X2>./?,2 PROCEEDINGS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. Vol. LVIII. FROM MAY 1922, TO MAY 1923. BOSTON: PUBLISHED BY THE ACADEMY. 1923. The Cosmos Press, Inc. CAMBRIDGE, MASS. CONTENTS. Page. I. Vision and the Technique of Art. By A. Ames, Jr., C. A. Proctor and Blanche Ames 1 II. The Boundary Problems and Developments Associated with a System of Ordinary Linear Differential Equations of the First Order. By George D. Birkhoff and Rudolph E. Langer . 49 III. Lichenes in Insula Trinidad a Prof essoreR. Thaxter Collecti. By Edward A. Vainio 129 IV. The Effect of Pressure on the Electrical Resistance of Cobalt, Alumi- num, Nickel, Uranium and Caesium. By P. W. Bridgman . 149 V. The Compressibility of Thirty Metals as a Function of Pressure and Temperature. By P. W. Bridgman 163 VI. A Revision of the Atomic Weight of Silicon. The Analysis of Sili- con Tetrachloride and Tetrabromide. By Gregory P. Baxter, Philip F. Weatherill and Edward W. Scripture, Jr. . 243 VII. The Chilean Species of Metzgeria. By Alexander W. Evans . 269 VIII. Some New Fossil Parasitic H ymenoptera from Baltic Amber. By Charles T. Brues 325 IX. Text of the Charter of the Academie Royale de Bslgique, Translated from the Original in the Archives of the Academie at Brussels. By A. E. Kennelly 347 X. On Double Polyadics, with Application to the Linear Matrix Equa- tion. By Frank L. Hitchcock 353 XI. Identities Satisfied by Algebraic Point Functions in N-Space. By Frank L. Hitchcock 397 XII. The Minimum Audible Intensity of Sound. By Clifford M. Swan . . 423 XIII. The Salamanders of the Family Hynobiidae. By Emmett Reid Dunn 443 XIV. An Example in Potential Theory. By O. D. Kellogg . . . 525 XV. The Typical Shape of Polyhedral Cells in Vegetable Parenchyma and the Restoration of that Shape following Cell Division. By Frederic T. Lewis 535 XVI. The Effect of Pressure upon Optical Absorption. By Frances G. Wick 555 IV CONTENTS. Page. XVII. Records of Meetings 575 Biographical Notices 597 Officers and Committees for 1923-24 617 List of Fellows and Foreign Honorary Members 61S Statutes and Standing Votes 639 Rtjmford Premium 655 Index 657 58-1 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 1.— February, 1923. VISION AND THE TECHNIQUE OF ART. By A. Ames, Jr., C. A. Proctor and Blanche Ames. Investigations on Light and Heat, published with aid from the Rumfobd Fund. (Continued from- page 3 of cover.) VOLUME 58. 1. Ames, A. Jr., Proctor, C. A., and Ames, Blanche. — Vision and the Technique of Art. pp. 1-47. 2S pis. February, 1923. S3.75. 2. Birkhoff, George D. and Langer, Rudolph E. — The Boundary Problems Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49- 128. In press. 3. Vainio, Edward A. — Lichenes in Insula Trinidad a Professore B. Thaxter Collecti. pp. 129-147. January. 1923. $1.00. 4. Bridgman, P. W. — The Effect of Pressure on the Electrical Besistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. $.75. 5. Bridgman, P. W. — The Compressibility of Thirty Metals as a Function of Pressure and Temperature, pp. 163-242. January, 1923. $1.70. 6. Baxter, Gregory P.. AVeatherh-l, Philip F. and Scripture. Edward W., Jr. — A Revision of the Atomic Weight of Silicon. The Analysis of Silicon Tetrachloride and Tetrabromide. pp. 243-26S. February. 1923. $.75. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 1.— February, 1923 VISION AND THE TECHNIQUE OF ART. By A. Ames, Jr., C. A. Proctor and Blanche Ames. Investigations on Ligbt and Heat, published with aid from the Rumford Fund. LMMtARY NEW VllftK BOTANICAL VISION AND THE TECHNIQUE OF ART. By A. Ames, Jr., C. A. Proctor and Blanche Ames. Presented by Louis Bell. Received Jan. 20, 1922. Presented Jan. 11, 1922. INTRODUCTORY. There are many well known instances where very successful use has been made by artists of certain of the laws of vision as a basis of technique. The "pointillist" technique of Pissaro and Monet is probably the best example. The artist Birge Harrison l has gone farthest towards recognizing the dependency of the technique of art on the laws of vision. He most forcefully and lucidly shows that a picture in its general form should be similar to our retinal impressions. Mr. Ames and his sister, Blanche Ames, who were painting together came to a similar con- viction in 1912. Attempts were made to paint pictures of this nature. The difficulty of analyzing the character of images of objects upon which the eye was not focused was at once encountered. This is without doubt due to the universal and probably immemorial human practice of looking directly at, i.e., focusing upon anything we desire to judge or analyze. Mr. Ames therefore undertook to determine scientifically the characteristics of the images of those objects upon which the eye is not focused in the belief that an intellectual concep- tion of the characteristics of such images would help in the visual recognition and analysis of them, and thus be an aid in the technique of art. He thought that the desired information could be obtained in a few weeks — at most a few months. The scientific data upon which such information is based, however, had not been worked out. This necessitated research work upon which he has been occupied up to °^the present. The data collected, although representing a consider- able advance, constitute hardly more than the preliminary steps towards definitely determining the characteristics of the retinal 1 "Landscape Painting" by Birge Harrison. Charles Scribner's Sons. 4 AMES, PROCTOR AND AMES. picture. Most of the data has been published in a paper by Mr. Ames and Dr. C. A. Proctor, entitled "Dioptrics of the Eye." 2 It is the purpose of the present paper to convey as clear an under- standing as possible of the nature of the retinal picture and to point out conclusions to which such an understanding leads us. PREFACE. A consideration of the analogy between our eye and a photographic camera is helpful for a general understanding of the subject. Broadly speaking the eye is like a camera, or more truly speaking cameras were made like the eye, the lens of the camera corresponding to the lens of the eye and the plate or film to the retina. As the character of a photograph depends upon the kind of lens and plate used, so the characteristics of our retinal picture depend first upon the nature of the lens of the eye and second upon the nature and sensi- tivity of the nerve structure, i.e., the retina upon which the image is formed. Certain photographic lenses are corrected.3 The detail in photo- graphs taken with such lenses is clear and distinct over the entire picture and free from distortion. Other photographic lenses which are not corrected produce pictures in which the detail is indistinct and distorted in varying degrees. The lens system of the eye is not corrected. The details in part of the image formed by it are clear, in other parts unclear and all more or less distorted. Furthermore, the retina, instead of being equally sensitive over its entire area as is a photographic plate, varies in sensitiveness in differ- ent parts. As a result of the effects of the lens system of the eye and the effects of the variable sensitivity of the retina the retinal picture has charac- teristics which make it markedly different from photographs taken 2 Journal Opt. Soc. of Amer., Vol. V, Jan. 1921. 3 The image of a scene formed bv a simple lens (called uncorrected), such as a spectacle lens or a magnifying glass, is not an exact reproduction of the scene itself. The detail of objects at the center of the picture is slightly softened and has chromatic edges. The detail at the sides of the picture is still more softened and objects at the side are bent and distorted. To make the detail at the center as well as at the sides perfectly sharp and to do away with distortion, so called corrected lenses were devised. Ihey consist of a combination of two or more simple lenses of determined character and separation. VISION AND THE TECHNIQUE OF ART. 5 with a corrected lens or even with an uncorrected lens. For though the eye has an uncorrected lens, the lack of corrections, so to speak, is markedly different from that existing in any known uncorrected photographic lens, and so produces a different effect. To give an understanding of the more specific characteristics of the retinal picture it is necessary to take up and describe the character- istics of the images formed of objects in different parts of the field of view, i.e., the scene at which we are looking. We shall take up the images of objects in the various parts of the field of view in the following order: First, the images of objects at which the eye is directly looking, or, in other words, those objects which lie in the line of vision and which are in sharp focus. See A, Figure 1. This will be covered in Chapter I, Distinct Vision. Figure 1. Diagram showing positions of various objects in the field of view. F B A C is the axes of vision or line of sight along which the eye is looking. A is the object on which the eye is focused. It will be imaged sharply on the fovea F. B and C are points on the line of sight inside and outside the point of focus. D is an object to one side of the line of sight. It will be imaged on the peripheral part of the retina at G. Second, the images of objects which are on the line of vision nearer and farther away than the object in focus (see B and C, Figure 1); Chapter II, Depth of Field Axial. Third, the images of objects which are not in the direct line of vision as D, Figure 1. These images will be described in Chapters III, IV andV. Chapter III, Depth of Field (Lateral), will deal with the character- istic imaging of objects not in the direct line of vision and at different distances from the eye. See D and H, Figure 1. Chapter IV, Distortion in Form, will deal with the distortion of the images of objects not in the direct line of vision. Chapter V, Peripheral Color Sensitivity, will deal with the change in the local color of images of objects not in the direct line of vision. In Chapter VI the effect of binocular vision will be considered in a general way. In Chapter VII the results will be summarized and discussed. 6 AMES, PROCTOR AND AMES. Chapter I. DISTINCT VISION. Distinct vision deals with the nature of retinal images of objects at which the eye is looking directly, i.e., that are in sharp focus. As we have a very definite conception of the appearance of objects at which we look directly it may hardly seem necessary to analyze the char- acteristics of the images of such objects. It is believed however that such an analysis will not only be instructive but will make it easier to understand the characteristics of the images of objects upon which the eye is not focused, which will be taken up later. Owing to the nature of the lens system of the eye the image on the retina formed by an object at which the eye is directly looking is not an exact copy of the object itself. The image is spread out and some- what diffused. This is due primarily to three factors; chromatic aber- ration; spherical aberration; and irregular astigmatism. These factors and their effects will be considered in the order given. CHROMATIC ABERRATION. Chromatic aberration causes light of different wave length or color to come to focus at different distances behind the lens, light of shorter wave length, i.e., towards the blue end of the spectrum focusing nearer the lens. This is shown in Figure 2. Figure 2. Diagram showing chromatic aberration of the eye. If we have three point sources of light at A, one red, one yellow, and one blue, the image of the blue source will be formed at (b) the image of the yellow source at (y), while the red image will be at (r). Figure 3 shows the shape of image bundles as formed in Mr. Ames' eye by point sources of red, yellow and blue light. These of course are much enlarged ; the lens system of the eye is to the left. It will be seen that the blue bundle lies to the left or nearer the lens VISION AND THE TECHNIQUE OF ART. 7 than the yellow bundle and that the yellow bundle lies to the left of the red. The eye normally focuses for yellow light, i.e., so that the small cross section of the yellow bundle falls on the retina. If the eye looks at red, yellow, and blue point sources at the same time the retina would be in the position relative to the three bundles as shown in Figure 3. It will be noted that the smallest cross section of the blue bundle lies in front of the retina, that of the red behind. St.l. Axi. Red A«» Ye Figure 3. Image bundles formed by light from red, yellow, and blue point sources. Their displacement right and left shows the Chromatic Aberration of the eye. The different distances from the retina at which the individual rays, in each bundle, cut the axis shows the Spherical Aberration. It will also be noted that where the red and blue bundles cut the retina they are much larger in diameter than the yellow bundle. This is also shown in Figure 4 which is a photograph of the magnified image of a point source of white light taken with a lens which has approxi- mately the same chromatic aberration as the eye. A white light source is composed of light of all wave lengths. The camera was focused to 8 AMES, PROCTOR AND AMES. get as sharp an image as possible of the yellow light and then the three pictures were taken, the top one through a red screen, the middle one through a yellow,4 the lower one through a blue. The top photo- graph therefore shows the image formed by the red rays in the white light source, the middle one that formed by the yellow rays, and the bottom one that formed by the blue rays. Colored images of the same relative difference in size are apparent to the eye looking at a point source through monochromatic screens or filters if the eye is kept focused for yellow. Without any screens or niters the eye receives an image of the nature obtained by combining the three images in Figure 4. It would have a bright center tending towards yellow in color, surrounded by larger and larger rings of shorter and longer wave lengths, the blue rings extending out farther than the red. A comparison of the image formed by the eye and that formed by a lens corrected for chromatic aberration is of interest. Such a lens is designed so that light of all wave lengths focuses at the same distance from the lens. This is shown in Figure 5 where it will be seen that the narrowest part of the bundles for light of different wave length, in- stead of being focused at different distances from the lens, all focus at the same distance. Figure 6 is a photograph of a point source of white light taken with a corrected lens. The photographs were made in the same way as those in Figure 4. It will be noticed that where the images taken through the red, yellow and blue screens with the un- corrected lens are all of different size similar images formed by a cor- rected lens are substantially all the same size. That is, this correction causes the images of an object in focus to be much sharper or clearer than those formed in the eye. We have been speaking so far only of point source objects. If the object is either a line or an edge, for instance a dark edge of a window against a light sky, the diffusion circles we have been describing take the form of diffusion edges. The distribution of color in these diffu- sion edges follows the same laws as govern that in the diffusion circles. This is shown clearly in Figure 7, which is a photograph taken through a lens having approximately the same chromatic aberration as the eye, 4 As the filters actually used in taking this and following pictures were East- man Kodak Co. films Red No. 25, Green No. 58 and Blue No. 48, it would be more exact to use the word "green" instead of "yellow." The difference in wave length between the green and yellow however is such that there would be no appreciable differences in the appearance of the photographs whether a green or yellow filter was used. The term yellow will therefore be used for sake of simplicity and clearness. RED. YELLOW. BLUE. RED. YELLOW. BLUE. Fig. 4. Fig. b. A.,> Red A.i. Yello A.,, "BL Figure 4. Three color photograph of a white light point source taken with lens having same chromatic aberration as the eye. Magnification 110 diame- ters. Figure 5. Image bundles which would be formed by light from red, yellow and blue point sources passing through a lens perfectly corrected for chromatic and spherical aberration. Figure 6. Three color photograph of a white light point source with a lens corrected for chromatic aberration. Magnification 110 diameters. Ill l> YELLOW BLUE Fig. Fig. 8. Figure 7. Three color photograph of black and white wedges taken with lens having the same chromatic aberration as the eye. The base of the wedges which were about six feet distant, measured \ inch. Figure 8. Same as Figure 7 taken with a lens corrected for chromatic aberration. VISION AND THE TECHNIQUE OF ART. 9 of a white wedge on a black background and a black wedge on a white background. The base of the wedges was one-fourth inch and their distance six feet from the camera. The top picture was taken through a red, the middle through a yellow, and the bottom one through a blue filter. The camera was focused to give a sharp image with the yellow filter. The top picture shows the kind of image that is formed on the retina by the red light, the middle one the kind of image that is formed by the yellow light and the bottom one the kind of image that is formed by the blue light. If you imagine these superimposed, which is what takes place on the retina, the combined picture will have a blue diffused edge extending over the black and a less wide red edge. The color of the white near the edges will be slightly yellowish due to the subtraction of the blue and red. With a lens free from chromatic aberration no such effect is pro- duced. This is shown by Figure 8, which is a photograph of the same objects taken at the same distance and in the same way, with a cor- rected lens. The images of all the wedges in this case are sharp and clear and of the same size. They will all exactly superimpose and no chromatic edges will be formed. Under ordinary circumstances unless the attention is especially called to them these chromatic rings and edges formed in the eye are not seen. This is due, it is believed, partly to the fact that the red rings overlie the green, which being complementary colors form white light, and to the fact that the blue is so spread out that it is relatively weak. However, if one looks carefully for these rings or edges they can be seen around an arc light at night, in the blue haze or halo on the dark background. The red and blue chromatic circles or edges can be seen separately by looking at a dark object, such as a window sash against a bright sky at a distance of three to six feet and shutting off the light from half the pupil by passing a card or piece of paper close to the eye, the edge of the paper being kept parallel to the window sash. One side of the frame will have a red orange edge, the other a bluish edge. If the card is brought in from the other direction the color of the edges will reverse. Without the card the colors overlie each other and become much less visible for the reason given above. However, once the phenomenon has been noticed a soft floating purplish edge becomes apparent even without the card. A very striking example of chromatic aberration, and one which gives a very good idea of its magnitude, is apparent when one looks 10 AMES, PROCTOR AND AMES. at the purple railroad switch lamps used, for example, by the Boston and Albany Railroad. At close range these lamps appear purple, but as one moves away the light appears to have a red center sur- rounded by a blue disk or halo; the farther off one goes the larger the blue halo appears. SPHERICAL ABERRATION. Another factor that has a bearing on the nature of the retinal image of an object on which the eye is focused is the spherical aberration mentioned above. In a lens free from spherical aberration the rays that come through different zones of the lens, for example those that come through the lens near its center and near its edge, all focus down to a point. This is shown in Figure 5 where rays near the outside of the cone and those near its center all go through one point at the apex of the cone. In a lens which has spherical aberration this is not true. The rays from different zones of the lens do not pass through the same point. This is shown irt Figure 3, where it will be seen that ray A, for instance, crosses the axis far to the left of ray B. Due to this fact the image of a point source formed in the eye will be larger and with softer edges than that formed by a corrected lens. IRREGULAR ASTIGMATISM. There is still another factor that has a bearing on the nature of retinal images of an object on which the eye is focused. That is irregular astigmatism. This term covers all irregularities in the shape and distribution of light in the image due to such things as opaque substances or irregularities of densities in the lens system. A most marked example of this factor is the star shape appearance, known to everyone, of a small source of light. If the eye were not subject to irregular astigmatism of some sort the image of a small source, though it might be affected by chromatic and spherical aberration, would be circular. Such retinal images, however, are always star shaped. SUMMARY. The three aforementioned factors, chromatic aberration, spherical aberration, and irregular astigmatism, cause the retinal image of an object upon which the eye is focused to have a characteristic appear- ance both as to the amount of detail which is visible and in the appear- ance of all edges. VISION AND THE TECHNIQUE OF ART. 11 From the point of view of the technique of art the question arises is it necessary that these characteristics be reproduced in the depicting of an object upon which the eye is focused? The care used by the better artists to paint broadly, i.e., not to get, even in the most detailed parts of their pictures, any more detail than is apparent to them at the distance at which they stand from their model or scene and their quite common practice of softening and treat- ing their edges, is evidence that to be technically satisfactory from an artistic point of view the detail of an object on which the eye is focused should be depicted with the same characteristics with which it is imaged upon the retina. This evidence is further supported by the fact, as will be shown later, that the characteristics of the images of objects upon which the eye is not focused, must be reproduced to bring out the appearance of depth and to make the pictures pleasing. Chapter II. DEPTH OF FIELD (AXIAL). In this chapter we will deal with the nature of images of objects that are in the line of vision nearer and farther away from the observer than the object upon which the eye is focused. The images of such objects will have markedly different charac- teristics from those of objects in focus. These characteristic differences are due primarily to two factors. First, Depth of Field of the lens system; second, Chromatic Aberra- tion. Figure 9. Diagram showing diffusion of images of point source objects not in focus. DEPTH OF FIELD. For sake of brevity and simplicity a full explanation of Depth of Field will not be gone into. It is evident however that an object which lies nearer or farther from the observer than the object upon which he is focused will be imaged not on the retina but behind it or in front of it. This is shown in Figure 9. 12 AMES, PROCTOR AND AMES. The image of point source object A upon which the eye is focused is imaged on the retina at (a), that of B behind the retina at (b), that of C in front of the retina at (c) . Where the image bundle which forms the image (b) cuts the retina it is a cone of considerable size. The object B will appear as a diffusion circle. The image bundle which focuses down to the image C in front of the retina is spread out again and also appears as a diffusion circle. If instead of a point source the objects at B and C are objects with edges the edges will have the well known appearance that is seen on an object that is out of focus, i.e., a pencil held near the eye while the eye is focused on a distant object. The nearer the objects B and C approach A the smaller will be the size of these diffusion circles and the more similar all their images be- come. The magnitude of this effect depends upon the focal length of the lens system and the size of the aperture. In the eye this means, broadly speaking, the length of the eye and the size of the pupil. It is very marked where either the objects in or out of focus are close to the eye but decreases as they are moved away and becomes im- perceptible when they all are at a distance of thirty feet or more. CHROMATIC ABERRATION. As was described in Chapter I the effect of chromatic aberration is to cause light of different wave length or color to focus at different distances from the lens. See Figure 2. If we move the blue point source at A, Figure 2, towards the eye, the eye being kept focused on A, it will cause the image of the blue source to move back towards the retina. A position B, Figure 10, will be found where a blue light will focus sharply on the retina while the eye is still focused on the point at A. If we move the red light away from the eye a similar position R, Figure 10, will be found where the red light will also be in focus. Figure 10. Diagram showing how images from sources of different colors situated at different distances from the eye can all be in focus at the same time. That is all three lights will be in focus at the same time although they are at very different distances from the eye. For example if the eye is focused on a yellow light at a distance of six feet it will see Q H K O J w O O ►J J w P j ffl Q IS O J W H ' — G o3 bt s 2 Oj w «-t— i 03 ^ O 03 bO .S C3 II -Lh 03 oj "S- s" d a S1 o o -h bfl 03 .3 .fh £ c3 03 03^ o3 s_ u 5^ u U CC ■P 1) 4i i—l •gH 0J_CC ^ 03 03 .a3 fa 03 >. 0) '^3 '+3 ^ OS ^ o3 o3 g £ 2 £ 5 c3 '5 S o -» ^ a."o O w.3 03 t, « £ O O • — . -*-3 0j — — i t- Oj Oj ht po objec it sou orrect orrect .2f r- -&° « 3 3 oi « OJ o o o ^2 S .c — — 1 "*"" 03 03 a a - & o J W J 0 H J * ffl _ac . ^ ^ c r J3 Oj 0j OJ hH a »j « ^ ^ rt 03 ^ 03 03 ^ > 03 +3 -t-J b£ "" Sg 332 2 0/ Oj OJ OJ *5 — " — .2 oi 5c Sd Sc f-i Jh J-, ■— < -^H ■-- iR 0J 73 oj CQ ° . g 03 c3 03 Op ¥ .03 Q3 a; Q3 2 fr-a a a a &8S&&& +3 I— 1 >-h 52 S cn c6 "* i— 1 a a o3 a a h *"" ' GUR ratit nific GUR GUR GUR b£ s " (-. aC-.1-1 J-1 ," fail oj ej[JHfefe oaS VISION AND THE TECHNIQUE OF ART. 13 sharply at the same time a blue light about three feet away and a red light at about twenty feet away. We described in Chapter I the characteristic chromatic diffusion circles and edges of an object in focus. We saw that a white light point source as at A, Figure 10, would form an image with a yellowish center surrounded by red and blue diffusion circles. A white light point source at B, Figure 10, however, will form quite a different image. The blue in the white light being in focus will, as we have just shown, form a sharp image. The yellow will be out of focus and form a yellow diffusion circle around the blue and the red will be still more out of focus and form a larger red diffusion circle which will extend outside that of the yellow. Figure 11, is a photograph of a point source of white light taken with a lens which has approximately the same chromatic aberration as the eye. The lens wasfocused to give a sharp image of the yellow rays in a white light source about six feet away. The pictures are of a white light point source about three feet away. As in Figure 4 the top image was taken through a red, the middle one through a yellow, and the bottom one through a blue filter. The combined image which is the appearance the white light point source would have to the eye is markedly different from that shown in Figure 4. A white light point source at R, Figure 10, will form a still different kind of image. The red in the white light being in focus will form a sharp image. The yellow will be out of focus and form a yellow diffusion circle around the red and the blue being still more out of focus will form a larger blue diffusion circle. Figure 12 is a photograph similar to those described in Figure 4 and Figure 11, but with the objects twenty feet away. The images formed are very different from those shown in Figure 4 and 1 1 . Images of white light point sources situated at other distances along the axis will vary from those shown above, their characteristics de- pending on the position of the fixation point and their distance from it. A lens corrected for chromatic aberration forms very different images of similar white light point sources. Figures 13 and 14 are photographs taken with a corrected lens. The photographs were made in the same way as those shown in Figures 11 and 12 of white light point sources in similar positions. As the point in the focus of a corrected lens is in focus for all colors so a point out of focus is out of focus for all colors and to the same extent. The diffusion circles for red, yellow and blue light in Figures 13 and 14 are therefore all about the same size. The combined image although enlarged and fuzzy will 14 AMES, PROCTOR AND AMES. appear white, i.e., without colored edges, no matter in what direction or distance the white light point source is from the point in focus. As described in Chapter I if instead of a point source edges are used, as a dark object against a light background, the above described diffusions make themselves evident in the form of chromatic edges. Figure 15 shows a photograph taken with a lens, which has approxi- mately the same chromatic aberration as the eye, in the same way and of the same objects as described in Figure 7. The lens was focused as described for Figure 11 for yellow at six feet, the objects being three feet away. The combined images are marked by the orange red edges extending over the dark and the blueness of the white along its edges due to the subtraction of the red and yellow light which has diffused over the black. Figure 16 is a similar photograph, the focus of the lens remaining the same, the objects being placed at a distance of about twenty feet. In this case the combined images are characterized by the green and blue extending over the dark and the redness of the white along its edges. It is regretted extremely that these pictures cannot be reproduced in color as they not only show these characteristics much more clearly, but are very beautiful. Figure 17 is similar to Figure 15 taken with a corrected lens. As would be expected from what has been said of Figures 13 and 14 the images taken through the different filters are all diffused to the same extent. As a result the composite picture shows diffused but no colored edges. Once one's attention has been called to them, these characteristic chromatic edges in the retinal images are very easily seen. The aberration of the red rays over the dark, characteristic of images of objects which lie inside the focus, was first noticed in the warm color of black specks on a windowpane viewed from a few feet when the eye is focused on the distant sky. All of the characteristic colored edges can be easily seen with the following arrangement. Against the white wall of a room or a piece of cardboard or sheeting put up at the distance of about twenty feet a black object, preferably one that comes down to fine black points, as iron grill work, black wedges of paper will do. At six feet distance put up any small object to focus on in line with the distant objects. At three feet distance in the same line put up another black object preferably like the first. If the eye is kept focused on the object at six feet the dark edges of the near object appear to have a reddish orange tinge next to which the light appears colder or bluer. The edges of the object at twenty feet, in fact the whole surface if the a a o a a j a i— i 60 a o a P j i—i 60 a a 0 a j a {3 J a 60 o x • a a 03 .a a -i« +3 O e3 60 « g . xi ?-a 03 8* - 03 03 SJ S O 60 a ° «) °x & «a;%- £:- - 03^03 o3 60 a ■ — i > 03 XI .X £ ° © exits 03 +3 - - . V X 0) ■+3 Q; •r1 s- SX § g -£ a- 48 £ 03 t- t- o X c3 >-2 03^ 03 a o -t3 £ 60 « .ax o 0) >,£ o3 03 03 X is o 03 6C§ -© — O 55 > 03 03 O ^ ^ — «- ^j X -^ M o8.t3 m_.2X > o K -^ _ -i-5 -^ a 6o a o 5 03 ^ X - « «5 — 033 !-_, 60.60 2 to 03 03 03 03 O O s s = 03 W W ^- o3 03 03 E-i jj-gqoq 1- a 'X o £ «'££ 0 a a p p Fig. 17a — Xo. 1. Fig. 17a — No. 2. VISION AND THE TECHNIQUE OF ART. 15 dark objects are narrow enough, appear very bluish while the lights next to the dark objects appear pale orange. Of course if the focus of the eye is not kept fixed on the central object these appearances will not be visible for they only exist on objects that are not in focus. It may take some practice to keep from changing the focus as the natural tendency is to focus upon the object which one is trying to analyze. The colored edges on both far and near objects are best seen when the objects are at the relative distances above described. Red edges on the dark become more evident with a more distant fixation and blue on the dark with a nearer one. Where both the fixation and the nearer object are twenty feet or more away both objects are so nearly in the same focus that the difference in colored edges is hard to distinguish. These facts lead naturally to the assumption that with a given fixa- tion the characteristic colored edges of objects nearer and farther than the fixation object inform us of their relative distance, that is, if an object has red edges we judge it to be nearer to us than the fixation object while if it has blue objects we judge it to be farther away. If this is so objects depicted in a picture with red edges should appear nearer than those with blue edges. This is exactly what was found to be the case as is shown by Figure 17a. It will be seen that the circles with the reddish edges appear to be on a nearer plane than those with the blue edges.5 SUMMARY. The effect of depth of focus which produces the fuzzy edges of objects nearer or farther than the object plane has been long recognized in photography. This is shown in the photographs in Figures 18 and 19. That in Figure 18 was taken with a small aperture, F 16, which causes objects at all distances to be imaged sharply, while that in Figure 19 was taken with a large aperture, F 4.5, which causes objects nearer and farther than the focus point to the imaged with a greater softening of edge. The greater effect of depth in the photograph in Figure 19 as compared with that in Figure 18 is very evident. Gleichen 6 goes into the subject at length and points out that a 5 The illusion is more apparent if one eye is closed. This prevents the functioning of our binocular depth perception by which we tend to recognize the true distance of both figures. 