, 5,, and if we look from below we see
a hollow leading upwards to the same perforation : this for brevity
we shall call the perforation pqi\ The diagram shows around Si
six hollows leading down to perforations, of which two are similar
to pqVy and the other three, of which p'gV indicates one, are similar
one to another but are dissimilar to pqr. If we bring S^ from
above into position to touch S^ S^ and 5,, its place thus found is
in the hollow pqr, and the places of all the solids in the layer
above that of the diagram are necessarily in the hollows similar to
pqr. In this case the solids in the layer below that of the diagram
must lie in the hollows below the perforations dissimilar to pgr, in
order to make a single homogeneous assemblage. In the other
case /S4 brought up from below finds its place on the under side of
the hollow pqr J and all solids of the lower layer find similar places :
while solids in the layer above that of the diagram find their
places in the hollows similar to pqr. In the first case there are
no bodies of the upper layer in the hollows above the perforations
similar to p'q'r'y and no bodies of the lower layer in the hollows
below the perforations similar to pqr. In the second case there
are no bodies of the upper layer in the hollows above the perfora-
tions similar to pqr, and none of the under layer in the hollows
below the perforations similar to p'q'r.

§ 27. Going back now to operation (1) of § 24, remark that
when the point of contact p is arbitrarily chosen on one of the two
bodies Si, the point of contact on the other will be the point on it
corresponding to the point or one of the points of S,, where its
tangent plane is parallel to the tangent plane at p. If S, is wholly

622 APPENDIX H.

convex it has only two points at which the tangent planes are
parallel to a given plane, and therefore the operation (1) is deter-
minate and unambiguous. But if there is any concavity there
will be four or some greater even number of tangent planes
parallel to any one of some planes, while there will be other
planes to each of which only one pair of tangent planes is parallel.
Hence, operation (1), though still determinate, will have a multi-
plicity of solutions, or only a single solution, according to the
choice made of the position of p. [Any change of configuration
of the assemblage in stable equilibrium causes expansion ! K ;
Holwood, May 23. 1896.]

Henceforth, however, to avoid needless complications of ideas,
we shall suppose our solids to be wholly convex ; and of some such
unsymmetrical shape as those indicated in fig. 12 of § 25, and
shown by stereoscopic photograph in fig. 13 of § 36. With or
without this convenient limitation, operation (1) has two freedoms,
as p may be chosen freely on the surface of Si ; and operation (2)
has clearly just one freedom after operation (1) has been performed.
Thus, for a solid of any given shape, we have three disposables, or,
as commonly called in mathematics, three ' independent variables,'
aU free for making a homogeneous assemblage according to the
rule of §22*

* The following is information regarding ratio of void to whole Rpaoe in heaps
of gravel, sand, and broken stones, extracted from books of reference for engineers :

Hurst's Pocket- Book.
Thames Ballast 20 cub. ft. per ton
Gravel coarse 19 ,, ,,
Sandpit 22 ,, ,,

Shingle 23

Hasweirs Pocket-Book .
Voids in a cub. yard of Stone broken to gauge of 2 -5" 10 cub. ft.

II i» »» »» n « 10*00 ,,

»» »» »» II II 1*0 11*33 ,,

Shingle 9

Thames Ballast (containing sand) 4*5

Law and BurrelPs Civil Engineering,
(Road Metal.)

*' A cub. yard of broken stone metal of an ordinary size — 2" or 2 J" cube — when
screened and beaten down in regular layers 6" thick con tains... 11 cub. ft. of inter-
spaces, as tested by filling up the metal with liquid.*'... Herr E. Bokeberg found

that in a cubic yard of loosely heaped broken stones the void space was 50 °l^ of

the whole.

>*

4i
41

ORIENTATION OF MOLECULES FOR CLOSEST PACKING. 623

§ 28. In the homogeneous assemblage defined in § 24 each
solid, Sly is touched at twelve points, being the three points of
contact with /Sj, /S,, S4, and the three 3*s of points on Si correspond-
ing to the points on iSj, S,, /S4, at which these bodies are touched
by the others of the quartet. This statement is somewhat difficult
to follow, and we see more clearly the twelve points of contact by
not confining our attention to the quartet iSi, Sg, S,, ^4 (convenient
as this is for some purposes), but completing the assemblage and
considering six neighbours around Si in one plane layer of the
solids as shown in fig. 12, with their six points prq'p'rq^'' of
contact with Si ; and the three neighbours of the two adjacent
parallel layers which touch it above and below. This cluster
of thirteen. Si and twelve neighbours, is shown for the case of
spherical bodies in the stereoscopic photograph of § 4 above. We
might of course, if we pleased, have begun with the plane layer of
which Si, Si, S4 are members, or with that of which S^, S,, ^4 are
members, or with the plane layer parallel to the fourth side SJS^^
of the tetrahedron : and thus we have four different ways of
grouping the twelve points of contact on Si into one set of six and
two sets of three.

§ 29. In this assemblage we have what I call ' close order * or
* close packing.' For closest of close packings the volume of the
tetrahedron (§ 24) of corresponding points of Si, S^, S^, and ^4
must be a minimum, and the least of minimums if, as generally
will be the case, there are two more different configurations for
each of which the volume is a minimum. There will in general
also be configurations of minimax volume and of maximum volume,
subject to the condition that each body is touched by twelve
similarly oriented neighbours.

§ 30. Pause for a moment to consider the interesting kine-
matical and dynamical problems presented by a close homogeneous
assemblage of smooth solid bodies of given convex shape, whether
perfectly frictionless or exerting resistance against mutual sliding
according to the ordinarily stated law of friction between dry hard
solid bodies. First imagine that they are all similarly oriented
and each in contact with twelve neighbours, except outlying
individuals (which there must be at the boundary if the assem-
blage is finite, and each of which is touched by some number of
neighbours less than twelve). The coherent assemblage thus

624 APPENDIX H.

defined constitutes a kinematic frame or skeleton for an elastic
solid of very peculiar propertiea Instead of the six freedoms, or
disposables, of strain presented by a natural solid it has only
three. Change of shape of the whole can only take place in
virtue of rotation of the constituent parts relatively to any one
chosen row of them, and the plane through it and another chosen
row.

§ 31. Suppose first the solids to be not only perfectly smooth
but perfectly frictionless. Let the assemblage be subjected to
equal positive or negative pressure inwards all around its boundary.
Every position of minimum, minimax, or maximum volume will
be a position of equilibrium. If the pressure is positive the
equilibrium will be stable if, and unstable unless, the volume is
a minimum. If the pressure is negative the equilibrium will be
stable if, and unstable unless, the volume is a maximum. Con-
figurations of minimax volume will be essentially unstable.

§ 32. Consider now the assemblage of § 31 in a position of
stable equilibrium under the influence of a given constant uniform
pressure inwards all round its boundary. It will have rigidity in
simple proportion to the amount of this pressure. If now by
the superposition of non-uniform pressure at the boundary, for
example equal and opposite pressures on two sides of the assem-
blage, a finite change of shape is produced : the whole assemblage
essentially swells in bulk. This is the ' dilatancy ' which Osborne
Reynolds has described* in an exceedingly interesting manner
with reference to a sack of wheat or sand, or an india-rubber bag
tightly filled with sand or even small shot. Consider, for example,
a sack of wheat filled quite full and standing up open. It is limp
and flexible. Now shake it down well, fill it quite full, shake
again, so as to get as much into it as possible, and tie the mouth
very tightly close. The sack becomes almost as stiff as a log of
wood of the same shape. Open the mouth partially and it be-
comes again limp, especially in the upper paiiis of the bag. In
Reynolds* observations on india-rubber bags of small shot his
'dilatancy* depends, essentially and wholly, on breaches of some
of the contacts which exist between the molecules in their configu-
ration of minimum volume : and it is possible that in all his cases

* Philosophical Magazine, Vol. xx., 1S85, second half-year, p. 469, and British
Association Report^ 1885, Aberdeen, p. 896.

WET SAND WHITENED BY PRESSURE OF GLASS PLATE. 625

the dilatations which he observed are chiefly, if not wholly, due to
such breaches of contact.

But it is possible, it almost seems probable, that in bags or
boxes of sand or powder, of some kinds of smooth rounded bodies
of any shape, not spherical or ellipsoidal, subjected persistently to
unequal pressures in different directions, and well shaken, stable
positions of equilibrium are found with almost all the particles
each touched by twelve others.

Here is a curious subject of Natural History through all ages
till 1885, when Reynolds brought it into the province of Natural
Philosophy by the following highly interesting statement: —
"A well-marked phenomenon receives its explanation at once
" from the existence of dilatancy in sand. When the falling tide
" leaves the sand firm, as the foot falls on it, the sand whitens and
" appears momentarily to dry round the foot. When this happens
" the sand is full of water, the surface of which is kept up to that
"of the sand by capillary attractions; the pressure of the foot
" causing dilatation of the sand more water is required, which has
"to be obtained either by depressing the level of the surface
"against the capillary attractions, or by drawing water through
"the interstices of the surrounding sand. This latter requires
" time to accomplish, so that for the moment the capillary forces
" are overcome ; the surface of the water is lowered below that of
"the sand, leaving the latter white or drier until a suflScient
" supply has been obtained from below, when the surface rises and
" wets the sand again. On raising the foot it is generally seen
** that the sand under the foot and around becomes momentarily
" wet ; this is because, on the distorting forces being removed, the
" sand again contracts, and the excess of water finds momentary
" relief at the surface."

This proves that the sand under the foot, as well as the surface
around it, must be dry for a short time after the foot is pressed
upon it, though we cannot see it whitened, as the foot is not
transparent. That it is so has been verified by Mr Alex. Gait,
Experimental Instructor in the Physical Laboratory of Glasgow
University, by laying a small square of plate-glass on wet sand on
the sea-shore of Helensburgh, and suddenly pressing on it by a
stout stick with nearly all his weight. He found the sand, both
under the glass and around it in contact with the air, all became
white at the same moment. Of all the two hundred thousand

T. L. 40

624 APPENDIX H.

defined constitutes a kinematic frame or skeleton for an elastic
solid of very peculiar propertiea Instead of the six freedoms, or
disposables, of strain presented by a natural solid it has only
three. Change of shape of the whole can only take place in
virtue of rotation of the constituent parts relatively to any one
chosen row of them, and the plane through it and another chosen
row.

§ 31. Suppose first the solids to be not only perfectly smooth
but perfectly frictionless. Let the assemblage be subjected to
equal positive or negative pressure inwards all around its boundary.
Every position of minimum, minimax, or maximum volume will
be a position of equilibrium. If the pressure is positive the
equilibrium will be stable if, and unstable unless, the volume is
a minimum. If the pressure is negative the equilibrium will be
stable if, and unstable unless, the volume is a maximum. Con-
figurations of minimax volume will be essentially unstable.

§ 32. Consider now the assemblage of § 31 in a position of
stable equilibrium under the influence of a given constant uniform
pressure inwards all round its boundary. It will have rigidity in
simple proportion to the amount of this pressure. If now by
the superposition of non-uniform pressure at the boundary, for
example equal and opposite pressures on two sides of the assem-
blage, a finite change of shape is produced : the whole assemblage
essentially swells in bulk. This is the * dilatancy ' which Osborne
Reynolds has described* in an exceedingly interesting manner
with reference to a sack of wheat or sand, or an india-rubber bag
tightly filled with sand or even small shot. Consider, for example,
a sack of wheat filled quite full and standing up open. It is limp
and flexible. Now shake it down well, fill it quite full, shake
again, so as to get as much into it as possible, and tie the mouth
very tightly close. The sack becomes almost as stiff as a log of
wood of the same shape. Open the mouth partially and it be-
comes again limp, especially in the upper parts of the bag. In
Reynolds' observations on india-rubber bags of small shot his
* dilatancy ' depends, essentially and wholly, on breaches of some
of the contacts which exist between the molecules in their configu-
ration of minimum volume : and it is possible that in all his cases

* Philosophical Magazine, Vol. zz., 1885, second half-year, p. 469, and BritUk
Association Report, 1885, Aberdeen, p. 896.

WET SAND WHITENED BY PRESSURE OF GLASS PLATE. 625

the dilatations which he observed are chiefly, if not wholly, due to
such breaches of contact.

But it is possible, it almost seems probable, that in bags or
boxes of sand or powder, of some kinds of smooth rounded bodies
of any shape, not spherical or ellipsoidal, subjected persistently to
unequal pressures in different directions, and well shaken, stable
positions of equilibrium are found with almost all the particles
each touched by twelve others.

Here is a curious subject of Natural History through all ages
till 1885, when Reynolds brought it into the province of Natural
Philosophy by the following highly interesting statement: —
"A well-marked phenomenon receives its explanation at once
" from the existence of dilatancy in sand. When the falling tide
" leaves the sand firm, as the foot falls on it, the sand whitens and
" appears momentarily to dry round the foot. When this happens
" the sand is full of water, the surface of which is kept up to that
"of the sand by capillary attractions; the pressure of the foot
" causing dilatation of the sand more water is required, which has
"to be obtained either by depressing the level of the surface
"against the capillary attractions, or by drawing water through
"the interstices of the surrounding sand. This latter requires
" time to accomplish, so that for the moment the capillary forces
" are overcome ; the surface of the water is lowered below that of
"the sand, leaving the latter white or drier until a suflScient
" supply has been obtained from below, when the surface rises and
" wets the sand again. On raising the foot it is generally seen
" that the sand under the foot and around becomes momentarily
" wet ; this is because, on the distorting forces being removed, the
" sand again conti*acts, and the excess of water finds momentary
" relief at the surface."

This proves that the sand under the foot, as well as the surface
around it, must be dry for a short time after the foot is pressed
upon it, though we cannot see it whitened, as the foot is not
transparent. That it is so has been verified by Mr Alex. Gait,
Experimental Instructor in the Physical Laboratory of Glasgow
University, by laying a small square of plate-glass on wet sand on
the sea-shore of Helensburgh, and suddenly pressing on it by a
stout stick with nearly all his weight. He found the sand, both
under the glass and around it in contact with the air, all became
white at the same moment. Of all the two hundred thousand

T. L. 40

626 APPENDIX H.

million men, women, and children who, from the beginning of the
world, have ever walked on wet sand, how many, prior to the
British Association Meeting at Aberdeen in 1885, if asked, "Is the
sand compressed under your foot ? " would have answered otherwise
than " Yes ! " ?

(Contrast with this the case of walking over a bed of wet
sea- weed !)

§ 33. In the case of globes packed together in closest order
(and therefore also in the case of ellipsoids, if all similarly oriented),
our condition of coherent contact between each molecule and
twelve neighbours implies absolute rigidity of form and constancy
of bulk« Hence our convex solid must be neither ellipsoidal nor
spherical in order that there may be the changes of bulk which
we have been considering as dependent on three independent
variables specifying the orientation of each solid relatively to rows
of the assemblage. An interesting dynamical problem is presented
by supposing any mutual forces, such as might be produced by
springs, to act between the solid molecules, and investigating
configurations of equilibrium on the supposition of frictionless
contacts. The solution of it of course is that the potential energy
of the springs must be a minimum or a minimax or a maximum
for equilibrium, and a minimum for stable equilibrium. The
solution will be a configuration of minimum or minimax, or
maximum, volume, only in the case of pressure equal in all
directions.

§ 34. A purely geometrical question, of no importance in
respect to the molecular tactics of a crystal but of considerable
interest in pure mathematics, is forced on our attention by our
having seen (§ 27) that a homogeneous assemblage of solids of
given shape, each touched by twelve neighbours, has three freedoms
which may be conveniently taken as the three angles specifying
the orientation of each molecule relatively to rows of the assemblage
as explained in § 30.

Consider a solid Si and the twelve neighbours which touch it,
and try if i* is possible to cause it to touch more than twelve of
the bodies. Attach ends of three thick flexible wires to any places
on the surface of Si ; carry the wires through interstices of the
assemblage, and attach their other ends at any three places of
A, B, Cy respectively, these being any three of the bodies outside

QEOMETRICAL PROBLEM OF EACH SOLID TOUCHED BY EIGHTEEN. 627

the cluster of Si and its twelve neighbours. Cut the wires across
at any chosen positions in them ; and round off the cut ends, just
leaving contact between the rounded ends, which we shall call
yy, g'g, h'h. Do homogeneously for every other solid of the
assemblage what we have done for S,. Now bend the wires
slightly so as to separate the pairs of points of contact, taking
care to keep them from touching any other bodies which they
pass near on their courses between Si a6d A, B, C respectively.
After having done this, thoroughly rigidify all the wires thus
altered. We may now, having three independent variables at
our disposal, so change the orientation of the molecules relatively
to rows of the assemblage, as to bring /y, g'g, and h% again into
contact. We have thus six fresh points of Si ; of which three are
f\ gf* A' ; ai^d the other three are on the three extensions of Si
corresponding to the single extensions of A, B, C respectively,
which we have been making. Thus we have a real solution of
the interesting geometrical problem : — It is required so to form
a homogeneous assemblage of solids of any arbitrarily given shape
that each solid shall be touched by eighteen others. This problem
is determinate, because the making of the three contacts f'f, g^g,
h\ uses up the three independent variables left at our disposal
after we have first formed a homogeneous assemblage with twelve
points of contact on each solid. But our manner of finding a shape
for each solid which can allow the solution of the problem to be
real proves that the solution is essentially imaginary for every
wholly convex shape.

