contains only vanishing elements this involves a
contradiction. Hence the hypothesis | * = ^ •
7;— li
Moreover, equating the coefficients of - we see that
A
P2R + P[ = RP*+ BPU
i.e.
it=i
whence it follows upon setting i = j that
(i)' _ v ;. AD
(30) p}r = ?b3kP%
fc-1
76 BIRKHOFF AND LANGER.
Since the other quantities under the sign of summation in this formula
have all been determined, pjj may be found by means of a quadra-
ture. When this has been done the matrix Pi has been completely
determined.
It should be observed that we have thus far required only that the
matrices R(x) and B(x) be continuous. To determine the elements of
Po however, we have first, for i =(= j, the equation
(2) — Pij + L "ik pkj
Pa - k=i y
7™ Ti
and since this formula implies the existence of pij, whereas it is seen
from (29) that p® has a derivative only if this is true of the matrices
R(x) and B(x), we cannot proceed to the determination of P> if R(x)
and B(x) are merely continuous.
Let us suppose then that R(x) and B(x) both possess continuous
derivatives up to and including those of order h ^ 1, but that perhaps
one of these matrices possesses no such derivative of order (k + 1).
If, in particular R(x) and B(x) possess infinitely many derivatives we
may take h = oo . The derivatives of Pi up to and including that of
order k are now seen to exist from formulas (29) and (30).
Equating formally the coefficients of -— [ in the identity (27) we
A
have
i.e.
1\R + iVi = iU\ + BPp-i,
n
Pij 17" 7»J — —pij t- OikPkj >
A-l
n
Pij + - °ik Pkj
whence p^ = — for i =f= j,
7/ - 7;
1
and again, equating the coefficients of— we have
A
PM+i R + P'M = PPM+i + BI\
BOUNDARY PROBLEMS AND DEVELOPMENTS. 77
i.e. vt^hi-y^+pf' = ?hkP%,
whence it is seen, upon setting i = j that
n
„>)' — y h r>>
Pjj UjkPkj >
/fc=l
namely that pyy can be determined by means of a quadrature if the
quantities on the right are known.
The determination of the elements of pjy by means of these formulas
depends, therefore, only upon a knowledge of the elements of PM_i
and upon the existence of PM_i . Moreover, it is seen that in general
PM possesses one less derivative than PM_i. Inasmuch as Pi has
already been determined, and was seen to possess k derivatives it is
clear that the matrices P^ for fj. = 0, 1,. . ., (k + 1), may be suc-
cessively determined, and that in general the matrix Pa-+i is merely
continuous.
If k is finite, we can, therefore, determine a differentiable matrix
S(x), given by
(31) S(x) B j Po(x) + * Pi(x) + - . . + jf Pk(x) | E(x)
which will satisfy the equation
S'(x) = {\R + B) S(x) + 4 W(*) " B(x) Pk(x)} E(x),
A
i.e. an equation of the form
(32) S'(x) = | \R(x) + B(x) + 1 1(;r, X) j S(z),
where Z(x, X) is a matrix each element of which is rational in X,
with coefficients continuous in x, given by power series in ( - ).
If on the other hand k = °° , as many terms of the infinite series
~fl> J?)
78 BIRKHOFF AND LANGER.
as are desired may be used. These series are not in general con-
vergent. Nevertheless the formal matrix
S(x)= Po(.r)+^ Pi(x)+ Ie(z)
(i)
cxry O) J pg) + Pi + |
which is found in this case, formally satisfies the equation (26).
With the convention that the term in -r is to be omitted if k = »
X*
formula (32) holds in all cases. S(x) will be called a formal matrix
solution of equation (2(3) regardless of whether k is, for the case in hand,
finite or infinite.
It should be observed that each plj is not wholly determined but
contains a single arbitrary constant of integration, independent of x.
This arbitrariness corresponds to the fact that any convergent power
series Cj in ( - ) with constant coefficients may be multiplied into each
column of S(x) without thereby destroying its property of being a
formal solution in the sense (32).
Using the notation [ipa]k, or [^0], for an expression of the form
where \p is bounded for | X | large, we have
X
S b.-dx
S(x)
It is clear that an alternative form is
(33)
)£(.r).
j(x) = J bji
where Bj(x) = J bjj dx .
a
By similar considerations the equation Z' = — Z{A\ + B) may
be transformed and a formal matrix solution T(x) for the resulting
equation
BOUNDARY PROBLEMS AND DEVELOPMENTS. 79
(34) Z'(x) = -Z(x) [R(x)\+ B(.r)}
be obtained. This T(x) has the form
(35) T(x) = (cie-™iW-BiM[8i3]X
and satisfies an equation of the form
T'(x) = - T{x) \\R(x) + B(x) +~M(x, X) |.
Moreover each row is a formal solution of the vector equation
(36) >Z'(x) = - -Z(x) {R(x) X + B(x)}.
Considering differentiation as merely a formal process defined by
the usual rules it is readily seen that the differentiation of the formal
matrices S(x) and T(x) is permissible. Hence we have
d
— TS = TS' + T'S
ax
= TUR + B+\l\s- T\\R + B+-kMls
= ~T{L-M}S.
Since
IS = I i ae WihlkCje WhjlkJ
we find upon differentiating, and removing the exponential factor,
( ( a(1) (T{2k) ) )
CiC,- J {X(7;-7i) + bjj-biil |«l7+ -^- + . . . + ^£- | j
80 BIRKHOFF AND LARGER
1 "
~r 2 [5ih]k \lhs — >"hs} [&sj\
\k fc,«=l
Equating the coefficients of X° we have
ct Cj { (y,— y{) o® + (6,-,- ha) 5r;} = 0,
from which it follows that a,} = 0 when i 4= j- Again, equating the
1
coefficients of - we have
X
(7~ H) «# + (hi- bu) c#+ *$' ^ 0,
from which it is seen, upon setting i — j that a# ' = 0, namely that
°m — constant. The relation shows on the other hand that a\f = 0
when i =£ j. Equating to zero successively the coefficient of each
individual power of - it is found in the same way that o\f = 0 when
A
* ^ j, oft — constant for /j, = 1, 2, . . . ,(k — 1). It follows that
= (c,-c7-5i,-[l]ifc_i) +—kE~l (an) E,
A
where the coefficients of [lk-_i are constants.
Now for any choice of the set of series c,- it is clearly possible to
choose a set Cj such that c; c, [lk--i = 1, j = 1,2,.. .n. The formal
solutions S(x) and T{x) corresponding respectively to these values of
Cj and Cj are closely related. They are called associated formal solu-
tions and satisfy the relation T(x) -itf(«) [5..]).
Section VI.
The relation of the formal solutions to the actual solutions.
It was observed in the preceding section that the formal matrix
S(x) either satisfies equation (26) only in an approximate sense (i.e.
satisfies (32)) or, if k = °° satisfies it only formally, since the elements
of S(x) are in that case infinite series which are not in general con-
vergent. S(x) is, therefore, not a matrix solution of equation (26), and
its significance requires further investigation.
Consider the actual matrix which is derived from S(x) by retaining
in the latter only the first (m + 1) terms of its elements, where m is any
positive integer not exceeding k. This matrix S(.r) may be written,
in accordance with formula (37),
S(x) = (,*r;(*)+B;(*) [g..]j
= P(x, X) E(x),
where P(x,\) = (eBW [8i3]m),
and is seen to be analytic in X. Since | f s>(x)5{;- 1 =(= 0, it follows that
I iS I =4= 0 for I X I > N. If the formal solution S(x) is written as
P(x, X) E(x), it is apparent that we have the formulas
Six) = \l - ^ [Q] P"1 J S(x),
S(x)= J 7+ ~{Q}P-l]s(x),
from which we obtain upon differentiating and substituting from the
equation (32), the relation
82 BIRKHOFF AND LANGER.
It is apparent from this that Six) is a solution of a homogeneous
differential matrix equation of the type
(39) S'(x) = J \R + B 4- ^ $(.r, X) j S(«).
The elements of $(a-, X) are power series in ( - ) with coefficients
which are continuous in x, and are seen to be convergent, since these
elements are rational in X.
But equation (26) can be written in the form
(40) Y'(x)= pl(.x)\ + B(x) +^*(x,A) j Y(x) -j-jjjSfeX) Y(x),
and considering this as a non-homogeneous equation we know, in
virtue of the developments of page 62, that its solutions, i.e. the solu-
tions of (26), are given by
Y(x) = S(x) C + h(x) T(t) | - ± Ht, X) 7(0 | dt,
i.e. by
(41) Y(x) = Six) C- -1 C S(x) fit) Ht, X) 7(0 dt,
where Tix) = S~*(x) and where the lower limit of integration, which
has been omitted, may be chosen at pleasure for each of as many parts
of the integrand as desired.
Substituting for 7(.r) in equation (41) its equivalent as given by
the form
(42) Y(.r) = Uix) S(x),
we have further
BOUNDARY PROBLEMS AND DEVELOPMENTS. 83
(43) U(x) = Six) CT(x) - — / Six) T(t)*(t, X) U(t)S(t) T(x) dt.
°-r»P(
Theorem: If the functions 74(0*), i — 1, 2, . . .11, satisfy the relations
arg iy jix) — 7i(a*)} = hi,-, i, j, = 1,2,. . .n, where each hij is a con-
stant, then there corresponds to each sector bounded by two adjacent
rays R\\{yj(x) — ji(x)} } — 0 a choice of the lower limits of integra-
tion which is such that for X within the sector and | X | > N, and for
any continuous matrix U(t), each element of the matrix
X
yp{x, X) = Cs(x) f{t) $it, X) Uit) Sit) Tix) dt
is less numerically than KM, where M is the largest numerical maxi-
mum attained by any element of U.
Proof: Writing
Six) = isvix)) Eix), fix) = E-^ix) (%(*)),
we have
+(*, X) = ( I fsih ix)^rh{x)-Thil)lthlit) «,lp(t, X) uvqit)
\ h, l, p, a, r-l J
sgrit)eMrrW-rr(x)\tr.^dt\
n 1 x
x/(7Aa)-7r«)}di (hr)
<*"%> it, x, x) dt/,
where, for | X | > N, | to ({,r) | < /c(*r)M for all ?' and j, /c (*r) being a
positive constant.
Consider a particular sector and any element
X
X/{7A«)-7r(f)}d{
ffr.
If #|X{7/,(£) — 7r(^)| ^ 0 for X within the sector and any £ it is so
for all £, and, provided that t ^ x, the integral
8 The notation R \*

0
for X within the sector and any £, it is so for all £, and, provided that
t ^. x, the integral
f Yin®-Tr<*>}« (Ar)/7.
b
is similarly bounded. Consequently if K/,r(X) is defined by the rela-
tions
Khr = a if R{\{yh(£) - yr(M)}} ^ 0, Khr = b otherwise,
the numerical value of each element of the matrix
+(x, X) = ( I J e* o^ dt\
n
is clearly less than KM, where K = (b — a) 2 fc( , for X within the
sector and [ X | > N. Q. E. D.