6 "Die Grundgesetze der naturgetreuen photographischen Abbildung." Halle 1910. "Uber Helligkeit und Tiefe inbesondere bei der naturgetreuen Photographischen Abbildung." Zeit, fur Wiss. Phot, Vol. 9, 1911, p. 241. 16 AMES, PROCTOR AND AMES. natural effect of depth can only be produced by a lens which has the same depth of field as the eye. He does not, however, deal with the effects of chromatic aberration. The depth effect due to it is believed to be much more marked than that due to simple depth of field. In simple depth of field the diffusion of edge due to an object being out of focus gives no indication as to whether the object is nearer or farther than the focus point. Chromatic aberration on the other hand causes objects beyond the focus point to be imaged in a characteris- tically different way from those inside. This in turn gives a real basis for monocular depth perception apart from the relative sizes of objects or disappearing or other perspective. As shown by Figure 17a this effect of depth can be obtained in a picture if the objects are de- picted with their characteristic chromatic edges. A marked effect of depth has thus been obtained in pictures painted by Blanche Ames. Many paintings by great masters have been looked at to find whether this characteristic edging has been made use of. The treating of edges as stated before is quite common, and in certain paintings, by Millet for instance, warm or reddish edges are found around near objects while objects in the distance are colder and bluer. It is not felt, however, that the evidence is sufficient to conclude that it has been used consistently. Chapter III. DEPTH OF FIELD (LATERAL). In this chapter we will deal with the characteristics of images of objects which are situated not on the line of vision. The characteristic form of such images is due primarily to an aberration called oblique astigmatism. (This should not be confused with corneal astigmatism.) Oblique astigmatism causes the rays of light, from a point source not on the axis, to focus into a ray bundle of complex form. As ex- plained above the ray bundle from a point source on the axis focuses in the form of a cone to a point where the rays cross to spread out into another cone. The ray bundle from a point source not on the axis forms a more complicated figure. In its pure form in a simple un- corrected lens it focuses first to a line, see aa, Figure 22, which lies in a position tangential to a circle about the axis of the lens. It then crosses and narrows in its long dimension and lengthens in its short Figure 18. Photograph taken with a small aperture, i.e. F 1G. Figure 19. Same view as in Figure 18 taken with a large aperture, i.e. F 4.5. VISION AND THE TECHNIQUE OF ART. 17 one until it becomes a line again, see bb, Figure 22, which is perpendicu- lar to the first line. The image of every point source not on the axis has this peculiar form. The farther the source from the axis the greater the separation between the two parts of the image which have the form of lines. If a sensitive plate or ground glass screen is placed behind the lens the form of the image that is apparent depends upon the position of the screen. If it is placed far back, i.e., to the left of bb, Figure 22, the image will be in the form of a radial oval, i.e., radial to a circle about the axis of vision, in the horizontal plane this would be horizontal; if at bb, in the form of a radial line; if half way between aa and bb, the form of a circle; if at aa, the form of a tangential line, i.e., tangen- tial a circle about to the axis of vision, in the horizontal plan this would be vertical; if still nearer the lens, in the form of a tangential oval. If the screen is held stationary relative to the lens, a similar Figure 22. Diagram showing characteristic shape of the image bundle of a point source not on the axis. change in form of image can be noted by moving the point source from a distance to a position near the lens, always keeping it at the same angular obliquity. If, instead of a point, a line source is used, a similar imaging takes place. Every point on the line source is stretched tangentially or radially, depending upon the position of the line source. It can be seen that if the stretching of the various parts of the line source is along the length of the line source itself, the image of the line will appear perfectly sharp and slightly elongated. That is, if a line source is tan- gential to the axis, its image will be sharp when the source is so posi- tioned that the part of the image that forms a tangential line, i.e., aa, Figure 22, falls on the screen. If it is in a position radial to the axis, its image will be sharp when that part of its image that forms a radial line falls on the screen. 18 AMES, PROCTOR AND AMES. In all simple lenses this characteristic image formation is more or less confused by coma, a one-sided blur, and by chromatic aberrations. The magnitude of these aberrations in the eye has not yet been measured. This may all be made a little clearer by a brief description of how Mr. Ames measured the oblique astigmatism in his own eye. A point in space is fixated with one eye, i.e., by focusing on a point of light, the line of vision and the focus of the eye is not allowed to vary. At one side of the line of vision (at an angle of obliquity of five degrees), three narrow tangential lines (in this case vertical) of yellow light were moved back and forth. It was found that when these lines were at a certain distance they could be distinguished as separate. At any other distance they could not be distinguished. The distance at which they could be distinguished was such as to cause that part of the image designated as aa, Figure 22, to fall on the retina which made the tan- gential lines appear most sharp. This point was found to be nearer than the fixation point. Similar points were found at varying degrees of obliquity from the axis where the three narrow lines appeared most sharp. In this way a surface in space was determined where yellow tangential lines were most visible. The shape of this surface which is called the primary astigmatic object field for yellow is shown in plan on Figure 23 by the solid lines extending from the point marked "fixation point" back towards the eye. In the same way a surface in space was determined where yellow radial lines were most visible. This is called the secondary astigmatic object field for yellow and is shown in plan by the dotted lines extended in shape of a ram's horns from the point marked "fixation point" outward. Figure 23. Corresponding fields for red, blue and green monochromatic light were found and are shown in Figure 23. If this fixation point is changed although these primary and second- ary astigmatic object fields keep their general shape they shift forward and backward. Heinrich 7 made a similar experiment with a black thread. He moved a black vertical thread (as he worked in a horizontal plane this would be a tangential line) which was placed to one side of the line of vision, back and forward until it appeared most like the thread on which the eye was fixiated. He found a field similar in shape and position to that which the writer found for the primary astigmatic field for yellow. 7 British Journal of Psych., Vol. 3, p. 66. VISION AND THE TECHNIQUE OF ART. 19 £. meters III seUo~. Figure 23. Position of primary and secondary astigmatic object fields for different colors. 20 AMES, PROCTOR AND AMES. This means that with any given fixation, at a given angle of obliquity, there are certain positions in space where tangential black and white lines appear most sharp and other positions where radial black and white lines appear most sharp. It also means that with the same fixation there are still other positions where tangential lines of a particular color will appear most sharp and still other positions where radial lines of the same color appear most sharp. The positions in space therefore where tangential and radial colored lines appear most sharp depend upon their color. The different position of the primary and secondary astigmatic fields for different colors has a further effect of causing the images of black and white objects to have characteristic chromatic edges due to their position in space relative to the fixation point. This can be best shown by the following photographs taken with a lens which has approximately the same oblique astigmatism and chromatic aberration as the eye. Figure 24 (b) is of a white light point source situated at C, Figure 23, i.e., in the secondary field for yellow at an angular obliquity of about eighteen or twenty degrees. As the lens is focused at the distance marked "fixation point" this is in the plane of the focus. The top picture shows the image formed by the red rays in the white light source; the middle that formed by the yellow; the bottom one that formed by the blue. As the point source is in the secondary field for yellow the yellow light is stretched in a radial direction. Being near the primary field for red the red is beginning to be stretched in a tangential direction. And being beyond the secondary field for blue the blue is stretched in a radial direction forming a diffused radial oval. Figure 24 (a) is a photograph of a white light point source situated at B; Figure 23, i.e., in the secondary astigmatic field of red light. The red light is therefore stretched in a radial direction. The point source being beyond the secondary fields for both yellow and blue light they are imaged as diffused radial ovals, the blue more diffused than the yellow. Figure 24 (c) is of a white light point source situated at D, Figure 23, i.e., near the primary field of red and yellow and the secondary of blue, consequently we see the red and yellow stretched in a tangential direction, the blue in a radial one. The images of white light point sources in similar positions formed by a lens corrected for oblique astigmatism and chromatic aberration are quite different. Figure 24 (e) is such a photograph of a white light point source at C, Figure 23, i.e., in the plane of the focus. The lens (a) RED. YELLOW. BLUE. RED. YELLOW. BLUE. (d) Figure 24. Three color photographs showing characteristic image forms of white light point sources at different distances, (a) (b) and (c) were taken with a lens having approximately the same chromatic aberration and astigma- tism as the eye. (d) (e) and (f) were taken with the sources at the same dis- tances with a corrected lens. Magnification 28 diameters. VISION AND THE TECHNIQUE OF ART. 21 being corrected for oblique astigmatism means that the image lines aa and bb, Figure 22, are brought together to a point on the image plane. There is therefore none of the characteristic stretching in tangential and radial directions which was so evident in the oblique images formed by the other lens. The lens also being corrected for chromatic aberration there is no substantial difference in the form of the images for the different colors. They are all imaged as small spots of light of about the same size. The marked difference in imaging from that which occurs in the eye as shown in Figure 24 (b) should be noted. Figure 24 (d) and (f) are photographs made with a corrected lens of white light point sources at B and D, Figure 23. Due to the correc- tions the images do not show any stretching in a radial or tangential direction and they are all the same shape and size for different colors. The combined images will therefore appear simply as diffused white spots. The marked difference between this imaging and that shown in Figure 24 (a) and (c) should be noted. It should also be noted here, that while in the imaging of point sources by the lens having substantially the same aberrations as the eye the images have characteristic forms and colored fringes due to their distances from the lens, in the imaging by the corrected lens, although there is a diffusion which may indicate that the point source is not in the object plane, there is nothing to indicate which side of the object plane it is or how far it is from it. In order to show the characteristic chromatic edges produced in the images of black and white objects, colored photographs of a black cross on a white background and of a white cross on a black background were taken with a lens having approximately the same chromatic aberration and oblique astigmatism as the eye, and also with a corrected lens. They are shown in Figures 25, 26, 27, 28, and 29. The photographs taken with the corrected lens will be considered first as they show the exact shape of the black and white crosses. Figure 28 was taken with a corrected lens of the crosses placed at C, Figure 23. The blue, yellow and red images are seen to be all of the same size and shape, the combined image will therefore be white and black with no chromatic edges. Figure 25 was taken with a lens having approximately the same chromatic aberration and astigmatism as the eye, of the crosses at C as in Figure 28. Being beyond both the primary and secondary astigmatic fields for blue the blue is generally diffused. Being in the secondary field for yellow the yellow horizontal or radial lines, both black and white, are relatively sharp, the vertical or tangential ones, 22 AMES, PROCTOK AND AMES. diffused. Being nearer the primary than the secondary field for red, the vertical lines are sharper than the horizontal ones. The chro- matic edges that exist from the combined figures can be visualized by the amount the different colored edges extend in the different direc- tions. It is regretted again that these figures cannot be reproduced in color which shows the effects much more clearly. The marked differ- ence in characteristic edges in this figure compared with Figure 28 is however plainly evident. Figure 29 was taken with the corrected lens of the crosses placed at B, Figure 23. A photograph of the crosses placed at D is so similar that it is not shown. Figures 26 and 27 were taken with a lens having approximately the same chromatic and astigmatic aberrations as the eye of the crosses placed at B and D, Figure 23. The characteristic diffusion and accentuation of vertical and horizontal lines for the different colors is evident and a little study will show that it takes place in conformity with the position of the primary and secondary object astigmatic fields for the different colors. The marked difference in characteristic imaging between these fig- ures and Figure 29 should be noted. In all these photographs the objects were at an angular obliquity of between eighteen and twenty degrees. If the angular obliquity had been less the characteristic imaging would be different due to the difference in the relative positions of the primary and secondary astig- matic object fields for the different colors. SUMMARY. The foregoing demonstrates that the retinal image of an object in space has characteristics both as to shape and colored edges due to the object's particular position in space relative to the observer and his fixation point. In other words with a given fixation the image of a particular object in the field of view has characteristics which are pecidiar to the image of an object at its particular angular obliquity and distance. That these characteristics are of sufficient magnitude to be recog- nized is evidenced by the fact that those doing research in this line have been able to discover and measure them. The accentuation of radial and tangential lines is observable in landscape views. This accentuation can be observed in Figure 30. Hold up the page so that vou can either look by its edge at some distant object or at the small o - O J J w s a 4 CC s - S.§f sit. EO 1. — y tx-r — -■ 43 g 00 S -*i _i. = -'x 1 1 I i n -z-z - ~ So.** M - w O — ~ A o SO tD N c 03 a r .2 "is 2fJ J3 "S J § 8 1 C —- -- O cc -:_•- .e !K J O m «i S — a) £ ~ ~ — — ^ ^ ->^ fallS t-- ill ^5 N "o - - " b i s B X o '~* !- O ^4 Sh i- C t- c3 j e3 "S-c .** - ? S«o oi 5. So feS VI w 03 o3 O) o> S £ C3 03 cccc 00 cu ~ 3 g « co fat) E . CO O 03 5 | .3E — r. = & $.22 •- cv ^ S CO CO W H P P a o EE z — T. £ I - — — P VISION AND THE TECHNIQUE OF ART. 29 and shows further many instances where this has been done by great artists. He shows further that satisfactory results can be gotten by introducing what practically amounts to a " barrel distortion." It is not evident, however, that he recognizes the fundamental principles on which curved perspective is based. The writers have made what can hardly be called more than a casual search and has found distortion in the following works.: Leonardo da Vinci "Last Supper" Puvis de Chevanne "St. Genevieve" Rembrandt Numerous instances Israels "The Day before Parting" n and "The valescent" and other pictures Millet "Cliffs of Gouchy"11 Turner Numerous instances Whistler Etching Venice Scene, "The Palaces" Venice Set 1880 De Hoogh "Interior of a Dutch House" n Van Vleet "Church Interior"11 Inness "The Greenwood" Con- First The lack of its consistent use in most cases causes one to believe that those who used it did so intuitively to make the picture "look right." Israels, however, used it so consistently that he probably was conscious of the law and the same is probably true of Rembrandt. The only example of it that has been found in the works of living artists is in Sir William Orpen's painting " The Peace Conference." As only a photographic copy of this picture has been seen there is a possibility that the distortion was in the photograph. Figure 36 is a reproduction of Israels's " The Day before Parting." The distortion is most evident in the tiling in the floor though it can be seen in various other parts of the picture. One would think that the representation of straight architectural features by a curved line coming next to the straight edge of the frame of the pictures as it often does would be very noticeable. It is not however. Not that it is not perfectly evident when one's attention is called to it, but it does not attract one's attention. Its evident effect in many cases is to prevent certain parts of the picture from looking as if they were falling out. In using distortion the detail, edges, etc., of the distorted features H These pictures are at the Boston Art Museum. 30 AMES, PKOCTOR AND AMES. should be depicted approximately in the way they would be imaged on the retina. This was shown by the obtrusive unnatural appearance of a painting which Blanche Ames and Mr. Ames made in about 1912 in which they put in the approximate distortion which exists in the eye but painted all the detail as it appeared while looking directly at it. In later pictures painted by Blanche Ames in which the detail of the distorted features approximated in its characteristics the way it is imaged upon the retina the distortion ceases to be noticeable and gives a pleasing and natural effect. Another interesting effect due to distortion results from the fact that objects away from the optical axis are imaged in smaller relative size than those near the axis. This effect is very evident in Figures 31(a) and 32(a). In Figure 31(a) taken with the eye lens the statu- ette is much larger relative to the books than it is in Figure 32(a), although the statuette in both pictures is the same size. Y\ith a larger angular field this relative difference in size is still greater. In the eye where the field is between four and five times that in the figures the effect is very marked. This is probably the reason why a distant mountain appears so much larger to us when we look at it than it does in a photograph taken with a corrected lens. Chapter V. SENSITIVITY OF THE RETINA. As stated in the introduction, the character of the picture we get on the retina is determined not only by the kind of image that is formed by the lens system of the eye but also by the nature of the sensitive surface upon which the image falls, that is by the sensitivity of the retina. To gain a thorough knowledge of our retinal picture it is necessary therefore to know the sensitivity of the retina in its various parts both to light and to color. Unfortunately relatively very little is known as to the sensitivity of the retina as a whole. Considerable is known about the sensitivity of the fovea, i.e., that part of the retina which is on the axis of vision, but very little definite knowledge exists as to the sensitivity of the peri- pheral parts. A great deal of work has been done on the limits of the color fields, i.e., as to the limit of obliquity at which different colors are visible. VISION AND THE TECHNIQUE OF ART. 31 The validity of that work, however, is put in question for reasons given in the article on "Dioptrics of the Eye" 12 by Dr. Proctor and Mr. Ames. As far as is known no quantitative measurements, with one exception which will be considered later, have been made of the color sensitivity of different parts of the retina. The late Dr. J. W. Baird and Mr. Ames undertook to make such measurements at Clark University in 1913, but found that the aberrations of the lens system of the eye would first have to be determined. This led to the work described in "Dioptrics of the Eye." Mr. Ames hopes to carry out these quanti- tative measurements later. loo. /\ 1/ ' \ 9/ 9o 8o ml 7o In /Is \ °\ \\ e \ M II **\ 11 3 \ 11 ~*\ 1 * \ 6o So 4o / /$ * s \ 1 V J f I J r \ "A \ 3\ \ * ' \ * So 1 1 1 1 1 \ 1 T^ 6 8 ^o 3«. 46 ** -*e 56 fco G4-66 Figuee 37. Curves showing sensitivity of the retina in various parts to light of different color. Dr. Baird and Mr. Ames did find that blue appeared much more saturated on the periphery than it did on the fovea. This is in con- formity with measurements made by Abney.13 He measured the sensitivity of the retina to lights of different wave lengths both at the fovea and at an angle of ten degrees. His results are shown in Figure 37. It will be seen that blue appears brighter at ten degrees than at 12 Loco Set. 13 "Researches in Color Vision," p. 94, Longmans Green & Co., 1913. 32 AMES, PROCTOR AND AMES. the fovea. The amount brighter that blue light of different wave length appears is shown in Figure 38. This greater peripheral intensity of blue is probably primarily due not so much to the difference in sensitivity of the retina in the two regions as to the absorption of blue light at the fovea by the yellow spot. The yellow spot lies over the fovea, covering an angular area of about six degrees horizontally and four vertically. Its effect is to absorb light of short wave lengths, i.e., blue. The effect on our retinal picture of this difference in sensitivity is to cause those parts which are outside the yellow spot to appear more Sensitiveness oi 1he T^e-Tiwi 10° from fovea to that" oi Mellow staot near tVvc \t of \iell< 1 f ova. a.. Cooo Figure 38. Curve showing ratio of sensitivity of the retina 10° from the fovea to that at the fovea. blue. This effect was very evident to Dr. Proctor and Mr. Ames while measuring the astigmatic fields for blue light, the blue light ap- pearing many times brighter when it was a few degrees off the axis than when looked at directly. The sensation is also commonly experi- enced in the falling off of the apparent blueness of something one sees out of the side of the eye when one turns to observe it directly. To approximate this effect photographically a yellow spot of approx- imately the proper absorption and of about six degrees in angular size was put in the middle of the blue focal plane filter which with a red and green filter was used to take three color photographs. The exposure VISION AND THE TECHNIQUE OF ART. 33 for blue was then made sufficiently longer so that the color rendering would be normal in the center of the picture while on the outer parts the blue would be stronger in approximate accordance with the results found by Abney. A lens having the approximate aberrations and distortion of the eye was used. The results were very interesting. The effect was two-fold; first: to render all the colors outside of the "yellow spot" bluer; second: to make more apparent the aberrations in the images formed by oblique rays and thus cause a greater softening of the outer parts of the picture. In the photographs in which the color rendering all over the picture was the same as we get in our yellow spot the aberrations of light from the blue end of the spectrum, although they existed, were so low in intensity as to produce no effect. When, however, the intensity of the blue light was made greater these aberrations became apparent. These effects are primarily chromatic and show very poorly in black and white reproductions. The difference in the apparent aberration however is evident in the photographs shown in Figures 39a, b, and c. These were taken with a lens having the approximate aberrations of the eye. In Figure 39 (a) no filter used. In Figure 39 (b) a filter having the approximate absorption of our yellow spot covered the entire picture. In Figure 39 (c), which represents the conditions we get in our eye, a similar filter covered an angular area corresponding to that of the yellow spot. It will be seen that the whole of the picture in Figure 39 (a) is much softer than that in 39 (b). This is due to the greater amount of the aberrated blue light which struck the plate in Figure 39 (a) and which is absorbed by the filter used in taking 39 (b). In Figure 39 (c) the center portions are sharp due to the local action of the yellow filter while the outer part is soft due to its absence. It is very evident that Figure 39 (c) which approximates the conditions we get on our retina is much more pleasing 14 than Figure 39 (a) or 39 (b). Figure 39 (d) is a photograph of the same scene taken with a corrected lens. This seems to be pretty conclusive evidence that the purpose of the yellow spot is to counteract the strong chromatic aberration in the eye by reducing the brightness of the light from the blue end of the spectrum so that it ceases to be apparent. 14 It is regretted that a very marked effect that is apparent in the original photograph is lost in the reproductions. 34 AMES, PROCTOR AND AMES. SUMMARY. The above facts show the marked effect that variations in sensitivity of the retina have on the nature of our retinal picture. The slightly brighter warmer centers in some of Corot's pictures suggest the effect produced by the yellow spot. But besides his work the only evidence that has been found that the above described effects have been made use of by artists is in their very common practice of rendering shadows in out-of-door scenes much bluer than the}' appear when one looks directly at them. As far as is known this has not been limited to the outer parts of their pictures. The blue appearance of shadows which are imaged on the side of the retina are, however, very easily seen, and as this effect holds true over the greater part of the field of vision it was probably found that pictures look better with blue shadows all over them than without any blue shadows at all. As has been stated our knowledge of the sensitivity of the retina is very limited. We already know, however, that our capacity to dis- tinguish detail away from the center of focus is largely due to the structure of the retina. It is probable that a further knowledge would give suggestions as to the laws which control the difference of local values of which we are conscious on the different parts of the retina. There are also of course other effects such as contrast, simultaneous and successive, and after images which must have a marked influence on our retinal picture. Their use in pictures raises the thought of the possibility of suggesting eye motion. Chapter VI. BIXOCULAR VISION. The fundamental idea in undertaking the research work which is the basis of this article was that pictorial art should be similar to our mental visual images, and, since our mental visual images are probably similar to our retinal pictures, valuable suggestions could be obtained from a knowledge of the characteristics of our retinal picture. Our mental visual impression, however, is not derived from a single retinal picture but from two, as we normally look with two eyes. The whole subject of binocular vision is too long and complicated to be considered here. It was believed, however, that some of the char- acteristics of binocular vision under particular conditions could be reproduced in a single picture such as a photograph or painting. Figure 39a. View taken with a lens having the same chromatic aberration as the eye. Figure 39b. Same view as in 39a taken with lens having the same chro- matic aberration as the eye and a local plane filter having the approximate absorption of the "yellow spot" covering the whole picture. Figure 39c. Same as 39b. The focal plane filter covering the area covered by yellow spot thus approximating the conditions we get in one eye. Figure 39d. Same view taken with a corrected lens and no filter. bb & O u ji o o fl o E ft p ST M I.S " 03 S| i > 2 03 : X P3 « t= P 0 - i— t — — si VISION AND THE TECHNIQUE OF ART. 35 The conditions chosen were where the background behind the object at the point of convergence was of an indeterminate nature such as a mass of branches or foliage. The absence of any marked contours under such conditions would not call for the suppression of parts of either retinal image. Leaving out the effect of ocular movement and the fusion of doubled images the brain under such conditions may be considered as receiving two superimposed pictures of the object field as seen from each eye. To reproduce this effect a camera was devised which, by means of a reflector and half silvered prism, produced superimposed pictures of the landscape as viewed from two points of view, — the distance between which was the same as that between the eyes. The detail' in these pictures superimposed where the axis of the two systems crossed, as the two monocular images do at the point of convergence. The de- tails in all other parts of the pictures were more or less doubled due to the parallax of the two systems. Figure 41 shows such a "binocular" photograph. Figure 42 shows an ordinary photograph of the same scene. The following characteristics will be noted in the "binocular" picture: First, there is a "broadening" of everything in a horizontal direc- tion. At the convergence point this is due to seeing more of the sides of an object. At other points in the scene it is due to the doubling in a horizontal direction resulting from the parallax. This effect of the "broadening" of a scene when viewed binocularly can be noticed by anyone by first observing the scene with one eye and then with two. Second, there is an increase in contrast values between the lights and darks in the objects at the convergence point relative to that in other parts of the picture. This is due to the fact that at the con- vergence point the darks and lights superimpose and so reinforce each other while in all other parts of the picture they tend not to superim- pose and so counteract each other. Probably some such effect as this exists in our binocular impression. Third, there is a doubling up of the images of objects not at the convergence point, the extent of the separation of the doubled images depending upon their distance from that point. The seeing of objects not at the convergence point in doubled images is supposed to be one of the factors that gives us our idea of relief, the extent of the doubling suggesting the distance of the object from the convergence point. The impression we receive on our mind from these doubled images is differ- ent from that shown in Figure 41 due to the modifying effects of the 36 AMES, PROCTOR AND AMES. antagonism of the visual fields which suppresses one set of images, and to other physiological factors. Our impression, however, is probably more like the effects shown in Figure 41 than like a monocular impres- sion, as is shown by the greater effect of depth, less flat appearance of Figure 41 as compared with Figure 42. SUMMARY. It is well known among artists that a different effect is produced from painting with one eye than with two and that to get satisfactory results two eyes must be used. There are unquestionably certain effects in the binocular impression that can be reproduced in a single photograph and still further effects that can be reproduced in a paint- ing, where factors such as antagonism of the visual fields can be dealt with. There are other effects due to ocular movements which cannot be reproduced in a single picture but which may be possible of repro- duction in motion pictures. Chapter VII. GENERAL SUMMARY AND DISCUSSION OF RESULTS. From the foregoing description of the characteristics of retinal images of objects in various parts of the visual field it is possible to determine fairly definitely the nature of the retinal picture as a whole. It can be described, in general terms, as being a picture in which ob- jects at the center of interest, or focus point, are depicted in consider- able detail, but not with microscopic detail. Objects in the field of Anew, nearer or farther from the observer than the center of interest, are depicted with less detail and with chromatic edges the color of which depends upon the position of the objects relative to the center of interest. Objects lying to one side of the line of vision are also less clearly depicted, the lack of clearness increasing with the angle of obliquity, the accentuation of detail and edges in such objects depending upon their position relative to the center of interest. Speaking generally this accentuation is in a tangential direction if the objects are situated nearer to the observer than the center of interest and in a radial direc- tion if they are on the plane with or behind it. Such oblique objects also have characteristic chromatic edges depending upon their position relative to the center of interest. VISION AND THE TECHNIQUE OF ART. 37 All oblique objects are distorted and changed in shape varying in amount with their obliquity. This distortion is shown in the bowing out in their central portions of straight lines which do not pass through the center of interest and a reduction in size of oblique objects. And finally the color of the picture in its outer parts is bluer than at its center. In our ordinary habit of vision, when looking at a scene, we focus on some particular part or object in it due to its special interest or beauty to us; we hold that focus