§ 35. Pausing for a moment longer to consider afresh the
geometrical problem of putting arbitrarily given equal and similar
solids together to make a homogeneous assemblage of which each
member is touched by eighteen others, we see immediately that it
is determinate (whether it has any real solution or not), because
when the shape of each body is given we have nine disposables for
fixing the assemblage : six for the character of the assemblage of
the corresponding points, and three for the orientation of each
molecule relatively to rows of the assemblage of corresponding
points. These nine disposables are determined by the condition
that each body has nine pail's of contacts with others.

Suppose now a homogeneous assemblage of the given bodies,
in open order with no contacts, to be arbitrarily made according

40—2

C2% ATTESZ-H. a.

V/ K.r z^Zir hi^.'i^KZ-y ^'.skl Tilaes ice iJK

5; iE.j :^i« »;r» :f Vcia* fci ^;-2*Z osiasks nc Tie tit: 93s :f

pc.iie-E. '.i vLizb. -.tlj » fi^•^ ^-il»es' sh. fee re*! Ei^«t aacva^B.
is wbc-ljj TOCT^i-

tkacrfcr^ it S 24 *::; i5. fciid rtjreae-iwd is ^rs. li asti 13.
ur»s presTt:^ bj ifc>r* i»*. ia:^'^! » -iTfe ■:«■ r-iJirr <i see fe«r

&«s of th*: primitiT-^ ;*>traL^ir ij which »«■ found in § **. «od
ank yoi only to thiijk of the two sfd*? of a p':ate of crr^tal panlkl
t« aoT one of them, that is '.o ^»y. an assemblage of soeh lajeis
a± those repr^^^oted ge«aietn>?aIlT in ng. H and shown in
5tftr«y.is«)pic view iti ti^. 13. If, as i* the ease »ith the sc4ids*

* Tb« Kliii qf th« pt^iosimirh uc T&fu:::> ia £=« plastei of Puis fne a

SUGGESTION FOR A TWIN-CRYSTAf.. 629

photographed in fig. 13, the under side of each solid is nearly
plane but slightly convex, and the top is somewhat sharply curved,
we have the kind of difference between the upper and under of the
two parallel sides of the crystal which I have already described to
you in § 21 above. In this case the assemblage is formed by letting
the solids fall down from above and settle in the hollows to which
they come most readily, or which give them the stablest position.
It would, we may suppose, be the hollows p'^V, not pqr (fig. 12,
§ 25), that would be chosen ; and thus, of the two formations
described in § 25, we should have that in which the hollows above
p'qV are occupied by the comparatively flat under-sides of the
molecules of the layer above, and the hollows below the apertures
pqr by the comparatively sharp tops of the molecules of the layer
below.

§ 37. For many cases of natural crystals of the wholly asym-
metric character, the true forces between the crystalline molecules
will determine precisely the same tactics of crystallization as would
be determined by the influence of gravity and fluid viscosity in the
settlement from water, of sand composed of uniform molecules of
the wholly unsymmetrical convex shape represented in figs. 12
and 13. Thus we can readily believe that a real crystal which is
growing by additions to the face seen in fig. 12 would give layer
after layer regularly as I have just described. But if by some
change of circumstances the plate, already grown to a thickness
of many layers in this way, should come to have the side facing
from us in the diagi-am exposed to the mother-liquor, or mother-
gas, and begin to grow from that face, the tactics might probably
be that each molecule would find its resting-place with its most
nearly plane side in the wider hollows under p'gV» instead of with
its sharpest corner in the narrower and steeper hollows under pqr,
as are the molecules in the layer below that shown in the diagram
in the first formation. The result would be a compound crystal
consisting of two parts, of differently oriented quality, cohering
perfectly together on the two sides of an interfacial plane. It
seems probable that this double structure may be found in nature,
presented by crystals of the wholly unsymmetric class, though it
may not hitherto have been observed or described in crystallo-
graphic treatises.

§ 38. This asymmetric double crystal becomes simply the

630 • APPENDIX H.

well-known symmetrical 'twin-crystal*' in the particular case in
which each of the constituent molecules is symmetrical on the
two sides of a plane through it parallel to the plane of our
diagrams, and also on the two sides of some plane perpendicular

B

Fig. 14.

to this plane. We see, in fact, that in this case if we cut in two
the double crystal by the plane of fig. 14, and turn one part
ideally through 180** round the intersection of these two planes,
we bring it into perfect coincidence with the other part. This
we readily understand by looking at fig. 14, in which the solid
shown in outline may be either an egg-shaped figure of revolution,
or may be such a figure flattened by compression perpendicular to
the plane of the diagram. The most readily chosen and the most
stable resting-places for the constituents of each successive layer
might be the wider hollows 'p'qr' : and therefore if, fi:'om a single

* ** A twin -crystal is composed of two crystals joined together in such a manner
** that one woald come into the position of the other by revolving through two right
** angles roand an axis which is perpendicular to a plane which either is, or may be,
*< a face of either crystal. The axis will be called the twin-axis, and the plane to
*' which it is perpendicalar the twin-plane.*' Miller's Treatise on Crystallography^
p. 103. In the text the word * twin-plane,' quoted from the writings of Stokea and
Rayleigh, is used to signify the plane common to the two crystals in each of the
oases referred to : and not the plane perpendicular to this plane, in which one part
of the crystal must be rotated to bring it into coincidence with the other, and which
is the twin-plate as defined by Miller.

Twinning in which each molecule turns {M, P. P. Vol. m. xcvii. §§ 60, 61) mast
also be considered. Here all the molecules are not necessarily of same orientation.
See 1. 12 from foot of § 37. E., Oct. 12, 1896.

Turning in the plane of the paper, as contemplated by Miller, could not bring
molecules into coincidence with big and little ends : but turning round the axis AB
does bring them into coincidence. K., Oct. 12, 1896.

SUCCESSIVE TWIN PLANES REGULAR OR IRREGULAR. 631

layer to begin with, the assemblage were to grow by layer after
layer added to it on each side, it might probably grow as a twin-
crystal. But it might also be that the presence of a molecule in
the wider hollow p'^V on one side, might render the occupation
of the corresponding hollow on the other side by another molecule
less probable, or even impossible. Hence, according to the con-
figuration and the molecular forces of the particular crystalline
molecule in natural crystallization, there may be necessarily, or
almoist necessarily, the twin, when growth proceeds simultaneously
on the two sides : or the twin growth may be impossible, because
the first occupation of the wider hollows on one side may compel
the continuity of the crystalline quality throughout, by leaving
only the narrower hollows pqr free for occupation by molecules
attaching themselves on the other side.

§ 39. Or the character of the crystalline molecule may be
such that when the assemblage grows by the addition of layer
after layer on one side only, with a not very strongly decided
preference to the wider hollows p'qY, some change of circumstances
may cause the molecules of one layer to place themselves in a
hollow pqr. The molecules in the next layer after this would find
the hollows p^qr occupied on the far side, and would thus have
a bias in favour of the hollows pqr. Thus layer after layer might
be added, constituting a twinned portion of the growth, growing,
however, with less strong security for continued homogeneousness
than when the crystal was growing, as at first, by occupation of
the wider hollows p'qV. A slight disturbance might again occur,
causing the molecules of a fresh layer to settle, not in the narrow
hollows pqr, but in the wider hollows p'qV, notwithstanding the
nearness of molecules already occupying the wider hollows on the
other side. Disturbances such as these occurring irregularly during
the growth of a crystal might produce a large number of successive
twinnings at parallel planes with irregular intervals between them,
or a large number of twinnings in planes at equal intervals might
be produced by some regular periodic disturbance occurring for a
certain number of periods, and then ceasing. Whether regular
and periodic, or irregular, the tendency would be that the number
of twinnings should be even, and that after the disturbances cease
the crystal should go on growing in the first manner, because of
the permanent bias in favour of the wider hollows p^^r^. These

632 APPENDIX H.

changes of molecular tactics, which we have been necessarily led
to by the consideration of the fortuitous concourse of molecules,
are no doubt exemplified in a large variety of twinnings and
counter-twinnings found in natural minerals. In the artificial
crystallization of chlorate of potash they are of frequent occurrence,
as is proved, not only by the twinnings and counter-twinnings
readily seen in the crystalline forms, but also by the brilliant
iridescence observed in many of the crystals found among a large
multitude, which was investigated scientifically by Sir Geoige
Stokes ten years ago, and described in a communication to the
Royal Society " On a remarkable phenomenon of crystalline reflec-
tion " (Proc. R /S., Vol. XXXVIII., 1885, p. 174).

§ 40. A very interesting phenomenon, presented by what was
originally a clear homogeneous crystal of chlorate of potash, €md
was altered by heating to about 245° — 248° Cent., which I am
able to show you through the kindness of Lord Rayleigh, and of
its discoverer, Mr Madan, presents another very wonderful case
of changing molecular tactics, most instructive in respect of the
molecular constitution of elastic solids. When I hold this plate
before you with the perpendicular to its plane inclined at 10°
or more to your line of vision you see a tinsel-like appearance,
almost as bright as if it were a plate of polished silver, on this
little area, which is a thin plate of chlorate of potash cemented
for preservation between two pieces of glass; and, when I hold
a light behind, you see that the little plate is almost perfectly
opaque like metal foil. But now when I hold it nearly perpen-
dicular to your line of vision the tinsel-like appearance is lost
You can see clearly through the plate, and you also see that very
little light is reflected from it. As a result, both of Mr Madan's
own investigations and further observations by himself. Lord
Rayleigh came to the conclusion that the almost total reflection
of white light which you see is due to the reflection of light at
many interfacial planes between successive layers of twinned and
counter-twinned crystal of small irregular thicknesses, and not to
any splits or cavities or any other deviation from homogeneousness
than that presented by homogeneous portions of oppositely twinned-
crystals in thorough molecular contact at the interfaces.

§ 41. When the primitive clear crystal was first heated very
gradually by Madan to near its melting-point (359° according to

madan's change of tactics by heat in chlorate of potash. 633

Camelly) it remained clear, and only acquired the tinsel appear-
ance after it had cooled to about 245° or 248° *. Rayleigh found
that if a crystal thus altered was again and again heated it always
lost the tinsel appearance, and became perfectly clear at some
temperature considerably below the melting-point, and regained it
at about the same temperature in cooling. It seems therefore
certain, that at temperatures above 248° and below the melting-
point, the molecules had so much of thermal motions as to keep
them hovering about the positions of pqr^ p'q'r' of our diagrams,
but not enough to do away with the rigidity of the solid ; and that
when cooled below 248° the molecules were allowed to settle in
one or other of the two configurations, but with little of bias for
one in preference to the other. It is certainly a very remarkable
fact in Natural History, discovered by these observations, that
when the molecules come together to form a crystal out of the
watery solution, there should be so much more decided a bias in
favour of continued homogeneousness of the assemblage than when,
by cooling, they are allowed to settle from their agitations in a rigid,
but nearly melting, solid.

§ 42. But even in crystallization from watery solution of
chlorate of potash the bias in favour of thorough homogeneousness

* ** A clear transparent crystal of potassium chlorate, from which the inevitable
" twin-plate had been ground away so as to reduce it to a single crystal film about
" 1 mm. in thickness, was placed between pieces of mica and laid on a thick iron
"plate. About 3 cm. from it was laid a small 'bit of potassium clilorate, and the
*' heat of a Bunsen burner was applied below this latter, so as to obtain an indication
*' when the temperature of the plate was approaching the fusing-point of the substance
" (369° C. according to Prof. Carnelly). The crystal plate was carefully watched
" during the heating, but no depreciation took place, and no visible alteration was
** observed, up to the point at which the small sentinel crystal immediately over the
" burner began to fuse. The lamp was now withdrawn, and wlien the temperature
*' had sunk a few degrees a remarkable change spread quickly and quietly over the
** crystal plate, causing it to reflect light almost as brilliantly as if a film of silver
** had been deposited upon it. No further alteration occurred during the cooling ;
" and the plate, after being ground and polished on both sides, was mounted with
" Canada balsam between glass plates for examination. Many crystals have been
"similarly treated with precisely similar results; and the temperature at which
" the change takes place has been deternuned to lie between 245° and 248°, by
" heating the plates upon a bath of melted tin in which a thermometer was im-
'* mersed. With single crystal plates no decrepitation has ever been observed,
" while with the ordinary twinned-plates it always occurs more or less violently,
"each fragment showing the brilliant reflective power above noticed." — Nature,
May 20, 1886.

634 APPENDIX H.

is not in every contingency decisive. In the first place, beginning,
as the formation seems to begin, firom a single molecular plane
layer such as that ideally shown in fig. 14, it goes on, not to make
a homogeneous crystal on the two sides of this layer, but probably
always so as to form a twin-crystal on its two sides, exactly as
described in § 38, and if so, certainly for the reason there stated.
This is what Madan calls the ''inveterate tendency to produce
twins (such as would assuredly drive a Malthus to despair)^";
and it is to this that he alludes as ''the inevitable twin-plate"
in the passage from his paper given in the footnote to § 41
above.

§ 43. In the second place, I must tell you that many of the
crystals produced from the watery solution by the ordinary process
of slow evaporation and crystallization show twinnings and counter-
twinnings at irregular intervals in the otherwise homogeneous
crystal on either one or both sides of the main central twin-plane,
which henceforth for brevity I shall call (adopting the hypothesis
already explained, which seems to me undoubtedly true) the
* initial plane.' Each twinning is followed, I believe, by a counter-
twinning at a very short distance &x)m it; at all events Lord
Rayleigh's observations •}- prove that the whole number of twinnings
and counter-twinnings in a thin disturbed stratum of the crystal
on one side of the main central twin-plane is generally, perhaps
always, even ; so that, except through some comparatively very
small part or parts of the whole thickness, the crystal on either
side of the middle or initial plane is homogeneous. This is exactly
the generally regular growth which I have described to you (§ 39)
as interrupted occasionally or accidentally by some unexplained
disturbing cause, but with an essential bias to the homogeneous
continuance of the more easy or natural one of the two con-
figurations.

§ 44. I have now great pleasure in showing you a most
interesting collection of the iridescent crystals of chlorate of
potash, each carefully mounted for preservation between two glass
plates, which have been kindly lent to us for this evening by
Mr Madan. In March, 1854, Dr W. Bird Herapath sent to

• Nature, May 20, 1886.

t Philosophical Magazine^ 1888, second half-year, p. 260.

SUCCESSIVE TWINNINGS EXPLAIN BRILLIANT COLOURS. 635

Prof. Stokes some crystals of chlorate of potash showing the
brilliant and beautiful colours you now see, and, thirty years
later, Prof. E. J. Mills recalled his attention to the subject by
sending him *'a fine collection of splendidly coloured crystals
" of chlorate of potash of considerable size, several of the plates
'' having an area of a square inch or more, and all of them being thick
" enough to handle without diflBculty." The consequence was that
Stokes made a searching examination into the character of the
phenomenon, and gave a short but splendidly interesting com-
munication to the Royal Society, of which I have already told
you. The existence of these beautifully coloured crystals had
been well known to chemical manufacturers for a long time, but
it does not appear that any mention of them was to be found in
any scientific journal or treatise prior to Stokes's paper of 1885.
He found that the colour was due to twinnings and counter-
twinnings in a very thin disturbed stratum of the crystal showing
itself by a very fine line, dark or glistening, according to the
direction of the incident light when a transverse section of the
plate of crystal was examined in a microscope. By comparison
with a spore of lycopodium he estimated that the breadth of this
line, and therefore the thickness of the disturbed stratum of the
crystal, ranged somewhere about the one-thousandth of an inch.
He found that the stratum was visibly thicker in those crystals
which showed red colour than in those which showed blue. He
concluded that '* the seat of the coloration is certainly a thin
twinned stratum " (that is to say, a homogeneous portion of crystal
between a twinning and a counter-twinning), and found that
"a single twin-plane does not show anything of the kind."