Assuming the limits chosen in the manner above, a $(x, X) corre-
sponding to each sector and to each U is uniquely determined. More-
over, we have
(44) U(x) = S(x) CT(x) - ± *(z, X).
A
Consider now the particular solution, Yo(x), of equation (26) which
satisfies the relation Y0(a) = S(a). Since S(x) as well as the coefficients
of equation (26) are analytic in X, Yo(x) is likewise analytic in X.
Moreover we know from page 56 that every solution of the equation is
of the form Y(x) = Y0(.v) D, where the elements of D are constants
with respect to x. Substituting this form of Y(x) into equation (41)
and fixing x and solving for C we obtain the relation
1 To.
C = r(.r„) | l'„(.ro) + ^ S(.r„) T(t) *(/, x) r»(0 tit { D,
BOUNDARY PROBLEMS AND DEVELOPMENTS. 85
n
i.e. ^ ,
ca — ■" Pih (>hj ,
h=\
where the quantities p;/, are analytic in X.
Inasmuch as Y( x) is a non-identicall y zero solution if D =(= 0 there
will exist such a solution for which C — 0 provided the determinant
| p^ | vanishes. If on the other hand this determinant does not
vanish then there exists a solution Y(x) ^ 0 corresponding to every
Suppose the determinant | pih\ = 0 . Then there corresponds to the
choice C = 0 a solution Y(x) for which the matrix Uix) defined by
Y(x) = U(x) S(x) satisfies the relation
i / » rx/{7A(f)-7r(f)}di \
But we know that for some x, say xo, and for some i, j, say io, jo,
uujo(xo) — M. Then since
\L et u>%?(xo,t)dt\ and the sum (10S) i.e. (107) reduces to
v
jc 2tt
It follows, therefore, that
*
(109) X 2= (Sjf ) - | J.
In consequence we have
s2to0—**<*-o).+ «,
i.e.
(110) |imS2f(a?)-=iJP(a5-0)-
/I. gfffr)-.
The treatment of this expression is parallel to that of S^(x) • and is
as follows.
Writing
b
J*-= JY{x){8f*)Z{t)R{t)F{t)-dt,
:*)•= hS^
we have from (101) Sf£(x)-= — / J*-dK.
Hence
(HI) S%(b)-=0,
118 BIRKHOFF AND LANGER.
and it is necessary to consider further only the case x =fc b. We have
6
X*7 k,m,p,q-l ( X )
r.^r-w,-B-i° smP +
b
(f *[*&>*&)+*&>-*&% yi(t)m dt) +
x
b
-ll Y „XJ rt(x)-rfc(«J + Bt(a;)-B.(Os** /,w /,ws / \
XW * i * 8*fc 7p(0 /P(0 [5*p ^f*W
X
+ 8ik^kp(t)]dt\
Each integrand in the last matrix of this equation is of the type
tpi(t, X) and accordingly the matrix is, by lemma 1 of type (e)/X.
Integrating by parts the elements of the first matrix on the right we
have
/«■ = I («« | - eMW(W}^)-^)/i(6 _ 0) +fi{x + 0) +
Therefore, J2# can also be expressed as a sum of matrices, the first
and third of which, namely
and
6
X
are of the type (e)/X on arc CM„, for each element either vanishes or
approaches the limit zero as | X | = a> . The second matrix of the
sum is directly
BOUNDARY PROBLEMS AND DEVELOPMENTS. 119
A A
Hen
ce
/,-- ~ ! (©*(* + (>) '+(«)}.
which integrated over CM„ gives
^/{(#')^ + o)- + w}f = |r(5
do' : 35~3 dl ~ 35~2 dJ2'
We must next evaluate the derivatives of <£.
r/<& a nb 1 db
lb7 ' 82 " 5"+i + 6" ~d8 '
*■
M HH • -g r-H | S
,2 M ^ 5 ^ 5 CU
<• S O-O-JChnhO S O O O iO £P b OCICH » COOO-fOOOO
3 .cSis" oooo ^oooofe '-'-' o o o o £ cooooooooo
1—1 ■ ' So-n jam .... . . . . r ....5
ffl ,» • "S a a c 0000 . oooojj . o o o o J? 00000000
< o M £ — ' + +++ K I ^ S + 1 ++^ > ++ I I + I + I
P-H JS CO h- 1 f_|
« W to cc
« a w w
W M H "-1
to w « «
«~ WfflHN c73 CO i-H CM O /T LO Ol LO O .5 ■* ©CO^NOllOH
0 a OOOOLO lOt^t^CO * NOC1CO Ji fflHCO"*'MtO'^00
♦iag-^ooLO r-iO'-H'^1 mooh ioO'ffflfflLO»io
*'" i E NNHLO MMCSN -^fl CO -"tf1 CO lONOCOClCOOOOO
_MCaDg3|Ttlr-ICOCO "* t~ -f Ol LOCOrHN OCOOCOlOCOCCCO
•5 > CO lO CI N ■* CM OS OJ t^ CO — < tH CO to tO N O CO Li ffl
►* CI"— I CM"— I t-H t-H t-H t-H r-H 1— It-It— I t-H t— I t-H t-H
t-hOOCOt-hOI^O
tO t-I r-l O Ll ^ M -t
HOOHMlflHUJffi
N O O C) -t «■ N CO
COlOCOOt-h-*t-hlO
COCNCON^COCOCO
0
CO
CO
0 c a
=
cm
O CO CO ^f
co in lq cm
iOOOOlO
MNOJCO
^ONOO
in 0 t-h 01
05 t-h r^ i-
MIONCS
00 t-h CO GO
COOMO
CO CO O O
co 0 02 CO
CM CI CO t^
t- OS CO CO
05 TH CO CM
0
T-H
II
bo
<3
j^co >
O lO 00 CO
t-H
1O1OCON
CM CM r« ■*
13
0 ,
Fracti
of
SiCl
CM CO 05 CO
T-H
t^ CO T-H O
t-h CM
T-H T-H CO CO
N
oONooiontociH
h- 00 O N N (O N
CM
CM
60
.£? e
si -^l.-^0 cofoooooo5>
|-1 ■«•< ^! h- 0-2.2 j '° <£> £r fc lo lo lo h <; h
^ P** 0.1 u -3 9 rH --I O O O O O O hn _
M ^ _~T3 OS Soooooooo * S
S 7. 8 £££" oooooooo.S-g
S j -C .S^-§ S oooooooo^S
2 n $ | *-a I + + + + + + I 12
be to
es cs
e« O00ON©^OOMN > i
o a fi n h oi o t o n > >
+i a 5 J N 00 M ■* N « fl O ^ ^
2 .a 3 B •— ioocooi— i i-i oj t»
[uj Ml g UCONOOlOMOtlH
£"* £ NOOONNtdNiC)
*m
OO
o>
■*
o
o
CM
iO
iO
z. a a
o
o
CO
CO
-f
CO
1 -
:."
•^ .* 3
•
•V
-H
■*
CO
CO
-.
■..-
CO
o
d
a
a)
»o
Ol
Oi
CI
CD
CO
oo
•-oa °
sc
cm
CO
o
00
o
CD
00
CO
oo
i—i
r^
o
i-H
II
bC
a
<
.2 2
'g'SM HM(NrfU3CCN(D
B 0"3 H IM W i1 iO to h 00
= H CM CM CM CM CM CM CM CM
S <
266
BAXTER,, WEATHERILL AND SCRIPTURE.
In the foregoing tables the analyses are arranged in the order in
which they were carried out. A better comparison of the various
fractions is obtained if the results are tabulated in the order of volatil-
ity. The following table gives the observed atomic weights in the
order of decreasing volatility for the different chloride samples as well
as for the bromide.
SiCl4 I
SiCl
,11
SiCl4
III
Fraction
At.Wt.
Fraction
At.Wt.
Fraction
At.Wt.
3
28.089
1
28
.099
1
28.089
6
28.114
7
28.099
13
28.090
9
28.112
13
28.097
16
28.031
12
28.129
SiCl
20
iIV
28
144
Si Br
21
4
28.152
Fraction
At.Wt.
Fraction
At.Wt.
1
2S.071
1
28 064
4
28.064
2
28.064
6
28.068
3
28.062
8
2S.067
4
28.054
10
28.063
5
28 . 056
11
28.064
6
28.06S
12
28.06.-)
7
28.054
14
28.071
8
28.093
In the first three series, with the exception of Fraction 16, Series III,
there is unmistakable evidence of slightly increasing atomic weight
with decreasing volatility. It is especially noticeable that Fraction 12,
Series I, Fraction 20, Series II, and Fraction 21, Series III, which were
the third from the last, the next to the last and the last fractions
respectively, all give values nearly alike and markedly higher than the
others. As explained on page 252 this is probably due to less volatile
chlorides of silicon which seem to be removed with considerable
difficulty. The figures indicate, however, that the process of separa-
tion was still taking place even in the most carefully and elaborately
distilled material, Sample III. In Sample IV the fractionation was
many times as effective owing to the use of fractionating towers in
nearly every distillation. The greater efficiency is obvious from the
fact that even before the distillation was half completed, extreme
fractions, Nos. 1 and 14, showed no indication of a difference in compo-
sition, and the uniformity of all the material from that point on is all
that could be desired. There seems to be no question therefore that
ATOMIC WEIGHT OF SILICON. 267
the results of the analyses of Sample IV should be given preference to
those of the first three samples of chloride.
The silicon tetrabromide, although distilled at the beginning of the
fractional distillation without the use of fractionating columns, ulti-
mately was repeatedly distilled in much the same way as Sample IV of
the chloride. Here also uniformity in composition over a wide range
of fractions was secured. The fact that the least volatile fraction,
No. 8, gave a distinctly higher result than the remainder, is counter-
balanced by the fact that the three residual fractions from distillations
immediately following, were apparently no different from the more
volatile fractions.
Furthermore, -the fourth chloride series and the bromide series
yielded almost identical results. Therefore the average of the mean
values from these two series, 28.063, is the most probable value which
can be derived from these data. It seems unlikely that further
fractional distillation would have affected this outcome.
It is interesting to compare the influence which various impurities
would have on the results. In the following table are given the effects
produced upon the apparent atomic weight of silicon by one-tenth of
one per cent of the more likely impurities.
SioCl6
+0.009
Si2Br6
+0.009
Si3Cl3
+0.014
Si3Br8
+0 014
Si2OCl6
+0.024
HBr
-0.024
HC1
-0.024
CBr,
-0.016
CC14
-0.016
TiBr4
+0 020
TiCb
+0.020
The bearing of the outcome of the foregoing work upon the isotopic
character of silicon is an interesting one. Aston 26 finds ample evi-
dence of the existence of two isotopes of this element, with masses
28 and 29. He then continues:
"The evidence of a silicon of atomic weight 30 is of a much more
doubtful character. Its presence is suggested by the lines 30, 49, 68
and 87, but the possibility of hydrogen compounds makes the evidence
somewhat untrustworthy, and no proof can be drawn from a second
order line 15, as this is normally present and is due to CH3. On the
other hand, if we accept a mean atomic weight as high as 28.3, the
relative intensity of the lines due to compounds of Si28 and Si29 indi-
cates the probable presence of an isotope of higher mass."