for a moment or two and then look at another center of interest or another or look away entirely. With each fixation of the eye a retinal picture of the kind just described is formed. We therefore receive on our retinas a series of such pictures. MENTAL VISUAL IMAGES. The question arises : What is the nature of the mental visual images which we have of actuality? Without doubt, our brain receives a series of impressions similar in character to our retinal pictures. But how are those impressions registered in our consciousness and memory. There are two general possibilities. One that our visual memory consists of a series of pictures similar in nature to our retinal pictures. The other that, by some mental process, these serial impressions are combined and form a memory impression similar to actuality as we know it exists intellectually, i.e., with the detail all over the picture sharp and clear and with no colored edges or distortion. Although there is no known psychological work on the analysis of mental visual images to substantiate the conclusion, what evidence there is indicates that our mental visual images consist of a series of images similar to our retinal pictures. This was the opinion of the well known psychologist, Dr. J. W. Baird, who mentioned as a reason for such belief the relatively definite character of the center and nebulous character of the outer parts of our mental visual images both when we are awake and when we are dreaming. Such a view is further substantiated by the fact that the indefiniteness in those parts of a scene that are not at the focus point and other characteristics of our retinal picture give a sense of depth and relief. A sacrifice of these characteristics would mean a sacrifice of effect of depth in our mental visual images which would seem most improbable. Furthermore the existence of a mental visual image of a scene similar in detail to the scene itself could only be based on a visual knowledge of all the detail in the scene. This could only be acquired by passing 38 AMES, PROCTOR AND AMES. the fovea or clear seeing part of the eye over every part of the scene which of course, is never done in the ordinary habits of vision. It also assumes some process of mental synthesis of particular parts of a series of impressions ; of the existence of such a process we have no evidence. It is believed that it can be concluded that our mental visual images of actuality consist of a series of images similar in general character to the picture we receive on our retina, or more accurately a series of com- bined pictures such as we receive on our two retinas. METHODS OF DEPICTING NATURE. In the arts, painting, drawing, sculpture, photography, our purpose is to depict nature. There are in general two ways in which this can be done. First, a reproduction of the actuality can be attempted. By this is meant as close a reproduction as possible of all the objects in the scene in every measurement and detail. In sculpture the well known wax figures do this most successfully, though much work in marble and clay does so very closely. In the pictorial arts it has been most closely approximated by photographs taken with a corrected lens. Many paintings in which all objects have been depicted in full detail, as they appear on the fovea of the eye when directly observed, also very closely approximate actuality. By this is meant that the objects are so placed in the picture and the details of the objects are so depicted that, if the picture is viewed from the proper distance, all the objects will lie in the same angular direction as they do in the scene itself; while the depicting of each object is such as to produce to the eye looking directly at it the same appearance that the object itself would produce were the eye looking directly at it. In a photograph this is accomplished by using a corrected lens ; that is, one which has been so designed and constructed that every object in the field is imaged with as great detail as possible and the images of objects in the image plane have the same relative lateral positions to each other as the objects themselves. The same result is accomplished in a painting or drawing in which the artist depicts every part of the scene as it appears to him while looking directly at it. Every object will then be represented in full detail whether near, far, or on one side of the field of view, and the depicted objects will lie in the same relative lateral positions to each other as the objects themselves. While such representations of nature are a reproduction of actuality in the accuracy with which the detail and relative lateral positions of VISION AND THE TECHNIQUE OF ART. 39 objects are depicted, they do not reproduce the positioning of objects situated at different distances. This is impossible because a picture is, of necessity, on a flat surface where everything depicted must be the same distance from the observer, whereas nature exists in tri-dimen- sional space.15 Due to this fact, the picture as a whole can never give the same effect to one looking at it as one gets from looking at nature. For in looking at one part of a natural scene, objects at all other dis- tances take on characteristic appearances due to their different dis- tances. In a picture where they are all reduced to one plane this is not true. The second way in which the depicting of nature can be attempted is, instead of trying to reproduce actuality itself, to attempt to repro- duce the impression that nature makes on the human consciousness, i.e., to reproduce mental visual images. The general characteristics of our mental visual images are, as it is believed it has been shown, similar to those of the retinal pictures. Such depicting of nature can be approximated photographically by means of a lens which produces the same characteristic imaging as the lens system of the eye, and a plate whose sensitivity over its various parts is similar to that of the retina. It can be approximated in paint- ings and drawings if the artist keeps focused on whatever he picks out as the center of interest and depicts everything else as it appears to him while keeping his eye focused on that point. Artists who have very clear and lasting mental visual images closely approximate it by copying those images directly without regard to actuality. As the retinal picture lies on a surface and as the canvas or paper on which it is depicted is also a plane surface, there does not seem to be the same fundamental limitations in reproducing it that exists in attempting to reproduce tri-dimensional actuality. In reproducing the subjective binocular impression in a single picture there do exist, however, the limitations which arise from the impossibility of repro- ducing those parts of the impression which we get from eye movement and motor impulses. For all artistic purposes, it is believed that the attempt should be to reproduce not the actuality but the impression which it makes on us. Three general facts may be given in support of this. First, the use by so many of the great painters of characteristics of the retinal picture which is the strongest evidence of the artistic value !5 Stereoscopic photographs do reproduce the effect of the third dimension. They are not however satisfactory from an artistic point of view, and as we are dealing only with single pictures will not be considered here. 40 AMES, PROCTOR AND AMES. of pictures of this type. Second, photographs taken with a lens which approximately reproduces the characteristics of the retinal picture are more pleasing than those taken with a corrected lens. Third, photographs taken with corrected lenses are the most perfect repro- duction of actuality which we have, much more accurate than can probably ever be accomplished with brush or pencil, yet this type of picture is admittedly a complete failure from an artistic point of view, indeed its failure seems to be due to the fact that it does reproduce actuality so accurately. The following three argumentative reasons are also given : First, the general accepted belief that artistic expressions are sub- jective. The purpose of the great artist is to make others see nature as he sees it. He could convey no more by reproducing actuality to those who look at his picture than they would get by looking at the scene itself. He has to put into his picture nature's impression on himself, the beauty and the truth he sees. Second, the subtle varia- tions and differences which cause him to see the scene beautifully are alterations in his mental visual images due to personal psychological factors. The depicting of such subtle differences could be much more easily accomplished in a picture which in its general type was similar to his subjective impression than one which was not. Third, the purpose of art is to awaken subjective associative processes in those who look at it. This is especially evident in portraiture. The natural way to cause us to recall our mental visual images or start a train of them in motion is to present to us a picture similar to them. When we look at a picture of this type we recognize that it is an attempt to reproduce not actuality but our impression of actuality. Where we see the objects farther away than the object in focus depicted without much detail we do not think that they in fact did not have detail in them or perhaps that an intervening mist existed which obscured the detail but we know that such objects were farther away than the object in focus. And similarly with the distortion of line on the side of the field of view, we do not think that a building, for instance, so depicted is in fact curved. We recognize that such is the character of our subjective impression of a building on one side of our field of view. That is we pass our fovea or the part of our eye that gives us distinct vision over the picture and recognize its various parts as being similar to our mental visual image, just as we can direct our attention to various parts of such an image. There is possibly a third way to attempt to depict nature, which may be considered a modification of the first method above described. VISION AND THE TECHNIQUE OF ART. 41 In such a picture, all objects not at the center of interest would be painted so that, when the center of interest in the picture was looked at, the picture as a whole would make an impression similar to that produced by the scene itself. Such pictures as photographs taken with corrected lenses attempt to do this but, as has been stated, fail in that they do not give a proper suggestion of depth. In this third type this deficiency might be met by accentuating the characteristic imaging of all objects not at the focus point to suggest their position in tri-dimensional space. For instance, near dark objects, in the line of focus, would be shown with red edges, distant ones with blue; and, on the sides of the picture, the radial accentuation of distant objects and the tangential accentuation of near objects would be shown. In making these peripheral accentuations it would have to be borne in mind that they would be modified by the oblique aberrations of the eye as they would lie on the peripheral part of the picture and be imaged on the periphery of the retina. Allowances would therefore have to be made. As the picture plane lies near the secondary astigmatic field the radial accentuations would have to be relatively slight and tangential accentuations relatively marked. No distortion would be put in such a picture as the eye itself would introduce it, if the picture was viewed from the proper distance. Granting that the proper accentuation of the radial and tangential and chromatic edges could be made, which would be very difficult, it is questioned whether such a picture would be satisfactory. The proper impression could only be produced when the center of interest of the picture was looked at. If any other part of the picture was looked at that part would not only appear like nothing ever seen before but the rest of the picture would then cease to produce the proper impression. THE RETINAL PICTURES AND ARTISTIC PHOTOGRAPHY. As has been stated cameras in their inception were copied after the eye. It can be argued that the value of photographs for pictorial purposes rests on two quite different bases. One that it lies in repro- ducing in black and white, or in photographs in color, the same general picture we get on our retina. The other that it lies in reproducing in light and shade detail and color the effects that exist in nature. On the first basis it can be said that the results, when looked at, are satisfactory because they appear to us similar to the impression we 42 AMES, PROCTOR AND AMES. have received. On the second basis it can be said they are satis- factory because in looking at them our eye is affected in the same way as it is affected by nature. In spite of the inherent impossibility already pointed out that the effect of objects in tridimensional space cannot be reproduced on a flat surface, it seems to have been the second basis that was followed in the evolution of the art of photography. This led to the develop- ment of corrected lenses which would image everything in the field of view in full detail with no distortion. After such lenses had been perfected it was found that the photo- graphs taken with them, although ha"ring unquestioned value for scientific and other purposes, were not satisfactory from an artistic point of view. Then followed the use of so called "soft focus" lenses, and various manipulations in printing and enlarging to get away from the hard full detail effect produced by corrected lenses. Those desiring artistic effects bought up old lenses such as were used in making daguerreo- types and had lenses designed in which various aberrations were left uncorrected. As a result there has of late years been a most marked advance in the artistic side of photography. The method of development has however been one of "cut and try" and as far as is known, with the exception of the work done by Gleichen mentioned above, no fundamental laws have been followed. In- numerable methods and lenses have been tried and only those pro- ducing pleasing effects have survived. It is believed that the advance in artistic effect has been due to the fact that the results obtained were more similar to the subjective im- pression and that future developments in photography on the artistic side will come from approximating as closely as possible the retinal pictures and mental visual images. THE RETINAL PICTURES AS THE BASIS OF THE TECHNIQUE OF ART. In the chapters describing the characteristics of retinal images of objects situated in different parts of the field of view examples were given of paintings by artists in which these characteristics appear. The lists do not mention a far greater number of works that are in general suggestive of the retinal picture without showing the special characteristics noted. Of these Corot is the best example. Whistler, Manchini and Abbot Thayer and many others are also examples. VISION AND THE TECHNIQUE OF ART. 43 This similarity to the retinal picture is shown in a tendency to accentu- ate the center of interest and lose detail elsewhere. This is especially evident in black and white work and in etchings. An explanation of this may be that it is easier to accomplish in black and white work. For in that part of the picture where it is desired to lose detail, detail is simply left out, the white paper suggesting blankness. Where color is used this cannot be done. Some color must be put on the canvas and then the difficulty arises of getting on the right colors in the right way. It is believed that anyone going through a gallery, with the points of views here set forth in mind, will be impressed by the fact that the works of many of the best men show a suppression of detail in those parts of the pictures which are not the center of interest. And there will not be the slightest question that, in most cases, the entire picture is not painted as it would appear if every part of the scene were looked at directly. It is also very interesting to note that it is almost a general rule that the early work of most of the great masters was, so to speak, tight and hard, photographic. There is a very good example of this in an early picture by Corot in the Boston Art Museum. Later their style or technique changes. Their work is done more broadly especially the outer parts and their pictures get a center of focus and, to use a stock term, compose. It cannot be questioned that the change is a departure from a photographic reproduction of the scene. But the question may still remain: what are the laws that govern this change in style or tech- nique? The commonly accepted belief is that, if there are any laws, they are purely aesthetic or psychical, and that the artist puts in and leaves out and changes solely according to the dictates of his personal taste. In view of what has been shown, especially the use by so many great painters of distortion, see Chapter IV, it is believed that the improvement in technique of these artists was due to a development of their vision. Consciously or unconsciously they approximated the scene as it would have appeared to them had they kept their focus upon the center of interest. The difficulty which arises in this method of painting is to know how to reproduce the impression that one gets from an object at which one is not looking. That the capacity to recognize and analyze such impressions can be developed is shown by fact that some of the char- acteristics of such impressions have been represented by numerous artists. That this is very difficult is shown by the fact with all the •1 I LMES, PROCTOB \M> amks. genius we nuist attribute to tin* great artists, there is, it is believed, no artist who lias intuitively recognized and depicted all the character- istics above mentioned; and in spite o\' the fad thai distortion, the most apparent of these characteristics, has in the past been used by many great artists it has been recognized and used as far as the writers know by hut one artist living today, i.e., Sir William Orpen. In fact SO little o\o we know as to what we see that the ordinary person, including art students and many painters, Ao not know that we see clearly only those objects upon which we focus. They believe that the whole oi the field of view is clear because when they are inter- ested in the question of the clarity of any part o\~ it they look directly at it and are uneonseious of the two movement. Granting that it is desirable tor the painter to he able to recognize and depict the character of his retinal impression it is believed that there is no question hut that he can be greatly helped by an intellectual know ledge of the characteristics of his retinal picture. The knowledge that la- never sees the whole of tin- scene with equal clearness will, after he has tried a few times, awaken his consciousness to tin- fact that he sees objects away from the center of focus less clearly. The knowledge that the characteristic edges and shapes of those objects not at tin- center of interest can he seen only if the eye is kept focused on the center oi interest will enable him to see these characteristics. The knowledge of distortion and the perception following therefrom will cause him to become conscious o( its existence. Similarly the knowledge of the characteristic chromatic edges of objects nearer and farther than the focus point, of the accentuation of radial and tangen- tial lines and o( the greater brightness of blue at a slight obliquity w ill lead to his conscious perception of these phenomena. Apart from the matter o( developing his perceptions, the knowledge of the char- acteristic imaging of objects in various parts of the field of view will enable him to produce an effect of depth without the use of, or to supplement, the means now used. It will also enable him to produce a natural center o( interest. The objection has been raised that such an intellectual knowledge would ho harmful to an artist by destroying the "innocence" of his eye. That is in his knowledge o( and expectation of seeing these things, he would see them where they o\o not exist, or would over- accentuate them. If the eye were "innocent" in the Brst place such an objection might have weight. But it is not. Take for instance the matter o( the detail which we see in objects. We know that all objects both those in focus and those nearer and farther are sharp and VISION AND THE TBCHM';i K OI ART. 45 clear, so we think we see them sharp and clear while in faet we see the objects out ;of focus with fuzzy edges. • And likewise in the matter of distortion. We know that the side of a building is straight. That knowledge destroys the innocence of our vision and makes US think we see it straight out the side of our < e when in fact we see it curved. "Innocence" is already gone. To restore truth to the eve, we have to learn that under eertain condition we sec chromatic edges on objects whieh arc in fact sharp and curved lines in place of straight ones. The same line of argument applies to the other characteristics of the retinal picture The development of art seems to have been a struggle to put down what we see and not what we know or think we see. The Egyptians for instance in drawing an eve as seen from in front on a face in profile were putting down not what they saw but what they knew. So they put in the same picture what they saw from two different points of view. We an- confident that today we paint things as we see them. But on consideration are we not making just as had a mistake when we paint an object in the foreground and one in the di tance with equal clarity, or when we paint straight lines on the side of the picture straight. It is just as impossible for us to see both near and di itant object-, equally sharp at the same time or to see straight lines on the side of our field of view as straight, as it was for the Egyptian to lee the eye from the front view in the face from the side view. The conclusion must be that the "innocence" of vi ion can be developed by intellectual sugge tion . Such suggestions which help the artist to know what he 3ees will also bring within the grasp of hi . intellectual consciousness, 30 that he can definitely perceive it, what formerly lie could only feel intuitively and will leave his intuition free to reach out for the subtler expre ion of truth and beauty . In the ultimate anal; 1 what the great artist expre e through his work i- ;i matter of his own mental and spiritual visual images of beauty and truth. Their character depends upon hi personalit; and p chology. Of necessity these factors will modify the character- istics of hi-, retinal pictures, to what extent it is impo ible of eour e to say. It is believed however that the technical structure of hi work based upon his retinal impressions. The similarity of hi- retinal una to those of the rest of mankind will insure a universality of under tand- ing and appreciation. It should he clearly understood that it is not mgge ted that the methods of depicting the various parts of a scene whieh have been described will in themselves make a great work of art. The are 46 AMES, PROCTOR AND AMES. simply matters of technique, the grammar so to speak of the expres- sion. The quality that makes a great work of art is what the artist expresses through his work. Masterpieces expressing great truth and beauties have been done with entirely different bases of technique, the work of the "Primitives" as compared with that of the Barbizon School, for instance. Genius will express itself through any technique, but certain techniques will give greater possibilities for depth and subtlety of expression than others just as a modern piano, or violin or full orchestra gives the musical composer more freedom than the shepherd's pipes. It is believed that everything that has been said applies to scuplture as well as to painting, taking into consideration of course that sculp- ture is tridimensional. Its usefulness in architectural drawings seems to be pretty conclu- sively proved by Figures 34 and 35. As to its application to archi- tecture itself, the writers do not feel qualified to speak. The belief that it is applicable can only be based on the assumption that in architecture we desire a subjective impression of a preconceived recti- linear arrangement. It would seem that under certain circumstances this might be desirable. The possibility immediately comes to one's mind that there maj' be a connection between the curves in Greek and Gothic architecture and the characteristics of our retinal image. Before closing it might be interesting to suggest a line of thought to which the determination of the more specific characteristics of our retinal pictures might lead. In the first place the characteristics of our retinal pictures are greatly affected by external physical condi- tions. Take for instance retinal pictures of a scene in the day time and at night. Due to the enlarged pupil with which the night scene would be viewed and the greater sensitivity of the retina to blue light, as shown by the left hand curve in Figure 37, the characteristics of our retinal picture of the night scene will be very different from those of the day scene. It is believed that there is no doubt that pictures depicting these characteristics would most strongly suggest the external physical conditions which give rise to them. Further, the eye is a most delicatelv sensitive organ and is without doubt, affected bv our bodily and likewise by our grosser emotional states. It is conceivable that the characteristics of our retinal pictures are also affected, and that specific bodily and emotional states are accompanied by specific changes in the characteristics of our retinal pictures. This in turn opens up the possibility that if the characteristics produced by a particular bodily or emotional state were depicted in a picture, the picture would suggest that particular state to those looking at it. VISION AND THE TECHNIQUE OF ART. 47 CONCLUSIONS. 1. Every object in space is imaged on the retina with character- istics of form, color, accentuation of line and chromatic edges, due to its particular position relative to the focus point. 2. These characteristics, of which the observer may or may not be conscious, suggest to him the position of the object in space relative to his fixation point. 3. The reproduction in a picture of these characteristics of images of objects causes the depicted objects to appear in the same relative positions that they occupied in space. 4. A pictorial representation of nature to be technically satisfactory from an artistic point of view should be similar to our subjective impression. It should not attempt to reproduce actuality. 5. Our subjective impressions are, in their general character, simi- lar to the pictures we receive on our retinas while holding one center of focus. 6. A pictorial representation of nature to be technically satis- factory from an artistic point of view should be similar in its general characteristics to the pictures we receive on our retinas while holding one center of focus. 7. An intellectual knowledge of the characteristics of the retinal image of objects not at the center of focus helps one to become visually conscious of such characteristics. Wilder Laboratory Dartmouth College, Hanover, N. H. VOLUME 57. 1. Kent, Norton A. and Taylor, Lucien B. — The Grid Structure in Echelon Spectrum Lines, pp. 1-18. December, 1921. $.75. 2. Lotka, Alfred J. — The General Conditions of Validity of the Principle of Le Chatelier. pp. 19-37. January, 1922. $.75. 3. Bridgman, P. W. — The Effect of Tension on the Electrical Resistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Louis. — Notes on the Early Evolution of the Reflector, pp. 67-74. February, 1922. $.50. 5. Bridgman, P. W. — The Effect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Bridgman, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. $1.25. 7. Pierce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Refraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjacent Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Franklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. 8.65. 10. Bennitt, Rudolf. — Additions to the Hydroid Fauna of the Bermudas, pp. 239-259. May, 1922. $.65. 11. Brues, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonididre. pp, 261- 288. 2 pis. May, 1922. $.85. 12. Thaxter, Roland. — A Revision of the End ogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda, pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. R. — Atmospheric Attenuation of Ultra- Violet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Romberg, Arnold. — The Ratio of the Calorie at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Robert H. — Studies on Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389-422. 2 pis. November, 1922. $1.65. 17. Thaxter, Roland. — Note on Two Remarkable Ascomycetes. pp. 423-436. 2 pis. September, 1922. $1.35. 18. Records of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members; Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 2 of cover.) PUBLICATIONS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Series, Vols. 1-4; New Series, Vols. 1-13. 16 volumes, $10 each. Half volumes, *5 each. Discount to booksellers 2oc; ; to Fellows 50%, or for whole sets 60c7. Vol. 11. Part 1. Centennial Celebration. 1SS0. pp. 1-104. 1^S2. §2.00. Part 2. No. 1. Ajrassiz, A. — The Tortugas and Florida Reefs, pp. 105-134. 12 pis. June, 1SS5. (Author's copies, June, 1 • 3.00. Part 3. Nos. 2-3. Searle, \. — The Apparent Position of the Zodiacal Light, pp. 135-157, and Chandler, S. C. — On the Square Bar Micrometer, pp. 15^-17-. October 1885 $1.00. Part 4. No. 4. Pickering, E. C— Stellar Photography, pp. 179-226. 2 pis. March . $1.00. Part 4. No. 5. Rogers, \Y. A., and Winlock, Anna. — A Catalogue of 130 Polar Stars for the Epoch of 1S75.0, resulting from the available Observations made between 1860 and 1SS5, and reduced to the System of the Catalogue of Publication XIV of the Astrono- mi=che Gesellschaft. pp. 227-300. June, 1S86. 75c. Part 5. No. 6. Langley, S. P , Young, C. A., and Pickering, E. C— Pritchard's Wedge Photometer, pp. 301-324. November, 1SS6. L " Part 0. No. 7. Wyman. M— Memoir of Daniel Treadwell. pp. 325-523. October. 7. S2.00. Vol. 12. 1. Sawyer, E. F.— Catalogue of the Magnitudes of Southern Stars from 0° to — 30° Declination, to the Magnitude 7.0 in< lusive. pp. 1-100. May, 1S92. S1.50. 2. Rowland, H. A.— On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-1S6. December. 1596. S2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniacea. pp. 187- 430. 20 pis. December. 1S96. $6.00. 4. Lowell, P. — New observations of the Planet Mercury, pp. 431—166. 8 pis June -. S1.25. 5. Sedgwick. W. T.. and Winslow, C. E. A.— I. Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II. Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. 13. 1. Curtiss. D. R. — Binary Families in a Triply connected Begion with Especia Beference to Hypergeometric Families, pp. 1-60. January. 1904. $1.00. 2. Tonks, O. S. — Brygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T.— The Spectrum of Hydrogen in the Begion of Extremely Short Wave-Length. pp. 121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H. — Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv November, 1906. $1.10. 5. Trowbridge, J.— High Electro-motive Force, pp. 1S1-215. pis. rxv-xxvii. Mav 1907. 75c. 6. Thaxter, B — Contribution toward a Monograph of the Laboulbeniacea. Part II. pp. 217-I6:». pis. xxviii-Ixxi. June, 190S. $7.00. Vol. 14. 1. Lowell, Percival. — The Origin of the Planets, pp. 1-16. pis. i-iv June 1913. 60c. 2. Fernald. W. E.. Southard, E. E.. and Taft, A E — Waverley Besearches in the Pathology of the Feeble-Minded . (Besear< h Series. Cases I to X.) pp. 17-12S rl- May 1918. $6.00. 3. Fernald, W. E., Southard, E. E., Cana\an, M. M., Baeder. O. J. and Taft, A. E.— Waverley Besearches in the Pathology of the Feeble-Minded. (Besearch Series, Cases XI to XX.) pp. 129-207. 32 pis. December, 1921. $6.50. PROCEEDINGS. Vols. 1-56, So each. Discount to booksellers 25^c; to Fellows 50^-.. or for whole sets 60^. The individual articles ma}- be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. $5.00. Complete sets of the Life and Works of Rumford. 5 vols., *25.00; to Fellows, $5.00. For sale at the Library of The American Academy of Arts and ::xces. 28 Newbury Street. Boston, Massachusetts. 58-2 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 2.— April, 1923. THE BOUNDARY PROBLEMS AND DEVELOPMENTS ASSOCIATED WITH A SYSTEM OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. By George D. Birkhoff and Rudolph E. Langer. (Continued from page 3 of cover./ VOLUME 58. Ames, A. Jb., Proctor, C. A., and Ames, Blanche. — Vision and the Technique of Art pp. 1-47 28 pis. February, 1923. $3.75. Birkhoff, George D. and Langer, Budolph E. — The Boundary Problems and Develop- ments Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49-128. April, 1923. $3.15. 3. Vainio, Eowaro A. — Lichenes in Insula Trinidad a Professore B. Tharter Collect i pp. 129-147. January, 1923. $1.00. 4. Bridgman, P. W. — The Effect of Pressure on the Electrical Besistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. $.75. 5. Bridgman, P. W. — The Compressibility of Thirty Metals as a Function of Pressure and Temperature, pp. 163-242. January, 1923. $1.70. 6. Baxter, Gregory P., Weatherill, Philip F. and Scripture, Edward W., Jr. — A Bevision of the Atomic Weight of Silicon. The Analysis of Silicon Tetrachloride and Tetrabromide. pp. 243-268. February. 1923. $.75. 7. Evans, Alexander W .— The Chilean Species of Metzgeria. pp. 269-324. March. 1923. $1.25. 8. Brues, Charles T. — Some New Fossil Parasitic Hymenoptera from Baltic Amber. pp. 325-346. March, 1923. $.65. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 2.