§ 45. A year or two later Lord Rayleigh entered on the
subject with an exhaustive mathematical investigation of the
reflection of light at a twin-plane of a crystal (Philosophical
Magazine, September, 1888), by the application of which, in a
second paper "On the remarkable Phenomenon of Crystalline
Reflection described by Prof. Stokes/' published in the same
number of the Philosophical Magazine, he gave what seems
certainly the true explanation of the results of Sir George Stokes's
experimental analysis of these beautiful phenomena. He came
very decidedly to the conclusion that the selective quality of the
iridescent portion of the crystal, in virtue of which it reflects

636 APPENDIX H.

almost totally light nearly of one particular wave-length for one
particular direction of incidence (on which the brilliance of the
coloration depends), cannot be due to merely a single twin-stratum,
but that it essentially is due to a considerable number of parallel
twin-strata at nearly equal distances. The light reflected by this
complex stratum is, for any particular direction of incident and
reflected ray, chiefly that of which the wave-length is equal to
twice the length of the period of the twinning and counter-
twinning, on a line drawn through the stratum in the directicm
of either the incident or the reflected ray.

§ 46. It seems to me probable that each twinning is essentially
followed closely by a counter-twinning. Probably three or four of
these twin-strata might suffice to give colour ; but in any of the
brilliant specimens as many as twenty or thirty, or more, might
probably be necessary to give so nearly monochromatic light as
was proved by Stokes's prismatic analysis of the colours observed
in many of his specimens. The disturbed stratum of about a one-
thousandth of an inch thickness, seen by him in the microscope,
amply suffices for the 5, 10, or 100 half wave-lengths required by
Rayleigh's theory to account for perceptible or brilliant coloration.
But what can be the cause of any approach to regular periodicity
in the structure sufficiently good to give the colours actually
observed? Periodical motion of the mother-liquor relatively to
the growing crystal might possibly account for it But Lord
Rayleigh tells us that he tried rocking the pan containing the
solution without result. Influence of light has been suggested,
and I believe tried, also without result, by several enquirers. We
know, by the beautiful discovery of Edniond Becquerel, of the
prisnjatic colours photographed on a prepared silver plate by the
solar spectrum, that ' standing waves * (that is to say, vibrations
with stationary nodes and stationary places of maximum vibration),
due to co-existence of incident and reflected waves, do produce
such a periodic structure as that which Rayleigh's theory shows
capable of giving a corresponding tint when illuminated by white
light. It is difficult, therefore, not to think that light may be
effiective in producing the periodic structure in the crystallization
of chlorate of potash, to which the iridescence is due. Still,
experimental evidence seems against this tempting theory, and
we must perforce be content with the question unanswered: —

TERNARY TACTICS IN LATERAL AND TERMINAL FACES OF QUARTZ. 637

What can be the cause of 5, or 10, or 100 pairs of twinning and
counter-twinning following one another in the crystallization with
sufficient regularity to give the colour: and why, if there are
twinnings and counter-twinnings, are they not at irregular intervals,
as those produced by Madan's process, and giving the observed
white tinsel-like appearance with no coloration ?

§ 47. And now I have sadly taxed your patience : and I fear
I have exhausted it and not exhausted my subject ! I feel I have
not got half-way through what I hoped I might be able to put
before you this evening regarding the molecular structure of
crystals. I particularly desired to speak to you of quartz crystal
with its ternary symmetry and its chirality*; and to have told
you of the etchingf by hydrofluoric acid which, as it were,
commences to unbuild the crystal by taking away molecule after
molecule, but not in the reverse order of the primary up-building ;
and which thus reveals differences of tactics in the alternate faces
of the six-sided pyramid which terminates at either end, some-
times at both ends, the six-sided prism constituting generally the
main bulk of the crystal. I must confine myself to giving you
a geometrical symbol for the ternary symmetry of the prism and
its terminal pyramid.

§ 48. Make an equilateral equiangular hexagonal prism, with
its diagonal from edge to edge ninety-five hundredths^ of its
length. Place a number of these close together, so as to make
up a hexagonal plane layer with its sides perpendicular to the
sides of the constituent hexagonal prisms : see fig. 15, and
imagine the semicircles replaced by their diameters. You see
in each side of the hexagonal assemblage edges of the constituent
prisms, and you see at each comer of the assemblage a face (not
an edge) of one of the constituent prisms. Build up a hexagonal
prismatic assemblage by placing layer after layer over it with the
constituent prisms of each layer vertically over those in the layer
below; and finish the assemblage with a six-sided pyramid by

* See footnote to § 22 above.

t Widmanstatten, 1S07. Leydolt (1S66), Wien, Akad. Ber, 16, 69, T. 9, 10.
Baamhaaer, Pogg, Ann, 138, 663 (1869); 140, 271; 142, 324; 146, 460; 160, 619.
For an account of these investigations see Mallard, TraiU de CrystaUograpkie
(Paris, 1884), Tome n. chapitre xvi.

X More exactly -9626, being } x cot 39'' 13' ; 8ee2§ 48.

638 APPENDIX H.

building upon the upper end of the prism, layer after layer of
diminishing hexagonal groups, each less by one circumferential
row than the layer below it. You thus have a crystal of precisely
the shape of a symmetrical specimen of rock crystal, with the
faces of its terminal pyramid inclined at 38° 13' to the faces of
the prism from which they spring. But the assemblage thus
constituted has * senary ' (or six-rayed) symmetry. To reduce this
to ternary symmetry, cut a groove through the middle of each
alternate face of the prismatic molecule, making this groove in
the first place parallel to the edges: and add a corresponding
projection or fillet to the middles of the other three faces, so that
two of the cylinders similarly oriented would fit together, with
the projecting fillet on one side of one of them entering the groove
in the anti-corresponding side of the other. The prismatic portion
of the assemblage thus formed shows (see fig. 15), on its alternate

Fig. 15.

edges, faces of molecules with projections and faces of molecules
with grooves; and shows only orientational differences between
alternate faces, whether of the pyramid or of the prism. Having
gone only so far from * senary' symmetry we have exactly the
triple, or three-pair anti-symmetry required for the piezo-electricity
of quartz investigated so admirably by the brothers Curie*, who
found that a thin plate of quartz crystal cut from any position
perpendicular to a pair of faces of a symmetrical crystal becomes

* J. and P. Curie and G. Friedel, Comptes Rendu9, 1SS2, 1883, 1886, 1892.

OPTIC AND PIEZO-ELECTBIC CHIBALITT IK QUABTZ.

639

PiAM or Top.

Plan or Top.

positively electrified on one side and negatively on the other
when pulled in a direction perpendicular to those faces. But this
assemblage has not the chiral piezo-
electric quality discovered theoreti-
cally by Voigt*, and experimentally
in quartz and in tourmaline by himself
and Rieckef, nor the well-known optic
chirality of quartz.

§ 49. Change now the directions of
the grooves and fillets to either of the
oblique configurations shown in fig. 16,
which I call right-handed, because the
directions of the projections are tan-
gential to the threads of a three-thread
right-handed screw, and fig. 17 (left-
handed). The prisms with their grooves and fillets will still all
fit together if they are all right-handed, or all left-handed,
fig. 18 shows the upper side of a hexagonal layer of an assemblage

Elcvation.
Fig. 16.

Elevation.
Fig. 17.

Fig. 18.

thus composed of the right-handed molecule of fig. 16. Fig. 15
unchanged, still represents a horizontal section through the centres
of the molecules. A prism built up of such layers, and finished

* ** Allgemeine Theorie der piezo- und pyroelectrischen Erscheinnngen an
Kiysiallen," W. Voigt, Knnigl, Gesellschaft der WUsentchaften zu G'dttingen, August
2, 1890.

t WUdemann^s Annalen, 1892, xlv. p. 923*

640 APPENDIX H.

at each end with a pyramid according to the rule of § 48, has aD
the qualities of ternary chiral symmetry required for the piezo-
electricity of quartz ; for the orientational differences of the
alternate pairs of prismatic faces; for the absolute difference
between the alternate pairs of faces of each pyramid which are
shown in the etching by hydrofluoric acid; for the merely
orientational difference between the parallel feuces of the two
pyramids; and for the well-known chiro-optic* property of quartx.
Look at two contiguous faces A, B oi our geometrical model
quartz crystal now before you, with its axis vertical You will
see a difference between them : turn it upside down ; B will be
undistinguishable from what A was, and A will be undistinguish-
able from what B was. Look at the two terminal pyramids, and
you will find that the face above A and the face below B are
identical in quality, and that they differ from the face above B
and below A, This model is composed of the right-handed
constituent molecules shown in fig. 16. It is so placed before
you that the edge of the prismatic part of the assemblage nearest
to you shows you filleted faces of the prismatic molecules. You
see two pyramidal faces; the one to your right hand, over B,
presents complicated projections and hollows at the corners of the
constituent molecules; and the pyramidal face next your left
hand, over A, presents their unmodified comera But it will be
the face next your left hand which will present the complex
bristling corners, and the face next your right hand that will
present the simple corners if, for the model before you, you
substitute a model composed of left-handed molecules, such as
those shown in fig. 17.

§ 50. To give all the (qualities of symmetry and an ti -symmetry'
of the pyro-electric and piezo-electric properties of tourmaline,
make a hollow in one terminal face of each of our constituent
prisms, and a corresponding projection in its other terminal
face.

§ 51. Coming back to quartz, we can now understand perfectly
the two kinds of macling which are well known to mineralogists
as being found in many natural specimens of the crystal, and
which I call respectively the orientational macling, and the chiral
macling. In the' orientational macling all the crystalline molecules

* QeneraUy miscalled * rotational.'

ORIENTATIONAL MACLING AND CHIRAL MACLING OF QUARTZ. 641

are right-handed, or all left-handed ; but through all of some part
of the crystal, each of our component hexagonal prisms is turned
round its axis through GO"" from the position it would have if the
structure were homogeneous throughout. In each of the two parts
the structure is homogeneous, and possesses all the electric and
optic properties which any homogeneous portion of quartz crystal
presents, and the facial properties of natural uncut crystal shown
in the etching by hydrofluoric acid ; but there is a discontinuity
at the interface, not generally plane between the two parts, which
in our geometrical model would be shown by non-fittings between
the molecules on the two sides of the interface, while all the
contiguous molecules in one part, and all the contiguous molecules
in the other part, fit into one another perfectly. In chiral macling,
which is continually found in amethystine quartz, and sometimes
in ordinary clear quartz crystals, some parts are composed of right-
handed molecules, and others of left-handed molecules. It is not
known whether, in this chiral macling, there is or there is not
also the orientational macling on the two sides of each interface ;
but we may say probably rwt ; because we know that the orienta-
tional macling occurs in nature without any chiral macling, and
because there does not seem reason to expect that chiral macling
would imply orientational macling on the two sides of the same
interface. I would like to have spoken to you more of this most
interesting subject; and to have pointed out to you that some
of the simplest and most natural suppositions we can make as
to the chemical forces (or electrical forces, which probably means
the same thing) concerned in a single chemical molecule of quartz
SiOsi and acting between it and similar neighbouring
molecules, would lead essentially to these molecules
coming together in triplets, each necessarily either right-
handed or left-handed, but with as much probability of
one configuration as of the other: and to have shown
you that these triplets of silica 3(Si02) can form a
crystalline molecule with all the properties of ternary p|g 19^
chiral symmetry, typified by our grooved hexagonal
prisms, and can build up a quartz crystal by the fortuitous
concourse of atoms. I should like also to have suggested and
explained the possibility that a right-handed crystalline molecule
thus formed may, in natural circumstances of high temperature,
or even of great pressure, become changed into a left-handed

T. L. 41

642 APPENDIX H.

crystal, or vice versa. My watch, however, warns me that I must
not enter on this subject

§ 52. Coming back to mere molecular tactics of crystals,
remark that our assemblage of rounded, thoroughly scalene tetra-
hedrons, shown in the stereoscopic picture (§ 36, fig. 13 above),
essentially has chirality because each constituent tetrahedron, if
wholly scalene, has chirality*. I should like to have explained
to you how a single or double homogeneous assemblage of points
has essentially no chirality, and how three assemblages of single
points, or a single assemblage of triplets of points, can have
chirality, though a single triplet of points cannot have chirality.
I should like indeed to have brought somewhat thoroughly before
you the geometrical theory of chirality; and in illustration to
have explained the conditions under which four points or two
lines, or a line and two points, or a combination of point, line and
plane, can have chirality: and how a homogeneous assemblage
of non-chiral objects can have chirality ; but in pity I forbear, and
I thank you for the extreme patience with which you have listened
to me.

* See footnote to § 22 above.

APPENDIX I.

ON THE ELASTICITY OF A CRYSTAL ACCORDING

TO BOSCOVICH*

§ 1, A CRYSTAL in nature is essentially a homogeneous
assemblage of equal and similar molecules, which for brevity I
shall call crystalline molecules. The crystalline molecule may
be the smallest portion which can be taken from the substance
without chemical decomposition, that is to say, it may be the
group of atoms kept together by chemical affinity, which con-
stitutes what for brevity I shall call the chemical molecule; or
it may be a group of two, three, or more of these chemical
molecules kept together by cohesive force. In a crystal of tartaric
acid the crystalline molecule may be, and it seems to me probably
is, the chemical molecule, because if a crystal of tartaric acid is
dissolved and recrystallised it always remains dextro-chiral. In
a crystal of chlorate of soda, as has been pointed out to me by
Sir George Stokes, the crystalline molecule probably consists of
a group of two or more of the chemical molecules constituting
chlorate of soda, because, as found by Marbachf, crystals of the
substance are some of them dextro-chiral and some of them levo-
chiral ; and if a crystal of cither chirality is dissolved the solution
shows no chirality in its action on polarised light ; but if it is
recr}'stallised the crystals are found to be some of them dextro-
chiral and some of them levo-chiral, as shown both by their
crystalline forms and by their action on polarised light. It is
possible, however, that even in chlorate of soda the crystalline
molecule may be the chemical molecule, because it may be that
the chemical molecule in solution has its atoms relatively mobile
enough not to remain persistently in any dextro-chiral or levo-

* From Proc, R. S, June 8, 1893.

t Pogg. Ann. Vol.xci. pp.482— 487 (1854); or Ann, de Chimie, Vol. xun. (lv.),
pp. 262—255.

41—2

644 APPENDIX I.

chiral grouping, and that each individual chemical molecule
settles into either a dextro-chiral or levo-chiial configuration in
the act of forming a crystal. See ''Molecular Tactics," Oxford
Lecture, § 52, reproduced as App. H in the present volume,

§ 2. Certain it is that the crystalline molecule has a chiral
configuration in ever}' crystal which shows cfairality in its crystal-
line form or which produces right- or left-handed rotation of the
plane of polarisation of light passing through it. The magnetic
rotation has neither right-handed nor left-handed quality (that is
to say, no chirality). This was perfectly understood by Faraday
and made clear in his writings, yet even to the present day we
frequently find the chiral rotation and the magnetic rotation of
the plane of polarised light classed together in a manner against
which Faraday's original description of his discovery of the
magnetic polarisation contains ample warning.

§ 3. These questions, however, of chiralit}* and magnetic
rotation do not belong to my present subject, which is m^^Iy
the forcive* required to keep a crystal homogeneously strained
to any infinitesimal extent firom the condition in which it rests
when no force acts upon it firom without. In the elements of
the mathematical theory of elasticity f we find that this forcive
constitutes what is called a homogeneous stress, and is specified
completely by six generalised force-components, /),, p^, /i,, ...,/3^,
which are related to six corresponding generalised components
of strain, «i, &., «3, ..., «i, by the following formulas: —

w = i(pi«i+p2«a+.-.+l^«g) (IX

where w denotes the work required per unit volume to alter any

portion of the crystal from its natural unstressed and unstrained

condition to any condition of infinitesimal homogeneous stress or

strain : —

dw dw

where t- , •••> ;t- denote differential coefficients on the supposition

* Thifl is a word introdoeed by mj brother, the late ProfesBor James Thomson,
to designate any system of forces.

t PkiL Trant, April 24, 1856, reprinted in Vol. m. Math, and Pkyu P^^en
(Sir W. Thomson), pp. Si— 112.

ONLY 15 COEFFICIENTS FOR SIMPLEST BOSCOVICHIAN CRYSTAL. 645

that w is expressed as a homogeneous quadratic function of

*'"d^' ••"*•" dp, ^'*^'

where , ,...,, denote differential coefficients on the supposi-
dpi aps

tion that w is expressed as a homogeneous quadratic function of

Pu ---,P9-

§ 4. Each crystalline molecule in reality certainly experiences
forcive from some of its nearest neighbours on two sides, and
probably also from next nearest neighbours and others. What-
ever the mutual forcive between two mutually acting crystalline
molecules is in reality, and however it is produced, whether by
continuous pressure in some medium, or by action at a distance,
we may ideally reduce it, according to elementary statical
principles, to two forces, or to one single force and a couple in a-
plane perpendicular to that force. Boscovich's theory, a purely
mathematical idealism, makes each crystalline molecule a single
point, or a group of points, and assumes that there is a mutual
force between each point of one crystalline molecule and each
point of neighbouring crystalline molecules, in the line joining the
two points. The very simplest Boscovichian idea of a crystal is a
homogeneous gi-oup of single points. The next simplest idea is a
homogeneous group of double points.