26 Phil. Mag., 40, 628 (1920).
268
BAXTER, WEATHERILL AND .SCRIPTURE.
It is obvious that the atomic weight derived in this paper is more
nearly in accord with the evidence yielded by mass spectra as regards
the proportion of the isotopic components of silicon, than the value
28.3 which has been in general use for some time. Assuming the
existence of appreciable proportions of only the two isotopes 28 and 29,
our result indicates that the ratio of lighter to heavier is about 14 to 1.
^Yith at least two isotopes of silicon, chlorine, and bromine, there
are evidently possible ten different tetrachlorides of silicon as well as
ten different tetrabromides, with molecular weights ranging from 168,
at intervals of one unit, to 177 in the case of the chloride, and from
344 to 353 in the case of the bromide. Of these, the two lightest,
Si28 (CI35 )4, 168, and Si28 (Br 79)4, 354, are presumably the most abun-
dant and the most volatile.
The question therefore may be raised as to whether partial separa-
tion may have occurred during the fractionation of the two liquids.
If, however, separation actually occurs at all during the fractiona-
tion, it certainly is to be expected that it will continue to take place
during the whole fractionation. The absence of any systematic
difference in composition among the fractions analyzed in the case of
Series IV and V, which covered a large proportion of the original
material, is strong evidence that no important separation actually
was taking place.
This is only to be expected from the fact that the conditions of dis-
tillation were far from ideal for the separation of isotopic substances,
for as Mulliken and Harkins 27 and others have pointed out, such a
separation is to be expected only when distillation takes place at ex-
tremely low pressures.
We are very greatly indebted to the Wolcott Gibbs and Bache Funds
of the National Academy of Sciences for generous assistance in pro-
viding the necessary apparatus and materials.
Summary.
1. Improved methods for fractional distillation out of contact
with the air at various temperatures have been devised.
2. The analysis of pure silicon tetrachloride and tetrabromide has
yielded the values 28.067 and 28.059 for the atomic weight of silicon.
The average value is 28.063 (Ag = 107.SS0).
27 Jour. Amer. Chem. Soc, 44, 143 (1922).
VOLUME 57.
1. Kent, Norton A. and Taylor, Lucien B. — The Grid Structure in Echelon Spectrum
Lines, pp. 1-18. December, 1921. $.75.
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Vol. 58. No. 7.— March, 1923.
CONTRIBUTION FROM THE OSBORN BOTANICAL LABORATORY.
THE CHILEAN SPECIES OF METZGERIA.
By Alexander W. Evans.
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CONTRIBUTION FROM THE OSBORN BOTANICAL LABORATORY.
THE CHILEAN SPECIES OF METZGERIA.
By x\lexander W. Evans.
LHtKAKV
NEW * examined. Under these circumstances the costa seems to be
poorly defined, and the bistratose or tristratose hand becomes evident
by careful focusing, the outlines of the cells in the superimposed
layers not corresponding.
The type material of .1/. decrescens shows male branches in abun-
dance but no female branches. The male branches, which seem to he
the only ventral branches present, have involute margins and are
strongly incurved, although the apex does not approach the base very
closely (FlG. 2, F). They measure about 0.45X0.3 nun. in well-
developed examples and are wholly destitute o( hairs, the only appen-
dicular organs developed being the slime-papillae (Fig. 2, G). The
alar cells are more delicate than those of a vegetative thallus and
average only 25 fj. in diameter. In some of the other specimens
studied a few female branches with calyptras were found. These
organs bore scattered hairs and attained a length in some cases of
3-3.5 nun., the diameter being 0.6 0.8 nun. Unfortunately the
CHILEAN SPECIES OF METZGEBIA. 283
female branches themselves were so old and battered that their dis-
tinctive features could not be determined.
The three synonyms included under M . decrescens may now be con-
sidered. The first, M. terricola, was based on two specimens, one
collected by Savatier on Desolation Island and the other by Dusen on
Newton Island. According to Stephani M. terricola shows a varia-
bility in the number of cortical costal cells, comparable with what is
found in M. decrescens. The wings of the thallus, moreover, are
revolute in much the same way, while the cells are said to average
about 54 X 36m, measurements which diverge but slightly from tho e
given for M. decrescens. The following represent the most important
differential characters indicated: the presence of a few setulae on the
ventral surface of the thallus and the occurrence of the marginal hairs
in pairs.
In Savatier's specimens, which may perhaps be regarded as the
type, the costa is essentially like that of the type specimen of M.
decrescens; and, although ventral hairs are sometimes present on the
costa, this is equally true of M. decrescem. The marginal hairs, more-
over, so far as the writer can determine, are invariably borne singly.
In Dusen's material some of the thalli are like Savatier's, but other
show crowded marginal cilia in pairs. The latter, however, are associ-
ated with costae that are bounded constantly, both dorsally and
ventrally, by only two row- of cortical cells, while the cells of the
revolute wings are considerably larger, averaging about. 70 X \0 p.
In the writer's opinion the thalli with the geminate marginal hairs
should be referred to M. hamata, although they evidently formed a
part of the material from which the description of M. terricola was
drawn. If these thalli are eliminated there is apparently nothing
whatever to distinguish M. terricola from .1/. decrescens. The writer
regrets that he has not seen Herzog's specimens of "M. terricola" to
which allusion has already been marie ('see page 27 1 .
The second synonym, M. longiseta, was based on a specimen from
the Strait- of Magellan, the collector's name not being given. In his
account of this species Stephani calls ait.ent.ion to the strongly con
thallus, the variable number of cortical co tal cells, the long marginal
hair- borne singly, and the alar cell- averaging about 54 X 36 n, four
characters which .1/. Umgiseta clearly -hare- with M. dt is. He
mention- also the fact that the wings are two or three cells thick mar
the costa. This, as has been shown, is another characteristic feature
of M. decrescens, although the original description doe-, not allude to
it. The differential characters of M. longiseta are apparently drawn
284 EVANS.
from the marginal hairs, which are described as "hamate," and from
the costa. The latter is said to be strongly convex dorsally and nearly
plane ventrally and to have a thickness of five cells. The dorsal
cortical cells, furthermore, are said to be convex and much larger than
the internal and ventral cortical cells, which are said to be subequal
in size. Unfortunately the type specimen does not support this
description very convincingly. The marginal hairs are very rarely
hamate, most of them being straight or irregularly curved and con-
torted; while the costa, as shown by cross sections, may project
ventrally slightly more than dorsally. The dorsal cortical cells, more-
over, measure about 50 fx in width, the ventral about 40 p., and the
internal cells, which may be in more than three layers, measure about
30 \x. These observations show that Stephani's differential characters
are far from constant, and yet with the withdrawal of these his descrip-
tions of M. longiseta and M. decrescens are almost identical.
Although the first two synonyms of M. decrescens are placed by
Stephani among the Pinnatae, the third, M. Duscnii, is placed among
the Furcatae. It is based upon material collected by Dusen on Deso-
lation Island and the three original specimens examined have all been
badly mixed with M. frontipilis. Stephani's description of M. Dnsenii
would seem to indicate that the species was much less variable in its
costal features than M. decrescens, since the cortical cells are said
to be in four rows both dorsally and ventrally. It is added that the
dorsal surface is convex and the ventral smooth, that costal hairs are
lacking, that the dorsal cortical cells are large and projecting, and that
the ventral cells are much smaller. The original material shows at
once the inconstancy of these features. Although some of the thalli
show four rows of cortical cells on both surfaces, deviations from this
number are frequent; some of the branches, for example, show only
two or three such rows, while five rows of ventral cells were observed
in at least one instance. Costal hairs, moreover, can be demonstrated
by careful search in spite of their great infrequency, and they are really
not much rarer than in the type of M. decrescens. The costal cells,
finally, show deviations from the description. In a series of sections
examined by the writer, the costa was found to be distinctly convex
ventrally, while the ventral cortical cells measured 38 ll in width and
were thus only slightly narrower than the dorsal cells, which measured
42 ju. Aside from the characters which have just been discussed the
description of M. Duscnii agrees in all essential respects with that of
M. decrescens, since the thallus is said to be strongly convex with
naked wings and long marginal hairs borne singly, while the alar cells
CHILEAN SPECIES OF METZGERIA. 285
are said to average about 45 X 36 ix. The Desolation Island speci-
mens are perhaps a trifle less robust than the type of M. decrescens
from the Aysen Valley, and the branches tend to be shorter, but these
differences are too slight and too inconstant to be of much significance.
It is interesting to note that Dusen's material from Newton Island,
No. 113, has been differently determined by Stephani at different
times. The specimen in the Boissier Herbarium bears the name
M. terricola, while those at Stockholm and Upsala bear the name
M . Dusenii. In the writer's opinion, as indicated above, these speci-
mens are clearly the same and represent M. decrescens. It might
appear from his determinations that Stephani recognized the identity
of his M. terricola and M. Dusenii and wished to supplant one name
by the other. Unfortunately this assumption is contradicted by his
published writings.
3. Metzgeria corralensis Steph.
Metzgeria corralensis Steph. Bull. Herb. Boissier 7: 933. 1899.
Metzgeria Lechleri Steph. op. cit. 942. 1899.
Specimens examined : without definite locality or date, Gay (Mont.,
as M . furcata, and listed under this name by Montagne, 14, p. 297) ;
Corral, no date, Krause (B., Type); Arique, no date, Lcchler 652 (M.,
unnamed but probably representing the type of M. Lechleri); Val-
divia, 1887-88, Hahn (B., as M. Licbmanniana and listed under this
name by Stephani, 19, p. 935); same locality, date and collector (B.,
apparently a part of the same collection as the preceding but bearing
a manuscript name) ; Osarno Volcano, date and collector's name not
given (M.).
Stephani places M. corralensis among the Pinnatae and describes
the thallus as remotely pinnate. At the same time he makes no allow-
ance for variability in the number of costal cells, as he did in M.
decrescens, stating definitely that the dorsal cortical cells are in four
rows and the ventral in eight. The type specimen shows that these
numbers are too rigid. Although the dorsal cortical cells are usually
in four rows (Fig. 3, A, B), the number really varies from two to five,
and the ventral rows are frequently fewer than eight (Fig. 3, B). In
spite of this variability a pinnate habit is no more apparent in M.
corralensis than in M. decrescens.
The species varies in color from a pale yellowish green to a dull
green and is fairly robust. According to Stephani the thallus some-
times attains a length of 4 cm. The width is mostly 1-1.5 mm. but
286
EVANS.
may be as much as 2.5 mm. in well-developed plants. The wings are
plane or somewhat convex and are mostly fifteen to twenty -five cells
across in the type material, although Stephani gives the width as
only twelve cells. According to his statements the alar cells measure
36X27a<, and these figures agree pretty closely with the general
average of 33 X 26 /jl, obtained from the five specimens listed above.
The cells, as he notes, are essentially thin-walled throughout, although
vague indications of trigones are sometimes present.