— April, 1923. THE BOUNDARY PROBLEMS AND DEVELOPMENTS ASSOCIATED WITH A SYSTEM OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. By George D. Birkhoff and Rudolph E. Langer. THE BOUNDARY PROBLEMS AND DEVELOPMENTS ASSOCIATED WITH A SYSTEM OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS OF THE FIRST ORDER tmiTntiKA - ■ i ff yMT By George D. Birkhoff and Rudolph E. Langer.i Introduction. It is the purpose of this paper to develop in outline the theory of a system of n ordinary linear differential equations of the first order containing a parameter and subject to certain boundary conditions. Toward this end the notation of matrices is used. For the convenience of the reader the paper opens with a brief review of the fundamentals of matrix algebra and the integration and differentiation of matrices. This is followed by an expository discussion of the homogeneous and non-homogeneous differential matrix equations of the first order. The major portion of the treatment is devoted, however, to the homo- geneous differential vector equation with a complex parameter in its coefficient, and to the system composed of such an equation and suitable boundary conditions. The solutions of the equation for large values of the parameter are discussed and the formal development of a vector of arbitrary functions into a series of solutions of the system is obtained. The paper closes with the proof of the convergence of this development under appropriate conditions, which, in the ordinary notation, establishes the possibility of simultaneously expanding n arbitrary functions in terms of the characteristic solutions of a prop- erly restricted differential system of the type n y'i(x) = 2 [aik(x)\ + bik(x)}yk(x), n 2 {aikyk(a) + &#*(&)} =0, i = 1, 2,. . .n2 When reduced to a single equation of the nth. order this includes as a special case the expansions obtained by Birkhoff in 1908. 1 Much of the material preceding the proof of convergence is due to Birk- hoff, having been developed by him in lectures at Harvard University in the fall of 1920. The reorganization of this material into its present form, the treatment of the irregular case, and the proof of convergence are due to Langer. 2 For other developments in this field and more complete references see the following papers in the Transactions of the American Mathematical Society: Birkhoff, On the Asymptotic Character of the Solutions of Certain Linear 52 BIRKHOFF AND LANGER. Section I. Definitions.3 An array of elements of the form On «i2 fli.-i 0,21 f'22 .... «31 Cl3„ Onl On2- in which the number of rows equals the number of columns is called a square matrix, and is denoted either by (oi;) or by A. Two such matrices are said to be equal when, and only when, every element of the one is equal to the correspondingly situated element of the other. The sum of two matrices, («»,-) and (&#), having the same number of rows and columns, is by definition the matrix fan -f ft*,-), from which it follows that matrix addition is both commutative and associative, i.e., A + B = B + A, and A + (B + C) = (A + B)+C. The product of two n rowed matrices A and B is defined by the identical equation M faid n -k=l aikbkj), on the basis of which it is easily verified that matrix multiplication is both associative and distributive, i.e. A(BC) = (AB)C, A(B + C) = ABA- AC. Differential Equations Containing a Parameter, and Boundary Value and Ex- pansion Problems of Ordinary Linear Differential Equations, vol. 9 (190S), p. 219 and p. 373. Wilder, Expansion Problems of Ordinary Linear Differential Equations with Auxiliary Conditions at More than Two Points, vol. 18 (1917), p. 415. Hopkins, Some Convergent Developments Associated with Irregular Boundary Conditions, vol. 20 (1919), p. 245. Hurwitz, An Expansion Theorem for a System of Linear Differential Equa- tions of the First Order (about to appear in vol. 22 (1921)). Langer, Developments Associated with a Boundary Problem not Linear in the Parameter (about to appear in vol. 22 (1921)). 3 For a more ample discussion of the theory of matrices see Bocher, M., In- troduction to Higher Algebra. New York; The Macmillan Co., 1907. BOUNDARY PROBLEMS AND DEVELOPMENTS. 53 That it is in general not commutative is a consequence of the fact that n n 2 dikbkj is in general not equal to 2 bikCikj. The rearrangement of the factors in a matrix product is, therefore, as a rule, not permissible. The multiplication of a scalar into a matrix has simply the effect of multiplying each element of the matrix by the scalar. Thus if k is a scalar, then k A = Ak = (A;at-,-). Conversely, any factor common to all the elements of a matrix can be factored from the matrix. Two special matrices must be mentioned, namely 0 = (0), the zero matrix, and / = (5;;), the unit matrix, where 5ty = 0 when i 4=j, 8u = 1 . These matrices satisfy respectively the relations AO = OA = 0 and AI = IA = A. The determinant formed from the elements of a matrix without changing the order of the array is called the determinant of the matrix. The alternative notations | an | or | A | for the determinant of the matrix (a»,-) will be used. Given a matrix A it follows that if | A \ ^z 0 then there exists a unique solution in the x's for each of the linear systems n 2 dik Xkj0 = 8ij0, i = 1, 2, . . .n, where jo ranges from 1 to n. This means that there exists a unique set of n2 quantities xa such that n 2 aik xkj = Sij, i, j, = 1, 2, ... n, i.e. there exists a unique matrix (a*i;) such that A-(xi3) = I. This matrix (xi}) is denoted by the symbol A-1 and is called the inverse of A. From its derivation it is seen to satisfy the relation AA~X= I. Either of the relations, AX = 0 or XA = 0, leads, on the assump- tion that | ^4 | =4= 0, to the conclusion X = 0, as is evident from the theory of the systems of linear equations to which the matrix equations 54 BIRKHOFF AND LANGER. are equivalent. It follows from the relations AA~1 = I and I A = AI, however, that AA~l A = IA = AI, i.e., A(A~lA - I) = 0. Hence if A is any matrix for which | A | :£ 0, it is seen that A-1 A -1=0, i.e., A~lA = I = AA~\ In accordance with the following definitions, namely and Ja(x) dx = (faM dxj a a dA(x) (daij(x)\ dx dx a matrix is seen to be integrable or differentiable if and only if this is true of each of its elements. It is also clear that if C is a matrix of dC constants, then — = 0, while for any product ax d dB dA n — AB = A— + —B. dx dx dx Section II. The equation Y'(x) = A(x) Y{x)A Consider any matrix of functions Y(x) which satisfies (i.e. is a solution of) the equation (1) Y'= AY. In accordance with the rules for determinants we have d\ Y dx 2/n 2/21 ■Vln ■yin yn\ ■ ■ ■ ynn + 2/n ■ • -2/ln 2/21 • • • ?/2n + ■ • + 2/nl ■ • • -2/nn 2/ii 2/21 ■2/ln ■ yin 2/nl •2/nn 4 For the general theory of matrix differential equations see Schlesinger, L., Vorlesungen fiber lineare Different ialgleichungen. Leipzig; B. G. Teubner, 1908. BOUNDARY PROBLEMS AND DEVELOPMENTS. 55 whence, substituting from the system y-j = 2 aikykj, i, j, = 1, 2,...w k=i (which is equivalent to equation (1)) we obtain d\ Y dx 2au- yk\. . .Sflu-?/fc, k k 2/21 ■l/2n l/nl ■Vnn + 2/n 2/m 2 a2kVki- ■ .2 aohVkn k k llnl 2/nn + ...+ 2/11 2/21 ■2/ln •2/2n 2 ankyki- ■ -2 ankykn k k d i.e. - | y I = an | F | + a22 | Y | + . . .+ann | F |. Let us suppose, now (i), that the elements of A(x), Y(x), and Y'(x) are each continuous in an interval a ^ x ^ 6, and (n), that at some point a: of this interval | F | =£ 0- Then throughout a neighborhood of the point in question d\Y\ — — — = 2 a^fc da-, \ I \ fc=l or, integrating, | F | = ce / 2 aadx Inasmuch as the right-hand side of this equation cannot vanish if c d£ 0, it is seen that the hypothesis that | F [ =£ 0 for some x leads to the conclusion that [ F | differs from zero for all x. Thus we infer the Theorem: If F(.r) is a matrix of functions which satisfies equation (1), while | F(.i-o) [ = 0, a ^ x0 ^ b, then | F(.r) \ = 0, a ^ x ^ b. A matrix of functions Y(x) of this type which satisfies equation (1) 56 BIRKHOFF AND LANGER. and whose determinant | Y(x) | ^ 0, is called a matrix solution of equation (1). We assume at least one such solution to exist. Theorem: If Y(x) is any (matrix) solution of equation (1), and C is any matrix of constants (for which | C | 4= 0), then Y(x), defined by Y{x) = Y(x)C, is also a (matrix) solution of equation (1). Proof: We have ~ =— YC dx dx dY ,(1C dx dx But by hypothesis — = A Y. Hence i.e. dY dx dY dx dY dx A YC, dY _ dx AY. Moreover, since the product of two matrices is derived in the same manner as that of two determinants, it follows that | Y | = | Y | | C \. Hence | Y | =j= 0, if | C | =J= 0 and | Y | =J: 0. Q.E.D. Theorem : Given any matrix of constants Y0 for which | }'0 | + 0> then there exists a matrix solution of equation (1) which for any preassigned x, say x = x0, reduces to the matrix F<>. Proof: It has already been shown that when Y(x) is a matrix solu- tion of equation (1), then Y(x)C is also such a solution, where C may be any matrix of constants Avhose determinant is not zero. Then in particular C may be chosen as the matrix }T-1(a-0) Y0, whereupon it follows that Y(x) Y~1(x0) Yo is also a matrix solution of equation (1). This solution, however, obviously reduces to the matrix Yo when x = .r0. Q. E. D. Theorem: If Y(x) is a matrix solution of equation (1) then Y(x)C is the most general solution of equation (1). BOUNDARY PROBLEMS AND DEVELOPMENTS. 57 Proof: Let Y(x) be any solution of equation (1) whatever. Then the relation Y(x) = Y(x)$(x) defines a matrix <£(#), and substitut- ing from this expression in equation (1) we have — F$ = A Y, dx d$ dY i.e. Y— + —* = AY$. dY But by hypothesis — = A 1 . d<$ Hence Y-- + AY* = AY$, d$> i.e. Y— = 0. dx dx Y_ dx dx Y dx d<£> Inasmuch as I Y(x) I =)= 0, it follows that — = 0, and $(x) = C. dx Q. E. D. A mere interchange of the roles played by the rows and the columns of the matrices involved transforms the discussion carried out thus far for equation (1) to the corresponding discussion for the equation Y'= YA. The facts in the two cases may, therefore, be established by precisely the same methods. The pair of related equations (1) Y' = AY, (2) Z' = - ZA, are said to be adjoint, either being the adjoint of the other. Theorem: If Y(x) is any matrix solution of equation (1), and Z(x) is any matrix solution of equation (2), then Z(x) Y(x) = C. Proof: From the relation A V-—Y 7— dx dx dx 58 BIRKHOFF AND LANGER. it follows, upon substituting from equations (1) and (2), that ~ZY= (- ZA) Y + Z(AY). dx Hence ^- ZY = 0, and Z(x) Y(x) =C. Q. E. D. dx Converse Theorem: If Y(x) is any matrix solution of equation (1), and the matrix Z(x) is defined by the relation Z(x) Y(x) = C, where | C | =(= 0, then Z(x) is a matrix solution of equation (2). Proof: We have jLzY=0, dx dZ dY i.e. — Y + Z— = 0. dx dx In virtue of equation (1), therefore, dZ — - Y + ZAY = 0, dx and it follows, since | T | =}= 0, that dZ dx Moreover, \Z\= \ Y1 \ • | C | 4= 0. Q. E. D. Any pair of solutions Y(x) and Z(x) of equations (1) and (2) re- spectively which satisfy the relation Z(x) Y(x) = /, are said to be associated solutions. Thus if Y(x) is any matrix solution of equation (1) the associated matrix solution of equation (2) is Z(x) = Y~l(x). A differential equation is said to be self-adjoint when and only when it is identical with its adjoint after interchange of rows and columns. A necessary and sufficient condition that the equation (1) be self- adjoint is readily seen to be that an = — «/j for all i and j. BOUNDARY PROBLEMS AND DEVELOPMENTS. 59 Section III. The equation Y'(x) = A(x)Y(x)+ B(x). The Existence Theorem: Given the equation (3) Y' = AY + B, where A(x) and B(x) are matrices of continuous functions, a ^ x ^ b, then there exists a unique matrix Y(x) whose elements are functions which are continuous together with their first derivatives, a ^ x ^ b, and which satisfies equation (3) as well as the condition Y(xo) = Yo, the matrix Yo being any prescribed matrix of constants. Proof: By means of the relation X Ym(x) = F0 + J U00 F«-i(<) + B(t)} dt it is possible to define the following infinite sequence of matrices Y\{x), Yi{x),. . . Ym(x), . . . which satisfy the relations Y[(x) = A(x) YQ + B(x)9 Y1(x0) = F0, Y'2(x) = A{x) Y,{x) + B(x), Y2(x0) = F0, Ym'{x) = A(x) Fm_!(.r) + B(x), Ym(*o) = F0. Then setting Ym(x) - Ym-i(x) = Um(x) in the identical equation Ym{x) = F0+ (Y1(x)-Yo)+(Y2(x)-Y1(x))+ . . .+(Ym(x)-Ym.i(x)) we have Ym{x) = F0+ Ui(x) + U2(x) +...+ Um{x). Moreover, Um (xo) = 0, while Um'(x) = Ym'(x) - Y^'ix) = A(x) {Ym^(x) - ym_2(.r)} = A (x) Um-i(x) . 60 BIRKHOFF AND LANGEE. Denoting by the symbol « the fact that each element of the matrix of constants on the right represents a value as large as the largest numerical maximum attained by any element of the matrix on the left, we have, since A(x) and Ufa) are continuous in the closed interval a ^ x fS b, Ufa)«{k), A (x) « (a) . From this it follows that Ufa) = A(x) Ux (x) = (xaa ugj «{nak) and integrating £72(.r)«(^c*|.r-.r0|). Likewise U'3 = AU2 = I Sfl«tty ) «( 2a- knot \ x — Xq \ ), Us(x) « hW |,r~,ro12), and similarly for I = 3, 4, . . . , / i x — x \l\ Ul+1(x) « IM - - '- «k [nab — a} I! I \ l! However the infinite series of matrices (% Unb^a\l\ a=o 11 converges to the matrix Hence the series k (eani-a—\). Y0 + 2 Ufa) converges uniformly throughout the interval a ^ x ^ b, and since its terms are matrices of continuous functions the matrix Y(x) which it BOUNDARY PROBLEMS AND DEVELOPMENTS. 61 represents is necessarily one with elements continuous for a ^ x ^ b. Now X Ym(x) = Yo+ f {A(t)Y„Ut) + B(t)} dt Xo by definition, and since the convergence of Ym(x) to Y(x) is uniform X lim Ym(x) = l'o+ f lim { .4(0 Ym^(t) + B(t)} dt, m=oo *J m= oo xo X namely, Y(x) = Y0 + J { A(t) Y(t) + B(t) } dt. xo Differentiating we see that the elements of Y'(x) are continuous, a ^ x ^ b, while Y'(x) = A(x) Y(x) + -B(.r). Hence the solution exists as stated. To prove the solution unique suppose that both Y(x) and Y(x) are solutions of equation (3) each reducing to the matrix F0 for x = Xo, i.e. Y\x) = A{x) Y(x) + B(x), Y(x0) = Y0, T(x) = A{x) Y(x) + B(x), Y(x0) = F0. Now Y(x) — Y(x) = D(x) is a matrix whose elements are continuous, a ^ x ^ b, and which satisfies the relations D'(x) = A(x) D(x), D(x0) = 0. Let any neighborhood of the point .To be chosen, say | x — Xq | ^ 5. Then, if d denotes the largest numerical maximum of any element of D(x) for any x of this neighborhood so that D(x) « (d), A(x) D(x) « (nad), and D'(x) « (nad). Hence D(x) « (nad | x — x0 \ ) « (?iadS). But for some x of the interval at least one element of D(x) is numeri- cally equal to d (by the definition of d), and for this element it follows that d ^ nadd, i.e. d { 1 - nad} ^ 0. 62 BIRKHOFF AND LANGER. However, since a and n are fixed numbers and 8 can be chosen so that 5< - this leads to a contradiction unless d — 0. It follows that not there always exists a finite interval throughout which d = 0 and hence throughout which D(x) = 0. Inasmuch as this implies that D(x) = 0, a ^ x ^ b, the solutions Y(x) and Y(x) must be identically the same. Q.E.D. The solution of equation (3) can be easily expressed in terms of the solutions Yh(x) and Z^(.r) of the homogeneous equations (1) and (2) respectively. Thus, multiplying both sides of equation (3) on the left by Zh{x) we have dY Zh — = ZhA Y + ZhB, dx which, in view of equation (2), can be written Zh —- = — — — Y + ZnB, dx dx i.e. 7- ZhY = ZhB. dx Integrating we obtain X ZhY = C + f Zh(t) B(t) dt, a while the multiplication of this equation on the left by Yh{x), the solution associated with Zh(x), yields X (4) Y(x)= Yh(x)C + j Yk(x) Zh(t) B(t) dt. a This is the general solution of equation (3). In consequence any particular solution Y(x) may be written X Y(x) = Yh(x) C + / Yh(x) Zh(t) B(t) dt. BOUNDARY PROBLEMS AND DEVELOPMENTS. 63 Subtracting this from equation (4), however, we obtain Y(x) = f(x) + Yh(x) { C - C), from which it is seen that if Y(x) is any particular solution of equation (3) the general solution is given by (5) Y(x) = Y(x) + Yh(x) C. If in particular C is taken as C = 0 in formula (4), the solution which is characterized by the fact that Y(a) — 0 is obtained. This is called the principal solution at x = a. Since the choice of a as a limit of integration is unrestricted, the expression for the principal solution at any chosen point is at hand. Section IV. The differential system Y'(x)- = A(x)Y(x).+ B(x)* WaY(a).+ WhY(b)-=0. A matrix in which any row (or column) is precisely like every other row (or column) is called a vector. That a particular matrix is a vector is indicated by means of a dot suitably placed in relation to the letter designating the matrix in question, the dot preceding in case it is a vector in which the rows are the same and succeeding in case it is a vector of identical columns. Thus A = {a,) = and (ad ai a2. . .an CL\ «2- •«n a i «2- • a„ «i Oi. • «i a2 02- .a2 a„ On ■On 64 BIRKHOFF AND LANGER. That • AB = C, and that AB- = C-, are facts readily established by direct reference to the rules for multiplication. A matrix (a) all of whose elements are the same is written ^4-. From the preceding statement it is clear that -AB- = C-. Given Wa and W b, any two constant matrices for which | \Va | =£ 0 and | W b | =fc 0, then the two corresponding homogeneous vector differential systems ,fis (r(*)- =A(x)Y(x)- W \waY(a)' +WbY(b)- =0 and ~ \-Z'(x)= - >Z(x)A{x) Ki} (■Z(a)Wa-1+ -Z(b)Wb-l= 0 arc said to be adjoint. Theorem: The number of linearly independent solutions of system (()) is always equal to the number of linearly independent solutions of system (7). Proof: It has been shown that if F(.r) is any matrix solution of the differential equation (1), the most general solutionis Y(x) =Y(x)C. From this, C = Y'l(.v) Y(z), and if the solution Y(x) is a vector Y(x) ■ then C will be a vector C-. The general solution of the differential equation in system (6) is, therefore, Y(x)- =T(.r)C-, and since the substitution of this in the boundary conditions gives (8) WaY(a)C- +WbY(b)C- = 0, Y(x)- is seen to be a solution of system (G) when and only when C- is a solution of the equation (8). Moreover, it is readily seen that a necessary and sufficient condition that a set of solutions of system ((>) be linearly independent is that the corresponding solutions of equation <8) be independent. Setting IF,, F(a) + Wb Y(b) = (p»y), equation (8) can be written in the form (9) 2Pikck = 0, i=l,2,...n. BOUNDARY PROBLEMS AND DEVELOPMENTS. 65 The number of linearly independent solutions of the linear algebraic system (9), (and hence of equation (8)) is, however, precisely equal to the difference between the number of equations, n, and the rank of the determinant | pa |. Thus when | Wa Y(a) + W b Y(b) | = 0, and is of rank (n — k), then there are precisely k linearly independent solu- tions Cy , Ci',. . .Ch', of equation (8), and correspondingly just k linearly independent solutions Yi(x)- = Y(x) d- i= l,2,...k, of system (6). Conversely if there are k independent solutions of system (6) the rank of the determinant is (n — k). Suppose then that system (0) is known to have just k linearly inde- pendent solutions. This means that the determinant vanishes and is of rank (n—k), and since | XiU X2 | is of the same rank as | U | when | Arx | =J= 0, and | X2 | 4= 0,5 it follows that | Z(a) Wa-' { Wa Y(a) + Wh Y(b) } Z(b) Wb~l | = 0 and is of rank (n — k), Z{x) being any matrix solution of equation (2). If in particular Zix) is chosen as the solution associated with Y(x) this reduces to the statement that | Zifl) Wa~l+ Z{b) Wb~l \ = 0, and is of rank (n — k), which implies that system (7) also has just k linearly independent solutions. Thus the theorem is proved. A system of the type (6) or (7) is said to be either compatible or in- compatible according as it does or does not admit of a solution not identically 0. Its compatibility is said to be A: -fold when the number of its linearly independent solutions is k. Consider the non-homogeneous system I Y'(x)- =A(x)Y(x)-+B(x)- K } \ WaY(a)- +WhY(b)- =0. Theorem: A necessary and sufficient condition that system (10) has a unique solution is that the corresponding homogeneous system (6) is incompatible. Proof: It was shown (formula (5)) that the general solution of equa- tion (3) is Y(x) ss Yh(x) C + Y(x) 5 Cf. Bocher, loc. cit., pp. 77-79. 66 BIRKHOFF AND LANGEIt. where Y(x) is any particular solution and Yh(x) is a matrix solution of the homogeneous equation (1). Hence the general solution of the differential equation in (10) is Y(x). = Yh(x)C- +T.r). The substitution of Y(x) • in the boundary conditions shows it to be also a solution of the system (10) provided only that C ■ satisfies WaY{a) + Wb Y(b) + { Wa Yh(a) +Wh Yh(h)} C- = 0. But this relation can be solved for C-, and uniquely determines C- when and only when | Wa Yh(a) + W h Yh(b) | z\z 0, that is when system (6) is incompatible. Q. E. 1). Assuming then that system (6) is incompatible it is possible to obtain by the following procedure a solution of the equations (10) which is symmetrical with respect to the ends of the given interval a ^ x ^ b. Let Yi(x)- and }'2(.r)- be any pair of solutions of the differential equation (11) }"(•*•)• =A(x) Y(x)- +lB(z): Then clearly their sum Fi(.r)- + Y2(x)- = Y{x)> is a solution of equa- tion (10). But by formula (4), applied to equation (11), a particular choice of Y\(x)- and Y->{x)- is seen to be X Yi{z). = hfYh(x)Zh(t)B(t)'dt X Yi(x)- = hfYh(x)Zh(t)B(t)-dt, b from which it follows that F(a)' = hi f Yh(x) Zh(t) B(t)-dt + j Yh(x) Zh(t) B(t)>dt la b is a particular solution of the differential equation (10). BOUNDARY PROBLEMS AND DEVELOPMENTS. 07 Defining G(x, t) by the relations f \ Yh(x) Zh(t) when /, < x G(x, t) = < [ — \ Yh(x) Zh(t) when t > x, we may write Y(x)> = / G(x,t) B(t)- dt. a In accordance with formula (5), therefore, the general solution of the differential equation (10) is given by the equation b Y(x)- = I' G(x, t) B (0- dt + Yh(x)C- . a The substitution of this form in the boundary conditions yields the equation J [WaQ(a,t) + WbGQ>,t)} B(t)-dt+{WaYh(a) + WbYh{b)}C- = 0 a for the constant vector C- . Multiplying by the inverse of the matrix A= \WaYh(a) + WhYh(b)\, (A-1 exists since system (6) is incompatible) we see that 6 C- = - A-1 J \WaG{a,i) + WbG (b, 0} B(t)>dt. It follows that the general solution of system (10) is, in terms of a matrix G(x, t) which is defined by the formula (12) G(x, t) = G(x, t) - Yh(x) A-1 { Wa G(a, t) + W \ Q{b, t) ) , given by the expression 6 (13) Y(x) = J G(x,t)B(t)- dt. 68 BIRKHOFF AND LANGER. The matrix of functions G(x, t) is known as the Gree?i's function for the homogeneous system (6). By a precisely similar method the Green's function H(x, t) for the related system (7) may be derived, the general solution of the non-homogeneous system f -Z'(x) = - -Z(x)A(x) + -D(x) { ' \ -Zip) Wa-1 + -Z(b) Wh-l=0 being given in terms of H(x, t) by the formula b (15) -Z{x) = I ■ D(x) H(x, t) dt. a Theorem: If G (x, t) and H(x, t) are the Green's functions for sys- tems (6) and (7) respectively, then G(x, t) + H(t, x) = 0, t^x. Proof:6 Let B- and D be any two vectors of the types indicated and consider the two systems r- = AY- +B-, WaY{a)-+WbY(b)-=0, ■Z' = --ZA+-D, -Z(a) Wa~l+ -Z(b) Wb~'= 0. Multiplying the differential equations respectively by Z on the left and Y • on the right and adding we obtain ■ZY-' + Z'Y- = ZB- + DY-, an equality which upon integration yields (16) 0 0 ZY- \ba = J -Z{t) B(t)-dt + J -D(x) Y(x)-dx. Now from the boundary conditions we have •Z(b) = - -Z{a) Wa~l Wb, Y(b)- = - JIV1 Wa Y(a)-, whence ■Z(b) Y(b)- = -Z{a) Y{a)-, i.e. ZY- b = 0. a 6 The proof by direct computation is not difficult though somewhat laborious. BOUNDARY PROBLEMS AND DEVELOPMENTS. 69 b But -Z(t) = J 'D(x)H(t,z)dx a b and Y(x)-= JG(x,t)B(t) ■dt from formulas (13) and (15). Hence equation (16) is equivalent to b b (17) f f -D(x) { H (t, x) + G(x, t) } B(t)- dx dt = 0. a a This result, having been obtained without reference to the nature of B' and -D must, moreover, hold for all possible choices of these vectors. We shall proceed to choose a particular set apposite to the proof in hand. Let any point (x0, t0) of the region a ^ x,t ^ b,x^ t, be arbitrarily chosen and draw the surrounding small rectangle As whose sides are t = t0 ± A t and x — x0 =*= A x. Then choose B • and • D so that bt (t) = 0, when I =(= j0, bio(t) 3= 0 in As, bJ0(t) = 0 outside of As, dk(x) = 0, when k =f= H, dio(x) =£ 0 in As, d{0(x) = 0 outside of As. For this choice equation (17) is equivalent to (18) J J dio(x) {hioJO(t, x) + giok(x, t)} bJ0(t) dxdt = 0, to- At aro-Az and inasmuch as dio(x) bio(t) =j= 0 in As, clearly {hioJO(t, x) + Qi0ja{x, t)) changes sign, i.e. vanishes somewhere in this region. Now let Ax and At approach zero. Then in the limit we have hi0j0(to, .T0) + #i0;0(.T0, *o) = 0. From this it is seen that G(x0, U) -\- H(t0, x0) = 0, and since (x0, t0) was any point not on the diagonal it follows that G(x, t) + //(/, x) = 0, x =j= t. Q. E. D. By direct reference to formula (12) it is readily seen that the Green's function possesses the following characteristics: 70 BIRKHOFF AND LANGER. i) The elements of G(x, t) are continuous in x except for x = t. Along this line (x = t) there is a jump of unit magnitude in the elements of the principal diagonal, i.e. G(x, x - 0) - G(x, x + 0) = I. ii) For any given t, G{x, t) satisfies equation (6) in x, except along the line x = t, i.e. dx iii) For any given t, G(x, t) satisfies the boundary conditions of system (6) in x, i.e. WaG(a, t) + Wh G(b, t) = 0. Conversely we have the Theorem: The dependence of G(x, t) upon the variables x, t is com- pletely determined by the characteristics (i) to (iii) above. Proof: Suppose G (x, t) is a matrix possessing the characteristics (i), (ii), and (iii). By (i), (ii), J(.r, t) defined by J(x, t) =G (x, t) - G(x, t) is continuous for all x, a ^ x ^ b. Moreover, J(x, t) satisfies (iii) and is therefore a solution of system (6) in x. But system (6) is incompati- ble by hypothesis. Hence J(x, t) = 0 and G(x, 0 = G(x, t). Q. E. D. It is readily verified that a further set of three characteristics which completely determines the dependence G(x, t) upon the variables may be obtained by interchanging x and / and replacing system (6) by system (7) in the discussion above. The fact established by this theorem should be carefully noted. While the choice of the pair of associated solutions F/, and Z/, on the right-hand side of equation (12) is not unique, yet the entire function G(x, t) is independent of that choice. BOUNDARY PROBLEMS AND DEVELOPMENTS. 71 Section V. The formal solutions of the equation Y'(x). = [A(x)\+ B(x)}Y(x).. Returning to the homogeneous equation let us consider the nature of the solutions when the matrix coefficient of Y • is made to depend linearly upon a parameter X which is free to take on all values in the finite complex X plane. To this end we shall study the equation (19) F'(ar). = { A(x) \+ B(x) } Y(x)-, where A(x) and B (x) are matrices of continuous functions, by making the assumption that the equation has a formal solution (20) F(.r). 9 eXiy(x)dX \l\(x)- + i I\(x)- + i P,(x). + ."..{ where a is any chosen constant. If n = 1, equation (19) can be directly integrated and is seen to have an actual solution of the form (20). The passage to formulas (45) can, therefore, be made directly in this case, and hence we shall assume in the intervening work that n ^ 2. X Setting / y(x) dx = Y(x), and substituting the form (20) in the a equation (19), we obtain the formal identity X7(.r>XrU) | Po(») • + \ Pi(x) • + . . . | + e"™ j Po(x) ■ + . . . | = { A (x)\ + B(x) ) ,xr(l) PQ(x) • + i P^z) ■ + ...(, from which it is seen, upon equating the coefficients of X, that y(x)P0(x)-=A(x)Po(x)-. This is satisfied by a vector Po(.r) • not all of whose elements vanish, when and only when (21) \aij(x) -Stn(x)\ =0, 72 BIRKHOFF AND LANGER. i.e. when y(x) satisfies equation (21). For any given x, say x = a-0, however, the left-hand side of (21) is a polynomial of degree n in 7(0-0). Hence the equation is satisfied by n roots 7i(.To), 72(.fo),- • -7n(^o). We shall assume that for the case in hand these roots can be grouped into the ?* functions 7i(.r), 72(2),- • -7n(-i'), which satisfy the three conditions !(i) ji(x) continuous, i = 1, 2,. . .n, a ^ x ^ b, (ii) yfa) ±vi(x) for j±i, (hi) 7<(*)=t:0, *=l,2,...n. Clearly the last condition can be fulfilled only if | A | 4= 0; we shall assume this to be the case. Consider now a change of the dependent variable in equation (19). Setting Y (x) ■ = (a-) Y (x) • , where 4>(.r) is a matrix whose elements are continuous as well as their first derivatives, a ^ x ^ b, and | <£> | d£. 0, the equation becomes *T- + $T-' = [A\ + B) 3>F-, that is y'. = {$-1 ^l$X + $-] B - 4>-] $'} y- It is evident here that if the coefficient of X is not originally / then no such change of variable has the effect of reducing it to I. In the subsequent work it is desirable to transform A(x) into the matrix R(x) given by R(x) = (5,-,-7,(a0). By such a transformation the formal solutions (20) are carried into others with the same functions ji(x) in the exponents. Since the must be solutions of the algebraic system n S aik (a:) | 41 0. Suppose | <£(.r) | = 0. Then the system of linear equations 2 (fik Vk = 0 admits of solution by means of a set of quantities V\, v2,... vn, at least one of which differs from zero, i.e. there exists a vector V • ^ 0, such that 3>F- = 0. But then A$V- = 0, and in view of the relation (i) it follows further that $i?T'- =0. By continued repeti- tion of this reasoning it can be shown likewise that &R V • = 0 for all values of k. Consequently 3>{c0I + ciR+c2R2+ . . . -f cn_i iT"1} V-=0. But on the hypothesis that 7, 4= li when j ^z i it is well known that the determinant 1 2 1 7i 7i 1 72 722 ■7i ■72 re-1 ra-1 ■7n n-1 + 0. 1 7n 7n2 Hence the system of equations Co+('i7i + (VYi2 + • ■ ■ +C«-l 7in_1 = &1 f o+ Ci7„ -f- c27„2 + • • • + C„_i 7in = kn, admits of a solution in the c's not all zero for any choice of the quanti- ties ki not all zero. Some choice of the set c0, C\. . .cn_\, can therefore always be made to satisfy the relation {c0J + ci R + . . . + cn_! Rn~1} = (Sa ki). 8 Cf. Bocher, loc. cit., pp. 47. 74 BIRKIIOFF AND LANGER. Then HSifkf) V- s 0, n i.e. S (pukii'i = 0, for all choices of the set fci, /t-2, . . kn. But by construction some v, say »,• is not zero. If then the A*'s are chosen so that ki = 0, i 4= jo, fcy0 =}= 0, it follows that contains only vanishing elements this involves a contradiction. Hence the hypothesis | (.r) | =0 is not tenable, and the$ in question fulfills condition (ii), i.e. | (.r) | =f= 0. By a change of variable, then, equation (19) can be given the form Y'' = \R\ + B\ Y ■ , where the matrix B, being given by the relation B — <£"' B$> — $_1$', is a matrix of continuous functions. Supposing this to have been done we may drop the dashes and consider the equa- tion in the form (24) Y'(x)- = [R(x)\ + B(x)\ IXr)-. Since we have found that the Y(x) ■ of (20) may be any one of the n vectors l'i(.r)- obtained by replacing T(x) by I\(.r) the further dis- cussion might well be carried through for each of these vectors indi- vidually. However, if the matrix E(x) is defined by the relation E(x)^(8ije^x))> it is readily seen that the matrices Pi(x) can be chosen so that the f column of the matrix (25) Y(x) E jPoW + i P1(x) +. . . | E{x) is precisely the general column of the vector Y,-(x)- Hence all cases are simultaneously treated by the consideration of those formal solu- tions of the matrix equation (2G) T(.r)= lR(x)\ + B(x)} Y(x) which have the form (25). Inasmuch as E'(x) = (\y,-(x) 8<}- exrJ(s)) = \R(x) E(x), the formal substitution of (25) in (26) gives the identity BOUNDARY PROBLEMS AND DEVELOPMENTS. 75 (27) |po+^Pi+j~P2+... j \re+ |p;+^p;+ ... j E = jxp + pj ^Po+1-p1+..lE. Equating the coefficients of X we obtain the relation P0R = PP0) (°) (0) whence it is seen that pfj = 0, when j 4= i. Equating the coefficients of X we obtain in similar manner the equation PrR + Po' = PPi+ PPo, i.e. (28) pS)7i + l$)' = 7 = ^ • 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 $(x, X) corre- sponding to each sector and to each U is uniquely determined. More- over, we have (44) U(x) = S(x) CT(x) - ± *(z, X). A Consider now the particular solution, Yo(x), of equation (26) which satisfies the relation Y0(a) = S(a). Since S(x) as well as the coefficients of equation (26) are analytic in X, Yo(x) is likewise analytic in X. Moreover we know from page 56 that every solution of the equation is of the form Y(x) = Y0(.v) D, where the elements of D are constants with respect to x. Substituting this form of Y(x) into equation (41) and fixing x and solving for C we obtain the relation 1 To. C = r(.r„) | l'„(.ro) + ^ S(.r„) T(t) *(/, x) r»(0 tit { D, BOUNDARY PROBLEMS AND DEVELOPMENTS. 85 n i.e. ^ , ca — ■" Pih (>hj , h=\ where the quantities p;/, are analytic in X. Inasmuch as Y( x) is a non-identicall y zero solution if D =(= 0 there will exist such a solution for which C — 0 provided the determinant | p^ | vanishes. If on the other hand this determinant does not vanish then there exists a solution Y(x) ^ 0 corresponding to every Suppose the determinant | pih\ = 0 . Then there corresponds to the choice C = 0 a solution Y(x) for which the matrix Uix) defined by Y(x) = U(x) S(x) satisfies the relation i / » rx/{7A(f)-7r(f)}di \ But we know that for some x, say xo, and for some i, j, say io, jo, uujo(xo) — M. Then since \L et u>%?(xo,t)dt\iV. Accordingly we have i.e. Y& -(*&)+!*£$) E(z). Since the matrices Y0(x) and D are analytic in X the same is true also of Y(x). It is seen, moreover, that | Y | =(= 0. Hence, when the functions yt(x) satisfy the conditions of the theorem on page S3 there exists in every sector of the type described above a matrix solution Y(x) whose elements are continuous in x and analytic in X, and are the same as those of S(x) to terms in— zi . But by construction the A elements of S(x) are the same as those of S(x) to terms in — . Hence A we have the Theorem: Given any formal solution S(x) of equation (26) in which R(x) and B(x) both possess derivatives up to and including those of order k, then in each sector of the complex plane within which none of the quantities ft{\(7A(.r) — yr(x)} } change sign there exists an actual matrix solution Y(x) which is continuous with its first derivative in x and analytic in X, and is throughout the sector identical with S(x) to . 1 terms in —r . Xi In virtue of this there exists in any sector of the type described a pair of actual associated matrix solutions of the form (45) Y(x) = (*!>(.)«/.) [3«|m) Z(x) = (e-xr,-(x)-B,-(x) [d..]k_j t provided only that the matrices R(x) and B(x) are differentiable k terms. In all cases the functions [5*y]fc_i which occur in the expres- sions for Y(x) and Z(.r) are, of course, respectively identical, to terms in l/X " with the series [5W] in the formulas (37) and (38). BOUNDARY PROBLEMS AND DEVELOPMENTS. 