§ 5. In the present communication, I demonstrate that, if
we take the very simplest Boscovichian idea of a crystal, a
homogeneous group of single points, we find essentially six
relations between the twenty-one coefficients in the quadratic
function expressing w, whether in terms o{ 8i, ...fS^or ofpi, ...,p^.
These six relations are such that incompressibility, that is to say
infinite resistance to change of bulk, involves infinite rigidity. In
the particular case of an equilateral* homogeneous assemblage
with such a law of force as to give equal rigidities for all directions
of shearing, these six relations give Sk = on, which is the relation

* That is to say, an assemblage in which the lines from any point to three
neighbours nearest to it and nearest to one another are inclined at 60° to one another ;
and these neighbours are at equal distances from it. This implies that each point
has twelve equidistant nearest neighbours around it, and that any tetrahedron of
four nearest neighbours has for its four faces four equilateral triangles.

646 APPENDIX I.

found by Navier and Poisson in their Boscovichian theory for
isotropic elasticity in a solid. This relation was shown by Stokes
to be violated by many real homogeneous isotropic substances,*
such, for example, as jelly and india-rubber, which oppose so great
resistance to compression and so small resistance to change of
shape, that we may, with but little practical error, consider them
as incompressible elastic solids.

§ 6. I next demonstrate that if we take the next simplest
Boscovichian idea for a crystal, a homogeneous group of double
points, we can assign very simple laws of variation of the forces
between the points which shall give any arbitrarily assigned value
to each of the twenty-one coeflScients in either of the quadratic
expressions for w.

§ 7. I consider particularly the problem of assigning such
values to the twenty-one coeflScients of either of the quadratic
formulas as shall render the solid incompressible. This is most
easily done by taking i^; as a quadratic function of j)], ..., jo,, and
by taking one of these generalised stress components, say p^, as
uniform positive or negative pressure in all dii'ections. This
makes 8^ uniform compression or extension in all directions, and
makes *i, ...,*6 five distortional components with no change of
bulk. The condition that the solid shall be incompressible is
then simply that the coefficients of the six terms involving p^ arc
each of them zero. Thus, the expression for %o becomes merely
a quadratic function of the five distortional stress-components,
Pii •••jj^flj with fifteen independent coefficients : and equations (3)
of § 3 above express the five distortional components as linear
functions of the five stress-components with these fifteen in-
dependent coefficients.

Added July 18, 1893.

§ 8. To demonstrate the propositions of § 5, let OX, 0 F, OZ
be three mutually perpendicular lines through any point 0 of a
homogeneous assemblage, and let a?, y, z be the coordinates of any
other point P of the assemblage, in its unstn\ined condition. As
it is a homogeneous assemblage of single points that we are now
considering, there must be another point P', whose coordinates
are — a?, — y, — -?. Let (x -|- Sa?, y + Sy, z + hz) be the coordinates
of the altered position of P in any condition of infinitesimal

DISPLACEMENT-RATIOS AND THEIR Si^UAKES AND PRODUCTS. 647

strain, specified by the six symbols e, /, g, a, 6, c, according to
the notation of Thomson and Tait's Natural PhUasophy, Vol. i.,
Ft. II., § 669. In this notation, e,f,g denote simple infinitesimal
elongations parallel to OX, OY, OZ respectively; and a, 6, c
infinitesimal changes from the right angles between three paitts
of planes of the substance, which, in the unstrained condition, are
parallel to {XOY, XOZ\ (YOZ, YOX), (ZOX, ZOY) respectively
(all angles being measured in terms of the radian). The definition
of a, 6, c may be given, in other words, as follows, with a taken as
example : a denotes the difference of component motions parallel
to OF of two planes of the substance at unit distance asunder,
kept parallel to YOX during the displacement ; or, which is the
same thing, the difference of component motioas parallel to OZ of
two planes at unit distance asunder kept parallel to ZOX during
the displacement. To avoid the unnecessary consideration of
rotational displacement, we shall suppose the displacement corre-
sponding to the strain-component a to consist of elongation
perpendicular to OX in the plane through OX bisecting YOZ,
and shrinkage perpendicular to OX in the plane through OX
perpendicular to that bisecting plane. This displacement gives
no contribution to &r, and contributes to £y and hz respectively
i^az and \ay. Hence, and dealing similarly with h and c, and
taking into account the contributions of c, /, g, we find

Sa: = ea? + J^ (6^ + cy)\

^y^fy-^h{ox + az)\ (4).

hz =gz + ^(ay + bx))

§ 9. In our dynamical treatment below, the following formulas,
in which ix)wers higher than squares or products of the infini-
tesimal ratios Sa:/r, Sy/r, Sz/r (r denoting OP) are neglected, will
be found useful.

Br xBx + ySy + zBz .Sx' + By' + Sz^ . /iJcSx-{-yBy + zSzy

Now by (4) we have

xBaj + ySy + zSz^ea^ + fy^+gz^ + ayz + bzx + cwy ...(6),
and

Baf'-{'Sf + Sz^ = ^ic" +/y + (/V

+ [i6c + (/+ sr) a] y-? + [^ca -\-(g + e)b] zx + [ia6 + (e +/) c]xy. . .(7).

648

APPENDIX I.

Using (6) and (7) in (5), we find

Sr

— = r-^{ex'-¥ff + gz^'\'at/z + bzx'{-cxi/)'\-q(e,f,g,a,b,c)...{Sl

where g denotes a quadratic function of e, /, &c., with coefficients
as follows : —

Coefficient of le' is -^— .

\

ia»

y + z' f^

/if

be

yz a?yz

ea

a?yz

eb

.zx a?z

\

(9)

and corresponding symmetrical expressions for the other fifteen
coefficients.

§ 10. Going back now to § 3, let us find w, the work per unit
volume, required to alter our homogeneous assemblage fi-om its
unstrained condition to the infinitesimally strained condition
specified by e, /, g, a, 6, c. Let ^ (r) be the work required to
bring two points of the system from an infinitely great distance
asunder to distance r. This is what I shall call the mutual
potential energy of two points at distance r. What I shall now
call the potential energy of the whole system, and denote by W,
is the total work which must be done to bring all the points of it
from infinite mutual distances to their actual positions in the
system; so that we have

F = i22<^(r) (10),

where S0(r) denotes the sum of the values of j>{r) for the
distances between any one point 0, and all the others; and
22^ (r) denotes the sum of these sums with the point 0 taken
successively at every point of the system. In this double sum-
mation ^(z') is taken twice over, whence the factor J^ in the
formula (10).

WORK AGAINST MUTUAL BOSCOVICHIAN FORCES. 649

§ 11. Suppose DOW the law of force to be such that <t>(r)
vanishes for every value of r greater than v\ where \ denotes
the distance between any one point and its nearest neighbour,
and V any small or large numeric exceeding unity, and limited
only by the condition that v\ is very small in comparison with
the linear dimensions of the whole assemblage. This, and the
homogeneousness of our assemblage, imply that, except through
a very thin surface layer of thickness i/X, exceedingly small in
comparison with diameters of the assemblage, every point ex-
periences the same set of balancing forces from neighbours as
every other point, whether the system be in what we have called
its unstrained condition or in any condition whatever of homo-
geneous strain. This strain is not of necessity an infinitely small
strain, so far as concerns the proposition just stated, although in
our mathematical work we limit ourselves to strains which are
infinitely small.

§ 12. Remark also that if the whole system be given as a
homogeneous assemblage of any specified description, and if all
points in the surface-layer be held by externally applied forces in
their positions as constituents of a finite homogeneous assemblage,
the whole assemblage will be in equilibrium under the influence
of mutual forces between the points ; because the force exerted on
any point 0 by any point P is balanced by the equal and opposite
force exerted by the point P' at equal distance on the opposite
side of 0.

§ 13. Neglecting now all points in the thin surface layer,
let N denote the whole number of points in the homogeneous
assemblage within it. We have, in § 10, by reason of the homo-
geneousness of the assemblage,

tX<t>(r)^Nt<l>(r) (11),

and equation (10) becomes

W = ^NX<t>(r) (12).

Hence, by Taylor's theorem,

STf = ii^{f (r)Sr + i<^"(r)S»-^} (13);

and using (8) in this, and remarking that if (as in § 14 below) we
take the volume of our assemblage as unity, so that N is the

650

APPENDIX I.

number of points per unit volume, SW becomes the «£; of §3;
we find

■¥r<f>'(r)q(e,f,g,a,b,c)

§ 14. Let us now suppose, for simplicity, the whole assemblage,
in its unstrained condition, to be a cube of unit edge, and let P be
the sum of the normal components of the extraneous forces applied
to the points of the surface-layer in one of the faces of the cube.

0

\

...r

...'-•- —

7^

x^

Fig. 1.

The equilibrium of the cube, as a whole, requires an e([ual aud
opposite normal component P in the opposite face of the cube.
Similarly, let Q and jK denote the sums of the nonnal components
of extraneous force on the two other pairs of faces of the cube.
Let T be the sum of tangential components, parallel to OZ, of the
extraneous forces on either of the YZ faces. The equilibrium of
the cube as a whole requires four such forces on the four faces
parallel to OY, constituting two balancing couples, as shown in
the accompanying diagram. Similarly, we must have four
balancing tangential forces /S on the four faces parallel to OX,
and four tangential forces V on the four faces parallel to OZ.

§ 15. Considering now an infinitely small change of strain in
the cube from (e, /, gr, a, 6, c) to {e + cfe, /+ df, g-^dg, a + da,
b-hdb, c + dc) ; the work required to produce it, as we see by

'[...(16).

ELASTIC ENERGY OF SIMPLE HOMOGENEOUS ASSEMBLAGE. 651

considering the definitions of the displacements e, f, g, a, b, c,
explained above in § 8, is as follows,

dw^Pde + Qd/+Itdg + Sda + Tdb+ Udc (15).

Hence we have

P = dw/de ; Q = dw/df; R = dw/dg ;

S = dw/da; T^^dw/db; V^dwjdc

Hence, by (14), and taking L, L to denote linear functions, we
find

P = ii\r2 [^'^"^ a-^L {ej. 5r, a, 6, c)l]

^ -^ I (17N

S = \NX ^'P yz + L (6,/ (/, a, 6, c)| J

and symmetrical expressions for Q, R, T, U,

§ 16. Let now our condition of zero strain bo one* in which
no extraneous force is required to prevent the assemblage from
leaving it. We must have P = 0, Q = 0, i2 = 0, S = 0, 7=0,
Z7=0, when e = 0, /=0, flr = 0, a = 0, 6 = 0, c = 0. Hence, by
(17), and the other four symmetrical formulae, we see that

...(18).

Hence, in the summation for all the points a?, y, z, between
which and the point 0 there is force, we see that the first term of

the summed coefficients in q, given by (9) above, vanishes in

every case, except those of ^ and ca, in each of which there is
only a single term; and thus from (9) and (14) we find

+ {he + ea) 2«r ^ + eh%m -

where - r<^' (r) + ^-^t^" (r) = tir (20).

* The consideration of the equilibrium of the thin surface layer, in these circum-
stances, nnder the influence of merely their proper mutual forces, is exceedingly
interesting, both in its relation to Laplace's theory of capillary attraction, and to the
physical condition of the faces of a crystal and of surfaces of irregular fracture.
But it must be deferred. [See App. J, |g 29» 30.]

652 APPENDIX L

The terms given explicitly in (19) suffice to show bysymmetiT
all the remaining terms represented by the * Ac"

§ 17. Thus we see that with no limitation whatever to the
number of neighbours acting with sensible force on any one
point 0, and with no simplifying assumption as to the law of
force, we have in the quadratic for w equal values for the
coefficients o( /g and ^a'; ge and ^b^; ef and ^c*; be and ea:
ca and eb; and ab and ec. These equalities constitute the six
relations promised for demonstration in § 5.

§ 18. In the particular case of an equilateral assemUage,
with axes OX, OY, OZ parallel to the three pairs of opposite
edges of a tetrahedron of four nearest neighbours, the coefficients
which we have found for all the products except ^,^, ^dearly
vanish; because in the complete sum for a single homogeneous
equilateral assemblage we have ±x, ±y, ±z in the t^mmetrical
terms. Hence, and because for this case

2«r5 = 2«rg = 2«r^, and 2«r?^ = 2«r ^ = 2«r^ ..,(21),

(19) becomes
a; = i a (6= +/» + fif=) + 13 (/5r +5r6 + 6/) + in (tt« + 6= + c»)...(22).

where ^^\N^m^, and J3 = n = i.V2t;j^^'f (23).

§ 19. Looking to Thomson and Tait's Xaiunil Philosophy,
§ (i95 (7)*, wc see that the 33 of that formula is now proved to
be, in our present simplest form of Boscovichian assumption, equal
to the n of our present formula (22) which denotes the rigidity-
modulus relative to shearings parallel to the planes YOZ, ZOX,
XOY: and that if we denote by r?i the rigidity-modulus relative
to shearing parallel to planes through OX, OY, OZ, and cutting
(OY, OZ), {OZ, OX), {OX, OY) at angles of 45°, and if k denote
the compressibility-modulus, wc have

}

* This formula in given for the case of a body which is wholly isotropic in respect
to elasticity moduluses ; bat from the investigation in §§ 681, 6S2 we see that our
present formula, (22) or (25), expresses the elastic energy for the case of an elastio
solid possessing cubic isotropy with unequal rigidities, Up n, in respect to these two
sets of shearings.

NAVIER-POISSON RELATION PROVED FOR SIMPLE ASSEMBLAGE. 653

and the previous expression for the elastic energy of the strained
solid becomes, for the case of cubic symmetry without any re-
striction,

+ n{a^ + h' + <f) (25).

§ 20. Comparing this with (22), for our present restricted

case, we find

3fe = 2n, + 37i (26).

This remarkable relation between the two rigidities and the
compressibility of an equilateral homogeneous assemblage of
Boscovich atoms was announced without proof in § 27 of my
paper on the "Molecular Constitution of Matter*." In it n
denotes what I called the facial rigidity, being rigidity relative
to shearings parallel to the faces of the principal cubef : and n^
the diagonal rigidity, being rigidity relative to shearings parallel
to any of the six diagonal planes through pairs of mutually
remotest parallel edges of the same cube. By (24) and (23) we
see that if the law of force be such that

2^^ = 32w3jf' (27),

we have n = 7ii, and the body constituted by the assemblage is
wholly isotropic in its elastic quality. In this cajse (26) becomes
3A; = on, as found by Navier and Poisson ; and thus we complete
the demonstration of the statements of § 5 above.

§ 21. A case which is not uninteresting in respect to Bosco-
vichian theory, and which is very interesting indeed in respect to
mechanical engineering (of which the relationship with Bosco-
vich*8 theory has been pointed out and beautifully illustrated
by M. Brillouin|), is the case of an equilateral homogeneous
assemblage with forces only between each point and its twelve
equidistant nearest neighbours. The annexed diagram (fig. 2)
represents the point 0 and three of its twelve nearest neighbours

♦ 1?. S, E, Proc,, July, 1889; Art. xcvn. of my Math, and Phyt. Papem,
VoL ni.

t That is to say, a cube whose edges are parallel to the three pairs of opposite
edges of a tetrahedron of foar nearest neighbours.

X ConfSrenees Scientifiquea et Allocutions (Lord Kelvin), tradnites et annoU*es;
P. Lagol et M. Brillonin : Paris, 1893, pp. 820—325.

654

APPENDIX I.

(their distances X), being in the middles of the near faces of the
principal cube shown in the diagram ; and* three of its six next-
nearest neighbours (their distances XV^X being at X, Y, Z, the
comers of the cube nearest to it ; and, at other comei*s of the
cube, three other neighbours K, Z, M, which are next-next-next-
nearest (their distances '2X). The points in the middles of the
three remote sides of the cube, not seen in the diagram, are
next-next-nearest neighbours of 0 (their distances XV3).

Fig. 2.

§ 22. Confining our attention now to O's nearest neighbours,
we see that the nine not shown in the diagram are in the middles
of squares obtained by producing the lines YO, ZO, XO to equal
distances beyond 0 and completing the squares on all the pairs of
lines so obtained. To see this more clearly, imagine eight equal
cubes placed together, with faces in contact and each with one
comer at 0. The pairs of faces in contact are four squares in
each of the three planes cutting one another at right angles
through 0; and the centres of these twelve squares are the
twelve nearest neighbours of 0. If we denote by X the distance
of each of them firom 0, we have for the coordinates x, y, z of
these twelve points as follows: —

V'V2'V2/' V' V2V2y' V'V2' V2/* V' V2'

-AY

V2/

W2'^V2)* ( V2''^V2J* W2''^' V2/* ( V2'^' '^ij

\ \

V2 ' V2

'^J' ( V2V2' V' W2'~V2' V' ("V2' \/2'V

(28).