In well-developed thalli the whole ventral surface, including both
the costa and the wings, is covered over with crowded hairs, giving
it a pubescent appearance. These hairs are mostly 0.1-0.3 mm. in
Fig. 3. Metzgeria coRRAUENSisXSteph.
A. Part of a thallus showing costa and adjoining cells of wings, dorsal
view, X 50. B. Transverse section of costa, X 100. C. Costa and
adjoining alar cells of a male branch, showing ventral hairs, X 100. D.
Gemma at time of separation, X 100. E. Germinating gemma, X 100.
A, D and E were drawn from a specimen collected on the Osarno Volcano;
B and C, from a specimen collected at Arique by Lechler.
length and 10-12 p. in diameter. Those along the margin, which are
essentially like the others, sometimes spread widely and sometimes
grow downward; they usually arise singly, as Stephani notes, but
twinned hairs may occasionallv be demonstrated. Between the
densely pubescent ventral surface, which is doubtless typical for the
species, and a smooth or nearly smooth condition, all gradations occur,
although it is doubtful if a thallus is ever smooth throughout.
In the specimens studied by the writer a few male branches are pres-
ent and are mostly 0.2-0.35 mm. in length by 0.25-0.4 in width. The
CHILEAN SPECIES OF METZGERIA. 287
wings are involute and the costa so strongly incurved that the apex
almost reaches the base, the branch thus acquiring a spherical or sub-
spherical form. x\ccording to Stephani the surface is smooth, but
this is rarely the case, from one to a dozen hairs being usually present
(Fig. 3, C).
If Stephani's descriptions of M . Lechleri and M. corralensis are com-
pared it will be seen that they agree in most important respects, even
though he places M. Lechleri among the Furcatac. The most impor-
tant differences that he brings out are derived from the costae and alar
cells, the features of which in typical M. corralensis have already been
discussed. According to his description the costa of M. Lechleri is
bounded both dorsally and ventrally by four rows of cortical cells,
while the alar cells have firm walls distinctly thickened at the angles.
The study of Lechler's Arique specimens in the Mitten Herbarium,
which agree in most respects with Stephani's description, brings out
the fact that the cortical costal cells are inconstant in number, just
as in M. corralensis. The alar cells, moreover, although slightly
thickened, do not show conspicuous trigones; in fact it is usually
difficult to make them out at all. Since the differences between the
species thus break down, and since the Arique specimens are essentially
like Krause's type, the writer feels convinced that the two species are
synonymous.
The importance of gemmae in distinguishing species of Metzgeria
has already been emphasized by the writer in another connection (4).
In M. corralensis the gemmae are dorsal and are borne on ordinary
vegetative branches, the growth of which is apparently unlimited.
As in M . crassipilis (Lindb.) Evans of the eastern United States (see 4,
p. 282) and other species having dorsal gemmae, many thalli are not
gemmiparous at all, while others produce the gemmae in great pro-
fusion. In the case of M. corralensis the early stages of development
have not been studied, but their adult features will be described.
At the time of separation (Fig. 3, D) the gemmae vary somewhat
in size but most of them are 0.18-0.27 mm. long and 0.16-0.24 mm.
wide; they may be orbicular, but the width is usually a little less or a
little more than the length. A gemma is six to eight cells across and
has a single apical cell. What may be described as the dorsal surface
is convex and usually shows from two to six short and rudimentary
hairs. The gemma bears in addition from three to eight marginal
hairs on each side, and these may be truly marginal or slightly dis-
placed to the ventral surface, which seems otherwise to be perfectly
smooth.
288 EVANS.
When a gemma germinates its apical cell continues (or resumes) its
activities and gives rise to a flat, strap-shaped thallus which tends to
be narrower than the gemma itself (Fig. 3, E) being often only four
cells wide. While this is going on the hairs on the gemma increase
somewhat in length, and similar superficial and marginal hairs appear
on the flat extension. The superficial hairs are always more numerous
on one surface than on the other and may be confined to one surface.
Sometimes the more hairy surface of the extension is continuous with
the hairy convex surface of the gemma and sometimes with the smooth
concave surface, these observations apparently showing that the
dorsiventrality of the gemma is not firmly fixed but that a reversal of
the dorsiventrality may take place at germination.
The presence of superficial hairs on the gemmae of M. corralcnsis
and on the young thalli to which they give rise are perhaps the most
distinctive features of these structures. Except for these peculiarities
the gemmae and young plants are much like those of M. crassipilis and
M. IAebmanniana. The latter species, in fact, is closely related to
M. corralcnsis, differing from it mainly in its greater size; and it is
therefore not surprising that specimens of the Chilean species have
been referred to M. Licbmanniana.
4. Metzgeria divaricata sp. now
Grayish or yellowish green, scattered or growing in depressed mats,
more or less firmly attached to the substratum: thallus prostrate,
repeatedly dichotomous but rarely branching ventrally, plane or
slightly convex, well-developed thalli mostly 0.6-1.2 mm. wide, the
forks mostly 2-8 mm. apart; costa bounded dorsally by two rows of
cortical cells and ventrally by four; wings mostly eight to fifteen cells
broad, the cells mostly 38 X 31 /*, the walls thin or slightly thickened
and sometimes with more or less distinct trigones and nodular inter-
mediate thickenings; hairs varying greatly in abundance; marginal
hairs in the hairiest and most characteristic plants occurring in
divaricate pairs, ventral hairs under these circumstances numerous
on the wings and especially on the costa; hairs averaging about 0.15
mm. in length and 10-12 jx in width, often branched at the apex and
acting as rhizoids: inflorescence dioicous: d" branches sometimes
borne in considerable abundance, subspherical, usually bearing on
the ventral surface from one to five scattered hairs, 0.33-0.36 mm. long
and 0.33-0.45 mm. in width: 9 branch broadly obcordate, 0.25-04.
mm. long and 0.45-06. mm. wide, hairs abundant along the margin
CHILEAN SPECIES OF METZGERIA. 289
and also scattered over the ventral surface, especially in the median
part; calyptra about 2 mm. long and 0.9 mm. wide, more or less hairy
throughout but especially in the upper half: gemmae rarely abun-
dant, marginal, borne on undifferentiated branches, oblong, flat or
nearly so, usually with crowded rudiments of marginal and sometimes
paired hairs slightly displaced to one surface.
Specimens examined: Chile, without definite locality or date,
Neger 68 (B., as M. conjugate,, and listed under this name by Stephani,
19, p. 951); near Santiago, 1882, Philippi 24 (B., as M. f areata, and
listed under this name by Stephani, 9, p. 941); Concepcion 1905-06,
Thaxter 90, G (H., Y.); San Antonio, Pudeto River, Chiloe, July,
1908, Halle & Slcottsberg 257 (U., as M. Leehleri, and listed under this
name by Stephani, 24, p. 10). No. 90, collected by Professor Roland
Thaxter, may be designated the type; No. 257, from Chiloe, is poorly
developed and somewhat doubtful.
In M . divaricata and the species that follow the structure of the costa
is far more constant than in M. frontipilis, M. decreseens and M. eor-
ralensis. This does not mean that an absolute constancy is to be
expected. In M. divarieata, for example, the ventral cortical cells
may be in five rows instead of four, even at some little distance from a
fork; it simply means that deviations from the typical numbers are
infrequent enough to be ignored.
As noted in the description the ventral hairs vary greatly in abun-
dance. In the more extreme development of these hairs the entire
ventral surface appears loosely pubescent, and the marginal hairs
occur between every two marginal cells. In typical cases these mar-
ginal hairs are paired and spread so widely apart that they form a
straight line perpendicular to the margin. As a rule the outer hair of
each pair is truly marginal and the inner ventrally displaced. Some-
times, however, the outer hair is slightly displaced too, and a sem-
blance of displacement is often brought about by the slight convexity
of the wing-margins. When a long series of these paired and divari-
cate marginal hairs is present the thallus acquires a very striking and
distinctive appearance (Fig. 4, A). Unfortunately the condition just
described is not always realized. Sometimes, for example, one part
of a thallus may be pubescent, while other parts produce hairs spar-
ingly or not at all. An entire thallus, in fact, may be sparingly hairy
throughout, and most of the marginal hairs present may be borne
singly. Even under such circumstances, however, a prolonged search
will usually bring to view an occasional pair of the characteristic
marginal hairs.
290
EVANS.
Marginal gemmae occur abundantly on some of the plants collected
flby Neger but are apparently absent from all the other specimens. The
•gemmiparous branches are essentially like the others and present no
evidence of limitation in growth. The gemmae are usually scattered,
although a crowded series is sometimes to be observed, and the
mother-cells of the gemmae arise directly from the marginal cells,
just as in M.furcata (4, p. 277). At the time of separation the gemmae
vary considerably in size, average
examples measuring perhaps 0.3-
0.4 mm. in length and 0.15-0.2
mm. in width. Most of them are
oblong in form, six to eight cells
across, and show an indistinct
stalk and a single apical cell.
Crowded rudiments of marginal
hairs, slightly displaced to one
surface, are usually present and
not infrequently show a paired
arrangement. Otherwise the gem-
mae are scarcely differentiated.
In germination (Fig. 4, B-D)
the young plant is at first noth-
ing more than a slightly narrower
extension of the gemma, although
in one somewhat older example a
rudimentary costa was present
with a wing three cells wide on
each side. No late stages of
germination have been observed.
The list of specimens cited
brings out the fact that M . divari-
cata, apparently on account of its
variability, has been confused
with three other species of Metz-
geria. In the structure of the
costa with its two rows of dorsal
and four rows of ventral cortical
cells it agrees with 31. conjugate and 31. furcata; in having ventral
hairs, sometimes produced in considerable abundance, it agrees with
M. corralensis. It is, however, amply distinct from all three species.
It differs from 31. conjugata in being dioicous and in having gemmae
Fig. 4. Metzgeria divaricata Evans.
A. Marginal portion of a thallus-
wing, ventral view, X 50. B-D. Ger-
minating gemmae, X 100. A was drawn
from the type material; B-D, from a
specimen collected in Chile by Neger,
No. 68.
CHILEAN SPECIES OF METZGERIA. 291
and ventral alar hairs, while it differs from M.furcata in having paired
marginal hairs. When strongly pubescent it resembles M. corralensis
rather markedly but is distinguished by the more definite structure
of the costa, by the occurrence of the marginal hairs in divaricate
pairs, and by the marginal gemmae.
5. Metzgeria patagonica Steph.
Metzgeria patagonica Steph. Bull. Herb. Boissier 7: 940. 1899.
Specimens examined: Newton Island, May, 1896, Dusen 24 (M.,
U., Type).
The following additional station may be cited from the literature:
Escapada Island, Skyring, Skottsberg (24, p. 11).
According to the original description of this well-marked species
the wings of the thallus are strongly decurved and often revolute, but
a supplementary note adds that the specimens are "etiolated" and
that the normal structure is to be found only on the younger " inno-
vations." In the material studied by the writer most of the thalli are
perfectly plane and only a few of the branches show revolute margins.