87 In the course of the deduction of forms (45) it was assumed through- out that n ^ 2. Direct integration of the equation, however, shows that in the case n = I there exist solutions which are of this form over the entire plane. Accordingly formulas (45) may be used in every case. The explicit forms yW=^r,(,)+B,.Wji,7 + ^>j), with (pij, \pij bounded, hold if R(x), B(x) are differentiable. Section VII. The characteristic values of the system Y'(xj. = {R(x)\+B(x)}Y(x). WaY(a)- + WbY{b)-=0. It was shown on page 65 that a necessary and sufficient condition that the vector system K*°} Wa Y(a) ■ + Wb Y(b) ■ = 0 (| Wa | * 0, | Wb | * 0), has a solution is that (47) | Wa Y(a) + Wb Y(b) \ = 0, Y(x) being any matrix solution of equation (26). Let us make the specific form of this condition in any sector of the X plane apparent by substituting a Y(x) which is analytic in X for | X | > N, and which has within the sector the form (48) Y(x) = (e*Vx)+B>(x) [Si}]) , as determined in the preceding sections. Choosing the a of formula (20) as a = a it follows that Y(a) = ([5l7]). Substituting from (48) in the determinant on the left of equation (47), (call it D{\)) we have 88 BIRKHOFF AND LAAGER. Z)(X) = I Wa ([*«]) + Wb (Cxr;(6)+B;(6)[5l7]) |, i.e. (49) B{\)=\W^} + [u^\c^^^\. In this as in subsequent formulas when the expressions in question are determinants the letters r and c rather than i and j are used to indicate row and column respectively. Due to the fact that those and only those values of X which satisfy equation (47) are characteristic values, i.e. values which yield solu- tions of system (46), equation (47) is known as the characteristic equation of the system in question. The introduction at this point of quantities 8^ and <5,-y will do much toward simplifying the further discussion. These quantities are de- fined by the relations ( = 5u when RiWjib)} ^ 0 ( = 0 when iJ{xr,-(6)} ^ 0 ( = 0 when RlW^b)} > 0, ! = 8{j when #{xry(6)} > 0. It is to be noted that 5# and 8jj are functions of arg X alone. Suppose now that the rays9 R{\Ti(b)} = 0, i = 1, 2, . . .n, are drawn in the plane of the parameter X and let one of the sectors bounded by a pair of adjacent rays of this kind, jR {xr *.-(&) } = 0, and * ** R{\Tl (&)} = 0, be denoted by 0, and the fact that R\\Ti(b)} 4= 0, i = 1, 2,. . .n, that Z)(X) takes, for any X in the interior of such a sector the form 8 A half-line issuing from the origin of the X plane will be referred to as a ray. BOUNDARY PROBLEMS AND DEVELOPMENTS. S9 (51) z>(\) = i [«(»>] c + [««] 5:; e*r^ +*cw |. Factoring from the determinant the exponential factors, any one of which occurs in each element of an entire column, the further alterna- tive form (52) D(\) = n e,xr*(6) +Bk(b) ]8tk I [«#] 8*cc + [«&>] 5 &-i ** cc is obtained. It becomes necessary at this point to differentiate be- tween certain types of conditions which inherently characterize any particular system of type (46). The conditions of the system will be said to be regular (i) if n = 1, or if, when n = 2, arg Vjib) 4: arg { =>= T{(b) } when j =1= i, and (ii) the boundary conditions are such that each of the determinants (53) W™ = | «,« 5|, farX within &VTJ It is readily seen from this that D(\) is given for any X in S by the formula (55) D(\) = | &?] $ + [*g2][$ + tm\**M+*M\. n ** Factoring from this determinant the product II rUr*(6) + B*(6)| S(t£) we have (56) D(\) = ri eM r*»> +**<*> ) 8 h? Z)(X), fc-i where 5(X) = I [«{?] $ + k^]{5^ > + 5C„} ^r„(6)+B,(» |. Now D(X) is seen to have elements consisting of a single term in every column but the v , the elements of that column being binomial. Con- sequently the expression of the determinant as the sum of two others is possible, i.e. D(K = | [«#>] 8[f + [«£>] S& | + | M?)lst - Sc„} + ** or, in view of relations (53) and (54), (57) D(\) = [JFH -f [JVT] er„x(w+B,(» BOUNDARY PROBLEMS AND DEVELOPMENTS. 91 Since the roots of D(\) =0 and those of D(\) = 0 are the same we have, therefore, as the characteristic equation (58) [W^v)] + [W{VT)} exr»(W+«.(»= 0 This yields, since W^v' =fc 0, the equation TJ-(w) Wv(b) + Bv(b) If -(">•) TW N, the same is readily seen to be true of e also. Consider now a small circle of fixed radius r drawn about the point for any given p. Then for a proper choice of origin in the X plane (see page 98) this circle lies entirely within the sector crM„ or N. Also if p is sufficiently large the point 1 \ , , -IV^ ) B,(b) + log— ^j + e +2 pm Tv(b) ( ° W lies within the circle for all X on the circumference. Consequently arg r ~ t^)\~ Bv{b) + ]og ~^ +^ + 2^| I increases by 2ir as X describes this circumference, and just one root of equation (59) is accordingly seen to lie within the circle. Since it is 92 BIRKHOFF AND LANGER. readily verified that every such root is a characteristic value it is seen that (59) determines such a value for every p which is sufficiently large, i.e. that for large values of X the characteristic values lie ap- proximately along a line parallel to the ray E{Xr„(&)} = 0, the dis- tance between two adjacent ones approaching as a limit the finite length 2x/ | T„(b) \, as j X | increases indefinitely. From the deriva- tion of this result it is seen moreover, that a similar sequence of characteristic values lies near each ray i^Xr^)} = 0, and that no further distribution of characteristic values exists. Case II. A Type of Irregular Conditions.10 Let us suppose now that the regularity condition (ii) is fulfilled, that n ^ 2, but that (60) argl\(&) =arg{±rj(6)} for the pair of values i = vu j = v2. ^Ye have then a case of irregular conditions, and while we shall restrict the discussion to the case when the relation (60) holds for only a single set of values i, j, the reasoning to be employed is typical and may be applied with equal success to the cases in which a greater number of the points Ti(b) are collinear with the origin of the complex plane. It is only for the sake of brevity that the simplest, rather than the most general case which results from dropping the regularity condition (i) is treated. A review of the discussion applied to the case of regular conditions readily shows that the methods employed there apply equally well to the case in hand and yield the same results in any sector of the X plane which does not contain the line RiXT^ (b)\ = 0. It is, therefore, necessary to consider here only the distribution of characteristic values in a sector S containing this line. Along the line in question neither the expressions for the elements of the v[h nor of the vf column of D(\) can be contracted by means of the relations (50). Accord- ingly it is found that D(\) takes the form * ** (61) Z)(X) = | [„«] «« + [«£>] {*« + 8CPi + 5C„J cxrc(6)+fic(6) | * ** throughout the entire sector S, 5^ and 5^ representing the quantities 8CC and 8CC for X on the ray R{\TnQ>) } = 0 which lies in S. Factoring 10 For the discussion of a differential system representing a different type of irregular conditions see Hopkins, loc. cit. BOUNDARY PROBLEMS AND DEVELOPMENTS. 93 from (61) the product II elxrV6)+iW 5u- we have k=i ** (62) b(X)- fie|Xr' and o-„T the sectors abutting on the ray R{\TVi(b)} = 0 in which fljxr,,^)} < 0 and > 0 respectively. It is necessary to consider the case in which arg I\ (6) = arg r„ (b) and that in which arg T„ (b) = arg { — r„ (b) } . Sub-case A. arg T„ (b) = arg TVi>(b). In this case the quantities i?{Xr^(ft)} and i?{Xr„ (b)} are of the same sign throughout the sector S, and the relations (64) * * + 5^ + 5C„S #* ** "CK. "CV, _ si") _ sCm") — °cc — 0ce 94 BIRKHOFF AND LANGER. are readily verified. Defining the determinants Wx and W2 by the formulas (65) * ** Wx = | u& [$? ~ BeVi\ + «« {8if + SeVi} | * ** we have Z)(X) = D(X) where, in view of (64), D(\) satisfies the equa- tion (56) and is given by (66) D{\) = [WM] + [Wj\ eXT»w+B»w+[W2] eXIV.W+«*»<» + [Tf"("T)] ex{rn(6)+r,2(6)}+JB„I(6)+Bn(6) Sub-case B. arg I\(&) = arg { - I\f(&)}. In this case #{Xr„2(&)} has throughout S the sign opposite to that of i?{XrVi(6)}, and it is found that ;(»"•) = s(") _ SW (67) + *w, = C = *£" + 5C ** ft? - 5„„ ■ ** ** Oce — Or.r. ~ 0, 'cc vcc Accordingly we have + [PFi]exl rvl(b)+v^(b) }+B„l(b)+By2(b) If now we define D(\) by the relation D(\) e-xr^b)-B^b) = D(\), it is readily verified on the basis of formulas (67) that D (X) again satisfies equation (56) while it is given in this case by (68) D(\) = [WM] + [Wj\ exr^b)+B^b) + [W2]e-XT"^b)-B^b) + [n*('"')]rxfr''l(6)_r"2(6)I+B"i(6)_B''2(6) . Let us suppose now that the notation has been so chosen that rv(b) |r„,(6)|^ |r„(6)|, and set TAb) r. Then upon dividing equations (66) and (68) through by [WM], (recall that WM 4= 0 by BOUNDARY PROBLEMS AND DEVELOPMENTS. 95 hypothesis) it is seen that the characteristic equation, i.e. D(\) = 0, is in each case of the form (69) 1 + N,xr"'(6) + [c2] /xr*'(6) + N *< 1+'l*r'.<*> = 0, where Ci, i = 1,2, 3, are complex constants, c3 ^ 0 by hypothesis, and r is a real constant r = 1 . Wilder n has shown that the roots of this equation are asymptotically represented by those of the equation (70) 1 + d 6>xr<"(6) + c2 /xr»(6) + c. e\1+r !xr"'(6) = 0, and has discussed the distribution of the roots of this equation. We shall proceed to this discussion, observing, however, first that when | r„ (b) | and | r„ (b) \ are commensurable a far simpler treatment is V possible. In that case r = — where y and q are integers and the equation (70) is an algebraic equation of degree (p + q) in e q Accordingly it has (p + q) roots, i.e. Ar„,(&) e a =«,-, J= 1,2,..., (p-{-q) from which it follows that (71) X* = j^{log«?- + 2M. In this case, therefore, the characteristic values which lie in sector S for | X | > N are asymptotically represented by a set of points which are 2qv spaced at intervals of length , r /, , , on (p + q) lines (not neces- sarily distinct) parallel to the line .ftfXr^ (b)} = 0. When r„2(fr) and Tn(b) are incommensurable no such simple treat- ment is possible. The distribution of the characteristic values may be obtained, however, by Wilder's procedure,12 which follows. Setting Xi\i(6) = z = x + iy we have as the equation (70) /(*)= 1 + clC* + c2erz + c3Cl1+rl2 = 0, 11 Wilder, loc. cit., p. 423. 12 Cf. Wilder, loc. cit., pp. 420-422. 96 BIRKHOFF AND LANGER. and it is readily verified that there corresponds to each choice of an arbitrarily small positive constant x» some value of .r, say x = X , such that (72) (i) |1 -f(-z)\ < X, (ii) I c3 ~ ~U^u I < X, for x g X. We shall assume that x is chosen sufficiently small to preclude the vanishing of /(z) outside or on the boundary of the region | x | ^ X. Now /(z) is analytic throughout the entire finite plane, and hence it is possible to find in any interval of the Y axis, however small, some point y = i/o which is such that the line y = y0 contains no zero of /(z). Let y = \\ and y = Y% be any two such lines and consider the rect- angle K bounded by them and the lines x = X and x = — X. We shall determine the number of zeros of/(z) within K by observing the increase in arg/(z) as z describes the perimeter. We have arg/(z) = sin-1 , where l\f(z)\ denotes the coeffi- I /(z) I cient of V — 1 in the expression for f(z). Moreover, for y = constant I {/(z) } has the form Hf(z)} =d1ex + d,erx + d3el^x, where the coefficients, (/,, i — 1, 2, 3 are real constants. The finite zeros of /{/(z)}, and hence those of , being the roots of the I /(z) I equation di + foW' + d*" = 0, are, however, separated by the finite zeros of the derivative of the left- hand member, namely by the roots of the equation f/2{r- 1} +d3rcx = 0. Since this equation is satisfied by at most one value of x it follows that ^— vanishes at most twice, and consequently that arg /(z) changes ! /(z) I BOUNDARY PROBLEMS AND DEVELOPMENTS. 97 by loss than Dir as .r varies between the limits .r = —A' and .r = + .V along a line // = constant. Because of the relation (,721). however, we know that for every 3 on line x = —X,f(z) lies within a circle of radius \ about the point a = lj and hence that as a moves along this line arg/(z) changes by less than 2 sin-1 \. Similarly relation (72ii) shows that argiA~) .u } changes by less than 2 sin"1 x ;is - moves along the line .r = A". From the identity arg/(z) = arggi1+r}* + arg -yj^j; it follows, therefore, that the increase in arg/(z) as : moves along the line x = X from y — \\ to // = ¥•: lies between (1 + r) j Y% — }\\ + '_' sin-1 x and (1 + r) J }•_> — 1\} — 2 sin-1 \. Consequently the in- crease in arg/(z) as z describes the perimeter pf K lies between (1+r) I Y2-Yi\ + 67r+4sin-1 Xand (1+r) ( F2- }\! -Gtt-4 sin^x, and accordingly the number of zeros located in the interior of K must lie between (1 + r) [Ys-Yx] +3 + - sin-1 X and 27T (l+r){Y2-Y1) Q 2 . ■ — o — — sin-1 x 27T 7T Since \ can be chosen arbitrarily small, however, this means that the number of characteristic values between any two lines y = Ci and y — Ci+ / is at least / — 3, and cannot on the other hand exceed 2- 1 + ', + ;, 27T Summarizing the results it is seen, therefore, that the characteristic values in sector S lie in a strip bounded by two parallels to the line R{\r„ {!>) ! = 0, and that they are so distributed throughout, this strip that for I X I > A no more than three lie between anv two lines which 2tt are at a distance (/ < from each other and are perpendicular 1+r l to the line ft{Xr„i(6)j = 0. 98 BIRKHOFF AND LANGER. Section VIII. The formal expansion of an arbitrary vector. It was shown in the preceding section that both under regular conditions and under the type of irregular conditions discussed the characteristic values for system (46) are numerable and cluster about the point X = °° . Denoting these values by X], X2, X3, . . . it is possible, therefore, to assign the subscripts in such manner that | \m | ^ | Xm+i |. Moreover then lim | |\„, | = oo . TO=0O Assuming that system (46) is simply compatible at the characteristic values there exists for each of these values just one solution of the system in question and just one solution of its adjoint system. These solutions for X = Xa will be designated by Y (x) • and -Z(k){x) respec- tively. Now if X = 0 is a characteristic value for system (46) let the para- meter be changed by setting X = X + c, c being a constant. Equation (46) then becomes Y'(x)- = {R(x)\+B(x)} Y(x)-, where B(x) = c R(x) + B(x). The characteristic values of the system thus modified are X = X&— c and it is clearly always possible to choose c so that X = 0 is not a characteristic value. No loss of generality is entailed, therefore, by the assumption, which will be made, that Xa- =£ 0 for any k. Writing the equation (46) for X = Xa- in the form Yw\x)< = B(x) Y(k\x)- +\kR(x) Y(k) (.r)- and considering this as a non-homogeneous equation we have from page 67 6 (73) Y(k\x)- = \kJa(x, t) R(t) Y(h\t)-dt, a where G(x, t) is the Green's function for the system BOUNDAEY PROBLEMS AND DEVELOPMENTS. 99 Y'{x)> = B{x) Y{x): WaY(a)- + WbYQ>)> =0. (74) „7 T//1, In precisely similar manner we have from the adjoint system 6 •Z{l\x) = -X?J-Z(/)(0i?(/)//(.r,/)^, a or, substituting from the relation G(t, x) = - H(x, t). b (75) -Z(l\t) = h f-Z«\x) R(x) G(x, t) dx. a Consider, now, the integral 6 b J = J j >Z{l\x) R(x) G(x, t) R(t) Y{k)(t)-dx dt. a a In view of relation (73) we have b \kJ = J-Zw(x) R(x) Y{k\x)-dx, a while it is seen from (75) that b \tJ= J.Z{l)(t)R(t) Y{k)(t)-dt a By subtraction, then (X,- \) J = 0, and it follows that 3 — 0 provided k^p I. But then \J = 0, i.e. (76) 0 j -Z(l\x) R{x) Y{k\x)-dx = 0, for k =Jz I. 13 Since X = 0 is not a characteristic value of system (46) system (74) is incompatible and the Green's function exists. 100 BIRKHOFF AND LANGER. It can, moreover, be easily shown (see page 107) that when the system (46) is simply compatible at every characteristic value, this is not true for k= I, i.e. J -Z{l\x) R{x) Y{l){x)-dx + 0, for any /. a Let us suppose now that an arbitrarily chosen vector F(x) • can be developed into a series of the form (77) F(x)- = £ ckY™(x)-. fc-i Multiplying both sides of this equation on the left by the vector •Z{l\x) R(x) and integrating term by term we have formally b b^ J-Z{l\x) R(x) F{x)-dx = £c,J -Z{l\x) R(x) Y{k\x)-dx, which in view of relation (76) reduces to b b (78) f-Z(1)(x) R(x) F(x)-dx = cj -Z{l\x) R{x) Y{l\x)-dx. Inasmuch as the matrix on each side of this equation is one all of whose elements are identical the equation may equally well be written b I 2 z{?(x)yh(x)fh(x)dx [ (1) 6 ct< I £z%>(x)yh(x)ynx)dxya), a whence 0 ci = — b f 'izV\x)yh(x)yil)(x)dx a BOUNDARY PROBLEMS AND DEVELOPMENTS. 101 Consequently we have 00 b /n 2 tf>(. h-l x)yh(x)fh(x) dx (79) F{x)-= I { f S «!*>(*) 7*(a) vPto dx -(*) Fw(a;)' If, therefore, F(.t) • may be developed into a series of form (74) which converges in such a manner as to legitimatize the processes above, then (79) is a necessary form for the series in question. Thus far it has been stipulated only that Y (x)- and -Z \x) be respectively solutions of system (46) and its adjoint for X = X*,. But each of these systems is homogeneous, and if Y (x) • and -Z (x) are any particular solutions then c Y (x)- and c-Z (x) are also solutions. Having chosen a definite pair Y (x) • and -Z( (x) we have then F(x) 00 z b 2 CZh h=l (x)yh(x)fh(x) dx /n h=i -(k) cz^(x)7h(x)y^(x)dx Hk) Y^(x) If in particular c is chosen so that b cf^-zik\x)yh(x)yik\x)dx= 1, h=\ and the vector c-Z , for this value of c is associated with Y • so that the choice of one implies the choice of the other, we have, on dropping the bars over the letters, 1 Y{k\x)-. (80) F(x)- = L fc-i f2zik)(x)yh(x)fh(x)dx In using this formula it must be remembered that -Z (x) is deter- mined as soon as the particular 1 (x)- is chosen. 102 BIKKHOFF AND LANGER. In a precisely analogous manner it may be found that a development of the type ■F(x) = Lck-Zik)(x), fc-i which converges in such manner that it may be integrated term by term after being multiplied by any of the vectors R(x) Y (%)•, must necessarily coincide with the expansion 00 k-1 )yh(x)y^(x)dx Zw(x). Inasmuch as the matrix G(x, t) is not a vector the methods outlined above do not apply directly to the problem of expanding the Green's function. We shall proceed, therefore, as follows. Let Gj(x, t) • denote the vector each of whose columns is the j col- umn of G(x, t) . From the relation (SO) , F(x) being replaced by Gj(x, t) • , it follows that for any value of t, Gj(x, t) • can be formally developed, the series obtained being (81) Gj(x,t) fc=i 6 d h-1 r) 7h(x) (I hi (x, 0 dx . Having obtained this formal development of the Green's function it is possible to state the Theorem: If the development (S3) converges to G(x, t) in such a manner that a uniformly convergent series is obtained by multiplying (83) on the right by any matrix of continuous functions and integrating term by term, then any vector F(x) • the elements of which are con- tinuous and have continuous first derivatives, and which satisfies the boundary conditions WaF(a)- +WbF(b)- = 0 is represented by a convergent development of the type (80). Proof: The relation F'(x)- = B(x)F(x)- +C(x)- defines the vector C(x) ■ , and inasmuch as F(x) • is then a solution of the non-homogeneous system Y\x)- =B(x) Y(x)- + #>(*). Hence gf (x, t) = c(*> yj»(*) z?\t) or r(*) (92) Gm(x,t)= — Y{k\x)--Z{k)(t), n the value of c( being as yet undetermined. Writing equation (46) in the form Ym'(x)> = \R(x) X + B(x)} Yw(x)-+ {X,- X} R(x) Y{k\x)-, and considering this as a non-homogeneous equation, we have BOUNDARY PROBLEMS AND DEVELOPMENTS. 107 Y{k)(x)- = - {X - X,} J G(x, t, X) R(t) Y{k\t)-dt. Inasmuch as this yields lim {X-X,} G{x,t,\) = G{k)(x,t), x=xfc o Y{k)(x)- = - J G{k\x,t)R(t) Y{k){t)-dt, and substituting from (92) we see that 6 Y{k)(x)- = - — Ym (x)--Z(k)(t)R(t) Y{k\t)-dt, *J 0. We shall consider the convergence of the contour integral in formula (95) as the contour of integration is taken successively as a larger and larger circle of the type C above. With the size of the circle the num- ber of characteristic values which it includes and hence the number of terms of the series which are summed by the integration may be in- creased indefinitely, the limit of the integral for | X | = °° , being, if it exists, the sum of the corresponding series. Let us recall the hypotheses already made concerning the functions 7i(x). We have (96) (i) 7i(x) * 0 (ii) 7i0*0 continuous a ^ x ^ b (page 72) (iii) if n ^ 2 y{(x) =j= y,{x) for i =f= j (iv) if 7i ^ 2 arg{7i(.r) — y,{x)} = //l7 (page 83) To these we shall add (97) arg 7»(.r) = Hi (a constant), i = 1, 2,. . .n. Concerning the vector to be expanded we shall assume that the elements fi(x), i = 1, 2, . . .n, consist in the interval (98) a ^ x ^ b, of only a finite number of pieces each of which is real, continuous, and has a continuous derivative. It will readily be seen from (96) and (97) that we have restricted each yj{x) to vary along a ray in the complex plane. Moreover, if n ^ 2 the dependence of y3(x) upon x must be such that the slope of the line joining any two of these points remains constant. This 110 BIRKHOFF AND LANGEK. means that every tt sided polygon with vertices at the points yj(xo\ j =1, 2, . . .n, a ^ Xq^ b, at most expands or contracts about X = 0 as .ro is allowed to vary. If the lines along which jj(.v),j = 1,2,...//, vary are all distinct, the conditions of the system are regular provided Wa and W b are suitably chosen. If, however, one or more sides of any of the polygons mentioned lie on a line through the point X = 0, we have the irregular ease discussed in section VII. In particular all the sides may lie on such a line, (for instance the quantities yj(x) may he all real) and it is upon this configuration that the irregular case in question bases its chief claim for interest. Since the condition (96 iv) is automatically fulfilled in this case the functional dependence of jj(x) upon .r is far less restricted than when the points y,(x) form the vertices of actual polygons. Substituting in formula (86) the value G(x, t) = ± Y(x) Z(t) we have G(x,t,\) = Y(x) | ± U + hA^\U\,Y«i) - WbY(b)}} Z(t), the upper sign holding for t < x and the lower sign for t > x. It is desirable in the following work to express G{x, f, X) in a somewhat different form. Upon setting + § I + 1 A~] { WaY(a) - WbY(b)} = (5*y) + U, multiplication by A yields I WaY{a) + i WhY{b) + I WaY{a) - \ WbY(b) = [WaY(a)+WbY(b)\ (4-)+Atf, which, in view of the relation (8$) + (5 ,y ) = /, reduces to WaY{a) (S**) - }VbY{b)(8*j) = AU. Hence U = A-1 WaY(a) (5?) - A-1 WbYQ>) («J). Now setting -i/ + ^A-'{irn}>) - wbY(b)) = -(5**) + r, BOUNDARY PROBLEMS AND DEVELOPMENTS. Ill it Is found by precisely the same method that V = U. In consequence we have for the Green's timet ion ( ( <& I (99) G(x, t, X) = Y(x) \\ or \ + A-1 WaY{fl) (s£) -A-1 WbY(b)(6%) r Z(t), where the upper form is to be chosen when t < z, and the lower one for t > x. By means of this formula G(x, t, Xj may be explicitly represented bj choosing as the solutions Y(x) and Z(<) a pair which are analytic in A and have the forms obtained in sections V and VI. It should be observed that G(x,t,\) is unique (see page 70), despite the fact that if n ^ 2 the choice of Y(x) and £(/) thus determined upon changes from anj one to any other of the sectors within which no quantities li\K\yJ.r) — y/xj\\ change sign, and despite the fact that the values of o,t and o,] change from any one to any other -eetor v^. From (95 we have, upon denoting by Sm(x) • the -um of those terms of the formal development of F(x)- which correspond to the charac- teristic values enclosed by the circle C in question, b (100) SJx>- ='f-l G(x, t, \) R(t) F(t)'dt rf\. C a Substituting in this the value of G(x, t, X) as found above in (99) it is 4 seen that Sm(x)- = £ S)2(x)", where the quantities S«(«)- are given t-i by the relation- 112 BIRKHOFF AND LANGER. «!?(*)• = 1-4 / I Y(x)(6*i)Z(t)R(t)F(t).dtd\ c im O a S%(x)- = L~ I f Y(.v)(dZ)Z(t)R(t)F(t)-dtd\ C SlTl (101) \ Cpv 0 Sff(*)-= 1^4 f TWA-] fwmY(a)QfS n V.irl *J *J )Z(t)R(t)F(t)-dtd\ C &TI Ciiv b &x)- = L^~. fY(x)A-i fjVbY(b)(8*j)Z(t)R(t)F(t)-dtd\ c 2-kiJ ° CM„ denoting any arc of circle C which lies within the sector a ^ and, if n ^ 2, upon which none of the quantities R {X{yt(a;) — 73-(a:)} change sign. If C^ abuts upon a ray bounding o ^ it shall either include or exclude the end point for which the quantity R {XT^b) } or the quantity R {\r„(7>)} vanishes according as the quantity in question is < 0 or > 0 within c The arc Cuv may, therefore, include one, both, or neither end point, and since the reasoning is precisely the same in each case (the sector can be split if both end points are included) we shall consider for the sake of concreteness that it includes one, namely that one which lies on the ray R {\T„(b)\ = 0. The symbol 2 indicates that the sum of the integrals over all arcs composing the circle C is to be taken. We shall proceed to evaluate each of the integrals above in turn, and in the course of this evaluation it will be convenient to refer to the facts established by the following lemmas. The notation | (p{x) \ N , (102) (ii) lim I t7 |X|=oo a Proof: Under the hypotheses (102) we have P-x lim / ipi(.r, X) dx — 0 uniformly, |X| = oo d a and | J Vl(x, X) dx | < ilf x, /3-X while it follows from these relations and the relation 0 P-x 0 I J N 1 .... lim | t \n) |X|=oo where 0X is a constant arbitrarily small but positive, then p2(X) — = 0, lim J_ f - d* Ixl=w 2™'* CQ/8 being that arc of the circle | X | = p which lies in the sector bounded by arg X = 6a and arg X = dp. 114 BIRKHOFF AND LANGER. Proof: Writing X = pet8 we have N> (104) (ii) lim tp3(x, X)| = 0 uniformlv for a ^x^ /?— x, |x|=» 0a ^argX ^ 00-0x, then /(X) = / -*& S*m(0 /,(*) 2(X)/X on arc CM„. Hence the entire matrix may be represented by M/\. The third matrix X - ^&- JVl^-W | ifi(t) eBi(l)~Bi(t) }dt) a is likewise one each of whose elements either vanishes or is, on arc C nV, of type <£>2/a by lemma 3. This matrix is, therefore, also of the type ((*-O).+ fo)) The integration of J\ • over the arc C„„ yields, therefore, by lemma 2 Cav V )f - = °^ (5Sf ]) F(x • (e), where wM„ is the angle subtended by CM„ at X = 0. A similar expression results from integration over each arc similar in character to C„„. Hence we have * sff(«) • = Z =c (ajr) ) f(x - o) • + (e). c 2tt Let us consider in detail the sum (107) L^(8\f). C 2ir BOUNDARY PROBLEMS AND DEVELOPMENTS. 117 Corresponding to each arc CM„ there exists another arc C^ - which is the reflection of CM„ in the point X = 0. Since C~; also subtends the angle coMJ, at the point X = 0 we can write the sum (107) equally well in the form (108) L Se («£ > + 5 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 2s (gjf >) = i J, C 27T as may be shown by applying again the argument by means of which relation (109) was established, we have i.e. (113) lim S<£(x)-=iF(x + 0)- TO=0O ui. sl2(z)-. Following the procedure used in the discussion of the preceding expressions, let J$- be defined by the equation 6 J3- = - Y(x) A-1 J WaY(a) (4*) Z{t) R(t)F(t)-dt. a Then S%(x)-= ~JJrd\. Now A-1=f^))16 \ D(\) '> 16 See note on page 104. 120 BIRKHOFF AND LANGER. and substituting for D(X) its value as given by equation (56) we have n 7T k-1 -\ \TAb)+BAb) | fiM Dji(\) z>(x) where Z)(X) is given for the case of regular conditions by formula (57) and for the case of irregular conditions discussed in section VII by either formula (66) or formula (68). A glance at these formulas shows, however, that in each case the expression for D(\) reduces, under the conditions for which the expression is valid to the form £(X) = W^ + on arc C'M„. Now every point of a circle C, and hence in particular of an arc C^ is at a distance which exceeds a fixed quantity from any characteristic value. However, by analogy with formula (59), for any point of C^ under regular conditions for which the function D is sufficiently small we have In other words the point in question will necessarily lie near one of the characteristic values. This stands in contradiction with the fundamental property of the circle C. Inasmuch as a similar relation holds on every arc of type C^, it follows that for every point of all circles C, D(\) exceeds, under regular conditions, some fixed positive constant 5i, i.e. (114) D(k) |= 0 for X on any circle C. While we have thus proved the relation (114) only for the case of regu- lar conditions it can be shown to hold equally well under the type of BOUNDARY PROBLEMS AND DEVELOPMENTS. 121 irregular conditions considered in section VII. In either case, there- fore, A-1 is uniformly bounded on all circles of type C. Proceeding, we have Y(x)A~u- ( l X ) D(X) fc=l ^iriW-iVWiKl+B^-BiWiii flji(X) 5(X) + f ex\rk(x)-rk(b)8Z-\+Bk(x)-Bk(b)5**k [ya-(3-0]g,fc(X) X D(X) k=l Since the last matrix of this expression is of type (e) we have or, more briefly, (115) F(a!)A-»= (,xlr*«-r*«M*« *+*i<*>-B.<6>5«Tt7(X)), where | rt7(X) | < Jfcf for | X | > Ar. ' Inasmuch as D(\) was seen to be of the form D(\) = WM + (f2 for X on arc CM„, we have for any X on this arc , v (wtf + n \ (wtf\ {Til) ~ \w^V^, + V = Vir^V + (^2)' or, denoting by 0 M" the matrix whose determinant is W{fi"\ (116) (m») (ry) = «<"" + fo). 17 Cf. Wilder, loc. cit. p. 422. 122 BIRKHOFF AND LANGER. We have, further, b f WaYifl) (8**) Z(t) R(t) F(t)-dt = r n ( Whkip)] «rm^v«- v« dmp + kl>mp(!)] dpqyg(t)Ut)dt) o A-l + 1 X A 1 X _e-^W-^»>/A (6 - 0) + /»(« + 0) + 6 , /e"UiW|pW«"B*(<)| 5*,; A(a + 0) + ^2) , may be established in similar manner by use of the relation Substituting for (t,-,) its value as given by formula (115) it is readily seen, therefore, that on arc CM„ 124 BIRKHOFF AND LANGER. 1 * -1 ** Jz(a) ■ = — (*&*) nM Wa (8\f) F(a + 0) • + fa) } A -1 ** —i ** Mb) • = — I 2) } . A It follows from this that 1 / j\ = Z — . J 00 — when -r ^ a> x ^ &- :(3) i 2iri r -1 O*)- \ 1 I * ** r/\ = L —J {-tif)QM Tra$?V(a+(». + (**)}- when .r = a 1 I ** -1 ** TV = I f-. J i - &?) &* wa (ajr>) F(« + o) • + fa) } =* c 2?ri r ** -1 ** ) I K3 = z - — $f ) " w° tif) . [ c ( 2tt ) H3) 0 when .r 41 +W/A (a + 0) + JeHThW-ThW}i {cBh(b)-Bh(0 m j J + W and multiplying this matrix by the matrix Y(x) A-1, as given by formula (115) it follows that Jv 1 £ gx| rl(x)-rl.(6)5ii ^B^-B^su T.k (X) w(«5 fc.a-i hh •^ dt -Mb-o) + 126 BIRKHOFF AND LANGER. The second matrix of the sum on the right, namely A \k.h-l ' is seen to be of the type {)/\, whereas the third, i.e. 1 X k,h=l ** ii 0 ^ is also of this type, as is apparent from the fact that by lemma 3, each of the integrals with respect to t which actually occurs in any element of the matrix is of type 2) = ^ (4) S2-1 IF6 (5*;) F(b - 0) • + ^ W, and J4(6) • = — [LCi ra(\) «$ *I» A (6 - 0) + ft j X U-,A=1 ~Y L (*J) tt-1 FP» (S^) F(6 - 0) • + i (ft), BOUNDARY PROBLEMS AND DEVELOPMENTS. 127 may be deduced, whereupon it follows that T l f / N ^ = n7T~-J vPv T when x =F «, & 4= o, S(m4)Cr) 2iri 'ia> , —J {- (S^fl0"0 JF6 $f) F(6-0) -+MJ^ C 9 J7Tt C M< when x = a. 0 Z7TI c A when x = b. Defining the matrices ^4 and Ki by the equations (119) -1 Ka= L ) I3f (aj*»>) q^> H'6 $f >) ( UP 41 — 1 '.IT it follows, therefore, that 0 (5[-f) ^ ir6 (5i-f ) , when x =(= o, .r =£ & (120) lim Sl£(x) • j = X4 F(6 - 0) • when 2 = a = K* F(b - 0) • when x = b. L A summary of the various results as contained in formulas (110), (113), (118) and (120) is seen, in virtue of (105) and (111), to yield lim Sm(x) ■ = \ Fix - 0) • + \ F{x + 0) • when x^a,x^b, TO— OO lim SM-= ± F(a + 0)- +KsF(a + 0)- + Ki F(b-0)-, TO" OO lim Sm(b)-= ±Fib-0)- + K-iFia+0)-+KiF(b-0)-, TO=0O In consequence we have the following Theorem : Given that L is any vector differential system of the type 128 BIRKHOFF AND LANGER. T(x)- = \A(x)\ + B(x)}Y(x)-, WaY(a)-+WbY(b)~ = 0, which can be reduced by a change of the dependent variable to a system of type (46) for which (a), R(x) and B(x) are continuous to- gether with their first derivatives (t),thefunctions 7j(a*), i = 1,2, . . .nt satisfy the relations (96) and (97), and (c), the condition (ii) on page 89 is fulfilled. Then the development in characteristic functions of the system L which is associated with any vector F(x) • whose elements satisfy condition (98), converges to \ F(x - 0) • + i F(x + 0) • when x 4= a, x + b, HaF(a + 0) • + JaF(b - 0) • when x = a, HbF(a + 0) • + JbF(b - 0) • when x = b, the four matrices of constants Ha, Ja, H b, and J b being explicitly determined by the matrices R(x), A(x), W„ and W b as stated. VOLUME 57. 1. Kknt, Norton A. and Taylor, Lucien B. — The Grid Structure in Echelon Spectrum Lines. pp. 1-18 December, 1921. $.75. 2. Lotka, Alfred J— The General Conditions of Validity of the Principle of Le Chatelier. pp. 19-37- January, 1922. $.75. 3. Bridgman, P. W. — The Effect of Tension on the Electrical Besistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Louis. — Notes on the Early Evolution of the Beflector. pp. 67-74. February, 1922. $.50. 5. Rridgman, P. W. — The Effect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Bridgman, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. $1.25. 7. Pierce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Refraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjaornt Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Franklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. $.65. 10. Bennitt, Rudolf. — Additions to the Hydroid Fauna of the Bermudas, pp. 239-259. May, 1922. $.65. 11. Brues, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonididw. pp, 261- 288. 2 pis. May, 1922. $.85. 12. Thaxter, Roland. — A Bevision of the Endogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda. pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. R. — Atmospheric Attenuation of Ultra-VioIet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Romberg, Arnold. — The Ratio of the Calorie at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Robert H. — Studies on Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389-422. 2 pis. November, 1922. $1.65. 17. Thaxter, Roland. — Note on Two Remarkable Ascomycetes. pp. 423—436. 2 pis. September, 1922. $1.35. 18. Records of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members; Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 2 of cover.) PUBLICATIONS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Series, Vols. 1-4; New Series, Vols. 1-14. 16 volumes, $10 each. Half volumes, $5 each. Discount to booksellers and Libraries 25%; to Fellows 50%, or for whole sets 60%. Vol. 12. 1. Sawyer, E. F. — Catalogue of the Magnitudes of Southern Stars from 0° to — 30° Declination, to the Magnitude 7.0 inclusive, pp. 1-100. May, 1S92. $1.50. 2. Rowland, H. A. — On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-186. December, 1896. $2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniaceae. pp. 187- 430. 26 pis. December, 1896. $6.00.- 4. Lowell, P. — New observations of the Planet Mercury, pp. 431-466. 8 pis. June, 1898. $1.25. 5. Sedgwick, W. T., and Winslow, C. E. A. — (I.) Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II.) Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. 13. 