SINGLE ASSEMBLAGE IN SIMPLE CUBIC ORDER. 655

§ 23. Suppose now 0 to experience force only from its twelve
nearest neighbours : the summations 2 of § 18 (23) will include
just these twelve points with equal values of tsr, which we shall
denote by w©, for all. These yield eight tenns to 2(a?*/r*), and
four to 2 (y^z^/i^)', and the value of each term in these sums is J.
Thus we find that

^^Nmo, and 13 = n = ^JV^Wo (29).

Hence and by (24), we see that

Wi = in (30).

Thus we have the remarkable result that, relatively to the
principal cube, the diagonal rigidity is half the facial rigidity
when each point experiences force only from its twelve nearest
neighbours. This proposition was announced without proof in
§ 28 of " Molecular Constitution of Matter* "

§ 24. Suppose now the points in the middles of the faces of
the cubes which in the equilateral assemblage are 0*8 twelve
equidistant nearest neighbours to be removed, and the assemblage
to consist of points in simplest cubic order; that is to say, of
Boscovichian points at the points of intersection of three sets
of equidistant parallel planes dividing space into cubes. Fig. 2
shows 0 ; and, at X, F, Z, three of the six equidistant nearest
neighbours which it has in the simple cubic arrangement.
Keeping X with the same signification in respect to fig. 2 as
before, we have now for the coordinates of O's six nearest
neighbours :

(\ V2, 0, 0), (0, \V2, 0), (0, 0, X V2),

(- X V2, 0, 0), (0, - X V2, 0), (0, 0, - X V2).

Hence, and denoting by w, the value of tsr for this case, we
find, by § 18 (23),

a = -nrt!ri and J3 = n = 0 (31).

The explanation of w = 0 (facial rigidity zero) is obvious when wo
consider that a cube having for its edges twelve equal straight
bars, with their ends jointed by threes at the eight comers, affords
no resistance to change of the right angles of its faces to acute
and obtuse angles.

* Math, and Pkyf, Papen^ Vol. in. p. 403.

656 APPENDIX I.

§ 25. Replacing now the Boscovich points in the middles of
the faces of the cubes, from which we supposed them temporarily
annulled in § 24; putting the results of § 23 and § 24 together;
and using (24) of § 19 ; we find for our equilateral homogeneoiu
assemblage, its elasticity-moduluses as follows : —

a = N(^o + tsTi)
J3 = n = ^Nmo

where, as we see by § 16 (20) above,

.(32).

1

, (33);

w, = X V2i^(X V2) - 2X«i?" (X V2)|

F(r) being now taken to denote repulsion between any two of the
points at any distance r, which, with ^ (r) defined as in § 10, is
the meaning of — <f>' (r). To render the solid, constituted of our
homogeneous assemblage, elastically isotropic, we must, by § 19
(24), have a - J3 = 2n, and therefore, by (32),

i«ro = 2cri (34X

By (33) we see that the distant forces contribute to «,, and
not to n.

§ 26. The l&st three of the six equilibrium equations, § 16
(18), are fulfilled in virtue of symmetry in the case of an equi-
lateral assemblage of single points whatever be the law of force
between them, and whatever be the distance between any point
and its nearest neighbours. The first three of them require in
the case of' § 23 that i^(X) = 0; and in the case of (24) that
-F(X\/2) = 0, results of which the interpretation is obvious and
important.

§ 27. The first three of the six equilibrium equations, § 16
(18), applied to the case of § 25, yield the following equation : —

Vi^(XV2) = -^(X) (35);

that is to say, if there is repulsion or attraction between each
point and its twelve nearest neighbours, there is attraction or
repulsion of \/2 of its amount between each point and its six
next-nearest neighbours, unless there are also forces between
more distant points. This result is easily verified by simple
sjmthetical and geometrical considerations of the equilibrium
between a point and its twelve nearest and six next-nearest

SCALENE HOMOGENEOUS ASSEMBLAGE OF POINTS. 657

neighbours in an equilateral homogeneous assemblage. The con-
sideration of it is exceedingly interesting and important in respect
to, and in illustration of, the engineering of jointed structures
with redundant links or tie-struts.

§ 28. Leaving, now, the case of an equilateral homogeneous
assemblage, let us consider what we may call a scalene assemblage,
that is to say, an assemblage in which there are three sets of
parallel rows of points, determinately fixed as follows, according
to the system first taught by Bravais*: —

I. Just one set of rows of points at consecutively shortest
distances Xj.

IL Just one set of rows of points at consecutively next-
shortest distances X,.

III. Just one set of rows of points at consecutive distances X,
shorter than those of all other rows not in the plane
of L and II.

To the condition X, > X, > X, we may add the condition that
none of the angles between the three sets of rows is a right angle,
in order that our assemblage may be what we may call wholly
scalene.

§ 29. Let A'OA, BOB, CVC be the primary rows thus
determinately found having any chosen point, 0, in common ; we
have

BO^OB^xX (36).

CV = OC = A,J

Thus A' and A are O's nearest neighbours ; and B and B, 0*8
next-nearest neighbours; and C" and C, O's nearest neighbours
not in the plane AOB. (It should be understood that there may
be in the plane AOB points which, though at greater distances
from 0 than B and B, are nearer to 0 than are C and C\)

§ 30. Supposing, now, BOO, BOC\ &c., to be the acute angles
between the three lines meeting in 0; we have two equal and

* Journal de VEcole Polytechniquet tome xix. cahier xxziii. pp. 1 — 12S ; Paris,
1S50.

T. L. 42

658 APPENDIX I.

dichirally similar* tetrahedrons of each of which each of the fear
faces is a scalene acute-angled triangle. That every angle in and
between the faces is acute we readily see, by remembering that
OC and OC are shorter than the distances of 0 from any other of
the points on the two sides of the plane AOBf.

§ 31. As a preliminary to the engineering of an incompressible
elastic solid according to Boscovich, it is convenient now to con-
sider a special case of scalene tetrahedron, in which perpendiculars
f from the four comers to the four opposite faces intersect in one
point. I do not know if the species of tetrahedron which fulfils
this condition has found a place in geometrical treatises, but I am
informed by Dr Forsjrth that it has appeared in Cambridge
examination papers. For my present purpose it occurred to me
thus : — Let QO, QA, QB^ QG be four lines of given lengths drawn
from one point, Q. It is required to draw them in such relative
directions that the volume of the tetrahedron OABG is a
maximum. Whatever be the four given lengths, this problem
clearly has one real solution and one only; and it is such that
the four planes BOC, CO A, AOB, ABC are cut perpendicularly by
the lines AQ, BQ, CQ, OQ, respectively, each produced through Q,
Thus we see that the special tetrahedron is defined by four
lengths, and conclude that two equations among the six edges
of the tetrahedron in general are required to make it our special
tetrahedron.

§ 32. Hence we see the following simple way of drawing a
special tetrahedron. Choose as data three sides of one face and
the length of the perpendicular to it from the opposite angle.
The planes through this perpendicular, and the angles of the
triangle, contain the perpendiculars from these angles to the
opposite faces of the tetrahedron, and therefore cut the opposite
sides of the triangle perpendicularly. (Thus, parenthetically, we
have a proof of the known theorem of elementary geometry that
the perpendiculars from the three angles of a triangle to the

* Either of these may be turned round so as to coincide with the image of the
other in any plane mirror. Either may be called a pervert of the other; as,
according to the usage of some writers, an object is called a pervert of another if
one of them can be brought to coincide with the image of the other in a plane
mirror (as, for example, a right hand and a left hand).

t See *" Molecular Constitution of Matter," § (45), (h), (i), Math, and Phys. Papert,
Vol. in. pp. 412-413.

SCALENE ASSEIIBLAQE GIVINQ AN INCOMPRESSIBLE SOLID. 659

opposite Bides intersect in one point.) Let ABC be the chosen
triangle and S the point in which it is cut by the perpendicular
from O, the opposite corner of the tetrahedron. AS, BS, CS,
produced through S, cut the opposite sides perpendicularly, and
therefore we find the point S by drawing two of these perpen-
diculars and taking their point of intersection. The tetrahedron

Fig. 8.

is then found by drawing through S a line SO of the given
length perpendicular to the plane of ABC. (We have, again
parenthetically, an interesting geometrical theorem. The per-
pendiculars from A, By C to the planes of OBC, OCA, OAB
cut OS in the same point; SO being of any arbitrarily chosen
length.)

§ 33. I wish now to show how an incompressible homogeneous
solid of wholly oblique crystalline configuration can be constructed
without going beyond Boscovich for material. Consider, in any
scalene assemblage, the plane of the line A'OA through any
point 0 and its nearest neighbours, and the line ROB through
the same point and its next-nearest neighbours. To fix the ideas,
and avoid circumlocutions, we shall suppose this plane to be
horizontal. Consider the two parallel planes of points nearest
to the plane above it and below it. The comer C of the acute-
angled tetrahedron OABC, which we have been considering, is
one of the points in one of the two nearest parallel planes, that
above AOB we shall suppose. And the corner C of the equal
and dichirally similar tetrahedron OA'BC is one of the points in
the nearest parallel plane below. All the points in the plane
through C are corners of equal tetrahedrons chirally similar to
OABC, and standing on the horizontal triangles oriented as BOA,

42—2

660 APPEXDIX L

All the points C in the nearest plane below are oomezs of tetim-
hedrons chirally similar to OA'RC placed downwanis oo tk
triangles oriented as ROA'. The Tolome oi the tetrahedroa
OABC is ^ of the volume of the parallelepiped, of which OA^
OB, OC are conterminoos edges. Henoe the sam of the Tohmei
of all the upward tetrahedrons having their bases in one phne
is ^ of the volume of the space between large areas of iheat
planes : and, therefore, the sum of all the chiiallj siisilar leiza-
hedrons, such as OABC, is | of the whole volame of the
assemblage through any larger space. Hence any hoiDogeiieoQi
strain of the assemblage which does not alter the T<diinie of the
tetrahedrons does not alter the volume of the solid. Let tie^stmtE
OQ, AQ, BQ, CQ be placed between any point Q within the
tetrahedron and its four comers, and let these tie-«tnits be
mechanically jointed together at Q. so that they may eitha* posh
or pull at this point. This is merely a mechanical way of stating
the Boscovichian idea of a second homogeneous assemblage; equal
and similarly oriented to the first assemblage and placed with
one of its points at Q, and the others in the other oorreqioiidiiig
positions relatively to the primary assemblage. When it is <ioiie
for all the tetrahedrons chirally similar to OABC^ we find fiwr
tie-strut ends at every point 0, or A^ or B^ or C^ Ux example;
of the primary assemblage. Let each set of these four ends be
mechanicaliy jointed together, Si^ as to allow either posh or palL
A model of the curious structure thus formed was shown at the
conversazione of the Royal Society of June 7, 1893. It is for
three dimensions of space what ordinary hexagonal netting is in
a plane.

§ 34. Having thus constructed our model alter its shape
until we find its volume a maximum. This brings the tetra-
hedron. OABC, to be of the special kind defined in § 31. Suppose
for the present the tie-struts to be absolutely resistant against
push and pulL that is to say. to be each of constant length. This
secures that the volume of the whole assemblage is unaltei^ by
any infinitesimal change of shape possible to it ; so that we have,
in &ct, the skeleton of an incomprets^ible and inextensible solid*.

* This resTilt vms giTcn for an cq^ulatcfml tfti&bcdioDal anembSaf» in | ^
of **Mokc:ilAr Cozi$sir::c -n of ^Utficr." lfa:L cmd PAy«. Ptptrt^ ToL eel pfL

21 COEFFICIENTS FOR ELASTICITY OF DOUBLE ASSEMBLAGE. 661

Let now any forces whatever, subject to the law of uniformity in
the assemblage, act between the points of our primary assemblage
(and, if we please, also between the points of our second as-
semblage; and between other pairs of points than the nearests
of the two assemblages). Let these forces fulfil the conditions of
equilibrium ; of which the principle is described in § 16 and
applied to find the equations of equilibrium for the simple case
of a single homogeneous assemblage there considered. Thus
we have an incompressible elastic solid; and, as in § 17 above,
we see that there are just fifteen independent coefficients in the
quadratic function of the strain-components expressing the work
required to produce an infinitesimal strain. Thus we realise the
result described in § 7 above.

§ 35. Suppose now each of the four tie-struts to be not
infinitely resistant against change of length, and to have a given
modulus of longitudinal rigidity, which, for brevity,, we shall call
its stiffness. By assigning proper values to these four stiffnesses,
and by supposing the tetrahedron to be freed from the two con-
ditions making it our special tetrahedron, we have six quantities
arbitrarily assignable, by which, adding these six to the former
fifteen, we may give arbitrary values to each of the twenty-one
coefficients in the quadratic function of the six strain-components
with which we have to deal when change of bulk is allowed.
Thus, in strictest Boscovichian doctrine, we provide for twenty-
one independent coefficients in Green's energy-function. The
d}niamical details of the consideration of the equilibrium of two
homogeneous assemblages with mutual attraction between them,
and of the extension of §§ 9 — 17 to the larger problem now before
us, are full of purely scientific and engineering interest, but must
be reserved for what I hope is a future communication.

APPENDIX J.

MOLECULAR DYNAMICS OF A CRYSTAL*

§ 1. The object of this communication is to partially realise
the hope expressed at the end of my paper of July 1 and July 15,
1889, on the Molecular Constitution of Matter \: — "The mathe-
matical investigation must be deferred for a future communication,
when I hope to give it with some farther developments" The
italics are of present date.

Following* the ideas and principles suggested in §§ 14 — 20 of
that paper (referred to henceforth for brevity as " M. C. M."), let
us first find the work required to separate all the atoms of a
homogeneous assemblage of a great number n of molecules to
infinite distances from one another. Each molecule may be a
single atom, or it may be a group of. i atoms (similar to one
another or dissimilar, as the case may be) which makes the whole
assemblage a group of i assemblages, each of n single atoms.

§ 2. Remove now one molecule from its place in the assem-
blage to an infinite distance, keeping unchanged the configuration
of its constituent atoms, and keeping unmoved every atom
remaining in the assemblage. Let W be the work required to do
so. This is the same for all the molecules within the assemblage,
except the negligible number of those (§ 30 below) which are
within influential distance of the surface. Hence ^nTT is the
total work required to separate all the n molecules of the assem-
blage to infinite distances from one another. Add to this n
times the work required to separate the i atoms of one of the
molecules to infinite distances from one another, and we have the
whole work required to separate all the in atoms of the given
assemblage.

♦ Proc» Roy, Soc. Edin., May, 1902.

f Proc. Roy, Soc, Edin., and Vol. m. of Mathetnatical and Phytical Papertf
Art. xcyii.

DYNAMICS OF DOUBLE HOMOGENEOUS ASSEMBLAGE. 663

Another procedure, sometimes more convenient, is as follows: —
Remove any one atom from the assemblage, keeping all the others
unmoved. Let w be the work required to do so, and let 2i/;
denote the sum of the amounts of work required to do this for
every atom separately of the whole assemblage. Thef total amount
of work required to separate all the atoms to infinite distances
from one another is i^^w. This (not subject to any limitation
such as that stated for the former procedure) is rigorously true for
any assemblage whatever of any number of atoms, small or large.
It is, in fact, the well-known theorem of potential energy in the
dynamics of a system of mutually attracting or repelling particles;
and from it we easily demonstrate the item ^nTT in the former
procedure.

§ 3. In the present communication we shall consider only
atoms of identical quality, and only two kinds of assemblage.

I. A homogeneous assemblage of N single atoms, in which the
twelve nearest neighbours of each atom are equidistant from it.
This, for brevity, I call an equilateral assemblage. It is fully
described in " M. C. M.," §§ 46, 50 . . . 57.

II. Two simple homogeneous assemblages oi )tN single atoms,
placed together so that one atom of each assemblage is at the
centre of a (quartet of nearest neighbours of the others.

For assemblage II., as well as for assemblage I., iv is the same
for all the atoms, except the negligible number of those within
influential distance of the boundary. Neglecting these, we there-
fore have Sw = Nw, and therefore the whole work required to
separate all the atoms to infinite distances is —

iiNw (1).

§ 4. Let (^ (JD) be the work required to increase the distance
between two atoms from JD to oo ; and let f{D) be the attraction
between them at distance JD. We have

/(D) = -^<f,(D) (2).

For either assemblage I. or assemblage II. we have

w = <f>(D) + <t>(iy) + (l>{iy')-h etc (3);

where JD, JD', JD", etc., denote the distances from any one atom of
all neighbours, including the farthest in the assemblage, which
exercise any force upon it.