At the same time the plane thalli can hardly be regarded as abnormal ;
they do not present the appearance of being etiolated, and the pres-
ence of female branches in some abundance shows that the plants are
by no means in a juvenile stage of development. The soluble yellow
substance, to which Stephani calls attention in a later paper (20, p. 20),
is very much in evidence when the specimens are soaked in water.
The thalli of M. patagonica are pale green, often deeply tinged with
yellow, and grow in depressed mats. The width is mostly 1-1.5 mm.
and the length may be as much as 3 cm. Measured in cells the wings
are usually fifteen to twenty-five cells across. Although ventral
branching sometimes occurs, dichotomous branching is far more com-
mon, the successive forks being mostly 1-5 mm. part.
Hairs are rarely abundant and many regions are nearly or quite free
from them. The marginal hairs are straight and seem to be invariably
borne singly. They are usually slightly displaced to the ventral sur-
face, tending to extend at right angles to the wings, but they may be
truly marginal and lie in the same plane as the wings. The hairs are
about 10 /x in diameter and rather short, the length being usually
only 0.1-0.12 mm. Although the wings are naked the costa some-
times bears loose and scattered clusters of hairs, essentially like the
marginal hairs but sometimes a trifle longer. Apparently in either
292 EVANS.
position a hair has the power of branching at the tip and acting as an
organ of attachment.
The costa shows the same structure as that of M. diraricata, being
bounded dorsally by two rows of cortical cells and below by four. The
alar cells, according to Stephani, measure 54 X 40 //, those near the
costa being 72 X 40 /x. The writer's measurements give an average of
41 X 33 n and do not indicate that the cells near the costa are appreci-
ably longer than the others. The cells throughout have rather firm
walls, which often show nodular intermediate thickenings as Stephani
notes, but the thickened angles that he likewise emphasizes are diffi-
cult to demonstrate.
No male branches have been seen by the writer and the original
description does not mention them. Female branches are often
abundantly produced, and it is a noteworthy fact that a female thallus
sometimes becomes gemmiparous shortly after it has borne the sexual
branches. Some of the latter are small and undeveloped, but most
of them are of fair size (Fig. 5, A), measuring perhaps 0.5-0.7 mm. in
length and 0.9-1 mm. in width. The outline, which is broadly orbicu-
lar with a deep apical indentation, does not show clearly without
spreading the branches out flat, owing to their strong concavity.
The margin shows crowded hairs borne singly, each representing the
outgrowth of a small cell situated between two larger cells, just as in
the normal vegetative thalli of most Metzgeriae. On the ventral
surface the thickened median portion bears a dense cluster of hairs,
and a few other hairs are scattered over the unistratose portion.
Mention has just been made of gemmiparous plants, although
Stephani does not allude to them. As a matter of fact the gemmae of
M. patagonica, which are marginal in position, yield some of the most
distinctive characters of the species. The gemmiparous branches are
at first scarcely modified but rapidly decrease in width after the forma-
tion of the gemmae has been initiated. When the wings have been
reduced to a width of four or five cells the growth of the branch comes
to an end. The reduction in the width of the wings is often accom-
panied by a simplification in the structure of the costa, the rows of
ventral cortical cells being only two or three. In the formation of
the gemmae their mother-cells are derived directly from the marginal
cells of the branch, without a preliminary cell-division. The gemmae
may be scattered or crowded, a long series of adjoining marginal cells
sometimes giving rise to a continuous row of gemmae. The latter
tend to appear in acropetal succession and yet show many exceptions
to this arrangement.
CHILEAN SPECIES OF METZGERIA.
293
At the time of separation the gemmae are flat and unistratose struc-
tures, orbicular to oblong in outline, broadening out abruptly from a
two-celled and often indistinct stalk, and showing a broad and rounded
apex with a single apical cell. They are mostly 0.25-0.3 mm. long and
0.18-0.25 mm. wide, being composed of six to eight indefinite rows of
cells. On each side six to eight hairs are usually present, and these
are commonly (but not invariably) arranged in pairs. The hairs
extend almost at right angles to the surface of the gemma and, when
paired, spread in opposite directions. The majority are strongly
Fig. 5. Metzgeria patagonica Steph.
A. Female branch, X 50. B. Germinating gemma,
figures were both drawn from the type material.
X 100. The
curved and might often be described as hamate. Only the earliest
stages of germination have been observed and in these the young
plants have simply repeated the features of the gemmae, except that
they have sometimes been a little narrower (Fig. 5, B). In other
words they have remained flat and unistratose thalli, showing no signs
of dorsiventrality and tending to produce a succession of paired and
divergent marginal hairs.
Marginal gemmae with hooked hairs have been described in M .
294 EVANS.
uncigera Evans of the West Indies and Florida (4, p. 273), a species in
which the vegetative thallus bears straight hairs arising singly.
Dorsal gemmae with hooked hairs have been described in two West
Indian species, M. dichotoma and M . vivipara Evans (4, pp. 285, 288),
in both of which the vegetative thallus bears straight marginal hairs,
again arising singly. In M. vivipara twinned hairs occur as a rare
exception, the hairs being usually borne singly; in the other two
species twinned hairs are apparently never found. M. patagonica
shows a new combination of characters — marginal gemmae with
curved or hooked hairs arising in pairs and a vegetative thallus with
straight marginal hairs arising singly. It is this unusual association
that separates the species most sharply from its allies.
Of course the structure of the costa allies M. patagonica with M . '
conjugata and M.furcata, as well as with the preceding species. In M.
conjugata, however, no gemmae are produced and the marginal hairs of
the thallus are normally borne in pairs ; in M. furcata the hairs of the
gemmae, if present at all, are straight and arise singly; while in M.
divaricata the marginal hairs of both thallus and gemmae often arise
in pairs but are straight. Aside from these differences M . patagonica
can be distinguished from M. conjugata by its dioicous inflorescence
and from the other two species by its lack of ventral hairs on the wings.
6. Metzgeria chilensis Steph.
Metzgeria chilensis Steph. Bull. Herb. Boissier 7: 937. 1899.
Specimens examined: Quinquina Island, near Concepcion, no
date, Duscn 179 (M., Type).
The following additional stations may be cited from the literature:
Clarence Island, Racoritza (22, p. 4) ; Quicavi, Chiloe, Skottsbcrg (24,
p. 10) ; Juan Fernandez, Skottsbcrg (24, p. 10) ; New Zealand, Colcnso
(19, p. 937). The Juan Fernandez specimen is clearly distinct from
the true M. chilensis; the other specimens have not been seen by the
writer.
The species was based on two specimens, one from Chile and the
other from New Zealand. The Chilean specimen is naturally to be
regarded as the type, but the original description was probably partly
drawn from the New Zealand specimen, since it does not agree in all
respects with Dusen's material.
The plants in the Mitten Herbarium are very fragmentary and grew
in a loose depressed mat in admixture with other bryophytes. The
CHILEAN SPECIES OF METZGERIA. 295
thallus is mostly 0.5-0.9 mm. wide and attains a length of 1—1.5 cm.
The wings, although described as almost revolute by Stephani, are
flat or even slightly concave and are mostly six to twelve cells wide.
The normal branching is dichotomous with the forks 2-10 mm. apart,
but ventral branching is not exceptional.
The marginal hairs vary greatly in abundance. In some places they
may be absent altogether; in other places, even on the same thallus,
they may be as numerous as the marginal cells, a single hair arising
between every two cells. In most cases the hairs are slightly displaced
to the ventral surface, but they may be truly marginal, and it is not
unusual for the apex to be branched and to act as an organ of attach-
ment. The longest hair seen was 0.3 mm. long but most of them were
0.1 mm. or less in length, the average diameter being about 10 fx.
The ventral surface of the wings is apparently wholly free from hairs,
but the costa bears them in loose clusters or scattered and is rarely
free from hairs for any great distance. These costal hairs are essen-
tially like the marginal hairs but tend to be a little longer.
The costa is bounded both dorsally and ventrally by two rows of
cortical cells, a type of structure found also in all the following species.
The alar cells average about 35 X 27 \x, although Stephani's measure-
ments gave 54 X 36 \i. The walls are slightly thickened and some-
times show minute trigones and occasional nodular intermediate
thickenings.
According to Stephani the inflorescence is dioicous. The type
specimen, however, is clearly autoicous, the male and female branches
often occurring in close proximity. The male branches are mostly
0.3-0.4 mm. long and 0.25-0.3 mm. wide and are ellipsoidal in form,
the margins being involute and the costa so strongly incurved that it
approaches the base without reaching it. Except for the slime-
papillae the surface is smooth. The female branches, which are more
or less concave and obcordate in outline, are mostly 0.4-0.45 mm. long
and 0.45-0.6 mm. wide. The marginal hairs grow out from small
cells but are not numerous; the ventral hairs may be restricted to a
cluster of six to twelve on the thickened median portion, but one to
three scattered hairs may be present also on the wings. No gemmae
have been observed.
The autoicous inflorescence will at once distinguish M. chilensis from
all the other Chilean species. It agrees in this unusual feature with
M. conjugata, but in that species the ventral cortical cells of the costa
are in four rows and the marginal hairs often in pairs. The only other
South American species to which an autoicous inflorescence has been
296 EVANS.
assigned is M. albinea Spruce, which further agrees with M. chilensis
in the structure of the costa. In M. albinea, however, the marginal
hairs are in pairs. Aside from the inflorescence M. chilensis ap-
proaches the following species very closely.
7. Metzgeria decipiens (Massal.) Schiffn. & Gottsche.
Metzgeria furcata /?. decipiens Massal. Nuovo Gior. Bot. Ital. 17: 256.
pi. 28, f. 36. 1885.
Metzgeria decipiens Schiffn. & Gottsche in Schiffner, Forschungsreise
"Gazelle" 44: 43. 1890.
Metzgeria glaberrima Steph. Bull. Herb. Boissier 7: 939. 1899.
Metzgeria nuda Steph. Kungl. Svenska Vet.-Akad. Handl. 469: 10. /. 3a.
1911.