1. Curtiss, D. R. — Binary Families in a Triply connected Region with Especial Reference to Hypergeometric Families, pp. 1-60. January, 1904. $1.00. 2. Tonks, O. S. — Brygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T. — The Spectrum of Hydrogen in the Begion of Extremely Short Wave-Length . pp. 121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H. — Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv. November, 1906. $1.10. 5. Trowbridge, J. — High Electro-motive Force, pp. 1S1-215. pis. xxv-xxvii. May, 1907. 75c. 6. Thaxter, B. — Contribution toward a Monograph of the Laboulbeniacea1. Part II. pp. 217-469. pis. xxviii-lxxi. June, 1908. $7.00. Vol. 14. 1. Lowell, Percival. — The Origin of the Planets, pp. 1-16. pis. i-iv. June, 1913. 60c. 2. Fernald, W. E., Southard, E. E., and Taf t, A. E. — Waverley Besearches in the Pathology of the Feeble-Minded. (Besearch Series, Cases 1 to X.) pp. 17-128. 20 pis. May, 1918. $6.00. 3. Fernald, W. E., Southard, E. E., Canavan, M. M., Baeder, O. J. and Taft, A. E. — Waverley Besearches in the Pathology of the Feeble-Minded. (Besearch Series, Cases XI to XX.) pp. 129-207. 32 pis. December, 1921. $6.50. 4. Pickering, Edward C, Parkhurst, J. A., Aitken, Bobert G., Curtis, Heber D., Mitchell, S. A., Alden, Harold L., Simpson, T. McN., Beed, F. W. — Photometric Magnitudes of Faint Standard Stars, measured at Harvard, Yerkes, Lick and McCormick Observatories, pp. 209- . April, 1923. $0.00. PROCEEDINGS. Vols. 1-57, $5 each. Discount to booksellers and Libraries 25%; to Fellows 50%, or for whole sets 60%. The individual articles may be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. $5.00. Complete sets of the Life and Works of Rumford. 5 vols., $25.00: to Fellows, $5.00. For sale at the Library of The American Academy of Arts and Sciences, 28 Newbury Street, Boston, Massachusetts. 58-3 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 3.— January, 1923. LICHENES IN INSULA TRINIDAD A PROFESSORE R. THAXTER COLLECTI. By Edward A. Vainio. (Continued from page 3 of cover. ) VOLUME 58. 1. Ames, A. Jr., Proctor, C. A., and Ames, Blanche. — Vision and the Technique of Art. pp. 1-47. 28 pis. In press. 2. Birkhoff, George D. and Langer, Rldolpb E. — The Boundary Problems Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49- 128. In press. 3. Vainio, Edward A. — Lichenes in Insula Trinidad a Professore R. Thaxter Collect i. pp. 129-147. January. 1923. SI. 60. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 3.— January, 1923. LICHENES IN INSULA TRINIDAD A PROFESSORE R. THAXTER COLLECTI. By Edward A. Vainio. LICHENES IN INSULA TRINIDAD A PROFESSORE R. THAXTER COLLECTI. By Edward A. Vainio. Received July 10, 1922. Presented by R. Thaxter. Note. The following enumeration, which Dr. Vainio has been so kind as to prepare for publication, includes a portion of a small col- lection of lichens made largely in the vicinity of Port of Spain, Trini- dad, B. W. L, from January to April, 1913. (R. T.) 1. Pertusaria commutata Mull. Arg. (Vain. Et. Lich. Bres. I, 1890, p. 105). Ad corticem arboris. Maraval Valley (No. 8). Ster. 2. Pertusaria torulosa Vain. Addit. Lich. Antill. (Ann. Ac. Scient. Fennicae, Ser. A, torn. VI, 1915), p. 31. Ad corticem arboris. La Seiva Valley (No. 100). 3. Placodium ochraceum (Nyl.) Vain. Lecanora aurantiaca var. ochracea Nyl. Et. Lich. de 1' Alger (1854), p. 325 (sec. Herb. Nyl.) Ad saxa corallina in Gasparee Island (No. 62). Fert. 4. Sticta Weigelii (Ach.) Vain. (Et. Lich. Bres. I, p. 105) f. Beauvoisii (Del.) Stizenb. Griibchenflecht. (1895), p. 133. Ad corticem arboris. Sangre Grande (No. 86). Apotheciis novel- lis. 5. Pannaria stylophora Vain. (Addit. Lich. Antill. p. 102) f. disserpens Vain. Lich. Ins. Philipp. Ill (Annal. Ac. Scient. Fenn., Ser. A, torn. XV, 1921), p. 9. Ad corticem arboris ( Theobroma cacao), Sangre Grande (No. 75). Fert. 6. Pannaria rubiginosa (Thunb.) Del. f. caesiocinerea Vain., 1. c, p. 12. Ad corticem arborum. Sangre Grande (Nos. 84, 87). Fert. 7. Coccocarpia ciliata (Bel.) Vain. Thallus parce isidiosus. Etiam in specim. orig. Collem. ciliati Bel. (Voy., p. 130), n. 30981 in herb. Nyl., thallus est parce isidiosus. Ad folia exarida arboris. St. Ann's Valley (No. 49). Ster. 132 VAINIO. 8. Heppia Trinitatis Vain. (sp. n.). Thallus foliaceus, dicho- tome aut partim subirregulariter crebre repetito-laciniatus, caespites circ. 10-25 mm. latos formans, laciniis 0.4-0.7 mm. latis, sublinearibus, superne leviter convexis, subtus canaliculato-concavis, superne et subtus laevigatus et cinereo-nigrieans, glaber, rhizinis nullis distinctis, sed hyphis hypothallinis penicillato-confertis, impure albidis passim substrato affixus, crass, circ. 0.220 mm., heteromericus, in parte superiore et inferiore zonam gonidialem continens, zona medullari gonidiis destituta, strato corticali destitutus. Gonidia cyanophycea, caerulescentia aut glaucescentia, cellulis subglobosis, diam. 0.005- 0.004 mm., concatenatis aut solitariis binisve, membrana tenuissima, vagina haud evoluta, heterocystis decoloratis, aeque crassis, mem- brana distincta. Habitu Phycias in memoriam revocans. Ad rupem in rivulis in silva. St. Ann's Valley, Trinidad (No. 36). Ster. 9. Physma byrsinum (Acb.) Miill. Arg. var. hypomelaena Nyl. (Vain. Lich. Ins. Philipp., Ill, p. 45) Ad corticem arboris. Sangre Grande (No. 85). Fert. 10. Leptogium azureum (Sw.) Nyl. var. laevior Vain. 1. c. p. 37. Ad corticem arboris. Sangre Grande (No. 85). Fert. 11. Cladonia pityrea (Floerk.) Fr. f. squamulij 'era Vain. Mon. Clad. Univ. Ill, p. 255. Ad plantas destructas. Four Roads (No. 35). Ster. 12. Bilimbia Maravalensis Vain. (sp. n.). Thallus crustaceus, sat tenuis, subverruculoso-inaequalis, sat continuus, cinereo-glauces- cens, esorediatus, hypothallo indistincto. Apothecia dispersa, aut partim aggregata, lat. 0.8-0.4 mm. late adnata, basi leviter constricta, disco piano aut vulgo demum convexo, carneo-pallido, nudo, leviter nitido, margine concolore, tenui, integro, demum vulgo excluso. Peri- thecium hyphis radiantibus, sat leptodermaticis conglutinatis, cavi- tatibus cellularum 0.002-0.003 mm. latis. Hypothecium decolora- tum, hyphis irregulariter contextis, sat pachydermaticis, conglutinatis, in parte superiore cellulis depressis, in seriebus horizontalibus dis- positis. Hymenium 0.060-0.070 mm. crassum, jodo sat dilute caeru- lescens. Epithecium fere decoloratum. Paraphyses arete con- glutinatae, tenues, apice vix incrassatae. Sporae 8:nae, decolores, oblongae aut elongatae, apicibus obtusis aut rotundatis, rectae aut leviter curvatae, 3-septatae, long. 0.012-0.017, crass. 0.0025-0.003 LICHENES IN INSULA TRINIDAD. 133 mm. Gonidia pleurococcoidea, globosa, diam. 0.005-0.006 mm., simplicia aut glomerulosa, membrana sat tenui. Subsimilis Bilimbiae sphaeroidi, sed sporis tenuioribus, thallo minus evoluto et reactione hymenii ab ea differt, ad Bacidias accedens. Ad corticem arboris. Maraval Valley (No. 28). 13. Bilimbia rufopunctata Vain. (sp. nov.). Thallus crusta- ceus, sat tenuis aut sat crassus, continuus, verruculoso-inaequalis et majore parte verruculis minutissimis (circ. 0.050 mm. latis) et granu- lis soredioideis, crebris subcontiguisque instructus, glaucescens vel stramineo-virescens, hypothallo indistincto. Apothecia numerosis- sima crebraque, lat. 0.4-0.3 mm., basi leviter constricta, disco piano aut demum convexo, rufo aut obscure rufescente, nudo, opaco, margine tenuissimo, subconcolore aut paullo pallidiore, integro, demum excluso, KOH solutionem rubescentem vel subroseam effundentia. Hypo- thecium dilute fulvo-rubescens. Epithecium fulvo-rubescens, hy- menium ceterum totum fulvescens, circ 0.040 mm. crassum, jodo persistenter caerulescens. Paraphyses arete cohaerentes, simplices aut parce ramoso-connexae furcataeve, membranis leviter gelatinosis cavitatibus 0.0012 mm. crassis, apice saepe leviter incrassatis. Sporae 8:nae, decolores, oblongae aut cuneato-oblongae, rectae, apicibus obtusis, 3-1-septatae, long. 0.009-0.014, crass. 0.002-0.0035 mm. Gonidia pleurococcoidea, globosa, diam. 0.006-0.009 mm., simplicia aut glomerulosa, membrana modice incrassata aut gelatinosa. Facie externa Bacidiam inundatam in memoriam revocans. Ad truncum putridum palmae. Four Roads (No. 48). 14. Sporopodium Thaxteri Vain. (sp. n.). Thallus crustaceus, tenuis, continuus, verruculis minutissimis (circ. 0.050-0.0S0 mm. latis), intus albis crebre adspersus, stramineo- aut albido-glaucescens, sat opacus, KOH flavescens, hypothallo albido, parum distincto. Apo- thecia lat. 0.8-0.4 mm., dispersa, basi constricta, disco piano, tenuiter pruinoso, partim demum leviter denudato et livido-rufescente, mar- gine sat tenui, haud prominente, albido, opaco, persistente, glabro, plus minus distincte verruculoso. Hypothecium superne rufescens, inferne dilutius coloratum. Parathecium rufescens, modice incrassa- tum, extus strato albido aeque crasso obductum. Hymenium 0.065- 0.110 mm. crassum, totum decoloratum, jodo persistenter caerulescens. Paraphyses arete cohaerentes, ramoso-connexae. Sporae singulae, decoloratae, oblongae, apicibus rotundatis, murali-divisae, cellulis numerosissimis, long. 0.046-0.066, crass. 0.015-0.018 mm., membrana modice incrassata, haud gelatinosa. 134 VAINIO. Proximum Sp. cupulifero Nyl. et Sp. mastophoro Vain. (Lich. Ins. Philipp. Ill, p. 91), quae sporis majoribus et disco obscuriore et thallo KOH non reagente ab hac specie differunt. Ad folia arborum. St. Ann's Valley (No. 56). La Seiva Valley (No. 60). Supra thallum parce etiam observavi Cyphellam (conf. Vain. Etud. Lich. Bres. II, p. 27). 15. Sporopodium glaucophaeopsis Vain. (sp. n.). Thallus crustaceus, tenuissimus, continuus aut subdispersus, laevigatus, stramineo-glaucescens, sat opacus, KOH flavescens, hypothallo indis- tincto. Apothecia lat. 0.5-0.4 mm., dispersa aut subaggregata, basi constricta, disco piano, pallido, pruinoso, margine sat tenui, haud aut leviter prominente, parum verruculoso, stramineo-albido, opaco, persistente. Hypothecium rufeseens. Hymenium circ. 0.0S0 mm. crassum, totum decoloratum aut epithecio pallido, jodo persist- enter caerulescens. Paraphyses ramoso-connexae. Sporae singulae, decoloratae, demum pallidae, oblongae, apicibus rotundatis, murali- divisae, cellulis numerosissimis, long. 0.05 6-0. 060 (-0.030), crass. 0.018 (-0.012) mm. Conidangia (pycnidia) prominentia, depresso-sub- globosa, albida. Conidia (stylosporae) decolorata, ovoideo-oblonga, altero apice rotundato, altero obtuso, recta, simplicia, long. 0.007, crass. 0.002 mm. Gonidia globosa, diam. 0.006-0.008 mm., tantum simplicia visa, membrana tenui, forsan pleurococcoidea. Sporopodium glaucophacodcs (Nyl. Lich. Guin. p. 19) sporis majori- bus margine apotheciorum laevigata et disco carneo-fuscescente, haud pruinoso ab hac specie differt. Ad folia arboris. Maraval Valley (No. 70). 16. Lecidea coronulans (Nyl.) Vain. {Lecanora granifera var. coronulans Nyl. in Fl. 1874, p. 72), var. Gaspareina Vain. (var. n.) Sporis subglobosis aut breviter ellipsoideis, long. 0.00S-0.011, crass. 0.005-0.007 mm. a Lcc. coronulanie (Nyl.) differt. Apothecia disco cinereo fuscescente aut fusconigro, margine verruculoso, verruculis albidis, gonidia continentibus, ceterum perithecium intus fuscofuli- gineum, hyphis conglutinatis. Hypothecium fuscofuligineum, hyphis conglutinatis. Thallus albidus aut albido-glaucescens, verruculis minutis, intus albis inspersus, KOH flavescens, verruculis fulvescen- tibus (in No. 44 reactio minus distincta). Ad hanc speciem non pertinet Lecanora coronulans Nyl. (Fl. 1876, p. 510). Ad corticem arborum. Maraval Valley (No. 44). Gasparee Island (No. 38). LICHENES IN INSULA TRINIDAD. 135 17. Lecidea fuscorubescens Nyl., Not. Lich. Port. Natal (1868) p. 8 (L. Natalensis Nyl. in Cromb. Lich. Challeng. Exp. 1878, p. 216). Sporae long. 0.007-0.013, crass. 0.007-0.008 mm. in No. 64; long. 0.007-0.0010, crass. 0.006-0.007 mm. in No. 51.— L. caliginosa Stirt. est autonoma species, disco metalloideo-caerulescente differens etiam e speciminibus nomine L. fuscorubesccntis in Nyl. Lich. Ins. Andam. (1873) p. 10 descriptis. Ad corticem arborum. St. Ann's Valley (No. 51). La Seiva Valley (No. 64). 18. Lecidea canoumbrina Vain. (sp. n.). Thallus crustaceus, tennis, subcontinuus, subgranuloso-inaequalis, glauco-vireseens, opa- cus, hypothallo albido vel indistincto. Apothecia dispersa, lat. 1.2- 0.7 mm., basi late adnata, parum constricta, disco depresso-convexo, cinereo-rnfescente aut rufo-fuscescente subnitido, margine tenui, haud prominente, fnscescente, integro, parnm conspicuo, demum excluso. Hypothecium pnrpnreum aut purpureo-fuligineum, KOH non reagens, superne ex hyphis erectis, conglutinatis formatum, inferne irregulari- ter contextis conglutinatisque. Perithecium intus purpureum, extus dilute coloratum pallidumve. Hymenium circ. 0.050-0.060 mm. crassum, totum pallidum, jodo persistenter caerulescens. Paraphyses arete cohaerentes, tubulo tenuissimo. Sporae 8-nae, distichae, deeolo- ratae, fusiformes-oblongae, apicibus obtusis, simplices, long. 0.007- 0.010 crass. 0.0025-0.003 mm. Proxima est L. canorubellae Vain. Lich. Bras. Exs. (1892), No. 32, quae sporis crassioribus et hypothecio testaceo ab ea differt. Ad corticem arboris. Maraval Valley (No. 19). 19. Lecidea cinereopallida Vain. (sp. n.). Thallus crustaceus, tenuis continuus, laevigatus, glaucescens, hypothallo indistincto. Apothecia dispersa, lat. 1.2-0.8 mm., basi constricta, sat late adnata, disco piano aut demum leviter convexo, cinereo-pallescente pallidove, parum nitido, haud pruinoso, margine sat tenui, leviter aut haud prominente, impure albido aut raro subcinerascente, persistente. Hypothecium inferius, albidum, ex hyphis irregulariter contextis, conglutinatis formatum, pars superior intense fuscescens, KOH non reagens. Perithecium albidum, ex hyphis subradiantibus aut irregu- lariter contextis, sat leptodermaticis, conglutinatis, cavitatibus cellu- larum oblongis aut ellipsoideis, 0.003-0.0015 mm. latis. Hymenium circ. 0.100-0.110 mm. crassum, jodo persistente caerulescens. Para- physes arete cohaerentes, simplices, membrana leviter gelatinosa, 136 VAINIO. cavitatibus tenuibus. Asci clavati. Sporae S-nae, distichae, de- colores, fusiformes-oblongae aut ellipsoideae aut rarius subglobosae, long. 0.009-0.014, crass. 0.006-0.008 mm. Gonidia pleurococcoidea, globosa, diam. 0.006-0.004 mm. Proxima L. declinii Nyl. (haud Bilimbia declinis (Tuck.)) sive L. declinandae Nyl. (Lich. Nov. Zel. p. 146), quae thallo crassiore, verru- culoso-rugoso ab eo differt. Ad corticem arboris. Maraval Valley (No. 42). 20. Pilocarpon glabrum Vain. (sp. n.). Thallus crustaceus, sat tenuis, areolatus, areolis 0.3-0.2 (0.35-0.15) mm. latis, planis aut depresso convexis, contiguis aut partim dispersis, albido-glaucescenti- bus, leviter nitidis, nee KOH, nee HCaC^C^ reagentibus, hypothallo indistincto. Apothecia vulgo dispersa, lat. 0.7-0.5 mm., basi leviter constricta, late adnata, disco piano, nigro nudoque aut vulgo tenuiter cinereo-pruinoso, opaco, margine sat tenui, primum diu leviter promi- nente, persistente, glabro, integro, laevigato, albo, opaco. Hypothe- cium fusco-fuligineum, basi usque ad substratum conice productum, KOH non reagens. Parathecium tenue, fusco-fuligineum. Perithe- cium albidum, ex hyphis crebre contextis, haud conglutinatis forma- tum, gonidiis destitutum. Hymenium 0.120-0.130 mm. crassum, jodo persistenter caerulescens. Epithecium olivaceo-obscuratum. Paraphyses ramoso-connexae, apicibus irregulariter contextis. Sporae 8 :nae, decolores, f usiformes, leviter curvatae, apicibus sat obtusis aut altero apice rotundato, 5-7 septatae, loculis cylindricis, long. 0.052- 0.080, crass. 0.008-0.013 mm. Gonidia globosa, tantum simplicia visa, diam. 0.006-0.010 mm., membrana tenui aut sat tenui, pleurococ- coidea, ut videtur. Supra folia arboris. La Seiva Valley (No. 59). 21. Gyalecta pachyspora Vain. (sp. n.). Thallus crustaceus, tenuissimus, continuus, laevigatus, glaucescens, hypothallo indistincto. Apothecia lat. 0.4-0.3 mm., basi constricta, disco piano aut primum subconcavo, fulvescenti-carneo aut carneo-pallido, nudo, margine concolore aut pallidiore, persistente, primum diu prominente, sat tenui, integro. Perithecium albidum, plectenparenchymaticum levi- ter pachydermaticum, gonidiis destitutum. Hypothecium decolora- tum. Hymenium 0.055 mm. crassum, jodo non reagens. Asci sub- cylindrici. Sporae 8:nae, distichae, ovoideae aut ellipsoideae, vulgo altero apice rotundato, altero obtuso, aut rarius apicibus ambobus obtusis, 1-septatae, long. 0.008-0.0017, crass. 0.004-0.008 mm. Goni- LICHENES IN INSULA TRINIDAD. 137 dia ad Heterothallum pertinentia, irregulariter ramosa, cellulis irregu- lariter oblongis aut difformibus, 0.004-0.005 mm. crassis. Sporis majoribus differt a G. epiphylla (Mull. Arg.) Vain. Lich. Ins. Philipp. Ill, p. 148. Supra folia arboris. Maraval Valley (No. 69). 22. Coenogonium interplexum Nyl. Obs. Coenog. p. 92 ; Vain. Addit. Lich. Antill. p. 131. Sporae long. 0.010-0.011, crass. 0.002- 0.0015 mm., 1-septatae. Gonidia cellulis 0.030-0.036 mm. longis, 0.018-0.020 mm. crassis. Ad ramos arbusculorum. Aripo Savanna, Cumuto (No. 32). Fert. 23. Tricharia Amazonum Vain. (sp. n.). Thallus tenuis aut tenuissimus, continuus, glaucescens, leviter nitidus, laevigatus aut demum leviter inaequalis, pilis ornatus nigris, dispersis, apicem versus sensim attenuatis, basi in bulbum nigrum, vulgo haud prominentem, incrassatis, hypothallo indistincto. Apothecia numerosa et sat crebra aut sat dispersa, orbicularia, 0.25-0.2 mm. lata, lecideoidea, promi- nentia, basi tota adnata, nee constricta, disco leviter concavo aut planiusculo, rufescente aut pallido-rufescente, nudo, margine tenui, prominente, integro, fuscescente, persistente. Hypothecium albidum. Perithecium margine dilute fuscescens, ceterum pallidum. Hymen- ium jodo non reagens; paraphyses ramoso-connexae, tubulis 0.0005 mm. crassis, gelatinam hymenialem percurrentes. Sporae 8:nae aut pauciores, etiam solitariae, decoloratae, oblongae, apicibus rotundatis, murali-divisae, cellulis numerosissimis, long. 0.065-0.066, crass. 0.016- 0.026 mm., membrana leviter gelatinoso-incrassata. Gonidia pleuro- coccoidea, globosa, tantum simplicia visa, diam. 0.005-0.007 mm. Tricharia melanothrix (Vain. Lich. Ins. Philipp. Ill, p. 159) huic speciei proxime est affinis, praesertim sporis minoribus, tantum soli- tariis ab ea differens. Pilis crebris, bulbisque prominentibus planta in Fee. Ess. Crypt. Ecorc. tab. Ill, f. 18 a et c delineata magis speciei Philippinae, quam Trinidadensi, similis est. Supra folia coriacea arboris, La Seiva Valley (No. 50). Fert. Hue verisimiliter etiam pertinet No. 47 in Maraval Valley sterilis collectus. 24. Thelotrema (Ocellularia) sublilacinum (Ellis) Vain. (comb. n.). Karstenia sublilacina Ellis in Smith Centr. Amer. Fungi, No. 49 (sec. Thaxter in litt.) Sacc. Syll. Fung. XIV (1899) p. 810. Thallus modice incrassatus, laevigatus aut partim rugosus vel verru- coso-inaequalis, glaucescens vel stramineo-glaucescens, KOH demum leviter rubescens, gonidia trentepohliacea continens, hypothallo indis- 138 VAINIO. tincto. Apothecia vulgo dispersa, disco 4-1 mm. lato, immerso, sub- lilacino-carneo aut carneo-pallido, tenuissime, pruinoso, piano aut concavo, orbiculari aut leviter irregulari, margine prominente, discum et thallum superante, intus albido, leviter lacerato aut subintegro, extus strato thallino plus minus lacerato cincto. Parathecium albi- dum. Hypothecium albidum, subtus linea fuscescente limitatum. Paraphyses simplices. Sporae 8:nae-4:nae, fusiformi-elongatae, api- cibus obtusis, septis transversis 16-19 (14-23 sec. Sacc. I.e.), decolo- ratae, jodo violascentes, long. 0.060-0.094 crass. 0.010-0.012 mm. loculis lenticularibus. Ad corticem arboris. St. Ann's Valley (No. 36). 25. Thelotrema (Ocellularia) platycarpella Vain. (sp. n.). Thallus sat tenuis, leviter subverruculoso-inaequalis, olivaceo-vir- escens, leviter nitidus, hypothallo indistincto. Apothecia partim nonnulla aggregata vel confluentia, disco 0.8-0.3 mm. lato, immerso, orbiculari aut leviter irregulari, piano, bene pruinoso, caesio, margine prominente, discum et thallum superante, tenui, albo, minute lacerato. Parathecium et hypothecium albida. Hymenium circ. 0.045 mm. crassum, columella nulla. Epithecium albidum. Paraphyses sim- plices. Sporae 8:nae, distichae, altero apice rotundato obtusove, altero acuto, decoloratae, jodo non reagentes, septis transversis 4-5, long. 0.012-0.014, crass. 0.005 mm., membrana haud gelatinosa. x\ffinis speciei praecedenti, sed omnibus partibus minoribus. Ad corticem arboris. Verdant Vale, Arima (No. 57). 26. Thelotrema (Phaeotrema) difforme (Tuck.) Vain. Lich. Ins. Philipp. Ill, p. 194 (Graphis subnivescens Nyl. Fl. lS86,p. 174, Lich. Guin., 1889, p. 27). Apothecia aggregata aut confluentia, orbicularia aut difformia, margine albo, disco aperto, obscure cinerescente, tenu- iter pruinoso. Perithecium album. Paraphyses simplices. Sporae 8:nae, monostichae, fumoso-iuscescentes, oblongae, apicibus obtusis, 3-septatae, loculis lenticularibus, long. 0.0011-0.013, crass. 0.005 mm. Thelotrema platycarpoidi Tuck, affine, et ad species inter Thelo- trema et Graphidem intermedias pertinet. Ad corticem Mangiferae. La Seiva Valley (No. 31). 27. Graphis (Phaeographina) chrysocarpa (Raddi) Eschw. Ad corticem arborum. La Seiva Valley (No. 52). Maraval Valley (No. 12). 28. Graphis dissimilis (Nyl.) Vain. (Gr. sculpturata f. dissimih's Nyl. Prodr. Fl. Novo-Granat.*Lich. Addit., p. 564). Thallus KOH LICHENES IN INSULA TRINIDAD. 139 rubescens. Perithecium tenue, latere fuligineum, basi albidum. Sporae binae aut raro solitariae, murali-divisae, obscuratae, long. 0.045-0.084, crass. 0.018-0.025 mm. Sporae saepe binae etiam in specim. orig. see. annot. Nyl. Ad corticem arborum. Maraval Valley (Nos. 13 and 15). 29. Graphis (Phaeographis) haematites Fee. Ad corticem arboris. La Seiva Valley (No. 71). 30. Graphis Sangrensis Vain. (sp. n.). Thallus bene evolutus, modice incrassatus aut sat tenuis, laevigatus, stramineo-glaucescens, leviter nitidus, KOH demum intense ferrugineo-rubescens, partim linea hypothallina nigricante limitatus. Apothecia elongata, long. circ. 4-7, lat. 0.3-0.4 mm., flexuosa curvatave, increbre dichotome vel varie ramosa, subsolitaria aut parce aggregata, leviter elevata aut subim- mersa, apicibus obtusis aut attenuatis. Discus dilatatus, planus aut leviter concavus, cinereus, pruinosus, margine albo, discum vulgo superante, cinctus. Perithecium tenue, subpallidum et partim fus- cescens, aut evanescens. Hymenium circ. 0.090 mm. crassum. Sporae 4:nae, cylindricae, apicibus obtusis aut rotundatis, fusces- centes, septis transversis 5, long. circ. 0.027, crass. 0.009 mm. Affinis Gr. medusaeformi Kremplh. (Vain. Etud. Lich. Bres. II, p. 115). Gr. inusta Ach. sec. herb. Ach. thallo tenuissimo, KOH haud distincte reagente, apotheciis brevioribus, tenuiter pruinosis et sporis 5-septatis, 8:nis ab his distinguiter. Ad corticem arboris. Sangre Grande (No. 88). 31. Graphis tricosa Ach. Ad corticem arboris. Maraval Valley (No. 93). 32. Graphis labyrinthica (Aeh.) Vain. var. quatuorseptata Vain. Lich. Ins. Philipp. Ill, p. 230. Sporae in eodem apothecio 3-4-septatae, long. 0.015-0.017, crass. 0.007 mm., fumoso-obscuratae. Ad corticem arboris ( Theobroma cacao). Sangre Grande (No. 78). 33. Graphis Feei (Messn.) Vain. 1. c. p. 229. Sporae 3-septatae, long. 0.014-0.017, crass. 0.007 mm., fumoso-fuscescentes. Ad corticem arborum ( Theobroma cacao etc.). Sangre Grande (No. 82 pro parte). Maraval Valley (No. 24). 34. Graphis difformis Vain. 1. c. p. 233. Vix nisi sporis 5-septatis a Gr. heteroclitica (Mont.) Vain, differt. Thallus, sicut etiam in specim. orig. KOH demum rubescens. Ad corticem arboris ( Theobroma cacao). La Seiva Valley (No. 53). 140 VAINIO. 35. Graphis regressa Vain. (sp. n.). Thallus modice incrassatus, glaucescens, KOH lutescens deindeque ferrugineo-rubescens. Sporae 3-septatae, long. 0.016-0.017, crass. 0.006-0.007 mm. Gr. difformis et Gr. regressa forsan sunt variationes Gr. hetero- cliticae, cui habitu similes sunt. Ad corticem arborum ( Theobroma cacao). Sangre Grande (No. 79). 36. Graphis dilatescens Vain. (sp. n.). Thallus modice incrassa- tus, sat laevigatus, glaucescens, leviter nitidus, KOH lutescens deinde- que ferrugineo-rubescens. Apothecia long. 18-1 mm. lat. 1.5-0.7 mm., simplicia aut dichotome vel irregulariter ramosa vulgo leviter flexu- osa, bene aut modice elevata, alt. 0.4-0.2 mm., apicibus vulgo rotun- datis, lateribus praeruptis aut demum constrictis, margine sat crasso, albido, thallino, sicut thallus reagente, discum vulgo superante, saepe etiam zeorino, margine interiore tenui, albo, rima e margine thallino sejuncta. Discus apertus, planus, caesio-cinerascens, tenuiter pruino- sus, transversim et longitudinaliter rimis et marginibus tenuibus albisque saepe tantum defecte et plus minus increbre divisus. Hypo- thecium crassum, fuscofuligineum. Parathecium fuscum, tenue aut partim evanescens aut ad superficiem apothecii deficiens. Epithe- cium fuscofuligineum. Sporae 8: nae, distichae, fumoso-fuscescentes, oblongae, subcylindrieae, apicibus rotundatis obtusisve, septis trans- versis 5, long. 0.018-0.022, crass. 0.007-0.009 mm. Proxima est Gr. Labuanae Nyl. (Vain. Lich. Ins. Philipp. Ill, p. 224), thallo crassiore et disco latiore ab ea differens. Ad corticem arborum {Theobroma cacao). La Seiva Valley (No. 54). Sangre Grande (No. 80). No. 54 pertinet ad f. constrictam Vain, apotheciis magis elevatis, basi constrictis dignotam. 37. Graphis (Geaphina) Maravalensis Vain. (sp. n.). Thallus tenuis, laevigatus, cinerascenti-albidus, parum nitidus, KOH non reagens, linea hypothallina nigricante limitatus. Apothecia long, circ. 4-1, lat. 0.45-0.25 mm., sat dispersa, curvata flexuosave, simpli- cia aut parce furcata, leviter prominentia, margine modice incrassato, KOH lutescente deindeque f errugineo-rubescente. Perithecium album, labiis apertis, amphithecio thallino obductis, latere haud praerupto. Discus apertus, leviter impressus, pallidus, sat tenuiter pruinosus. Hymenium jodo dilute caerulescens. Epithecium albidum. Sporae 4-2:nae decoloratae, jodo violascentes, murali-divisae, cellulis numer- osis, long. 0.040-0.060, crass. 0.014-0.021 mm., strato gelatinoso indutae. Proxima est Gr. subobtectae Nyl. (Lich. Ins. Andaman, p. 18), LICHENES IN INSULA TRINIDAD. 141 quae apotheciis crebris, epithecio obscurato et sporis majoribus ab ea recedit. Gr. affinissima et Gr. tetraphora Nyl. habitu ab ea magis differunt. Ad corticem arboris. Maraval Valley (No. 94). 38. Graphis exsolvens Vain. (sp. n.). Thallus modice incrassa- tus aut sat tenuis, sat laevigatus, glaucescens aut ad ambitum partim subroseus, opacus, nee KOH nee jodo reagens, fragilis, inferne strato hypothallino tenui, nigro crebre contexto, totus obductus, demum substrato saltern passim laxe adhaerens. Apothecia subdendroideo- aggregata aut dispersa, elongata aut brevia, long. 7-0.5, lat. 0.55-0.3 mm., dichotome ramosa aut simplicia, leviter prominentia, margine modice incrassato, KOH demum leviter subfulvescente. Perithecium album, labiis apertis, amphithecio thallino obductis, latere praerupto. Discus apertus, leviter impressus, planus, livido-pallescens, pruinosus. Hymenium jodo dilutissime caerulescens. Epithecium dilute oliva- ceum. Sporae solitariae, decoloratae, jodo caerulescentes, murali- divisae, cellulis numerosissimis, long. circ. 0.120, crass. 0.040-0.050 mm., membrana haud gelatinosa. Habitu subsimilis Gr. fissurinoideae (Nyl.) Vain. (Fl. Koh Chang, p. 360), quae thallo jodo caerulescente ab ea differt. Ad ramos arborum. Maraval Valley (No. 14). 39. Graphis collosporella Vain. (sp. n.). Thallus tenuis, laevi- gatus, albido-glaucescens aut cinerascenti-albidus, KOH lutescens deindeque fulvescens aut aurantiaco-subfulvescens arete adnatus, linea hypothallina nigricante limitatus. Apothecia sat crebra, long. 2.2-0.4, lat. 0.25-0.3 (-0.5) mm., leviter curvata fluxuosave, simplicia aut raro furcata, leviter prominentia, margine sat tenui, KOH lutes- cente deindeque aurantiaco-subrubescente. Perithecium album, labiis demum apertis, amphithecio thallino obductis, basim versus sensim dilatatis aut demum sat praeruptis. Discus demum apertus, angus- tus, lat. 0.15-0.2 mm., planus, leviter impressus, livido-pallescens, tenuiter pruinosus. Hymenium jodo dilute caerulescens. Epithe- cium impure pallidum. Sporae 4:nae aut binae, decoloratae, jodo caerulescentes, murali-divisae, cellulis numerosissimis, long. 0.050- 0.062, crass. 0.014-0.020 mm., strato gelatinoso tenui aut sat crasso vulgo indutae. Affinis G. collosporae Vain. (Addit. Lich. Antill. p. 153). Ad corticem arborum. Maraval Valley (Nos. 9-10). La Seiva Valley (No. 97). 142 YAINIO. 40. Graphis (Scolaecospora) rufula Mont. (Vain. Lich. Ins. Philipp. Ill, p. 259). Ad corticem arboris. St. Ann's Valley (No. 33). 41. Graphis timida Vain. (sp. n.). Thallus hypophloeodes, macula olivaceo-pallida indicatus, hypothallo nigricante partim limi- tatus. Apothecia sat crebra, long. 0.4-1.2 (-0.3) mm., lat. 0.3-0. 4mm., substrato erumpentia, vulgo recta, labiis subalbidis, conniventibus, fragmenta substrati continentibus aut substrato obductis, clausis aut demum rima angusta disjunctis, basim versus sensim dilatatis. Discus rimaeformis. Perithecium album. Sporae 8:nae, distichae, decolo- ratae, altero apice rotundato, altero obtuso, septis transversis 3, long, circ. 0.016, crass. 0.006 mm., jodo non reagentes (sed tantum novellae visae) . Gr. timidulam Nyl. (Flor. 18S6, p. 174) facie externa in memoriam revoeans. Ad corticem arboris. Maraval Valley (Xo. 30). 42. Graphis anguilliformis Tayl. (Vain. Addit. Lich. Antill. p. 156) var. infecunda Vain. (nov. v.) Sporis binis aut solitariis, long. 0.082-0.9S, crass. 0.013-0.016 mm., 17-19-septatis a f. typica differt, ceterum ei etiam habitu omnino similis. Ad corticem arboris. Maraval Valley (Xo. 92). 43. Graphis subcaesia (Nyl.) Vain. Lich. Ins. Philipp. III,p.254. Thallus subcaesio-albidus, opacus, KOH non reagens (similis in specim. orig. n. 7163 in herb. Nyl.). Perithecium fuligineum, dimidiatum. Sporae 8:nae, decoloratae, septis transversis 10-6, long. 0.017-0.038, crass. 0.007-0.010 mm., jodo caerulescentes. Ad corticem arborum. Maraval Valley (Xos. 11 and 17). Sangre Grande (Xo. 83). 44. Chiodecton (Stigmatidiopsis) seriale Ach. Ad Corticem arboris. St. Ann's Valley (Xo. 55). 45. Chiodecton (Byssophorum) cineritium (Ach.) Vain. var. coraUina Vain. Lich. Antill. Elliott (Journ. of Botany 1S96), p. 29. Ad corticem arboris. Verdant Vale, Arima (Xo. 61). 46. Arthonia (Arthothelium) Candida (Krempelh.) Vain. (comb, n.). Myriostigma candidum Krempelh. Lich. foliic. (1874), p. 22, Lich. Beccari Born. (X. Giorn. Bot. Ital. II, 1875), p. 45. Arthothelium Mull. Arg. Lich. Beitr. (Fl. 1890) no. 1544. LICHENES IN INSULA TRINIDAD. 143 var. hypocreoides (Ferd. & Winge) Vain. Myxotheca hypocreoides Ferd. & Winge in Bot. Tidskr. XXX (1910), p. 212 (sec. Thaxter); Sacc. Syll. Fung. XXII (1913) p. 5S2. Thallus tenuis, continuus, albido-glauceseens, opacus, laevigatus, hypothallo albido partim anguste limitatus. Apothecia sat erebra, difformia aut subrotundata, lat. 1.5-0.5 mm., leviter piominentia, valde depresse subeohvexa, alba, opaca, immarginata, submembranacea. Hypothecium albidum, tenue. Hymenium totum albidum, impellucidum, jodo eaerulescens. Paraphyses ramoso-eonnexae ramosaeque, baud gelatinosae. Asci vulgo subglobosi, diam. circ. 0.064-0.080 mm., primum membrana bene inerassata. Sporae S:nae aut 4:nae, decoloratae, oblongae, apicibus rotundatis aut subobtusis, saepe curvatae, murali-divisae, cellulis numerosissimis, long. 0.051-0.064, crass. 0.018 mm. Sec. Ferd. & Winge 0.064-0.074, crass. 0.018-0.020 mm. Gonidia ad Trentepoh- liam pertinentia, cellulis concatenatis aut pro parte simplicibus, globosis aut subellipsoideis, 0.008-0.010 mm. crassis, membrana sat tenui. Ab A. Candida (Krempelh.) et A. cardinali (Krempelh.) apotheciis minoribus differt et forsan est autonoma species. In pagina superiore et inferiore Hymenophyllacearum et foliorum variarum arborum etc. Maraval Valley (No. 91) et vulgatim per insulam. 47. Arthonia (Euarthonia) thamnocarpa Vain. (sp. n.) Thal- lus sat tenuis, continuus, laevigatus aut parum distincte verruculo- sus, albidus, leviter nitidus, KOH lutescens, hypothallo indistincto. Apothecia long. 10-3 mm., dichotome repetito-ramosa, disco immerso, nigro, nudo, vix 0.1 mm. lato, immarginato. Hypothecium albidum. Hymenium praesertim latere jodo eaerulescens, epithecium suboliva- ceo-fuscescens. Paraphyses apice incrassatae. Asci clavati. Sporae 8:nae, distichae, decoloratae, ovoideo-oblongae, apicibus rotundatis, 3-septatae, Ioculis sat aequalibus autjoculo apicis crassioris reliquis duplo longiore, membrana haud gelatinosa, long. 0.016-0.017, crass. 0.005-0.006 mm., jodo non reagentes. Ab A. dispartibili Nyl. apotheciis multo longioribus et sporis minori- bus differ t. Ad corticem arboris. Sangre Grande (No. 89). 48. Astrothelium conicum Eschw. (in Mart. Fl. Bras. Lich., 1833, p. 163) var. pallida Mull. Arg. (Pyr. Cub. 1885, p. 382). Ad ramos arboris. Aripo Savanna, Cumuto (No. 1). 144 vaixio. 49. Thelenella (Phyllobathelium) epiphtlla (Mull. Arg.) Vain. Etud. Lich. Bres. II (1890) p. 216. Supra folia arboris. La Seiva Valley (No. 45). 50. Thelenella Thaxteri Vain. (sp. n.). Thallus cinereo glauces- cens, nitidus, tenuis, primordiis pycnidiorum crebre inspersus, hypo- thallo nigricante partim limitatus. Apothecja sat crebra, verrucas hemisphaericas, basi haud constrietas, thallo obductas, formantia, saepe 2-3 confluentia aut simplicia. Peritbecium fuscofuligineum, basi plus minus late albidum, KOH solutionem flavo-virescentem efTundens. Nucleus depresso-subglobosus. Parapbyses breves, crass. 0.002 mm., simplices, periphysibus licbenum similes. Sporae 8: nae, murali-divisae, cellulis numerosis, demum pallidae, jodo non reagentes, long. 0.064-0.084, crass. 0.020-0.022 mm., membrana baud gelatinosa. Gonidia ramosa, cellulis 0.003-0.004 mm. crassis, ellip- soideis aut irregulariter oblongis. Sporis majoribus et verrucis apotbeciorum basi nee constrictis nee praeruptis a Th. epiphylla differt, sed forsan non est autonoma species. Supra folia coriacea arboris. La Seiva Valley (No. 46). Var. heterogena Vain. (var. n.). Ad corticem arboris crescens, vix nisi statione a Th. Thaxteri differt. Peritheeium basi albidum, cete- rum fuscofuligineum, KOH solutionem flavo-virescentem effundens. Parapbyses breves, periphysibus similes, simplices. Sporae in ascis numero variabiles, decoloratae, murali-divisae, cellulis numerosissimis, long. 0.078-0.102, crass. 0.01S-0.03 mm., membrana haud gelatinosa, jodo non reagentes. Gonidia trentepohliacea, ramosa, cellulis globo- sis aut ellipsoideis aut irregularibus concatenatis, 0.003 (-0.006) mm. crassis, membrana sat tenui. Ad corticem arboris. La Seiva Valley (No. 66). 