664

APPENDIX J.

§ 5. To find as many as we desire of these distances for
assemblage I. look at figs. 1 and 2. Fig. 1 shows an atom A, and
neighbours in one plane in circles of nearest, next-nearest, next-
next-nearest, etc. Fig. 2 shows an equilateral triangle of three
nearest neighbours, and concentric circles, of neighbours in the
same plane round it. The circles corresponding to r^ and r, of
§ 7 below, are not drawn in fig. 2. In all that follows the side oi
each of the equilateral triangles is denoted by X.

§ 6. All the neighbours in assemblage I. are found by aid of
the diagrams as follows : —

Fig. 1.

(a) The atoms of the net shown in fig. 1. The plane of this
net we shall call our "middle plane." Let lines be drawn per-
pendicular to it through the atom A, and the points marked b, c,
to guide the placing of nets of atoms in parallel planes on its two
sides.

(6) Two nets of atoms at equal distances X,\/| on the two

DISTANCES OF NEIQHBODRS IN SINGLE AiiSEHBLAOE. 665

aides of the " middle plane." These nets are so placed that an
atom of one of them, say the near one as we look at the diagram,
is in the guide line b ; and an atom of the for one is in the guide
line c

(c) Two parallel jiets of atoms at eijual distances, 2W|, on
the two sides of the " middle plane," so placed that an atom of the
near one is in the guide line c, and an atom of the far one is in
the guide line b.

(d) A third pair of parallel planes at equal distances, SXVi,
from the " middle plane," and each of them having an atom in
guide line A.

{e) Successive triplets of parallel nets with their atoms

Fig. 2.

cyclically arranged Abe, Abe... at greater and greater distances
from A on the near aide of the paper, and Acb, Adi ... at greater
and greater distances on the far side.

I 7. Let 5i, qt, q^ ... be the radii of the circles shown in fig. 1,
and r,, r„ r^... be the radii of the circles shown in fig. 2; and

666 APPENDIX J.

for brevity denote X\/f by k. The distances from A of all the
neighbours around it are : —

In our "middle plane": 6 each equal to fi ; 6, 9,; 6, g,; 12, }<;

"1 S's ) • • • •

In the two parallel nets at distances k from middle : 6 each
equal to V(«' + n«) ; 6, V(«' + rf) ; 12, V(^' + V) ; 12, V(^* + r}) \
6, \/(/c»+V); 12,V(^'« + r.«); 6, V(^4-r,»).

In the two parallel nets at distances 2/c from middle : the same
as (5) altered by taking 2/c everywhere in place of k.

In the two parallel nets at distances %k, from centre : the same
as (-4) altered by taking \/(9^+ ?i')» \/(9/c'+ q<}\ etc., in place of
?i, 921 etc.

In nets at distances on each side greater than %k : distances of
atoms from A, found as above, according to the cycle of atomic
configuration described in (e) of § 6.

§ 8. By geometry we find

?! = >-; 9, = V3X=l-732\; j3 = 2\;
g, = ^7X = 2-646\ ; 5'5 = 3\:
r^ = ViX = -STTX ; r, = 2 VJX = I154X
r, = v|x = l-527\ ; n = V^^X = 2-082X
r5 = 4ViX=2-308X; r«= V-^X = 2-517X
r, = 5 V^X = 2-887X.

§ 9. Denoting now, for assemblage I., distances from atom A
of its nearest neighbours, its next-nearests, its next-next-nearests,
etc., by i)i» A. A» etc., and their numbers by j\, j',, j',, etc., we
find by ^ 7, 8 for distances up to 2X, for use in § 12 below,

2)j = X, 2)2 = 1-414X, A = 1-732X, A = 2X,
j, = 12; j. = 6; j, = 18; j, = 6.

§ 10. Look back now to § 5, and proceed similarly in respect
to assemblage II., to find distances from any atom j1 to a limited
number of its neighbours. Consider first only the neighbours
forming with A a single equilateral assemblage : we have the same
set of distances as we had in § 9. Consider next the neighbours
which belong to the other equilateral assemblage. Of these, the
four nearest (being the comers of a tetrahedron having A at its
centre) are each at distance \ \/§X, and these are -4*s nearest
neighbours of all the double assemblage II. Three of these four
are situated in a net whose plane is at the distance \ \/|X on one

,(4X

SCALENE ASSEMBLAGE IN EQUILIBRIUM IF INFINITE. 667

side of our " middle plane" through A, and having one of its
atoms on either of the guide lines 6 or c. The distances from A
of all the atoms in this net are, according to fig. 2,

\/(iV^ + n')> VdV'^' + r,«), etc (6).

The remaining one of the four nearests is on a net at distance
i ^/i^ froiif^ our " middle plane," having one of its atoms on the
guide line through A, The distances from A of all the atoms in
this net are, according to fig. 1,

f\/«X. \/(A'^' + 7i'X Vdlr^ + ^aO. etc (6).

All the other atoms of the equilateral assemblage to which A
does not belong lie in nets at successive distances «, 2#c, Stc, etc.,
beyond the two nets we have already considered on the two sides
of our "middle plane"; the atoms of each net placed of course
accoitling to the cyclical law described in (e) of § 6.

§ 11. Working out for the double a.ssemblage II. for A's
nearest neighbours according to § 10, we find four nearest neigh-
bours at equal distances J\/S^ = '613\; twelve next-nearests at
equal distances X ; and twelve next-next-nearests at equal distances
\/-y-X= 1173\. These suffice for §12 below. It is easy and
tedious, and not at present useful, to work out for D^, D^, D^, etc.

§ 12. Using now §§ 9, 11 in (3) of § 4 we find,—

for assemblage I.,

w = 12<^ (\) + 6<^(1-414\) + 18<^(l-732\)4- 6<^(2\) + ...
for assemblage II.,

ti; = 4<^0613X) + 12<^(\) + 12<^(l-173\)4-...

These formulas prepare us for working out in detail the practical
dynamics of each assemblage, guided by the following statements
taken from §§ 18, 16 of " M. C. M."

§ 13. Every infinite homogeneous assemblage of Boscovich
atoms is in equilibrium. So, therefore, is every finite homogeneous
assemblage, provided that extraneous forces be applied to all
within influential distance of the frontier, equal to the forces
which a homogeneous continuation of the assemblage through
influential distance beyond the frontier would exert on them.
The investigation of these extraneous forces for any given homo-
geneous assemblage of single atoms — or groups of atoms as

668

APPENDIX J.

explained above (§ IV— constitutes the Boecovich equilibiinm-
theorv of elastic solids.

It is wonderful how much towards explaining the cnrstallo-
graphy and elasticity of solids, and the thenno-elastic properties
of solids, liquids, and gases, we find ; without assuming, in the
Bo6Ci)viehian law of force, more than one transition from
attraction to repulsion. Suppose, for instance, that the mutual
force between two atoms is zero for all distances exceeding a
certain distance /, which we shall call the diameter of the sphere
of influence ; is repulsive when the distance between them is < {^;
xero when the distance is = f; and attractive when the distance is
> f and < /.

§ 14. Two different examples are represented on the two
curves of fig. 3, drawn arbitrarily to obtain markedly diverse con-
ditions of equilibrium for the monatomic equilateral assemblage
L, and abo for the diatomic assemblage IL The abscissa (x)

■* *

1

Fig. 3L

of each diagram, reckoned from a zero outside the diagnun oo the
left, represents the distance between centres of two atoms; the
ordinates {y\ represent the work required to separate them from

this distance to x . Hence — -^ represents the mutual attraction

at distance x. This we see by each curve is — x (infinite repuisioo)
at distance 1*0. which means that the atom is an ideal hard
ball of diameter ll). Fca- distances increasing from ID the force
is repulsive as &r as 1*61 in curve 1, and 1*55 in curve ± At
these distances the mutual force is zero ; and at greats* distances
up to I'S in curve 1. and 1*9 in curve 2. the finrce is attmclive.
The force is zero for ail greater distances than the last mentioDed

WORK-CURVES FOR TWO ASSEMBLAGES.

669

in the two examples respectively. Thus, according to my old
notation, we have f = 1'61, / = 1'8 in curve 1; and f = 1*55, /= 1'9
in curve 2. The distances for maximum attractive force (as shown
by the points of inflection of the two curves) are 1*68 for curve 1,
and 1*76 for curve 2.

According to our notation of § 4 we have y = 0 (D), i{ x = D
in each curve.

§ lo. The two formulas (7), § 12, are represented in fig. 4 for
curve 1, and in fig. 5 for curve 2; with a? = X for Ass. I., and
^=s-613\ for Ass. II. In each diagram the abscissa, ^, is distance
between nearest atoms of the assemblage. The heavy portions of

JVt

1

/

/w/orAss.I.

,

W

y/tu/orAssM.

V

/

V*' ^

\

^

a

•9

10

/

^n '*

1-3 i-4' IS ■-«

1-7

Ifl

Fig. 4. Law of Force according to Curve 1.

the curves represent the values of w calculated from (7). The
light portions of the curves, and their continuations in heavy
curves, represent 4<^ (a?) and 1 2<^ (a?) respectively in each diagram.
The point where the light curve passes into the heavy curve in
each case corresponds to the least distance between neighbours at
which next-nearests are beyond range of mutual force. All the
diagrams here reproduced were drawn first on a large scale on
squared paper for use in the calculations from (7); which included
accurate determinations of the maximum and minimum values of
w and the corresponding distances between nearest neighbours in

670

APPENDIX J.

each assemblage. The corresponding densities, given in the last
column of the following table of results, are CEdcuIated by the
formula »Jt X.' for assemblage I., and 2 V2/V for assemblage II.;

Fig. 5. Law of Force aooording to Cure 2.

^'density*' being in each case number of atoms per cube of the
unit of abscissas of the diagram. This unit is (§ 14) equal to the
diameter of the atom. For simplicity we assume the atom to be
an intinitely hard ball exerting (§ 13) on neighbouring atoms, not
in contact with it, repulsion at distance between centres less than
f and attraction at any distance between f and /.

§ 16. To interpret these results, suppose all the atoms of the
as>emblage to be subjected to guidance constraining them either
to the equilateral homogeneousness of assemblage I^ or to the
diatomic homogeneousness of assemblage IL. with each atom of
one constituent assemblage at the centre of an equilateral quartet
of the other constituent assemblage. It is easy to construct
ideally mechanism bv which this mav be done : and we need not

s • •

occupy our minds with it at present. It is enough to know that
it can be done. If the system, subject to the prescribed constrain-
ing guidance, be left to itself at any given density, the condition
for equilibrium without extraneous force is that ir is either a
maximum or a minimum : the equilibrium is stable when ir is a
maximum, unstable when a minimuuL It is interesting to see the

8TABIUTIES OF MONATOMIC AND DIATOMIC ASSEMBLAGES. 671

two stable equilibriums of assemblage I. according to law of force
1, and the three according to law of force 2 ; and the two stable
equilibriums of assemblage II. with each of these laws of force.

AMexublage I.

Assemblage 11.

Dtotancet be-
tween centret
of nearest
atoms for
maximum and
minimum
Talues of IT.

Maximum and

miniumm

values of ir.

Densities.

Distances be-
tween centres
of nearest
atoms for
maximum and
minimum
values of w.

Maximum and

minimum

values of ie.

Densities.

1

Law of Force according to Curve 1.

116
1-23
1-61

8*28 (max.)

5-22 (min.)

14-76 (max.)

•904
•759
•338

100
110
1-61

1152 (max.)

•76 (min.)

4-92 (max.)

•652

■ -490

•158

Law of Force according to Carve 2.

1-00
107
1-22
1-28
1-63

11-58 (max.)
3-78 (min.)

10-44 (max.)
9-36 (min.)

15-60 (max.)

1-414

1-146

•774

•671

•393

100
115
153

12-36 (max.)
0-16 (min.)
5-20 (max.)

•652
•433
•184

§ 17. But we must not forget that it is only with the specified
constraining guidance (§ 16) that we are sure of these equili-
briums being stable. It is quite certain, however, that without
guidance the monatomic assemblage would be stable for the small
density corresponding to the point vi of each of the diagrams,
because for infinitesimal deviations each atom experiences forces
only from its twelve nearest neighbours, and these forces are each
of them zero for equilibrium. It may conceivably be that each of
the maximums of Wy whether for the monatomic or the diatomic
assemblage, is stable without guidance. But it seems more pro-
bable that, for assemblage I. and law of force 2, the intermediate
maximum m' (close to a minimum) is unstable. If it is so, the
assemblage left to itself in this configuration would fall away, and
would (in virtue of energy lost by waves through ether, that is to
say, radiation of heat) settle in stable equilibrium corresponding
to the maximum vi (single assemblage), or either of the maxi-
mums m" (single assemblage), or m'" (double assemblage). It is

672 APPENDIX J.

also possible that for law of force 1 the maximum m for the single
assemblage is unstable. If so, the system left to itself in this
configuration would fall away and settle in either of the configu-
rations w (single assemblage) or m" (double assemblage). Or it
is possible that with either of our arbitrarily assumed laws of force
there may be stable configurations of equilibrium with the atoms
in simple cubic order (§ 21 below): and in double cubic order; that
is to say, with each atom in the centre of a cube of which the
eight corners are its nearest neighbours.

§ 18. It is important to remark further, that certainly a law
of force fulfilling the conditions of § 13 may be found, accordiDg
to which even the simple cubic order is a stable configuration;
though perhaps not the only stable configuration. The double
cubic order, which has hitherto not got as much consideration as
it deserves in the molecular theory of crystals, is certainly stable
for some laws of force which would render the simple cubic order
unstable. Meantime it is exceedingly probable that there are in
nature crystals of elementary substances, such as metals, or frozen
oxygen, or nitrogen, or argon, of the simple cubic, and double
cubic, and simple equilateral, and double equilateral, classes. It
is also probable that the crystalline molecules in crystals of com-
pound chemical substance are in many cases simply the chemical
molecules, and in many cases are composed of groups of the
chemical molecules. The crystalline molecules, however consti-
tuted, are, in crystals of the cubic class, probably arranged either
in simple cubic, or double cubic, or in simple equilateral, or double
equilateral, order.

§ 19. It will be an interesting further development of the
molecular theory to find some illustrative cases of chemical
compound molecules (that is to say, groups of atoms presenting
different laws of force, whether between two atoms of the same
kind or between atoms of different kinds), which are, and others
which are not, in stable equilibrium at some density or densities
of equilateral assemblage. In this last class of cases the molecules
make up crystals not of the cubic class. This certainly can be
arranged for by compound molecules with law of force between
any two atoms fulfilling the condition of § 13; and it can be
done even for a monatomic homogeneous assemblage very easily, if
we leave the simplicity of § 13 in our assumption as to law of force.

STABILITY NOT SECURED BY POSITIVE MODULUSES. 673

§ 20. The mathematical theory wauts development in respect
to the conditions for stability. If, with the constraining guidance
of § 16, w is either a maximum or a minimum, there is equi-
libriam with or without the guidance. For w a maximum the
equilibrium is stable with the guidance ; but may be stable or un-
stable without the guidance. A criterion of stability which will
answer this last question is much wanted ; and it seems to me that
though the number of atoms is quasi-infinite the wanted criterion
may be finite in every case in which the number of atoms exerting
force on any one atom is finite. To find it generally for the equi-
librium of any homogeneous assemblage of homogeneous groups,
each of a finite number of atoms, is a worthy object for mathe-
matical consideration. Its difficulty and complexity is illustrated in
§§ 21, 22 for the particularly simple case of similar atoms arranged
in simple cubic oixler ; and in ^ 23-29 for a still simpler case.

§ 21. Consider a group of eight particles at the eight corners
of a cube (edge \) mutually acting on one another with forces all
varying according to the same law of distance. Let the magni-
tudes of the forces be such that there is equilibrium ; and in the
first place let the law of variation of the forces' be such that the
equilibrium is stable, fiuild up now a quasi-infinite number of
such cubes with coincident corners to form one large cube or a
crystal of any other shape. Join ideally, to make one atom, each
set of eight particles in contact which we find in this structure.
The whole system is in stable equilibrium. The four forces in
each set of four coincident edges of the primitive cubes become
one force equal to the force between atom and atom at distance \.
The two forces in either diagonal of the coincident square faces of
two cubes in contact make one force equal to the force between
atoms at distance X V^* "^^^ single force in each body-diagonal
of any one of the cubes is the force between atom and atom at
distance X \/3. The three moduluses of elasticity (compressibility-
modulus, modulus with reference to change of angles of the square
faces, and modulus with reference to change of angles between
their diagonals) are all easily found by consideration of the dynamics
of a single primitive cube, or they may be found by the general
method given in " On the Elasticity of a Crystal according to
Boficovich*." (In passing, remark that neither in this nor in other

• Proc. R,S.L. Vol. liv. June 8, 1893. App. I.
T. L. 43

674 APPENDIX J.

cases is it to be assumed without proof that stability is ensured by
positive values of the elasticity moduluses.)