Specimens examined : Valdivia, 1887, Hahn (S.); Corral, 1905-06,
Thaxter If, 2c, 78, 110, 122, 124, 138, IP (H., Y.) ; valley of the Aysen
River, 1897, Dusen 283 (Moll., as M. glaberrima, and listed under this
name by Stephani, 20, p. 20) ; Puerto Chacabuco, 1908, Halle 256 (St.,
as M. glaberrima, and listed under this name by Stephani, 24, p. 10);
near the mouth of the Rio Pudeto, Chiloe, 190S, Halle 256 (St., as
M. glaberrima, and listed under this name by Stephani, 24 p 10);
Guaitecas Islands, 1897, Dusen 394 (M., Moll., St., as M. glaberrima,
and listed under this name by Stephani, 20, p. 20); Port Gallant,
Straits of Magellan, 1896, Dusen (N. Y., St., as M. glaberrima); Tues-
day Bay, Straits of Magellan, 1876, Naumann (S., Y., listed by
Schiffner, 16, p. 43) ; Grappler Bay, Straits of Magellan, 1893, Douglas
(H., Y.); Rio Azopardo, Tierra del Fuego, 1896, Dusen 71 (U., as
M. glaberrima, and listed under this name by Stephani, 21, p. 10);
Rio Olivia, Tierra del Fuego, 1902, Skottsberg (St., as M. glaberrima,
and listed under this name by Stephani, 23, p. 9); Cape Horn and
Hermite Island, Hooker (M., as M. furcata, and listed under this name
by Hooker and Taylor, 8, p. 480) ; near Basil Hall, Staten Island, 1882,
Spegazzini (Massal., Y., Type of M. furcata 6. decipiens). The
following three specimens from the Falkland Islands have likewise been
examined: Port Stanley, 1902, Skottsberg (St., as M. glaberrima, and
listed under this name by Stephani, 23, p. 9); same locality, 1905,
Thaxter (H., Y.); near Port Stanley, 1907, Skottsberg 356 (U., type of
M. nuda).
The following additional stations from the literature may be cited:
Wellington and Desolation Islands, Savatier; and Hoste Island, Hya-
des (2, j). 246, as M. furcata var. 3. decipiens).
The following stations for M. glaberrima may likewise be cited:
CHILEAN SPECIES OF METZGERIA.
297
near Puerto Varas, Dusen (20, p. 20) ; Skyring and Dawson Island,
Skotisbcrg (24, p. 10); Desolation Island, Dusen (21, p. 10); Ushuaia,
Tierra del Fuego, Skotisbcrg (23, p. 9). Also the following stations
beyond the boundaries of Chile : New Zealand and Australia, several
collectors (19, p. 939); Antipodes Islands (24, p. 10).
As here understood M. decipiens is probably the commonest and
most widely distributed Metzgeria in Chile. It exhibits a great deal
of variation in size and particularly in width, in the number and dis-
tribution of its hairs and in the measurements of its alar cells. It
shows, however, the following apparently constant features; a flat or
slightly convex thallus; a costa bounded both dorsally and ventrally
by two rows of cortical cells ; a lack of ventral alar hairs ; and a lack of
gemmae. Another feature almost as constant is the presence of
ventral vegetative branches. It is of course difficult to establish the
absolute constancy of any characters in so variable a genus as Metz-
geria, especially characters of a negative nature, but the writer has
found no exceptions to the four first enumerated after a detailed study
of the numerous specimens cited.
The plants are pale yellowish green and are sometimes scattered
but usually form depressed and layered mats of considerable extent.
They are frequently found on trees but are by no means restricted to
such localities; in rare instances, in fact, they are epiphyllous in habit.
The living portion of a thallus is usually 1-2 cm. long, while the width
is mostly 0.8-1.2 mm. These figures represent the mean averages
obtained from six specimens. The narrowest thallus seen, however,
was only 0.2 mm. wide, while the widest was l.S mm. Measured in
cells an average wing is usually thirteen to seventeen cells across; a
very narrow wing, however, may be as little as two cells and a very
wide one as much as twenty-seven cells. The ventral branches are
sometimes so abundant that they largely replace the normal branches.
When the latter occur to the usual extent the successive dichotomies
are mostly 1-3 mm. apart. A ventral branch broadens out abruptly
from a narrow stalk-like base and quickly acquires a normal width,
often in fact just beyond the margin of the higher axis. Sometimes
the branch spreads widely or obliquely; sometimes it grows in the
same direction as the higher axis. Under the latter circumstances
the axis is usually soon limited in growth; and, if the process is
repeated, a more or less definite sympodium may be the result.
Hairs occur in two positions — along the margin and on the ventral
surface of the costa. The marginal hairs (Fig. 6, A) are by far the
more numerous and are sometimes very abundantly produced. In
298 EVANS.
other cases, however, a prolonged search is necessary before any hair*
at all can be demonstrated, and there are many intermediate condi-
tions between these extremes. A thallus, in fact, may produce hairs
abundantly in one part and be hairless or nearly so in another. "When
the marginal hairs are crowded a single hair usually arises between
every two marginal cells, but sometimes the hairs arise in pairs more
or less frequently. ^Yhen borne singly they are either truly marginal
or slightly displaced to the ventral surface. The hairs are usually
straight and measure 0.15-0.3 mm. in length by 10-12 // in width.
In rare instances they are branched at the apex and act as organs of
attachment. Costal hairs are usually exceedingly rare, and in many
individual thalli none at all can be demonstrated, as Stephani notes
under M. glaberrima. When they occur they are either scattered or
in small irregular clusters and are essentially like the marginal hairs.
The alar cells vary considerably in size (Fig. 6, A-D), not only in
different thalli but also (in some cases at least) in different parts of
the same thallus. In Spegazzini's material from Staten Island, for
example, the cells in most places averaged about 4S X 36 n, while a
branch of a thallus yielding these higher measurements in its other
parts had cells averaging only 35 X 29 /*. Taking the mean averages
of fourteen specimens the cells measure about 38 X 29 n, the highest
average being 48 X 26 ll and the lowest 31 X 22 ji. Stephani's
measurements of M. glaberrima, 36 X 36 ll, agree closely with the
general average. It must of course be remembered that individual
alar cells may deviate rather widely from these average measure-
ments. The cells have thin or slightly thickened walls, and trigones
are either minute or absent altogether.
The male branches of M. *

*»a-EtE,, [i, k, = 1,2,-" n]. (25)
It is clear that a double polyadic depends on n", that is N2K scalar
elements. Thus double dyadics in ordinary space, of which the idem-
factor (22) is a special case, would yield in general 81 terms if it were
necessary to expand completely, which fortunately it is not. A
double dyadic is of course a tetrad ic. If need arose to emphasize the
2/v-adic character of a double polyadic and at the same time to depict
the scalar elements, use could be made of a system of adjacent squares
like (6). In the applications which follow, however, the scalar ele-
ments of a double polyadic are to be thought of as a binary assemblage
or square array. For example if K = 2 and N = 2, with i and j for
unit vectors, the order of the fundamental dyads may be agreed upon
by taking Ex = ii, E2 = ij, E3 = ji and E4 = jj. The corresponding
scalars mpq will then be arranged as
mn, W12, mi3, mu
>»21, m22, W23, ?»24 (26)
m,3i, m32, rttzz, w?34
Vli\, 7»42, ?/?43, ?»44
where mpq is the coefficient of EpEg in the double dyadic 2?nPgEpEg.
Comparing with the four-subscript arrangement (6), we see that the
upper left hand square of (6) would correspond to the first row of (26),
the upper right hand square of (6) to the second row of (26), and so on.
Returning to our fundamental concept of a double polyadic as a
sum of dyads whose antecedents and consequents are polyadics, we
next define the A'-tuple dot product or, in terms of the scalar elements of the polyadics,
will be ~Zmisbsk summed on s. This agrees in form with the law of
multiplication of matrices of order n, a result which might have an-
ticipated; for if vectors are analogous to polyadics we should expect
matrices to behave like double polyadics, just as matrices of order N
behave like ordinary dyadics. The analogy holds of course only so
long as we are concerned with formal laws possessed in common by
the two algorisms.
By virtue of this analogy, however, it is evident without further
proof that any double polyadic *

*' • •> , • ••,
R„ such that
{fP -of) : Ri = 0, (i= 1,2,- --,n). (39)
The polyadics Rr • R„ may by analogy be called the axes of the double
polyadic p with respect to dot product.
Again, we may say that a double polyadic which can be expressed
as the sum of I (and no fewer) properly chosen dyads AM has n—l
degrees of nullity. To get the simplest criterion of the number of
degrees of nullity, we may, following Gibbs and Wilson,6 introduce
cross products of polyadics by the law
AXB = - BAA (40)
and it is then possible to follow reasoning parallel to that of the paper
referred to in Note 3, so far as concerns the number of degrees of
nullity, by use of double powers of *

i and

* " ' ', *

*, • ••, '; hence it is sufficient to prove (58) as written.
ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 373
Proof. Let every *

* ^2 : Ay, • • • , >i + a22w22) (6117^11612^12 +
62iw2i + 622>m22) (64)
If instead we use the method of cubic determinants we shall first
form the square array of the fourth order belonging to the double
dyadic AM, which is
aii.mii, auwi2, an?»2i, anw22
Ol2/»H, «i2?M12, 012W21, Ol2?»22
«21»'ll, «2]?»12, 02177121, 02i??722
a22mn, a22?»)2, 022m21, a22^»22
(65)
with a similar array for BN. We then take the sum of all cubic
determinants along the main diagonal, using the two arrays as simi-
larly placed layers. The first of these will be
376
HITCHCOCK.
ai2>nn, «i2'"i2
biinn, bnrin
bunn, b12ni2
(66)
which develops into
aiimn&i2?h2 — ai2Wu6uWi2 — anWi2&i2»n + «i2»?i2&n??ii (67)
There are six of these groups of terms, or 2-4 in all. The scheme of
each group is given by
\ClrsMpq, drs''lrs
pq1lpq>
'pg" rs
0 r s'rl pqj 0 r s7l r £
(68)
which develops into
(IpqfdpqO rs^rs (Irs^^pqOpqflrs d pqW rs® rsWpq ~T~ ClrsMrsVpqtlpq \t)JJ
and the summation is performed by giving to the number-pairs pq
and rs unlike values chosen from the pairs 11, 12, 21, and 22. Com-
paring with (64) it is easy to verify that the two methods agree.
If we use the method of polyadic determinants and adopt the dyads
ii, ij, etc. as operands we first form the dyadics AM : ii, AM : ij, etc.
treating BN in a similar manner. We thus have two rows of a de-
terminant of the fourth order whose elements are dyadics. The other
two rows are alike and given by I : ii, / : ij, etc. that is by ii, ij, etc.
The result
Aran, Ara.12, Amii, A»^2
Bnu, B//12, B»2i, B«22
ii , ij , ji , jj
ii ij ji jj
(70)
should by (56) be the double of the required scalar. Developing by
the definition of Art. 8 the leading term is [A.mn, Brir2, ji, jj] which
denotes the ordinary determinant of the fourth order
anmu, ai2ran, «2i»'ii, «22'»n
&11"»12 > &127&12 , b»l)l 12 , ^22»22
0 , 0 , 1 , 0
0 , 0 , 0 , 1
(71)
which develops into an?»n^i2»i2— «i2'»ii^n"i2- The next term may be
taken as — [Amu, Bni2, jj, jii which doubles the two scalar terms
already found. It is easy to see that developing (70) by two-row
minors after Laplace's method yields the scalar terms in a fashion
similar to the method of cubic determinants.
ON
DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 377
The fact that any scalar (36) is unaltered by changing all the double
dyadics into their conjugates, and is also unaltered by changing the
order of double dyadics, appears in the double symmetry of (64). A
further and quite distinct fact could not have been brought out so
long as our notation represented poly adics as formal vectors : a scalar
formed from double dyadics, like (64), is unaltered if all the dyadics
A, M, B, N are replaced by their conjugates. This is equivalent to
interchanging the two subscripts throughout the expression on the
right of (64); whereas changing the double polyadics into their con-
jugates is equivalent to everywhere interchanging a with m and at the
same time b with n; and changing the order of the polyadics is equiva-
lent to everywhere interchanging a with b and at the same time m
with ?;.