51. Thelenella (Microglaexa) elaeophthalma Vain. (sp. n.). Thallus sat tenuis, continuus verruculis crebre inspersus, cinereo-glau- cescens, nitidus, hypothallo nigricante limitatus. Apothecia sat crebra, verrucas formantia hemisphaericas, 0.8-0.6 mm. latas, basi sat praeruptas, haud constrietas, sat laevigatas, olivaceo- aut cinereo- glaucescentes, vertice circiter 0.2 mm. lato fusco-nigricante, haud prominente, instructas, ostiolo minutissimo. Peritheeium ceterum albidum, tenue, strato thallino obductum. Paraphyses simplices crass. 0.0015 mm., gelatinam abundantem percurrentes. Sporae 8: nae distichae, decoloratae, jodo non reagentes, murali-divisae, cellulis numerosissimis, long. 0.065-0.090, crass. 0.024-0.026 mm., membrana 0.0025-0.006 mm. crass. LICHENES IN INSULA TRINIDAD. 145 Facie externa Porinis similis. Ad corticem arboris. La Seiva Valley (No. 65). 52. PSEUDOPYRENULA (TRYPETHELIUM) ANNULARIS (Fee) Milll. Arg. (Vain. Addit. Lich. Antill. p. 196). Ad corticem arborum. Aripo Savanna, Cumuto, (Nos. 6 and 90). 53. Pseudopyrenula (Bathelium) tropica (Ach.) Mull. Arg. (Vain. Etud. Lich. Bres. II, p. 210). Ad corticem arboris. Maraval Valley (No. 27). 54. Pseudopyrenula (Polymeria) calospora Mull. Arg. (Pyr. Cub. p. 409) var. rhodocheila Vain. (var. n.). Thallus sat crassus, glaucescens, leviter inaequalis. Apothecia immersa, margine ostiolari saepe annulum 0.2 mm. latum, baud aut leviter prominentem, roseum formante. Perithecium fuligineum. Paraphyses ramoso-connexae. Sporae decoloratae, elongatae, septis transversis 14-18 divisae, apices versus loculis diminutis, long. 0.090-0.160, crass. 0.018-0.024 mm. Ad corticem arboris. La Seiva Valley (No. 63). 55. Pyrenula (Melanotheca) aggregata Fee (Mull. Arg. Pyr. Feean. p. 18). Ad corticem arboris. Maraval Valley (No. 26). 56. Pyrenula (Eupyrenula) Maravalensis Vain. (sp. n.). Thallus hypophloeodes, macula olivaceo-pallida indicatus, hypo- thallo nigricante limitatus. Apothecia sat dispersa, verrucas for- mantia subconoideo- aut hemisphaerico-depressas, 0.6-0.5 mm. latas, nigras, nudas, leviter nitidas. Perithecium subhemisphaericum dimi- diatim fuligineum, latere extus acutato, haud membranaceo dilatato. Sporae 8:nae, monostichae, fuscescentes, subfusiformes, apicibus obtusis, 3-septatis, loculis apicalibus valde minutis, long. 0.021-0.025, crass. 0.010 mm. Ad corticem ramorum arboris. Maraval Valley (No. 20). 57. Pyrenula novemseptata Vain. (sp. n.). Thallus epiphloeo- des, sat tenuis, sat laevigatus, substramineo-glaucescens, leviter niti- dus, KOH subfulvescens deindeque rubescens, hypothallo nigricante partim limitatus. Apothecia sat crebra, 1-0.7 mm. lata, verrucas formantia conoideo-hemisphaericas, nudas, parum nitidas, margine ostiolari leviter conoideo-prominente. Perithecium conoideo-hemi- sphaericum, fusco-fuligineum, basi tenue. Sporae fuscescentes, fusi- 146 VAINIO. formes, apicibus anguste obtusis, septis transversis 7-9, apices versus loeulis diminutis, long. 0.036-0.060, crass. 0.016-0.020 mm. Ad ramos arborum. Aripo Savanna, Cumuto (No. 3). 58. Pyrenula aspistea Ach. (Vain. Fl. Koh Chang Lich. p. 379). Ad corticem arboris. Maraval Valley (No. 16). 59. Pyrenula subconfluens Vain. (Etud. Lich. Bres. II, 1890, p. 202; Lich. Bras. Exsic. no. 470). Apothecia pro parte confluentia, strato thallino tenuiter velata. Perithecium integrum, hemisphaeri- cum. Sporae 3-septatae, long. 0.021-0.024, crass. 0.010-0.011 mm., loeulis apicalibus paulo minoribus. Ad corticem arboris (Theobroma cacao). Sangre Grande (No. 76). 60. Strigula difformis Vain. (sp. n.). Thallus plagulas 8-27 mm. latas formans, laciniatus, laciniis glaucescenti-albidis, leviter nitidis, glabris, leviter verruculoso-inaequalibus, bene discretis, iteratim dichotome aut digitatim ramosis, leviter convexis, haud sulcatis, majore parte sublinearibus et 0.1-0.15 mm. latis, passim apicibus subspathulatis vel cuneato-dilatatis et 0.2-1.3 mm. latis, hypothallo indistincto aut dilute fumoso-obscurato tenuissimoque. Apothecia dispersa, neque numerosa,diam. 0.5-0.6 mm., verrucas subhemisphaeri- cas formantia, thallino-velata, apice dilute obscurata. Perithecium dimidiatim fusco-fuligineum. Asci cylindrici. Sporae 4:nae, mono- stichae, oblongae aut fusiformi-oblongae, rectae aut leviter curvatae, apicibus obtusis aut subrotundatis, 1-septatae, constrictae, decolora- tae, long. 0.012-0.015, crass. 0.0045-0.006 mm. Gonidia phycopelti- dea, glabra, cellulis elongatis, 0.002 mm. latis aut angustioribus, in membranam connatis. Sir. argyronema Mull. Arg. (Pyrenoearp. Cub. p. 379) sporis majoribus et laciniis subsulcatis ab hac specie differt. Supra folia coriacea arboris. Maraval Valley (No. 21). Var. Arimensis Vain. (var. n.). Sporae 8:nae, monostichae, sub- fusiformes, apicibus obtusis aut sat acutis, rectae, saepe leviter con- strictae, long. 0.014-0.015, crass. 0.0035-0.0045 mm. -Non sit auto- noma species. Supra folia coriacea arboris. Verdant Vale, Arima (No. 58). 61. Porina granulifera Vain. (sp. n.). Thallus modice incrassa- tus, continuus, hypothallo nigricante limitatusr cinereo-glaucescens, nitidus, leviter verruculoso-inaequalis, demum verruculis minutissi- mis, subsorediosis, crebre inspersus, hypothallo nigricante limitatus. LICHENES IN INSULA TRINIDAD. 147 Apothecia sat crebra, verrucas formantia mammaeformes, 0.7-0.4 mm. latas, demum basi praeruptas aut le\issime constrictas, amphi- thecio thallino, thallo concolore, laevigata aut leviter verruculoso obducta, margine ostiolari nigro, maculam aut verruculam leviter prominentem, 0.2-0.15 mm. latam formante. Perithecium apiee fuscescens, ceterum pallidum. Sporae fusiformes, apicibus acutis, decoloratae, septis transversis 3-7 divisae, loculis sat aequalibus, long. 0.032-0.036, crass. 0.005-0.000 mm., membrana haud gelatinosa. Proxime affinis P. isidiophorae Vain. (Addit. Lich. Antill. p. 203) et P. mastoideae (Ach.) Vain. (Etud. Lich. Bres. II, p. 222). Ad corticem arboris. Maraval Valley (No. 29). 62. Porina epiphylla Fee var. praestans (Nyl.) Vain. (Lich. Ins. Philipp. Ill, p. 305). Supra folia coriacea arborum. Maraval Valley (Nos. 22-23). 63. Arthropyrenia (Anisomeridium) infernalis (Mont.) Mull. Arg. (Lich. Beitr. in Fl. 1884, No. 879) var. rhynchostoma Vain, (var. n.). Margine ostiolari demum bene excrescente denudatoque a forma typica hujus speciei differens. Thallus modice incrassatus, cinereo-glaucescens. Perithecium fuligineum, integrum. Paraphyses ramoso-connexae. Asci clavati, long. circ. 0.200, crass. 0.040 mm. Sporae 8:nae, distichae, decoloratae, ovoideae, 1-septatae, cellula inferiore multo breviore, long. 0.039-0.050, crass. 0.018-0.020 mm. Ad corticem arboris. Sangre Grande (No. 73). 64. Didymosphaeria megalospora Vain. (sp. n.). Apothecia vulgo dispersa, verrucas depresso-hemisphaericas demum formantia. Perithecium hemisphaericum, integre fuligineum, lat. circ. 0.6-0.8 mm., basi saepe anguste membranaceo-dilatato. Paraphyses crass, circ. 0.0007 mm., ramoso-connexae. Sporae binae, dilute fumoso nigricantes, oblongae aut ovoideae, rectae aut leviter curvatae, api- cibus obtusis aut rotundatis, 1-septatae, medio saepe constrictae, long. 0.058-0.066, crass. 0.020-0.026 mm., membrana leviter incras- sata. Ad fungos pertinat, gonidiis destituta, sed passim parce Trentepohl- iae in cellulis substrati observantur. Ad corticem laevigatum arboris. St. Ann's Valley (Nos. 99 and 68). VOLUME 57. I. Kent. Norton \. and Taylor, Lucien B. — The Grid Structure id Echelon Spectrum Lines, pp. 1-18. December, 1921. $.75. S. Lotka. Alfred J. — The General Conditions of Validity of the Principle of Le Chatelier. pp. 19-37 January, 1922. $.75. 3. Bridgman. P. W. — The Effect of Tension on the Electrical Besistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Louis. — Notes on the Early Evolution of the Beflector. pp. 67-74. February, 1922. $.50. 5. Bridgman, P. W. — The Effect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Bridgman, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. $1.25. 7. Pierce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Befraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjacent Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Franklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. $.65. 10. Bennitt, Budolf. — Additions to the Hydroid Fauna of the Bermudas, pp. 239-259. May, 1922. $.65. 11. Brues, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonididse. pp, 261- 28S. 2 pis. May, 1922. $.85. 12. Thaxter, Boland. — A Bevision of the Endogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda. pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. R. — Atmospheric Attenuation of Ultra- Violet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Romberg, Arnold. — The Ratio of the Calorie at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Robert H. — Studies on_ Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389-422. 2 pis. November, 1922. $1.65. 17. Thaxter, Roland. — Note on Two Remarkable Ascomycetes. pp. 423-436. 2 pis. September, 1922. $1.35. 18. Records of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members; Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 2 of cover.) PUBLICATIONS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Series, Vols. 1-4; New Series, Vols. 1-13. 16 volumes, $10 each. Half volumes, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. Vol. 11. Part 1. Centennial Celebration. 1880. pp. 1-104. 1882. $2.00. Part 2. No. 1. Agassiz, A. — The Tortugas and Florida Reefs, pp. 105-134. 12 pis. June, 1885. (Author's copies, June, 1883.) $3.00. Part 3. Nos. 2-3. Searle, A. — The Apparent Position of the Zodiacal Light, pp. 135-157, and Chandler, S. C. — On the Square Bar Micrometer, pp. 158-178. October, 1885. $1.00. Part 4. No. 4. Pickering, E. C. — Stellar Photography, pp. 179-226. 2 pis. March, 1886. $1.00. Part 4. No. 5. Rogers, W. A., and Winlock, Anna. — A Catalogue of 130 Polar Stars for the Epoch of 187.5.0, resulting from the available Observations made between I860 and 1SS5, and reduced to the System of the Catalogue of Publication XIV of the Astrono- mische Gesellschaft. pp. 227-300. June, 1886. 75c. Part 5. No. 6. Langley, S. P., Young, C. A., and Pickering, E. C— Pritchard's Wedge Photometer, pp. 301-324. November, 1886. 25c. Part 6. No. 7. Wyman, M. — Memoir of Daniel Treadwell. pp. 325-523. October. 1887. $2.00. Vol. 12. 1. Sawyer, E. F.— Catalogue of the Magnitudes of Southern Stars from 0° to —30° Declination, to the Magnitude 7.0 inclusive, pp. 1-100. May, 1892. $1.50. 2. Rowland, H. A. — On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-186. December, 1S96. $2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniaceae. pp. 187- 430. 26 pis. December, 1896. $6.00. 4. Lowell, P. — New observations of the Planet Mercury, pp. 431-466. S pis June, 1S98. $1.25. 5. Sedgwick. W. T., and Winslow, C. E. A. — (I.) Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II.) Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. 13. 1. Curtiss, D. R. — Rinary Families in a Triply connected Region with Especial Reference to Hypergeometrie Families, pp. 1-60. January, 1904. $1.00. 2. Tonks, O. S. — Rrygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T. — The Spectrum of Hydrogen in the Region of Extremely Short Wave-Length, pp. 121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H. — Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv November, 1906. $1.10. 5. Trowbridge, J. — High Electro-motive Force, pp. 181-215. pis. xxv-xxvii. May, 1907. 75c. 6. Thaxter, R. — Contribution toward a Monograph of the Laboulbeniaceae. Part II. pp. 217-469. pis. xxviii-lxxi. June, 1908. $7.00. Vol. 14. 1. Lowell, Percival. — The Origin of the Planets, pp. 1-16. pis. i-iv. June, 1913. 60c. 2. Fernald, W. E., Southard, E. E.. and Taft, A E. — Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases I to X.) pp. 17-128. 20 pis. May, 1918. $6.00. 3. Fernald, W. E„ Southard, E. E.. Canavan, M. M., Raeder. O. J. and Taft, A. E. — Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases XI to XX.) pp. 129-207. 32 dIs. December, 1921. $6.50. PROCEEDINGS. Vols. 1-56, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. The individual articles may be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. $5.00. Complete sets of the Life and Works of Rumford. 5 vols., $25.00; to Fellows, $5.00. For sale at the Library of The American Academy of Arts and Sciences. 28 Newbury Street, Boston, Massachusetts. 58-4 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 4.— January, 1923. THE EFFECT OF PRESSURE ON THE ELECTRICAL RESISTANCE OF COBALT, ALUMINUM, NICKEL, URANIUM, AND CAESIUM. By P. W. Bridgman. (Continued from page 3 of cover VOLUME 58. Ames, A. Jr., Proctor, C. A., and Ames^ Blanche. — ■'. Vision and the Technique of Art. pp. 1-47. 28 pis. In press. Birkhoff, George D. and Langer, Rudolph E. — The Boundary Problems Associated with a System of Ordinary Xifiear JJitlererftial Equations of "the First Order, pp. 49- 128. In press. Vainio, Edward A. — Lichenes in Insula Trinidad a Professore R. Thaxter CoIIecti. pp. 129-147. January, 1923. $1.60. Bridgman, P. W. — The Effect of Pressure on the Electrical Resistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. S.75. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 4.— January, 1923. THE EFFECT OF PRESSURE ON THE ELECTRICAL RESISTANCE OF COBALT, ALUMINUM, NICKEL, URANIUM, AND CAESIUM. By P. W. Bridgman. THE EFFECT OF PRESSURE ON THE ELECTRICAL RESISTANCE OF COBALT, ALUMINUM, NICKEL, URANIUM, AND CAESIUM. By P. W. Bridgman . Presented October 11, 1922. Received October 18, 1922. Introduction. In former numbers of the Proceedings of the American Academy 1 I have given data for the effect of pressure on the resistance of 39 of the elements. It is my intention to keep this work as up-to-date as possible by the revision of the previous data or the inclusion of new substances; this paper is in the nature of such a supplement to the former work. The first three of the metals above have been previ- ously investigated, but I have now been able to obtain them in a state of considerably higher purity than formerly, so that a redetermination of the pressure effect was worth while. The pressure coefficients now found for these purer samples are of the order of 10% higher than the previous values. Measurements on uranium have not been made before; the specimen I had was presumably not of very great purity, but since even the sign of the coefficient was not known, and since as a general rule impurity does not greatly affect the pressure coefficient, it was of interest to make and record the measurements. The work on ceasium is not yet complete, but enough has been done to establish the existence of surprising phenomena for this element, and since it may be some time before I shall be able to complete the work, it seemed that announcement of the chief fact should not be delayed. I have found that caesium has a new polymorphic modification under high pressure, of smaller volume than the ordinary modification, but that the resistance of this new modification increases instead of de- creases under pressure. Since caesium is the most compressible of the metals, this result may be of significance in suggesting what may happen to all metals at pressures sufficiently extreme. Detailed Data. Cobalt. The former sample of cobalt was obtained from Dr. Herbert T. Kalmus who had prepared it in the course of an investiga- tion of the properties of metallic cobalt for the Canadian Government. This was one of the impurer of the samples which he prepared, but 152 BRIDGMAN. since it was the only one he had left, I was glad to have the chance to make the measurements. I have now obtained indirectly, through the kindness of Professor C. C. Bidwell of Cornell University, one of the purest of Kalmus's specimens. Since the preparation of cobalt in the form of wire is a matter of some difficulty, involving the swaging instead of drawing of the metal in the early stages, and since there seems to be no commercial use for pure cobalt, it is not likely that material other than that of Kalmus will be available for some time. The analysis of this sample was: Co 99.73, Fe 0.14, Ni 0.00, S 0.019, Si 0.02, and C 0.09. The total impurity is seen to be 0.27%, against 1.30% of the former sample. The diameter of this piece was 0.10 cm. and the length about 6 cm. Its compressibility had been previously measured; for this purpose it had been annealed and straightened by rolling between red hot iron plates. For these measurements of resistance, current and potential terminals were soft soldered near the ends, and measurements made with the potentiometer. The technique was exactly like that used with those metals formerly measured with the potentiometer, and has been fully described in the previous paper. Because of the low re- sistance of this sample and the smallness of the pressure coefficient, the individual readings did not have as great regularity as those on the previous sample, and I did not think it worth while to try for the temperature coefficient of the pressure coefficient by making readings at different temperatures. Measurements were made at 30° over the pressure range of 12000 kg/cm2. The accuracy was not high enough to detect departure from linearity with pressure. Discarding one point, the average deviation of the observed points from a straight line was 2.0% of the maximum pressure effect. The average pressure coefficient over the range of 12000 kg. was -0.06934, against -0.06S65 of the former sample. The purer sample has the numerically larger coefficient. This seems to be true in the majority of cases, although there is no general rule here as in the case of the temperature coefficient. The resistance was measured at 30° and 75° at atmospheric pressure. Assuming the relation between temperature and resistance to be linear, as it was for the former sample, the temperature coefficient between 0° and 100° is 0.00439, against 0.00365 for the other sample. The highest value which I find listed for cobalt is 0.0033.2 The higher coefficient of the new sample is what would be expected because of its greater purity. PRESSURE COEFFICIENT OF RESISTANCE. 153 Aluminum. This was a specimen of quite unusual purity, of the following analysis: Si 0.014, Fe 0.007, Cu 0.003, Al (by diff.) 99.970. I extruded it from a diameter of 1.27 cm. to 0.081 cm., from this diameter drew it through steel dies to a diameter of 0.051 cm., annealed it at a temperature of 350°, wound it non-inductively on one of the bone cores used in previous work, made connections to the current and potential leads with spring clips at the two ends, and made the measurements in the regular way on the potentiometer. The length was such as to give an initial resistance of about 0.2 ohms. The regular procedure in making the measurements was followed. Several preliminary seasoning applications of pressure were made, and runs were made at 0°, 30°, 50°, and 95°. The run at 75° failed because of a minor accident. At 0° the average arithmetical deviation of the observed points from a line (no discards) wTas 0.23% of the maximum pressure effect, at 30° (one discard) 0.13%, at 50° (two dis- cards) 0.6%, and at 95° (one discard) 0.17%. The deviation from linearity was symmetrical about the mean pressure within the limits of error, so that the changes of resistance can be represented by a second degree expression in the pressure. In addition to these pressure measurements, the temperature coefficient at atmospheric pressure was determined by measurements at 0°, 30°, 50°, 75°, and 95° in this order, and then the measurements at 50° and 0° were repeated. The repeated readings agreed with the original ones within the limits of setting the slider of the bridge wire, showing no temperature hysteresis. TABLE I. Aluminum. Temp. °C. Resistance Pressure Coefficient Average At 0 kg. At 12000 kg. 0-12000 kg. Maximum Deviation from Linearity Pressure of Maximum Deviation 0 1.0000 -.054489 -.063954 -.054128 -.00111 6000 25 1.1167 4418 4043 4135 95 6000 50 1 . 2334 4364 4087 4129 86 6000 75 1.3501 4336 4029 4088 103 6000 100 1 . 4668 4278 3879 3990 126 6000 154 BRIDGMAN. The results are collected in Table I and Figure 1. The method of presenting these results is the same as that of the previous papers. The average pressure coefficient between 0 and 12000 kg. is that number which multiplied by 12000 gives the change of resistance pro- duced by 12000 kg. pressure as a fractional part of the resistance at atmospheric pressure and the temperature in question. The instan- 1 /nn\ taneous coefficients at 0 and 12000 kg. are — ( — R\dp, resistance at the pressure and temperature in question. The maxi- mum deviation from linearity is in fractional parts of the resistance at 0° and atmospheric pressure. As an example, suppose that it is required to find the resistance of aluminum at 50° at 6000 kg. in terms of its resistance at 0° and atmospheric pressure as unity. The average coefficient at 50° to 12000 is — 0.054129, and the initial resistance at where R is the 40° 60° Temperature 40° 60° 80° Temperature Al uminum Figure 1. Aluminum, results for the measured resistance. The devia- tions from linearity are given as fractions of the resistance at 0 kg. and 0° C. The pressure coefficient is the average coefficient between 0 and 12000 kg. 50° is 1.2334. If the change of resistance with pressure were linear, the decrease of resistance under 0000 kg. would be 1.2334 X 6000 X 0.054129 or 0.03056. But the change of resistance is not linear, but as the sixth column shows, there is a deviation from linearity at 6000 of 0.000S6, giving for the total decrease of resistance under 6000 0.03142, and for the actual resistance at 50° under 6000, 1.2020. Compared with the results for the previous sample, the pressure coefficient of this is in general higher by eight or nine per cent. As a function of temperature, the pressure coefficient of this new sample shows a very flat maximum near the lower end of the temperature range, and from here on decreases. The pressure coefficient of the other sample decreased linearly over the entire temperature range. A decrease of the coefficient with increasing tempera tine is not what one might at first expect, but its reality seems vouched for by independent PRESSURE COEFFICIENT OF RESISTANCE. 155 measurement on two different samples. The maximum deviation from linearity of this new pure sample is not as simple as that of the less pure one; the former decreased linearly with rising temperature, whereas this new sample shows a minimum. The temperature behavior at atmospheric pressure is worth comment. In the first place, the strict linearity with temperature is not usual. In the second place, the high value of the temperature coefficient of this piece as compared with the former sample, or the values of other observers, is to be noticed. The establishment of this high value removes aluminum from an apparently unique position with regard to its temperature coefficient. It is well known that practically all metals have a higher temperature coefficient in a condi- tion of higher purity. Aluminum appeared to be the only exception, since the published values for samples of increasing purity have ap- parently become smaller with increasing purity.3 But the purest of these previous samples had about 0.1% impurity. It is now seen that the apparent anomaly disappears in the region beyond 0.1%, the initial trend being reversed, and at sufficiently high purities the temperature coefficient increases with increasing purity, as it does for all other metals, and furthermore, the coefficient of the purest metal is higher than that of any of the impurer specimens. Nickel. This was of exceptionally high purity, and I owe it to the kindness of Mr. I. B. Smith, of the Research Laboratory of Leeds and Northrup Co. I have already published data for the effect of tension on resistance,4 and pressure on thermal conductivity 5 of this same nickel. Data for the compressibility are to be published shortly in These Proceedings. Hitherto it has not been possible to obtain in this country nickel of purity higher than that corresponding to a tempera- ture coefficient of resistance of 0.0049. The coefficient of this was 0.00634, and is higher than any other published value except 0.00683 by Niccolai.6 Except for the value of the temperature coefficient, I have no chemical analysis to indicate the purity. The material was drawn by Leeds and Northrup to wire 0.0127 cm. in diameter, was annealed to redness, and then was double covered with silk insulation by the New England Electrical Works. It was wound non-inductively on a small glass core, seasoned at 135° for six hours, and subjected to three preliminary applications of 12000 kg. The initial resistance at 0° was about 115 ohms. The measurements were made on the Carey Foster bridge regularly used in measuring the changes of resistance of samples with high resistance. The regular series of pressure measurements was made, at 0°, 25°, 156 BRIDGMAN. 50°, 75°, and 99°, and at the same temperature intervals there were also made two sets of readings of resistance as a function of tempera- ture at atmospheric pressure, which agreed within the sensitiveness of setting the slider of the bridge. The accuracy of the pressure readings was as follows: at 0° the average arithmetical departure from a smooth curve (no discards) was 0.41% of the maximum pressure effect, at 25° (no discards) 0.32%, at 50° (one discard) 0.16%, at 75° (no discards) 0.34%, and at 99° (no discards) 0.24%. The average departure from linearity at the maximum was 0.9% of the maximum pressure effect. TABLE II. Nickel. Temp. °C 0 Resistance Pressure Coefficient Average At 0 kg. At 12000 kg. 0 - 12000 kg. Maximum Deviation from Linearity Pressure of Maximum Deviation 1.0000 -.O5I8SO -.051819 -.051S30 -.000149 6000 25 1 . 1443 1899 1S24 1843 19S 6000 50 1 . 2975 1915 1S27 1851 2ls 6000 75 1 . 4607 1925 1830 1S58 297 6000 100 1 . 6345 1934 1832 1859 347 6000 The numerical results of the measurements are reproduced in Table II and Figure 2. The method of computation and presentation is the same as that used in the preceding papers. Compared with the previous results on less pure nickel, the pressure coefficient of this is on the average about 15% higher, again verifying the observation that in most cases impurity depresses the pressure coefficient, but by a less amount than the temperature coefficient. The temperature coefficient of this piece is 0.00634, against 0.00487 of the previous sample, or an increase of 30%. The pressure coeffi- cient of this new sample increases with rising temperature, as did that of the other sample, but the increase is much less rapid, and is not linear, becoming less rapid at the higher temperatures. The devia- tion from linearity of this new sample increases linearly with rising temperature, whereas that of the less pure sample at first passed PRESSURE COEFFICIENT OF RESISTANCE. 157 through a flat minimum, and then increased at the higher tempera- tures. The departure from linearity of this new specimen is further- more symmetrical about the mean pressure, so that it can be repre- sented by a parabolic formula, whereas that of the former sample was unsymmetrical, and there was a progressive shift of the pressure of maximum deviation with rising temperature. Uranium. I am indebted to the kindness of Dr. A. W. Hull of the General Electric Company for a sample of this metal. This had been prepared with considerable difficulty for an investigation of its proper- ties with a view to a possible commercial use. It turned out that its properties were not promising, so that no more will be prepared. This sample is probably unique, and I was most fortunate to obtain it. It was furnished in the form of rolled sheet, about 0.05 cm. thick, 40° 60° Temperature 1 00° 2 0° Nickel 40° 60° Temperature Figure 2. Nickel. Results for the measured resistance. The deviations from linearity are given as fractions of the resistance at 0 kg. and 0° C. The pressure coefficient is the average coefficient between 0 and 12000 kg. 0.7 cm. wide, and 4.5 cm. long. In this form the compressibility was measured, and I am reporting the results in another place. There were flaws in the specimen, which did not affect the compressibility measurements, since it was exposed to pressure all over, but which would have interfered with measurements of its electrical resistance. A small homogeneous sliver was cut from the sheet about 0.79 cm. wide and 2.2 cm. long. Uranium is very difficult to work, and the sliver had to be cut with a steel disc charged with diamond powder; I am indebted to the skill of Mr. David Mann for preparing the specimen. The specimen was mounted for measurement with the potentio- meter. It is not possible to solder leads to it. Fine grooves were filed around each end of the specimen, and a special spring clamp arrangement made by which the current and potential leads were 158 BRIDGMAX. pressed tightly into the grooves. The potential terminals were 1.43 cm. apart, and the current terminals 2.0 cm. The small resistance of the specimen and the nature of the connections prevented measure- ments of the highest accuracy. Furthermore, it is probable that this specimen was not of high purity, because its temperature coefficient of resistance between 0° and 100° was found to be only 0.00230. In view of the impurity and somewhat low accuracy I made no attempt to find the temperature coefficient of the pressure coefficient, but made the pressure run at a single temperature, 30°. The resistance was found to decrease under pressure, as is normal for most metals. The sign of the pressure coefficient was the most important fact to be iblished by the measurement. The position of uranium as the heaviest of the elements at the end of the periodic table would have given particular interest to a possible positive pressure coefficient of resistance. The average arithmetical deviation of the observed points from a smooth line 'no discards was 1.6* ",- of the maximum pressure effect. It was not possible to detect any deviation from the linear relation between pressure and resistance, and the average pressure coefficient over the range 0 to 12000 kg. was found to be — .05436. The specific resistance of uranium seems not to be recorded in the literature. The specific resistance of this sample at 0° was 76.0 X 10-6, which is high for a metal, being of the order of magnitude of the re- sistance of liquid mercury or bismuth. Caesium. As already mentioned in the introduction, the results on caesium are preliminary, but because of their interest it seems worth while to briefly describe them. The preparation of pure caesium and it> manipulation requires some practise, and I have not yet achieved final success. My original purpose in measuring the resistance of caesium was to search for a more pronounced drop in the temperature coefficient of resistance at high pressures than was found in the case of potassium. It would be expected that there would be such a phenome- non here because of the chemical similarity of caesium and potassium, and because of the much greater compressibility of caesium. This search failed, however, because of the entrance of a new polymorphic form at high pressures. The material for the measurements was obtained from the Foote Mineral Co. of Philadelphia. I am also indebted to the kindness of Professor Baxter and Professor G. X. Lewis for other samples, but I was not successful in the manipulation of these. The two samples from the Foote Mineral Co. were provided in glass tubes sealed under oil. One of the samples was apparently somewhat purer than the PRESSURE COEFFICIENT OF RESISTANCE. 159 other; its melting point appeared to be sharper and its yellow color not so pronounced. These samples were transferred to capillaries of thin glass provided with four platinum terminals sealed through the glass, two for current and two for potential leads. The measurements were made in the regular way on the potentipmeter. The details of the method of transferring to the capillary need not be described; it is possible to improve it. Measurements were made on the first and purer sample with a view to establishing the temperature coefficient over the range 0 to 12000 kg. In order to avoid melting at atmospheric pressure, the temperature range of the measurements was low, from 0° to 16°. Over this range of pressure and temperature the most puzzling results were found. At low pressures the resistance apparently decreased greatly with increasing pressure, which was the result expected, but at higher pressures the resistance increased again. After a day's run, the apparatus was taken apart in the search for trouble, and the glass capillary was found cracked, so that air got at the caesium and de- stroyed it. The crack in the capillary had no effect on the measure- ments, however. The temperature coefficient of this first and purer sample between 0° and 16° was found to be 0.0054, which is high, and evidence of good purity. The second sample, as already stated, seemed to be less pure initially, and during the manipulation received further impurity because of the accidental access of a small quantity of air to the inside of the appa- ratus during the transfer to the capillary. This heightened the yellow color and dirtied the metal so that it left a yellow scum on the glass, whereas the first sample had run as cleanly through the glass as clean mercury. However, the impurities introduced in this way were obviously non-metallic, and there is no reason to expect that this impurity introduced any essential change in the phase relations of the different polymorphic forms. With the information received from the first sample it was possible to direct the measurements on the second sample much more intelli- gently, and to definitely establish the more important features. In the first place it was established that caesium has a new modification at high pressures, that the transition point is sharp, as it should be for a true transition, and that the transition pressure varies with the temperature. Two measurements were made of the transition pres- sure at 0°. The first of these established that the equilibrium point was contained between two pressure limits 140 kg. apart, and that the mean point was 1960 kg. The second measurement shut the transi- 160 BRIDGMAN. tion within limits 100 kg. apart, at a mean pressure of 1990 kg. We take as the most probable transition pressure at 0°, 1980 kg. At 17° the transition was shut between two limits 400 kg. apart, mean 3260 kg. The transition curve is therefore of the normal type, rising to higher temperatures at higher pressures, and the phase stable at the higher temperature has the greater volume. Unless the transition line has exceptionally great curvature, it should be possible to find the 1.35 1.30 1.25 1.20 4) U 3 1 15 0> 1.10 1.05 1.00 .95 mi in ill II is mm ||||||||||i||||||||||| |i||ii-mtttttt Hi"!"' 