§ 22. Now while it is obvious that our cubic system is in
stable equilibrium if the eight particles constituting a detached
primitive cube are in stable equilibrium, it is not obvious without
proof that this condition, though sufficient, is necessary for the
stability of the combined assemblage. It might be that though
each primitive cube by itself is unstable, the combined assemblage
is stable in virtue of mutual support given by the j#ining of eight
particles into one at the corners of the cubes which we have put
together.

§ 23. The simplest possible illustration of the stability question
of § 20 is presented by the exceedingly interesting problem of the
equilibrium of an infinite row of similar particles, free to move
only in a straight line. The consideration of this linear problem
we shall find also useful (§§ 28, 29 below) for investigation of the
disturbance from homogeneousness in the neighbourhood of the
bounding surface, experienced by a three-dimensional homogeneous
assemblage in equilibrium. First let us find a, the distance, or
one of the distances, from atom to atom at which the atoms must
be placed for equilibrium ; and after that try to find whether the
equilibrium is stable or unstable.

§ 24. Calling /(i)) (as in § 4) the attraction between atom

and atom at distance D, we have for the sum, P, of attractions

between all the atoms on one side of any point in their line, and

all the atoms on the other side, the following finite expression

having essentially a finite number of tenns, greater the smaller

is a:

/(o) + 2/(2a) + 3/(3a) + =P (8).

Hence a, for equilibrium with no extraneous force, is given by the
functional equation

/(a) + 2/(2a) + 3/(3a) + =0 (9);

which, according to the law of force, may give one or two or any
number of values for a : or may even give no value (all roots
imaginary) if the force at greatest distance for which there is
force at all, is repulsive. The solution or all the solutions of this
equation are readily found by calculating from the Boscovich
curve representative of /(D) a table of values of P, and plotting

boscovich's curve.

676

them on a curve, by formula (8), for values of a from a = / (the
limit above which the force is zero for all distances) downwards
to the value which makes P = — oo , or to zero if there is no
infinite repulsion. The accompan3ring diagram, fig. 6, copied

Fig. 6.

from fig. 1 of Boscovich's great book*, with slight modifications
(including positive instead of negative ordinates to indicate
attraction) to suit our present purpose, shows for this particular
curve three of the solutions of equation (8). (There are obviously
several other solutions.) In two of the solutions, respectively,
Aq, A\ and Ao, A'\ are consecutive atoms at distances at which
the force between them is zero. These are configurations of
equilibrium, because AoB, the extreme distance at which there is
mutual action, is less than twice AqA\ and less than twice A^A".
In the other of the solutions shown, ^o, ^i, j1,, A^^ A^, il,, A^
are seven equidistant consecutive atoms of an infinite row in
equilibrium in which ^5 is within range of the force of A^^ and
A^ is beyond it. The algebraic sum of the ordinates with their
proper multipliers is zero, and so the diagram represents a solution
of equation (9).

* Theoria Philosophiao Nataralis redacts ad Qnicam legem viriam in natara
eziBtentiiun, auotore P. Bogerio Josepbo Boscovioh, Sooietatis Jesu, nuno ab ipso
perpolita, et auota, ac a plurimis prfBcedentimn editionam mendis expargata.
Editio Yeneta prima ipso aaotore prssente, et oorrigente. Yenetiis, iiDOOLxnx.
Ex ^^ypographia Bemondiniana superiorum permlBsu, ac privilegio.

43—2

676 APPENDIX J.

§ 2o. In the general linear problem to find whether the equi-
librium is stable or not for equal consecutive distances, a, let
(as in § 4) ^ (i>) be the work required to increase the distance
between two atoms from D to oo. Suppose now the atoms to
be displaced from equal distances, a, to consecutive unequal
distances —

...a-i-Wi_„ a + Mi-i, a + Wi, a + t«t+i, a + t/<+s, (10).

The equilibrium will be stable or unstable according as the
work required to produce this displacement is, or is not, positive

for all infinitely small values of... Ui^^ v^, Vi+i, Its amount

is TTo— TT; where W denotes the total amount of work required
to separate all the atoms from the configuration (10) to infinite
mutual distances.

According to § 2 above W is given by

W = 1^{...+Wi^i + Wi + Wi+^'^...) (11);

where

Wt=0(a + ^^i) + 0 (2a + i^i_i + 1^) +0(3a-f iii_, + Mi_i + Ui) +...
+ <^ (a + tii^i) + 0 (2a + iii+i + i^+2)+<^(3a + Wf+i + Ui+^ + Uf+,)4- ...
+ (12).

Expanding each term by Taylor s theorem as far as terms of the
second order, and remarking that the sum of terms of the first
order is zero for equilibrium* at equal distances, a, and putting
<^" (D) = -/' (i)), we find

Tro-}f = i2{/'(a)(a,« + ttVi)

+/' (2a) [(i^-i + UiY + (Ui^, + Ui^y]

+ etc. etc. etc. etc.} ...(13);

where S denotes summation for all values of f, except those corre-
sponding to the small numbers of atoms (^ 28, 29 below) within
influential distances of the two ends of the row.

§ 26. Hence the equilibrium is stable if /'(a),/' (2a), /'(3a),
etc., are all positive; but it can be stable with some of them

* It is interesting and instractive to verify this analytically by seleoting aU the
terms in W which contain u^, and thus finding -j— . This equated to zero, for zero
values of ...tij_i, u^, u^+i,... gives equation (9) of the text.

EQUILIBRIUM IN NEIGHBOURHOOD OF ENDS OF ROW OF ATOMS. 677

negative. Thus, according to the Boscovich diagram, a condition
ensuring stability is that the position of each atom be on an up-
slope of the curve showing attractions at increasing distances.
We see that each of the atoms in each of our three equilibriums
for fig. 6 fulfils this condition.

§ 27. Fig. 7 shows a simple Boscovich curve drawn arbitrarily
to fulfil the condition of § 13 above, and with the further simpli-
fication for our present purpose, of limiting the sphere of influence
so as not to extend beyond the next-nearest neighbours in a row
of equidistant particles in equilibrium, with repulsions between
nearests and attractions between next-nearests. The distance, a,
between nearests is determined by

/(a) + 2/(2a) = 0 (14),

being what (9) of § 24 becomes when there is no mutual force
except between nearests and next-nearests. There is obviously
one stable solution of this equation in which one atom is at the
zero of the scale of abscissas (not shown in the diagram) and its
nearest neighbour on the right is at ^, the point of zero force with
attraction for greater distances and repulsion for less distances.
The only other configuration of stable equilibrium is found by
solution of (14) according to the plan described in § 24, which gives
a = '680. It is shown on fig. 7 by Ai, Ai^i, as consecutive atoms
in the row.

§ 28. Consider now the equilibrium in the neighbourhood of
either end of a rectilinear row of a very large number of atoms
which, beyond influential distance from either end, are at equal
consecutive du^tances a satisfying § 27 (14). We shall take for
simplicity the case of equilibrium in which there is no extraneous
force applied to any of the atoms, and no mutual force between
any two atoms except the positive or negative attraction /(D),
But suppose first that ties or struts are placed between con-
secutive atoms near each end of the row so as to keep all their
consecutive distances exactly equal to a. For brevity we shall
call them ties, though in ordinary language any one of them
would be called a strut if its force is push instead of pull on the
atoms to which it is applied. Calling ^i, A^, ^, ... the atoms at
one end of the row, suppose th^ tie between Ai and J., to be

678

APPENDIX J.

removed, and ^i allowed to take its position of equilibrinm.
A single equation gives the altered distance AiA^, which we shall
denote by a-\- iXi. Let an altered tie be placed between A^ and
A^ to keep them at this altered distance during the operations
which follow. Next remove the tie between A^ and A^, and find
by a single equation the altered distance a + i^. After that
remove the tie between A^ and A^ and find, still by a single equa-

Fig. 7.

tion, the altered distance aH- la^, and so on till we find iiCj or ^x^ or
,a?i, small enough to be negligible. Thus found, iX^, ^x^, i^,...i^»
give a first approximation to the deviations from equality of
distance for complete equilibrium. Repeat the process of removing

VARIATIONS OF DENSITY SEVEN OR EIGHT NETS FROM SURFACE. 679

the ties in order and replacing each one by the altered length as
in the first set of approximations, and we find a second set
1^19 1^1 s^s'*«* Go on similarly to a third, fourth, fifth, sixth ...
approximation till we find no change by a repetition of the process.
Thus, by a process essentially convergent if the equilibrium with
which we started is stable, we find the deviations from equality of
consecutive distances required for equilibrium when the system is
left free in the neighbourhood of each end, and all through the
row (except always the constraint to remain in a straight line).
By this proceeding applied to the curve of fig. 7 and the case of
equilibrium a = 680, the following successive approximations were
found : —

+ •018

^1

's

^4

^6

^«

«7

let Approximation .

-•009

+ •004

-002

+ •001

-•001

•000

2nd

+ 026

-•014

+ 007

-003

+ •002

3rd

+ -031

-•018

+ 009

-006

+ •003

4th

+ 034

-•020

+ •011

-•006

5th

+ 036

-•022

+ •012

-•007

6th

+ •037

-023

+ 013

1

7th

+ •038

-024

8th

+ •039

Thus our final solution, with a = "680, is

a:, = + -039, a^a = - 024,
a?, = + 013, a:4 = - 007,
a:,= 4.-003, a?« = --001,
av=000.

§ 29. It is exceedingly interesting to remark that the devia-
tions of the successive distances from a are alternately positive
and negative, and that they only become less than one-seventh per
cent, of a for the distance between Aj and A^, Thus, if we agree
to neglect anything less than one-seventh per cent, in the distance
between atom and atom, the influential distance from either end
is 7a, although the mutual force between atom and atom is null at
all distances exceeding 2'2a.

§ 30. If, instead of /(D) denoting the force between two atoms
in a rectilinear row, it denotes the mutual force between two

680 APPENDIX J.

parallel plane nets in a Bravais homogeneous assemblage of single
atoms, the work of §§ 27, 28 remains valid ; and thus we arrive at
the very important and interesting conclusion that when there is
repulsion between nearest nets, attraction between next-nearests,
and no force between next-next-nearests or any farther, the dis-
turbance from homogeneousness in the neighbourhood of the
bounding plane consists in alternate diminutions and augmenta-
tions of density becoming less and less as we travel inwards, but
remaining sensible at distances from the boundary amounting to
several times the distance from net to net.

APPENDIX K.

ON VARIATIONAL ELECTRIC AND MAGNETIC SCREENING*

§ 1. A SCREEN of imperfectly conducting material is as
thorough in its action, when time enough is allowed it, as is a
similar scfeen of metal. But if it be tried against rapidly varying
electrostatic force, its action lags. On account of this lagging, it is
easily seen that the screening effect against periodic variations of
electrostatic force will be less and less, the greater the frequency
of the variation. This is readily illustrated by means of various
forms of idiostatic electrometers. Thus, for example, a piece of
paper supported on metal in metallic communication with the
movable disc of an attracted disc electrometer annuls the at-
traction (or renders it quite insensible) a few seconds of time
after a difference of potential is established and kept constant
between the attracted disc and the opposed metal plate, if the
paper and the air surrounding it are in the ordinary hygrometric
condition of our climates. But if the instrument is applied to
measure a rapidly alternating difference of potential, with equal
differences on the two sides of zero, it gives very little less than
the same average force as that found when the paper is removed
and all other circumstances kept the same. Probably, with
ordinary clean white paper in ordinary hygrometric conditions,
a frequency of alternation of from 50 to 100 per second will
more than suffice to render the screening influence of the paper
insensible. And a much less frequency will sufHce if the atmo-
sphere surrounding the paper is artificially dried. Up to a
frequency of millions per second, we may safely say that, the
greater the frequency, the more perfect is the annulment of
screening by the paper; and this statement holds also if the
paper be thoroughly blackened on both sides with ink, although

♦ Proc, R,S,, April, 1891.

682 APPENDIX K.

possibly in this condition a greater frequency than 50 to 100
per second might be required for practical annulment of the
screening.

§ 2. Now, suppose, instead of attractive force between two
bodies separated by the screen, as our test of electrification,
that we have as test a faint spark, after the manner of Hertz.
Let two well insulated metal balls, A, B, he placed very nearly
in contact, and two much larger balls, E, F, placed beside them,
with the shortest distance between E, F suflScient to prevent
sparking, and with the lines joining the centres of the two pairs
parallel. Let a rapidly alternating difference of potential be
produced between E and F, varying, not abruptly, but according,
we may suppose, to the simple harmonic law. Two sparks in
every period will be observed between A and B. The inter-
position of a large paper screen between E, F, on one side, and
Af B, on the other, in ordinary hygrometric conditions, will
absolutely stop these sparks, if the frequency be less than, per-
haps, 4 or 5 per second. With a frequency of 50 or more, a clean
white paper screen will make no perceptible difference. If the
paper be thoroughly blackened with ink on both sides, a fre-
quency of something more than 50 per second may be necessary ;
but some moderate frequency of a few hundreds per second will,
no doubt, suffice to practically annul the effect of the interposition
of the screen. With frequencies up to 1000 million per second,
as in some of Hertz's experiments, screens such as our blackened
paper are still perfectly transparent, but if we raise the frequency
to 500 million million, the influence to be transmitted is light,
and the blackened paper becomes an almost perfect screen.

§ 3. Screening against a varying magnetic force follows an
opposite law to screening against varying electrostatic force. For
the present, I pass over the case of iron and other bodies possess-
ing magnetic susceptibility, and consider only materials devoid
of magnetic susceptibility, but possessing more or less of electric
conductivity. However perfect the electric conductivity of the
screen may be, it has. no screening efficiency against a steady
magnetic force. But if the magnetic force varies, currents are
induced in the material of the screen which tend to diminish
the magnetic force in the air on the remote side from the varying
magnet. For simplicity, we shall suppose the variations to follow

ELKCTRO-CONDUCnVE PLATE SCREENS MAGNETIC CHANGES. 683

the simple harmonic law. • The greater the electric conductivity
of the material, the greater is the screening effect for the same
frequency of alternation; and, the greater the frequency, the
greater is the screening effect for the same material. If the
screen be of copper, of specific resistance 1640 sq. cm. per second
(or electric diffusivity 130 sq. cm. per second), and with frequency
80 per second, what I have called the ** mhoic effective thickness*"
is 0*71 of a cm.; and the range of current intensity at depth
n X 0'71 cm. from the surface of the screen next the exciting
magnet is €~" of its value at the surface.

Thus (as €* = 2009) the range of current-intensity at depth
2'13cm. is ^ of its surface value. Hence we may expect that
a suflSciently large plate of copper 2 J cm. thick will be a little
less than perfect in its screening action against an alternating
magnetic force of frequency 80 per second.

§ 4. Lord Rayleigh, in his " Acoustical Observations " (Phil.
Mag.y 1882, first half-year), after referring to Maxwell's statement,
that a perfectly conducting sheet acts as a barrier to magnetic
force {Electricity and Magnetism, §665), describes an experiment
in which the interposition of a large and stout plate of copper
between two coils renders inaudible a sound which, without the
copper screen, is heard by a telephone in circuit with one of the
coils excited by electromagnetic induction from the other coil,
in which an intermittent current, with sudden, sharp variations
of strength, is produced by a " microphone clock " and a voltaic
battery. Larmor, in his paper on " Electromagnetic Induction
in Conducting Sheets and Solid Bodies " {Phil, Mag,, 1884, first
half-year), makes the following very interesting statement: — "If
" we have a sheet of conducting matter in the neighbourhood of
" a magnetic system, the effect of a disturbance of that system
" will be to induce currents in the sheet of such kind as will tend
" to prevent any change in the conformation of the tubes [lines]
"of force cutting through the sheet. This follows from Lenz's
" law, which itself has been shown by Helmholtz and Thomson to
" be a direct consequence of the conservation of energy. But if
"the arrangement of the tubes [lines of force] in the conductor
"is unaltered, the field on the other side of the conductor into
"which they pass (supposed isolated from the outside spaces by

* MaiK and Phy$, Papers, Vol. iii. Art. cii, § 35.

684 APPENDIX K.

'* the conductor) will be unaltered. Hence, if the disturbance is
** of an alternating character, with a period small enough to make
"it go through a cycle of changes before the currents decay
"sensibly, we shall have the conductor acting as a screen.

"Further, we shall also find, on the same principle, that a
" rapidly rotating conducting sheet screens the space inside it irom
" all magnetic action which is not symmetrical round the axis of
" rotation."