More generally a polyadic which is a sum of terms of the form
aia2- • &k where the A' factors are vectors in space of N dimensions
may be regarded as one of a set of K! polyadics obtained from one
another by making like permutations of factors in every term. A
glance at the definition (10) of multiple dot product is sufficient to
show that all scalars (36) are unaltered when all the antecedents and
consequents of every double polyadic have the factors of all their
terms permuted in the same manner. If each polyadic consists of a
single polyad and we apply the definition of Art. 5, the truth of the
proposition is evident, hence by the distributive principle holds
universally.
Examples like (64) or (69) may be generalized in three ways, ac-
cording as we increase N, the dimensions of the space, K, the order of
the polyadics, or p, the order of the scalars.
If N increases we still have double dyadics and the form of (69) is
unaltered, but the summation is performed over a larger group of
number-pairs, namely 11, 12, 21, 13, 31,- • •, NN.
If A' increases the number of subscripts increases but (69) still con-
sists of 4 scalar terms and the summation is over pairs of unlike
number-triplets, quadruplets, etc.
If p increases the order of the determinant (68) increases, likewise
the number of terms in (69). Thus if we keep N = 2, K = 2, but
make p = 3 there are 4 cubic determinants whose sum is ((AL, BM,
CN))s or 72 scalar terms in all.
If we have double dyadics in 3 dimensions with p = 2, ((AM, BN))s
will be the sum of 36 expressions of the same form as (69) or 144 scalar
terms.
It is usually possible to neglect the dimensionality of the space and
378 HITCHCOCK.
the number of terms in the double polyadic, frequently also the order
of the polyadics. Thus ((AM, BN))S is symbolic of ((*

2 is a new double dyadic. If we regard v>V and

+ •••+ (-l)m*J"= 0 (89)
nor to show that the star idemf actor I" is identical with 2F,F{. The
whole of Art. 3 might be repeated at this point, replacing multiple dot
product by star product and E by F. The coefficients m*p in (89)
will be formed as in Art. 5, replacing multiple dot by star product.
We shall also have a set of scalars
( ' I a'b> cd' I c'd- ef' I e'f))s, symbolically,
= aa'bb'cc'dd'ee'ff- aa'bb'[ce'c'e] [d'-fd-f]
- cc'dd'[ea'e'a] [f'bfb'] - ee'f-f'[a-c'a'-c] [b'dbd']
+ (ab' : c'd) (cd' : e'f) (ef' : a'b) + (ab' : e'f) (cd' : a'b) (ef' : c'd)
(121)
by direct application of the definition of Art. 5, and by grouping the
dot products in the second, third, and fourth terms after the manner
of (119). The first or leading term is evidently the same symbolically
as A i$B i$A jsB j$A ksB ks- The next three terms taken together are
symbolically — ZAisBiS(AjAk)s{B3Bk)s- In transforming the fifth
term the procedure, as in all cases, will be to group into one factor
those vectors which correspond to prefactors A in the original equa-
tion, and into another factor those which correspond to the post-
factors B. Thus
(ab'rc'd) (cd':e'f) (ef':a'b) = (ac'ce'ea') (b'dd'ff'b)
= (aa'ee'cc')s(bb'dd'ff')s
= (AiAtAMBiPiBk)8 (122)
It is especially worthy of remark that the order of the matrices in one
factor is the reverse of that in the other, a consequence of the trans-
formation (112). When there are only two matrices in each factor,
as in (119), it was of no importance whether we wrote (AC)s or (CA)s,
for these are equal: the scalar of the product of two matrices is inde-
pendent of their order, because (aa'-cc')s = ac'a'-c = (cc'-aa').
But by similar reasoning we see that the scalar of the product of several
matrices depends on their cyclic order, as is well known.
In the same way the last term of (121) becomes (AiAjAk)s(BiBkBj)s.
Collecting results, the complete statement for m3 is
6ra3 = AiSAjSAksBisBjSBks— AiSBiS(AjAk)s(BiBk)s —
AjsBjsiAkAMBkBih
- AksBkS(AiAMBiBi)s+ (AiAkA])s(BiBjBk)s +
(AiAjAMBiBrfds (123)
which is to be summed over all possible sets of three terms from the h
terms of the original equation (111), allowing repetitions for the reason
ON DOUBLE POLYADICS THE LINEAR MATRIX EQUATION. 389
already noted under ?»2; that is i, j, k — 1, 2,- •, h. Such a summa-
tion will hereafter be referred to as summation over the extent of the
equation,12 and is implied in every symbolic equation.
In general we may express mp symbolically by the equation
p!mp= ((aib'i | a'ibi, a2b'2 1 a'2b2, • • • , apb'p | a,'pbp))s (124)
Expanding by the definition of Art. 5, the leading term is the product
of p factors of the form arb'r : a'rbr, because the leading term is made
without interchanges among the consequents. These factors are the
same as ar-a'rbr-b'r and symbolically the same as ArsBrs- Thus if
we have a set of p subscripts i, j, h, ■ ■ ■ , r, s, t, the leading term in the
development of p!mp is
AisBisAjsBjsAksBks- • • ArsBrsAssBssAtsBts (125)
which is to be summed over the extent of the equation (11).
The other terms in the expansion of (124) are obtained by making
all possible interchanges among the consequents, according to the
definition of Art. 5. Therefore so long as we maintain the polyadic
notation any term is a product of factors of the form arb'r : a'sbs,
which is the same as ar-a'sb'r-bs. The vectors ar and a's might
correspond to different matrices, likewise b'r and bs. Hence such a
factor cannot in general be translated into matrix notation if con-
sidered by itself. For it is of the essence of this transformation that
every vector ar be associated with its mate a'r and likewise every br
with b'r; then ara'r is symbolically equivalent to Ar and brb'r to Br.
If, however, by a simple interchange of a pair of consequents in
the polyadic expression, we obtain a pair of factors (arb'r : a'sbs)
(asb's : a'rbr), these may be developed as in (118) and (119) and yield
the two factors (ArAs)sXBrBs)s- Such factors occurred already in the
second term of w2 and in the second, third, and fourth terms of m$.
If three consequents change places among themselves, we obtain a
product of three factors of the form (arb'r : a'sbs) (asb's : a,'tbt)
(a(b'( : a'rbr). It is important to notice that the order of subscripts in
the vectors a is the same as that in the vectors b but the order of accents is
reversed. It follows that when we develop after the manner of (122)
the vectors a yield the product of dot products ar- a'sa^a^a*, a'r,
while the vectors b yield the product bVbsbVb^b';, br, where the
order of subscripts is the same, but the accents are on the first vector in
each dot product, instead of the second. These factors are equivalent
in matrix form to (ArAtAs)s and (BrBsBt)s, where the order of sub-
390 HITCHCOCK.
scripts in the postfactors B is the same as in the polyadic term, but the
order of subscripts in the prefactors A is the reverse of the order in the
postfactors B. For in the polyadic expression b's follows its mate bs,
and b'( follows b(, and if we rewrite the factor in the form (brbV
bsb's-b/b'^s then b', also follows its mate br, whence directly the
matrix factor (BrBsBt)s- But in the polyadic expression as it stands,
each a', when we follow the same steps, precedes its mate on account
of the reversal of the order of accents. Thence follows the reversal
of order of matrices in the corresponding factor. Transformations
of this type already occurred in the last two terms of ?»3-
All the italicized statements in the above discussion are true no
matter how many consequents have changed places among themselves.
We may therefore write a general rule for the formation of m p.
Rule for forming p!mP where mp is the coefficient of 6P in the
Hamilton-Cayley equation for 6.
Let there be p subscripts i, j, k, • • • , r, s, t, each of which may have
any value from 1 to h. Choosing a particular set of values for these
subscripts, we form a group of pi terms as follows: the leading term
is AisBisAjsBjsAksBk-s- ■ -^rsBrsAssBssAtsBts- The other terms
are formed from the leading term by first interchanging the post-
factors B in all ways, while the prefactors A are at first left fixed in
position. If a particular postfactor B is left in position, it yields in
the corresponding term a factor Bs and its pref actor yields As pre-
cisely as in the leading term. If a pair of postfactors as Br and Bq
change places, there results in the corresponding term a factor (ArAg)s
(BQBr)s- If three postfactors, as Bk, Br, and B„ change places so that
their new order is Br, Bs, Bk, there results in the corresponding term a
factor (AkAsAr)s(BrBsBk)s where the order of prefactors is the reverse
of the final order of the postfactors. In general if any group of post-
factors change places among themselves so that their final order is
BkBr- • -BsBj
there results in the corresponding term a factor (A jA s • • • A rA k)s
(BkBr- • -BsBj)s where the order of prefactors is the reverse of the
final order of postfactors.
The group of p! terms thus obtained is to be summed over the extent
of the linear matrix equation.
18. Character of the Coefficients as Algebraic Polynomials.
It is evident from the form of the leading term that in every term
of the expansion of mp will be found p prefactors and p postfactors,
ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 391
whether alike or different. Hence mp is homogeneous and of degree
2p in the scalar elements of the given matrices which define 6.
Furthermore mp is non-homogeneous and of degree p (in general) in
the scalar elements of any particular matrix. For if the p subscripts
which occur in a group of terms are all alike, the p prefactors and the
p postf actors are all alike.
Again, each group of terms, corresponding to a particular choice of
p subscripts, is homogeneous in the scalar elements of each of the
matrices which occur in the group. The degree in any one matrix is
determined by the number of times its subscript has been chosen in
making up p subscripts. The degree in A r is the same as that in Br.
We thus see that each group of terms formed by the above rule is
made up of terms essentially unlike the terms of all the other groups,
each group, in other words, constitutes a distinct polynomial in the
scalar elements of certain of the given matrices. Within the group,
however, various terms may be like one another, and simplifications
may occur, particularly when special values are assigned to some of
the matrices ; for example, when some of them are allowed to become
the idemfactor.
19. The Equation of Extent Unity and Order Two.
As an example, closely allied to that of Art. 10, let us take all given
matrices to be of the second order, and, to begin with, let h = 1, so
that the linear matrix equation reduces to the simple form
AxB = C (126)
and can be at once solved as x = A~]CB~\ Forming the mp by our
rule we have mi = AsBs which is (an + 022) (&11 + 622)- Next
2m2 = A2sB*s - (A*)s(B*)s.
The scalar (A2)s may be expressed in terms of simpler quantities by
using the Hamilton-Cayley equation for the matrix A. Let the
coefficients in this equation be As and A"s so that
A2 = ASA - A" SI (127)
where / is the identical matrix of the second order. Taking the scalar
of both sides of the equation we have
(A*)s = Ah - 2A"S (128)
and a similar equation for (B-)s- Substituting in the expression for
2m2 we find on simplifying
392 HITCHCOCK.
m, = AhB"S - 2A"sB"s + A"sBh (129)
Since the matrices are of the second order, A" s and B"s are the de-
terminants of their respective matrices; hence on substituting,
w.2 = (an+ o22)2 (611622 — &12621) — 2(an022 — 012021) (611622 — 6)2621)
+ («ll022 — 0]202l) (6ll + 622)2
= (a2n+ a222) (611622- 612621) + (62n+ 6V) {ana-22 - anan)
-\- 2011022611622 — 2012021612621 (130)
Again
6m3 = ^3sB3s - 3^SB5(.42)S(52)S + 2(A*)S(B3)S (131)
by the rule. But A3 = As A2 - A"SA hence
(A*)s = Ass - 3ASA"S (132)
by (128), and similarly for B. Substituting values in (131) from (128)
and (132) to get rid of scalars of powers of matrices we find
m3 = AsBsA"sB"s (133)
= (On -f" 022) (611 + 622) («lia22 — ai202l) (611622 — 612621)
and finally 24m4 = A*SB*S - 6A*SB*S(A>)S(B% + 3U2)25(52)2S
+ SAsBs(As)s(B*)s - Q(A*)s(B*)s (134)
But A4 = /1^43 - A"SA- hence
(^4)g = ^ _ iA-sA"s + 2.4'V (135)
so that by eliminating the scalars of powers from (134)
mi = A"s>B"s- (140)
Now in the present example d-C = A-CB\ 03C = ASCB\ etc. Hence
x = O-'C = 4„lB„2[AsBsA"8B"8C - (A*bB"b- 2A"8B"s
S + A"sB-s)ACB + AsBsA-CB- - ASCB*] (141)
Simplifying by the aid of the Hamilton-Cayley equations for A and B,
—^—rlAsBsC - BSAC - ASBC + ACB]
A SB s
.v =
= -TT^r^sI - A)C[BSI -B} = A-*CB~\
A sB s
checking with the known solution.
on double polyadics — the linear matrix equation. 393
20. The Equation of Extent Two and Order Two.
The following considerations will suggest how the work of solution
may be arranged when the equation is of higher extent. We have
seen that any coefficient mp is a sum of groups of terms. Each group
is a homogeneous polynomial. Suppose a choice of subscripts in
which i occurs a times, j occurs b times, etc. Let the development of
terms, under the rule, corresponding to this choice of subscripts, be
denoted by G(iajb- • ■). Each choice of subscripts will occur a number
of times equal to the coefficient of the corresponding term in the
expansion of
(i + j+k+---+r + s+ t)*>. (142)
Thus the entire development may be systematically carried out.
For example, take
AixBi+ A1xBj=C (143)
and form m±. With the above notation we shall have
m4 = G^) + iG(Pj) + 6G(W + 46'(y3) + G(j») (144)
By the preceding example we have G(z4) = A"fsB"h and similarly
for G'O'4). By the rule we have
G(Pj) = AsiSA3sB\sB]S - SAWAiAda&isiBiBds
- 3AiSAjS(A%)sBlSBjS(B\) + 3(^)5(^)5(^)5(^)3
+ QAisiA^A^sBisiB^B^s + 2AjS(A\)sBjS(B\)s
- 6 {A*iAi)8(B*&,)a (145)
Brevity will be gained in notation, while nothing is lost in explicitness,
if, in such expressions, we indicate only the matrices A which are
pref actors in the original equation, remembering that the order of
subscripts among postfactors, when more than two matrices are
multiplied, is the reverse of that for prefactors. With this under-
standing we may also omit the letter A and the subscript S. Thus
(145) may be abbreviated
G(Pj) = t^-3^(v)-3ii(^)+3(?) (ij)+Wi2J)+2J(i3)-GttsJ) (145a)
As a check, if Aj and B, are replaced by the idemf actor of the second
order, G(i3j) should reduce to 6AsBsA"sB"s because w4 becomes the
determinant of A( )B + ( ), and the terms of the third degree in A
394 HITCHCOCK.
must be the coefficient m3 for the example of the last article. In fact
we then have, remembering Is= 2,
G(i3j) = 4AssBss-3A*sB3s-12As(A2)sBs(B*)s+3As(A>)sBs(B>)s
+QAs(A>)sBs(B%+8(A*)s(B3)s-e>(A*)s(B*)s
which is the same as (131).
Developments like (145) are the same in form no matter what the
order of the matrices involved. They may in general be simplified
by the use of the Hamilton-Cayley equations for A and B.
G(ijz) may be obtained from G(i3j) by interchange of subscripts.
For the middle term of (144) we have
G(»V) = i2f~ i2(f) ~ f(i2) ~ 4{/(y) + (i2) (j2) + 2(y)2+ 4i(ij2)
+ AjiPJ) - 2(ijij) - 4(i2j2) (146)
which by letting Aj and Bj be / of the second order should become
2A2sB2s ~ 2(A2)s(B2)s as is easily verified.
In simplifying, use is to be made of the identity
{AiA1)s= AiSAjS-A*iAj (147)
where the last term is the star product of A i and Aj as already defined.
Since this scalar is an invariant of the two matrices it may well be
abbreviated A*^. Collecting and reducing results we find
vU = A"\sB"\s+ A"iSB"iSA*iiB*ii+ A" iSA" jSB*2 ,-,-
+ ffWiaA** ~ 2A"lSA"lSB"iSB"jS + A" iSB"jSA*nB*i}
+ A"2jSB"2jS (148)
and by similar processes,
m3 = AisA"isBisB"iS+ AiSBjSB"isA*ii+ AiSBisA,,i8B*ii
— A jsB jsA" isB" is + AjsBjsA" jsB"js + AjsBisB"jsA*ns
+ AiSB3SA"isB*ij - AiSBiSA"jSB"jS (149)
ma= A2iSB"iS - 2A"iSB"iS+ A"iSB2iS + AigAjsB*^ - A*^*^
+ BiaBiaA+a + A2]SB"lS - 2A"jSB"]S + A"]SB2]S (150)
hji= AiSBis+ A]SBjs (151)
whence (143) is completely solved.
ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 395
1 This symbolic method is due to H. B. Phillips: Some invariants and co-
variants of ternary collineations," American Journal of Mathematics, 36, 1914.
2 Hamilton, Lectures on Quaternions, 1853, Cayley, A Memoir on the theory
of matrices, 1858.
3 E. B. Wilson, On the theory of double products and strains in hyperspace.
Conn. Acad. Trans. 14, 1908.
4 C. L. E. Moore and H. B. Phillips, The dyadics which occur in a point-space
of three dimensions, Proc. Amer. Acad, of Arts and Sci. 63, 1918.
5 For the elementary theory, Bocher's Algebra may be consulted.
6 See note 3.
7 For the laws of p-way determinants see Amer. Journal of Math. 40, 1918,
by Lepine Hall Rice.
8 Compare Joly's Appendix to Hamilton's Elements of Quaternions.
9 This result is an extension of Hamilton's invariant property of his coeffi-
cients.
10 Elements of Quaternions, 2nd Ed., Vol. I, Art. 348.
11 In a series of papers over many years. They are all, I think, listed in the
bulletins of the Quaternion Association. See, in particular, Wien. Ber. 112,
1903, pp. 645, 1091, and 1533.
12 The term "extent" is due to Sylvester. I have elsewhere given a sketch
of the present method in its relation to the work of Sylvester. Proc. Nat.
Acad, of Sci. 8, April, 1922.
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Vol. 58. No. 11.— May, 1923.
IDENTITIES SATISFIED BY ALGEBRAIC POINT
FUNCTIONS IN N-SPACE.
By Frank L. Hitchcock.
LIBRARY
NEW YORK
BOTANICAL
CiAKl>€N
IDENTITIES SATISFIED BY ALGEBRAIC POINT
FUNCTIONS IN N-SPACE.
By Frank L. Hitchcock.
TABLE OF CONTENTS.
Page.
1. The fundamental identity 399
2. Geometric meaning of the coefficients 402
3. The coefficients as eliminants of K-adics 402
4. The method of standard sets 403
5. The method of factoring . 407
6. Application to determinants whose elements are linear polynomials 412
7. Rule for constructing these determinants 417
8. The method of reduplication 421
1. The Fundamental Identity.
Let F(x) denote a homogeneous polynomial of degree K in N vari-
ables X\,x 2, K > 1).
If N = 3, A' = 2, we have n = 6, q = 2, Q = 3. If either AT or K be
larger, we have Q > q -\- 1 .
We may now prove that, regarding a relations of form (35) as
equations satisfied by a particular selection of g 4- 1 of the C's, not
more than q of these equations can be independent. For consider
the m + AT — 2 points ag+2, ag+3)- • -a„. These are the a's which
410 HITCHCOCK.
correspond to the terms absent from (35). Let E\, Ei, ••■Eq+\hz
polynomials of degree K — 1 which vanish, respectively, at q + 1
different selected groups of m points chosen from the above m + AT — 2
points. Let bi, b2, • • -bg+i denote those linear vector functions of x
of form (34) into which, in each case, enter the a's not occurring in
the corresponding E. Corresponding to each choice of E and corre-
sponding b we may write an identity of the form (35). Thus
bi*aiEi(ai)Ci + • • • + bi'a,+i£i(ag+i)C9+i = 0
b2-a1E2(a1)C71 + \- b2«a9+1E2(aa+i)Cg+1 = 0 (37)
bg+i,ai/ig+i(ai)Ci-r • • • -f- bg+i,ag+i£g+i(ag+i)(yg+i — 0
We may regard these as q + 1 linear equations satisfied by the q + 1
C's which enter. We know that these C's are actual polynomials of
degree K in x. Hence the equations cannot be independent, as was
to be demonstrated.
On the other hand we may in general select q such equations which
shall be independent. For consider the coefficients of Cy in the suc-
cessive equations: they are of the form b^a/F^ay), and may be
thought of as polynomials of degree K in a;. They all vanish if ay
be made to coincide with any one of the points ag+2- • a„ or x, by
hypothesis, that is at m + N — 1 points. In general n polynomials
may be linearly independent. Hence of these polynomials n —
(m + AT — 1) may be linearly independent. But this number is q,
as was to be shown. A specific rule for selecting the q equations will
be given below.
Corollary to Theorem IV. The following corollary to theorem
IV will be essential in the applications made below. We suppose
given a standard set of polynomials P](x),- • -P„(x) based on a set
of vectors ai, • • -an as already defined. We now adjoin AT — 3 other
vectors which need not be distinct from the others, and which we may
call an+i • • -etc.; and write as in (342)
bi = [xa/,.a;- • •] (37^)
where the A' — 2 vectors a*, a;, • • • which occur in b; are precisely
those a's (out of the total number of n -f- AT — 3), which do not occur
in P^ It is evident that bi«y will be a linear polynomial in x. Write
Li(x) = bi-fr (37B)
where y is an arbitrary vector. The corollary may now be stated:
ALGEBRAIC POINT FUNCTIONS IN N -SPACE. 411
Of the polynomials of a standard set, not fewer than n — m — t can
be connected by a relation of the form
2c