Mr Willoughby Smith's experiments on " Volta-electric induc-
tion," which he described in his inaugural address to the Society
of Telegraph Engineers of November, 1883, aflForded good illus-
trations of this kind of action with copper, zinc, tin, and lead,
screens, and with different degrees of frequency of alternation.
His results with iron are also very interesting : they showed, as
might be expected, comparatively little augmentation of screening
effect with augmentation of frequency. This is just what is to
be expected from the fact that a broad enough and long enough
iron plate exercises a large magnetostatic screening influence ;
which, with a thick enough plate, will be so nearly complete that
comparatively little is left for augmentation of the screening
influence by alternations of greater and greater frequency.

§ 5. A copper shell closed around an alternating magnet
produces a screening effect which on the principle of § 3 we may
reckon to be little short of perfection if the thickness be 2J cm.
or more, and the frequency of alternation 80 per second.

§ 6. Suppose now the alternation of the magnetic force to be
produced by the rotation of a magnet M about any axis. First,
to find the effect of the rotation, imagine the magnet to be repre-
sented by ideal magnetic matter. Let (after the manner of Gauss
in his treatment of the secular perturbations of the solar system)
the ideal magnetic matter be uniformly distributed over the
circles described by its different points. For brevity call / the
ideal magnet symmetrical round the axis, which is thus con-
stituted. The magnetic force throughout the space around the
rotating magnet will be the same as that due to /, compounded
with an alternating force of which the component at any point
in the direction of any fixed line varies from zero in the two
opposite directions in each period of the rotation. If the copper
shell is thick enough, and the angular velocity of the rotation

ROTATING CONDUCTOR SCRESENS STEADY MAGNETIC FORCE. 685

great enough, the alternating component is almost annulled for
external space, and only the steady force due to / is allowed to
act in the space outside the copper shell.

§ 7. Consider now, in the space outside the copper shell, a
point P rotating with the magnet M. It will experience a force
simply equal to that due to M when there is no rotation, and,
when M and P rotate together, P will experience a force gradu-
ally altering as the speed of rotation increases, until, when the
speed becomes sufficiently great, it becomes sensibly the same as
the force due to the symmetrical magnet /. Now superimpose
upon the whole system of the magnet, and the point P, and the
copper shell, a rotation equal and opposite to that of M and P.
The statement just made with reference to the magnetic force
at P remains unaltered, and we have now a fixed magnet M and
a point P at rest, with reference to it, while the copper shell
rotates round the axis around which we first supposed M to
rotate.

§ 8. A little piece of apparatus, constructed to illustrate the
result experimentally, is submitted to the Royal Society and
shown in action. In the copper shell is a cylindric drum, 1*25 cm.
thick, closed at its two ends with circular discs 1 cm. thick. The
magnet is supported on the inner end of a stiff wire passing
through the centre of a perforated fixed shaft which passes
through a hole in one end of the drum, and serves as one of
the bearings; the other bearing is a rotating pivot fixed to the
outside of the other end of the drum. The accompanying sections,
drawn to a scale of three-fourths full size, explain the arrange-
ment sufficiently. A magnetic needle outside, deflected by the
fixed magnet when the drum is at rest, shows a great diminution
of the deflection when the drum is set to rotate. If the (triple
compound) magnet inside is reversed, by means of the central
wire and cross bar outside, shown in the diagram, the magneto-
meter outside is greatly affected while the copper shell is at rest ;
but scarcely affected perceptibly while the copper shell is rotating
rapidly.

§ 9. When the copper shell is a figure of revolution, the mag-
netic force at any point of the space outside or inside is steady,
whatever be the speed of rotation ; but if the shell be not a figure
of revolution, the steady force in the external space observable

686

APPENDIX K.

when the shell is at rest becomes the resultant of the force due
to a fixed magnet intermediate between M and / compounded

PC4

with an alternating force with amplitude of alternation increasing
to a maximum, and ultimately diminishing to zero, as the angular
velocity is increased without limit.

EXPERIMENTAL ILLUSTRATION. 687

§ 10. If Jlf be symmetrical, with reference to its northern and
southern polarity, on the two sides of a plane through the axis of
rotation, / becomes a null magnet, the ideal magnetic matter in
every circle of which it is constituted being annulled by equal
quantities of positive and negative magnetic matter being laid
on it. Thus, when the rotation is sufficiently rapid, the magnetic
rforce is annulled throughout the space external to the shell. The
transition from the st^^y force of M to the final annulment of
force, when the copper shell is symmetrical round its axis of
rotation, is, through a steadily diminishing force, without alter-
nations. When the shell is not symmetrical round its axis of
rotation, the transition to zero is accompanied with alternations
as described in §8.

§11. When M is not symmetrical on the two sides of a plane
through the axis of rotation, / is not null; and the condition
approximated to through external space with increasing speed
of rotation is the force due to /, which is an ideal magnet
symmetrical round the axis of rotation.

§ 12. A very interesting simple experimental illustration of
screening against magnetic force may be shown by a rotating
disc with a fixed magnet held close to it on one side. A bar
magnet held with its magnetic axis bisected perpendicularly by
a plane through the axis of rotation would, by sufficiently rapid
rotation, have its magnetic force almost perfectly annulled at
points in the air as near as may be to it, on the other side of
the disc, if the diameter of the disc exceeds considerably the
length of the magnet. The magnetic force in the air close to
the disc, on the side next to the magnet, will be everywhere
parallel to the surface of the disc.

APPENDIX L.

ON ELECTRIC WAVES AND VIBRATIOMS IN A SUBMARINE

TELEGRAPH WIRE*.

§ 1. To simplify our problem, by avoiding the interesting
subject of alternating electric currents of electricity in a solid
conductor dealt with in §§ 9, 19, 29 — 35 of Art. cii. of my
Mathematical and Physical Papers, Vol. ill., I suppose for the
present the central conductor and the surrounding sheath to be
exceedingly thin copper tubes ; so thin that the electric current
carried by each is uniformly distributed through its substance,
with the highest frequency of alternation which we shall have to
consider. To simplify farther, by avoiding the exceedingly com-
plex question of electric currents in the water above the cable and
wet ground below it, I for the present suppose the outer sheath to
be perfectly insulated. This supposition will make exceedingly
little difference in respect to the solution of our problem for such
frequencies of alternation as are used in submarine signalling; but
it makes a vast difference and simplification for the very high
frequencies, up to those of the vibrations constituting light, which
must be considered. For brevity I shall call the system of two
conductors, with air or gutta-percha or other insulating material
between them, the cable. For simplicity we shall suppose the
cable to be laid straight ; and shall specify any place in the cable
by X, its distance from any chosen point of reference 0 in the axis
of the inner conductor. But all our calculations will be applicable
though the cable be not laid straight; provided its radius of
curvature everywhere is very large in comparison with the radius
of the sheath in cross-section ; and provided x is distance from 0
measured along the length of the cable.

* This is the Appendix promised in the footnote on Lecture lY., page 45, of the
present volume. A partial statement of results was given in Nichors Cyclopedia
(1860) under title "Electricity, Velocity of," republished Mathematical and
Physical Papers, Vol. n. Article lxxxi.

ELECTROKINEMATIC CONDITIONS. 689

§ 2. Let r and / be the radii of the inner conductor and the

sheath ^ (1);

12, the sum of the resistances of the two conductors per
unit of length of each (2);

c, the quantity of electricity on unit length of each
conductor when the diflFerence of potentials be-
tween them is unity (3);

lly the electromagnetic quasi-inertia of a circuit made
ideally by metallically connecting the inner con-
ductor with the sheath at two cross-sections with
the length I between them large in comparison
with ?•' — r (4);

7, the current in either of the two conductors at time
t and place x (5);

<!>, the diflFerence of potentials between the two con-
ductors at time t and place x (6);

qdxy the quantity of electricity on a length dx of
either conductor at time t and place x (7).

In all these items of electric reckoning we shall use electro-
magnetic measure. It is to be borne in mind that in every part
of the cable the electric currents in the central conductor, and in
the sheath are equal and opposite : also that the total quantities
of electricity at every instant on the smaller convex surface of
the central conductor, and the larger concave surface of the
sheath, are equal and opposite : and that there is no electricity
on the inner surface of the central conductor, and no electricity
on the outer surface of the sheath. The notation (3) above,
in terms of my original definition * of electrostatic capacity, means
that c is the electrostatic capacity, reckoned in electromagnetic
measure, of unit length of the central conductor; regarded as if
it were part of a Leyden phial. By (3) and (7) we have

q = o<l> (8).

§ 3. Changes of electrification of the opposed surfaces of the
two conductors can only take place through more electricity

* «0n the ElectroBtatioal Capacity of a Leyden Phial and of a Telegraph Wire
insulated in the Axis of a Cylindrical Conducting Sheath.** First published in
Phil, Mag, 1S55, Ist half-year; republished in my volume of Electrostatics and
Magnetism, §g 61—56.

T. L. 44

690 APPENDIX L.

flowing into, than flowing out of, any portion. This electro-
kinematic principle gives us the following equation :

dt dx ^^^•

The djmamical equation of our problem is as follows :

^^'t'-t <i»^

Here the first term of the first member expresses the ohmic
anti-electromotive force ; the second term the inertial anti-
electromotive force. Eliminating q and if> from these three
equations we find

^t^'%-\%- <»^

§ 4. For small frequencies, such as those of submarine cable
signalling, the ohmic resistance dominates; and the inertial
resistance is imperceptible: thus we have

<^t <>»^

§ 5. For very high frequencies the inertial resistance dominates ;
and the ohmic is ultimately almost imperceptible : thus we have

""^ dt^-'da^ ^^^^•

§ 6. According to the principles set forth in Part III., " Elec-
tricity in Motion*/* of a paper "On the Mechanical Values of
Distributions of Electricity, Magnetism, and Galvanism f," pub-
lished originally in the Proceedings of the Olasgow Philosophical
Society for January 1853, we find // most readily by imagining
the temporarily closed circuit described in (4) of § 2 to be divided
into a very great number of parts by meridional planes through
the axis of the two cylindrical conductors, and imagining any one
of these parts to be removed to any position outside, while the
currents are kept constant in all the parts : and calculating the
work done in this movement against the attractive force between

* Mathematical and Physieal Papers^ Vol. i. pp. 630 — 688.

t By this ill-obosen word I then meant electric currents in closed cironits.

VELOCITY OF ELECTRIC WAVES IN A SUBMARINE CABLE. 691

the portion of the circuit removed and the remainder. The full
synthetic working-out of this plan gives

Vlog (r'/r) (14)

for the total work spent in taking the whole circuit to pieces
separated by infinite distances from one another. Hence, for the
quasi-iriertia of the current, per unit length of cable, in the circuit
before dissection, we have

/ = 21og(r7r) (15).

§ 7. By Electrostatics and Magnetism^ § 55, we have for c

in electrostatic measure iA;/log (r'/r) ; where k is what Faraday

called the specific inductive capacity of the insulating material

between the core and the sheath. Now if we denote by N the

number of electrostatic units in the electromagnetic unit of

electric quantity, it is also the number of electrostatic units in the

electromagnetic unit of potential. Hence N^ is the number of

electrostatic units in the electromagnetic unit of capacity : and we

have accordingly

c = N-'kl2log(r/r) = klN^I (16).

Using this in (11) we find

I dt'^ dt^ k did' ^^'^'

and (13) becomes

dt'^ k dx^ ^^•^^^

which shows that the propagational velocity of the wave is more
and more nearly equal to Nj/s/k, the higher the frequency; subject
to the restriction of § 8. Numerous and varied experimental in-
vestigations have shown, probably within ^ per cent., that N is
equal to 3 . 10^^

§ 8. Our demonstration [see § 2 (4)] involves essentially the
supposition that the wave-length is long in comparison with the
thickness / — r of the dielectric between the outer and inner
conductors.

§ 9. A complete solution of (17) is

7 = 8inm^(il€P< + 5€p'0 (18),

where m denotes 27r divided by the wave-length, and

-i?V[(5)'-?»-] <'»)^

692 APPENDIX L.

which is real when

mNly/k<R/2I (20).

But when (20) is not fulfilled, the radical in (19) is zero or
imaginary; and, instead of (18), we may take, as a convenient
solution for standing waves,

7 = sin7na;sin^y/|^-^m»-(27)j^'*^ (^l)-

Ttus expresses a subsidential oscillation, with ratio of subsidence
per unit of time, €~**/^; which is independent of the wave-length ;
a very important and interesting result. What we call the
period* of the oscillation is

^V4?--(S)'] <^^>^

which becomes more and more nearly equal to (2vlm)l(N/y/k\
the greater is m, subject to the restriction of § 8.

§ 10. The complicated subsidential law expressed by (18),
with two ratios of subsidence, each dependent on the wave-length,
one of tbem with ratio diminishing, the other with ratio increasing,
when the wave-length is diminished from infinity, is also very
interesting. It is interesting to see how, with very small values
of m (very great wave-lengths), the solution blends with the
solution of (12) above; and how, with the greatest values of m
fulfilling (20), the solution (18) blends into (21).

§ 11. As a single example take the Atlantic Cable of 1865,
for which we had

R = 23000 c.G.s. per cm. of length ; r'/r N 3-S ; whence / = 1-76

(23);

i = 3; i\r/VA;= 1-732. 10l^ c = 1894.10-« (24).

By (23) we find ^R/I = 6534. Hence, when (20) is not ful-
filled, the ratio of extinction for 1/6534 of a second is e"^ and
for a millionth of a second it is €-*o«»»*, or approximately '99347.
And the limiting condition between fulfilment and non-fulfilment
of (20) is, by (24), m J 3*772 . 10-^ ; or, if \ denote the wave-length,

* This is according to common asage ; but it is not strictly correct, because
subsiding vibratory motion is not periodic.

ELECTRIC VIBRATIONS IN ONE KILOMETRE OF ATLANTIC CABLE. 693

\ < 167 kilometres. Hence, if we take a length of anything less
than 167 kilometres of the cable, and somehow manage to give it
initially a sinusoidal distribution of current, varying through its
length firom zero at one end, through maximum aud minimum,
to zero at the other end, and leave it to itself; the electric distri-
bution will subside vibrationally to zero. A millionth of a second
from the beginning the vibrational amplitude will be '99347 of its
initial value ; and in 1/6534 of a second it will have subsided to
€~* of its initial value. Instead now of the impracticable manage-
ment to begin with a sinusoidal distribution of current, apply a
voltaic battery to maintain, for a very short time, say a hundred-
thousandth of a second, a difference of potential between sheath
and interior conductor at one end, and leave it to itself. This
will give us an initial disturbance represented, according to
Fourier, by a sum of sinusoidal currents, each zero at both ends
of the cable. Every one of these components will subside vibra-
tionally, all with the same ratio of subsidence in the same tima
The periods of the different sinusoidal components will be those
expressed by (22) ; with, for m, the successive values 27r/X., 2 . 27r/\,
3.27r/\, ..., where \ denotes the length of the cable. If this
length of the cable is 16*7 kilometres, the second term within the
brackets of (22) will be only 1/100 of the first, and for this or any
shorter length it may be neglected. Thus, for a length of one
kilometre, the period of the gravest sinusoidal vibration, as we
see by (24) and (22), is 10»/l-732 . 10'« or '577 . lO"' of a second.
The ratio of subsidence in this time is €~'***" ; so that 26^ periods
of vibration will be performed before subsidence to e~\ of the
initial value.

§ 12. Consider, in conclusion, how thin the copper must be to
fulfil the first simplifying condition of § 1, when the vibrational
period is so short as '577 . 10""' of a second. This is 1/2166 of
1/80, the period for which I found 714 cm. as the mhoic thick-
ness* of copper of specific resistance 1610 c. G.s. Hence mhoic
thickness for period '577 .10-' is •714/V2166, or 0153 cm. Now
without going farther into the theory of the diffusion of alternate
currents of electricity through a metal, we may safely guess that,
if the thickness of the sheet is anything less than one-third of the

* Mathematical and Phytieal Papers, Vol. m, Art. en, § 85 (May, 1S90).

694 APPENDIX L.

mhoic thickness for any particular period, the current will be
nearly uniform through the whole thickness. Hence, to satisfy the
first simplifying condition of § 1, the copper need not be thinner
than 1/200 of a cm. Taking this as the thickness, let us find
what r and r\ with the ratio r jr = 3'3, must be to make the sum
of the resistances in a cm. of the inner tube and a cm. of the outer
tube equal to 23000 c. G. s. If we take the specific resistance of
copper as 1610 square centimetres per second, our equation to
determine r and / is

1610.200
27r

(r-^4) = 2^^^« (^^>'

by which we have r = 2'9 and r' = 9'6. It must be noted that it
is only for the gravest fundamental vibration that 1/200 cm. thick-
ness of the metal would be small enough to give the law and rate
of subsidence determined above.

CASIBRIDOB : PRINTED BY J. AND C. F. CLAY, AT THE UNIVEBSITY PRESS.

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