'TTTTTiT 1 Hit |j]|ll|||Jl||||l4^|4"|"|"'" ::||: ^fffifjjPffijf Mttmtttmlitll::::fF:J: 3 4 5 - 6 7 -4- ill 111 1 1 l^-H-t--!--!— i--M--.=--=-J — 1-, 8 9 0 1 12 Pressure, Kg. / Cm Caesium 2X 103 Figure 3. Caesium. The resistance at 0° C. of the new high pressure modification of caesium as a function of pressure in terms of its resistance at 0° C and 3000 kg/cm2 as unity. new modification at atmospheric pressure at some temperature below 0°; linear extrapolation gives —26° as the atmospheric transition temperature. The electrical properties of the high pressure modification were found to be abnormal in that the resistance increases with increasing pressure. This makes the sixth metal now known with this property, the others being bismuth, antimony, calcium, lithium, and strontium. PRESSURE COEFFICIENT OF RESISTANCE. 161 It was possible to make fairly satisfactory measurements of the re- sistance of the new modification at 0° from the transition pressure out to 12000 kg. On the first application of pressure the results were irregular. This is due to the cracking of the glass capillary under pressure and the consequent change in the geometrical configuration of the caesium due to its extreme softness, but after the seasoning produced by the initial application of pressure the results were regular and would repeat. (The results obtained with the first sample were irregular for the same reason, but no results were found with the first sample that were inconsistent with those given by the second.) The experimental results are reproduced in Figure 3 plotting the measured resistances in terms of the resistance at 3000 kg. as unity. The curve shown was obtained with increasing and decreasing pressure; the points alternately are those with increasing or decreasing pressure. The curve is the same in character as that obtained for the five other abnormal metals in that the curvature is upward, or the resistance increases at a continually increasing rate at the higher pressures. This is most important as suggesting what the mechanism of conduc- tion may be, and is just what would be expected on the basis of theoret- ical considerations which I have already described. The magnitude of the pressure coefficient is high; between 11000 and 12000 kg., the average pressure coefficient of resistance is 0.000493, which is about the same as that of strontium, which has the highest coefficient of the abnormal metals hitherto measured. At the transition point there is a discontinuity in resistance in the normal direction, that is, the high pressure phase, or the phase with the smaller volume, has the smaller resistance. I know of no exception to this rule. At 0° the resistance of the high pressure modification at the equilibrium pressure is 0.407 that of the low pressure form. I am indebted to my assistant Mr. I. M. Kerney, for help in making many of the readings. The Jefferson Physical Laboratory, Harvard University, Cambridge, Mass. 1 P. W. Bridgman, Proc. Amer. Acad. 52, 573-646, 1917; 56, 61-153, 1921. 2 G. Reichardt, Ann. Phys. 6, 832, 1901. 3 L. Holborn, Ann. Phys. 59, 145-169, 1919. 4 P. W. Bridgman, Proc. Amer. Acad. 57, 41-66, 1922. 5 P. W. Bridgman, Proc. Amer. Acad, 57, 77-126, 1922. 6 G. Niccolai, Phys. ZS. 9, 367, 1908. VOLUME 57. 1. Kent, Norton \.. and Taylor, Lucien B. — The Grid Structure in Echelon Spectrum Lines, pp. 1-18. December, 1921. $.75. ?. Lotka, Ar.PBED J. — The General Conditions of Validity of the Principle of Le Chatelier. •pp. 19-37 January. 1922. $.75. Bridgman, P. W. — The Effect of Tension on the Electrical Resistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Loujs. — Notes on the Early Evolution of the Reflector, pp. 67-74. February, 1922. $.50. 5. Bbidgman, P. W. — The Effect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Brjdguan, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. SI. 25. 7. Piebce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Refraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjacent Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Fbanklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. $.05. 10. Bennitt, Rudolf. — Additions to the Hydroid Fauna of the Bermudas, pp. 239—259. May, 1922. $.65. 11. Brt.es, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonididse. pp, 261- 288. 2 pis. May, 1922. S.85. 12. Thaxter, Roland. — A Revision of the Endogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda. pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. R. — Atmospheric Attenuation of Ultra- Violet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Romberg, Arnold.— The Ratio of the Calorie at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Robert H. — Studies on Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389—422. 2 pis. November, 1922. $1.65. 17. Thaxter, Roland. — Note on Two Remarkable Ascomycetes. pp. 423—436. 2 pis. September, 1922. $1.35. 18. Records of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members; Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 2 of cover.) PUBLICATIONS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Series, Vols. 1-4; New Series, Vols. 1-13. 16 volumes, $10 each. Half volumes, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. Vol. 11. Part 1. Centennial Celebration. 1880. pp. 1-104. 1882. $2.00. Part 2. No. 1. Agassiz, A. — The Tortugas and Florida Reefs, pp. 105-134. 12 pis. June, 1885. (Author's copies, June, 1883.) $3.00. Part 3. Nos. 2-3. Searle, A. — The Apparent Position of the Zodiacal Light, pp. 135-157, and Chandler, S. C. — On the Square Bar Micrometer, pp. 15S-178. October, 1885. $1.00. Part 4. No. 4. Pickering, E. C. — Stellar Photography, pp. 179-226. 2 pis. March, 1886. $1.00. Part 4. No. 5. Rogers, W. A., and Winlock, Anna. — A Catalogue of 130 Polar Stars for the Epoch of 1875.0, resulting from the available Observations made between 1860 and 1885, and reduced to the System of the Catalogue of Publication XIV of the Astrono- mische Gesellschaft. pp. 227-300. June, 1886. 75c. Part 5. No. 6. Langley, S. P., Young, C. A., and Pickering, E. C. — Pritchard's Wedge Photometer, pp. 301-324. November, 1886. 25c. Part 6. No. 7. Wyman, M. — Memoir of Daniel Treadwell. pp. 325-523. October. 1887. $2.00. Vol. 13. 1. Sawyer, E. F. — Catalogue of the Magnitudes of Southern Stars from 0° to — 30° Declination, to the Magnitude 7.0 inclusive, pp. 1-100. May, 1892. $1.50. 2. Rowland, H. A. — On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-186. December, 1896. $2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniaceae. pp. 187- 430. 26 pis. December, 1896. $6.00. 4. Lowell, P. — New observations of the Planet Mercury, pp. 431—466. 8 pis June, 1898. $1.25. 5. Sedgwick, W. T., and Winslow, C. E. A. — (I.) Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II.) Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. 13. 1. Curtiss, D. R. — Binary Families in a Triply connected Region with Especia Reference to Hypergeometric Families, pp. 1-60. January, 1904. $1.00. 2. Tonks, 0. S. — Brygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T. — The Spectrum of Hydrogen in the Region of Extremely Short Wave-Length. pp. 121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H. — Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv November, 1906. $1.10. 5. Trowbridge, J. — High Electro-motive Force, pp. 181-215. pis. xxv-xxvii. May, 1907. 75c. 6. Thaxter, R. — Contribution toward a Monograph of the Laboulbeniacese. Part II. pp. 217-469. pis. xxviii-lxxi. June, 1908. $7.00. Vol. 14. 1. Lowell, Percival. — The Origin of the Planets, pp. 1-16. pis. i-iv. June, 1913. 60c. 2. Fernald, W. E., Southard, E. E., and Taf t, A. E. — Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases I to X.) pp. 17-128. 20 pis. May, 1918. $6.00. 3. Fernald, W. E., Southard, E. E., Canavan, M. M., Raeder, 0. J. and Taft, A. E. — Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases XI to XX.) pp. 129-207. 32 oh. December, 1921. $6.50. PROCEEDINGS. Vols. 1-56, $5 each. Discount to booksellers ,25%; to Fellows 50%, or for whole sets 60%. The individual articles may be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. $5.00. Complete sets of the Life and Works of Rumford. 5 vols., $25.00; to Fellows, $5.00. For sale at the Library of The American Academy of Arts and Sciences. 28 Newbury Street, Boston, Massachusetts. 58-5 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 5. — January, 1923. THE COMPRESSIBILITY OF THIRTY METALS AS A FUNCTION OF PRESSURE AND TEMPERATURE. By P. W. Bridgman. Investigations on Light and Heat made and published with aid from the Rumfobd Fund. (Continued from page 3 of cover.) VOLUME 58. 1. Ames, A. Jb., Proctob, C. A., and Ames, Blanche. — ■ Vision and the Technique of Art. pp. 1-47. 28 pis. In press. 2. Bibkhoff, George D. and Langeb, Rudolph E. — The Boundary Problems Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49- 128. In press. 3. Vatnio, Edwabd A. — Lichenes in Insula Trinidad a Professore R. Thaxter Collecti. pp. 129-147. January, 1923. $1.00. 4. Bbtdgman, P. W. — The Effect of Pressure on the Electrical Resistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. $.75. 5. Bbidgman, P. W. — The Compressibility of Thirty Metals as a Function of Pressure and Temperature, pp. 163-242. January, 1923. $1.70. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 5. — January, 1923. THE COMPRESSIBILITY OF THIRTY METALS AS A FUNCTION OF PRESSURE AND TEMPERATURE. By P. W. Bridgman. Investigations on Light and Heat made and published with aid from the Rumford Fund. THE COMPRESSIBILITY OF THIRTY METALS AS A FUNCTION OF PRESSURE AND TEMPERATURE. By P. W. Bridgman. Presented October 11, 1922. Received October 18, 1922. TABLE OF CONTENTS. Page. Introduction . . 166 The Absolute Compressibility of Iron 169 Compressibility of Metals other than Iron 175 Method 175 Metals Crystallizing in the Cubic System 181 Tungsten 181 Platinum 182 Molybdenum 184 Tantalum 185 Palladium 186 Nickel 186 Cobalt 188 Nichrome 189 Gold 189 Copper 191 Uranium ... 192 Silver 193 Aluminum 194 Germanium 195 Lead 197 Thallium 198 Cerium 199 Calcium 199 Strontium 201 Lithium 202 Sodium 202 Potassium 204 Metals Crystallizing not in the Cubic System 208 Magnesium 209 Bismuth 210 Tin 211 Antimony 213 Cadmium 214 Zinc 216 Tellurium 217 Discussion of Results 218 Theoretical Considerations 222 Summary 241 1U0 BRIDGMAN. Introduction. Our present knowledge of the compressibility of metals is mostly due to the work of Richards l and his collaborators, and to Adams Williamson and Johnston.2 (I shall abbreviate reference to these authors by A. W. J.) Richards has measured the compressibility of a large number of metals over a pressure range of 500 kg. at room temperature. A. W. J. have measured the compressibility of a smaller number of metals over the much wider pressure range of 10000 kg/cm2, but again only at room temperature. The work of Richards, there- fore, does not enable us to find either the pressure or the temperature variation of compressibility, and although the work of A. W. J. gives valuable information as to the variation of compressibility with pressure, they have themselves recognized that the pressure variations so found are not accurate. Recent theoretical work, in particular that of Born3 is now bringing within the reach of the possibility of computation the compressibility of substances in terms of their crystalline structure. Born's theory of the compressibility of substances of the type of sodium chloride is far enough advanced to give an expression for the variation of com- pressibility with pressure. It seems therefore that the time is ripe for a more careful experimental examination of the question of the com- pressibility of the metals, although we may not have as yet a satis- factory theory of the metallic state itself. The experimental work of this paper consists of a determination of the compressibility of 30 metals over a pressure range of 12000 kg/cm2 at 30° and 75°. This range is sufficient to give the pressure and temperature coefficients of the compressibility. The question of experimental accuracy is an important one here. It is well known that compressibility is one of the harder quantities to measure experimentally, so that all the more is it difficult to measure the pressure or the temperature coefficient of compressibility. The use of high pressure is indispensable here, for by increasing the magni- tude of the effects to be measured it is possible to attain the necessary accuracy. This is particularly true with respect to the pressure coefficient of compressibility ; the accuracy with which this may be determined increases as the square of the pressure range. Other things being equal, therefore, it should be possible with the pressure range of this work to determine the pressure coefficient 570 times as accurately as possible over a pressure range of 500 kg. cr> COMPRESSIBILITY OF METALS. 107 A new method of measuring compressibility has been developed for this work, which has very considerable advantages, both of speed and accuracy, over previous methods. As compared with previous methods, the accuracy of this is much increased by the fact that all the corrections are very much less. The method essentially involves the measurement of the difference of linear compressibility between the substance in question and pure iron. If the absolute linear com- pressibility of iron is known, we have at once the linear compressi- bility of the substance in a definite direction, and if equal compressi- bility in all directions is assumed, the true volume compressibility may be computed. It is to be noticed that over the pressure range used here, the compressibility cannot be found by simple multiplica- tion by the factor three, but a correction has to be applied, which in some cases may rise to the order of ten per cent. The assumption that the substance is equally compressible in all directions is applicable only when the material is amorphous or when it belongs to that one of the crystalline systems enjoying this property, namely the cubic. This assumption is true for the majority of metals, but it is definitely not true for several, which may have very materially different compressibilities in different directions. For these substances it is necessary to measure the linear compressibility in several directions in order to get the true cubic compressibility. This is at once a dis- advantage and an advantage; the determination of the average cubic compressibility becomes more complicated than by the methods employed by Richards or A. W. J., but on the other hand the average cubic compressibility is not a datum of much significance in these cases, whereas it is possible by the use of the present method to obtain a complete description of the behavior under pressure. For substances of this type it is necessary to have single crystals. The growth of such crystals involves a technique in itself; in this paper only the beginnings of the attack on this subject are made. I have shown for a number of non-cubic crystals that there may be very great differences of linear compressibility in different directions, and have made a promising beginning at a method of obtaining metals with a uniform crystalline orientation throughout the entire mass. The further examination of this important question must be left for future work. ■With regard even to those metals which are known to crystallize in the cubic system it is not safe to assume without some examination that the linear compressibility is the same in every direction, for few masses of metal are unicrystalline, but there are also regions of 168 BRIDGMAN. amorphous cementing material. In a number of cases I have tried, therefore, to justify this assumption. A number of years ago I measured the compressibility of commercial iron in the form of boiler plate along and across the direction of rolling,4 and could find no difference in the linear compressibility. I have now repeated the same examination for rolled copper, and again could find no difference. Also, in several cases I have measured the compressibility, first in the shape of the unworked casting, and then after the casting had been extruded to a considerably smaller diameter, so that the crystalline grains were much distorted by working, and presumably considerable amorphous material was introdviced. The general result of this work is that no perceptible change of linear compressibility is produced in this way, and it is highly probable that the assumption is correct that the ordinary cast or worked forms of those metals that crystallize in the cubic system have the same compressibility in every direction. The correctness of this assumption may further be checked by a comparison of the results obtained by my method with those obtained by the other methods, which are essentially methods for measuring the cubic compressibility. The absolute linear compressibility of iron has been previously measured by me at two different temperatures.5 I did not succeed in finding any departure of the compressibility from constancy over a pressure range of 10000 kg., but I did find an increase of compressi- bility at the higher temperature. This early work was unsatisfactory to me for two reasons. In the first place the theoretical considerations of Born would suggest that the departure of compressibility from constancy at high pressures should be large enough to measure even for a metal as little compressible as iron, and in the second place I could not help feeling that my temperature coefficient was too high. I therefore have made a fresh experimental attack on this question, greatly improving the experimental method, and have been able to definitely find and measure with some accuracy the change of com- pressibility of iron with pressure, and to show that the temperature coefficient is several times smaller than previously found. This experimental work falls into two parts; first the redetermina- tion of the absolute linear compressibility of iron, and second the determination of the relative linear compressibility of iron and 29 other solid metals. COMPRESSIBILITY OF METALS. 169 The Absolute Compressibility of Iron. The method adopted is in many respects like that previously used. An iron rod approximately 30 cm. long is placed inside a heavy cylinder and exposed to hydrostatic pressure exerted by a fluid filling the cylinder. The relative change of length of the rod and the cylinder is measured, and at the same time the change of length of the cylinder is measured at outside points. From these two the absolute change of length of the iron rod under pressure may be computed. The only assumption is that the change of length of the cylinder externally, where it can be measured, is the same as that internally, where it is in contact with the rod. This assumption is entirely justified if the cylinder extends some distance beyond the end of the rod, so that warping of the cross section due to end effects is negligible, and in any event the assumption is of little importance because the correction for the change of length of the cylinder is only a few per cent of the entire effect. The essential difference between the old and the new method is in the means of measuring the relative change of length of the rod and the cylinder. Previously the rod carried a collar which bore against a fixed shoulder in the cylinder and whose relative displacement on the rod was measured after every application of pressure. This involved a complete setting up and disassembly of the apparatus for a single reading at any pressure. The present method is a continuous reading one. Attached to the end of the rod is a fine wire of high resistance alloy. This wire slides over a contact fixed to the cylinder. The sliding contact is made one potential terminal and another potential terminal is attached to a point of the wire. Current is passed through the wire independently of the potential terminals, entering the wire at one end and leaving at the other. The difference of potential between the fixed and the sliding point on the wire is measured for a given current, so that the resistance and hence the length of the wire between terminals may be calculated, and hence the relative motion of the rod and the cylinder. The reading is continuous, and after every change of pressure a new reading may be obtained in a few min- utes, or as soon as the heat of compression is dissipated. Further, there is no error from the constant possibility of introducing minute particles of dirt every time the apparatus is opened, as there was previously. The essentials of the apparatus are shown in Figure 1. The high 170 BMDGMAN. resistance wire B (which is of nichrorne) is mounted in a steel plunger A which is kept pressed by a spring against the upper end of the bar at S. The wire B slides over a contact at C which is attached to the plate D insulated by mica washers at E from the rest of the cylinder. At F the second potential terminal is soldered to the wire B, and at the end G is attached the current terminal. The other terminal is the cylinder itself, to which the lower end of B is grounded. The three wires from C, F, and G are attached at the upper end of the Figure 1. The slicing electrical contact device for measuring the absolute linear compressibility of iron. cylinder to a three terminal plug, of exactly the same design as used in previous work on electrical resistance under pressure. The plate D is kept rigidly in position by springs shown in section, which are com- pressed as the three terminal plug is screwed into place. There are various mechanical details not shown in the drawing to facilitate the attaching of the connections and the general assembly. The speci- men S is fixed with respect to the cylinder at the lower end. S ends in a disc which is tightly pressed against a shoulder of the cylinder by a spring (which of course must be stiffer than the spring around A) which is compressed by the connections to the pressure producing apparatus at the lower end of the cylinder. The resistance between the points C and F was measured on the same potentiometer as was used in previous measurements of the COMPRESSIBILITY OF METALS. 171 effect of pressure on resistance, and needs no further description. The sensitiveness of the electrical arrangements was such that a motion of the wire of 1.5 X 10-6 cm. could be detected. Of course there were always extraneous disturbances, so that an accuracy cor- responding to this sensitiveness could not be obtained, but it was nevertheless possible to measure very small motions with a highly gratifying accuracy; these measurements would have been entirely impossible without some such method. This method of measuring small displacements would seem to be applicable in a number of other places. The external change of length of the cylinder was measured some- what differently from formerly. This was done previously with a microscope, and there were difficulties at the higher temperature (50° in the former work). The arrangement of the pressure producing part of the apparatus was such that it was now much more convenient to use the rocking mirror device familiar to many engineers. A mirror was attached to a diamond sectioned staff, which could rock between two bars, one attached to the upper end and the other to the lower end of the cylinder. The motion of the mirror was read with a telescope and scale, and the elongation of the cylinder could at once be calcu- lated in terms of the dimensions of the various parts. The conven- tional devices of three point contacts and springs to ensure freedom from back-lash, etc., were used, and the apparatus functioned satis- factorily in every particular. Two measuring devices were used, attached at opposite sides of the cylinder, so that any error due to any slight Bourdon spring action of the cylinder under pressure (there usually is such action in appreciable amount) was eliminated. The method of mounting the mirrors at a distance of 30 cm. or so from the cylinder and transmitting the motion to them by bars allowed the cylinder to be placed in a temperature bath, and the temperature to be controlled by a thermostat and stirrer, as in all these experiments. The stirring of the water produced only a slight mechanical shaking of the mirrors, too slight to introduce perceptible difficulty in making the readings. It is obvious that there are various corrections to be applied to the readings as directly obtained. The most important of these is for the effect of pressure on the resistance of the nichrome wire B. This was determined by direct experiment, and is very low; in fact this, as well as the high specific resistance, was the reason for choosing this material for the wire. The resistance between terminals fixed to the wire was found to decrease linearly with pressure, the decrease for 12000 kg/ cm2 172 BRIDGMAN. being 0.45%. This is much less than for any of the pure metals, and is seven times less than for manganin. It is possible to find other nickel-chromium alloys that have an even smaller coefficient, but this was so small that it was not necessary to go out of the way for other material. Other corrections, not as important as the above, are for the temper- ature coefficient of resistance of nichrome wire, and for the difference of compressibility between the wire and iron. Both these corrections were determined by direct experiment; the determination of the compressibility will be described later. Although there are several corrections to be applied, it is a unique advantage of this method that the corrections are very small, and that the greater part by far of the measured effect is the final effect that is wanted. The corrections just discussed for the change of resistance of the nichrome wire, etc., did not attain as much as 2% of the measured effect, and the correction for the external change of length of the cylinder was only 1.5%. On the other hand it frequently happens that the corrections in such methods as used by Richards or A. W. J. may be two or three times as large as the final effect wanted. It is evident that the exact uniformity of cross section of the wire B is of vital importance if the changes of resistance are to give slight departures of compressibility from linearity -with pressure. To test the uniformity of the wire a special arrangement was made consisting of two knife edges a fixed distance apart, mounted to slide along the wire, which was stretched along a meter stick. A constant current was passed along the wire, and the difference of potential between the two knife edges measured on the potentiometer. The wire was ex- ceedingly uniform. In the first length tried, the extreme variation in a length of 20 cm. was less than 1/3000. The variation in a length of a millimeter or less was of course much smaller, and was obviously far too small to introduce any appreciable error. The iron used for the measurement of linear compressibility was cut from a bar of American ingot iron. This iron is very pure, containing only 0.03% impurity, and was considerably purer than the piece of boiler plate whose compressibility was previously measured. The resistance under pressure of a piece cut from this same material has been previously measured.6 The dimensions of the compressibility specimen were approximately 6 mm. in diameter and 30 cm. long. It was annealed at a red heat after machining. The containing cylinder was especially made for this experiment and was of chrome vanadium steel from the Halcomb Steel Co., sold COMPRESSIBILITY OF METALS. 173 by them under the name of Type D. After machining it was heat treated by quenching from 930° for a few seconds in water and then finishing the quenching in oil. It was pressure seasoned before the measurements by an application of 12500 kg., and then machined to final size. No perceptible change of dimensions was produced by this preliminary treatment. The measurements of the external change of length of the cylinder, which were made at the same time as the measurements of the change of relative length of cylinder and bar, were entirely satisfactory. The relation between pressure and elongation was linear within the limits of error, which were not more than a few per cent, and there was no perceptible hysteresis, showing that the elastic limit had not been exceeded. However, the absolute value of the elongation was ab- normal in that it was only half that which would be computed from the ordinary elastic constants of steel, assuming perfect homogeneity. It is evident that the method of heat treatment must have introduced considerable internal stress. Measurements were made at two temperatures, 30° and 75°, and at both temperatures the runs were repeated. The two runs at each temperature agreed within the limits of error. The magnitude of the effect was sufficient to give a displacement of 40 cm. on the wire of the potentiometer. The regular procedure was followed in making readings, which were at intervals of even thousands of kilograms with increasing pressure and at the odd intervals on decreasing pressure, the maximum being 12000. There was no perceptible hysteresis. At each temperature 27 readings were taken. Two of these had to be discarded, the observed points lying off a smooth curve by slightly over 1 cm. of the potentiometer wire; the average arithmetical devia- tion of the remaining 52 points from smooth curves was 0.6% of the maximum effect. The departure from linearity was perfectly well marked and unmistakable; at the middle of the range it amounted to 0.32%, and was sensibly the same at both temperatures. This deviation from linearity is thus twice as great as the average error of a single observation ; hence it is evident that no high degree of accuracy can be claimed for the deviation from linearity (which determines at once the pressure coefficient of compressibility), and certainly one is not entitled to more than two figures in the pressure coefficient of compressibility. With regard to the temperature coefficient of compressibility, the change with temperature was certainly established to be much smaller than had been previously found, but again in view of the smallness of 174 BRIDGMAJST. the effect no great accuracy can be claimed for the temperature coefficient itself, and in fact not more than one significant figure could be obtained here. After applying all corrections, including one due to the cubic com- pressibility not being exactly three times the linear, the following formulas were found to give the change of volume produced by any pressure p, expressed in kilograms per square centimeter. AV At 30° — = - 10-7 (5.87 - 2.1 X 1(H p) p I o AV At 75° -jr = - 10-7 (5.93 - 2.1 X 1Q-5 p) p. f 0 The T'o of these formulas is in each case the volume under atmos- pheric pressure at 30°; the difference between the atmospheric volume at 30° and 75° would produce a change of somewhat less than one unit in the last place. The fairest comparison of these results with those previously found is to be obtained by computing from the above formulas the average compressibility to 10000 kg., since the range of the previous work was 10000 kg., and no departure was found from linearity. The above formulas give as the average to 10000, 5.6G X 10-7 at 30° and 5.72 X 10~7 at 75°. These are materially smaller than the previous values, which were 5.83 X 10-7 at 0° and 6.01 X 10~7 at 50°. Probably part of the difference is to be explained by the greater purity of the present specimen, it being consistent with a number of my other observations that alloying iron increases its compressibility, but doubtless the important part of the difference is to be ascribed simply to the im- provement in the method; certainly the present determination of the temperature effect is to be preferred to the previous one. The new value for the compressibility of iron will not affect at all the compressibilities given by Richards, which involve my previous value for iron indirectly through the value for mercury, but will lower the absolute values of compressibility given by A. W. J. by 0.25 X 10-7, since their values also depend on my value for iron. The fact that the compressibility of iron is now found to decrease somewhat at high pressures will also of course somewhat increase the estimate of A. W. J. of the decrease of compressibility with pressure. 1 dV The instantaneous compressibility, which I define as 77 — -, may be found from the above formulas by differentiation. The change of COMPRESSIBILITY OF METALS. 175 compressibility with pressure is thus seen to be about 8.5%. One would not suspect so large a change from the slight apparent departure from linearity of the curve plotting change of length against pressure. Compressibility of Metals Other than Iron. method. The general idea of the method is the same as that used for the absolute linear compressibility of iron, namely a wire is attached to the specimen, which slides over a contact fixed to a comparison piece, and the relative motion of the wire is obtained from a measurement of the potential dif- ference between the sliding contact and the contact fixed to the wire. In this case, however, the com- parison piece is not the external cylinder itself, but is another piece of iron within the cylinder which is exposed to hydrostatic pressure all over, and which therefore experiences only a uniform compression. The amount of this compression is determined by the measurements on iron just described. Several forms of apparatus were used, depending on the numerical value of compressibility and the shape in which the specimens could be obtained. For those substances whose compressibility is near to that of iron, an apparatus with a lever was used, by which the relative motion of the specimen and the surrounding iron was magnified about seven times. A sketch of the essential parts is shown in Figure 2. The specimen shown at S was, when convenient, made in the form of a turned bar 6 mm. in diameter. Over each end was slipped a well fitting steel cap, provided with steel points. At the lower end one of these points rested against the containing envelope of iron, and at the upper end the other point bore against the short arm of the lever. This lever had three point support, one at A, and two, one behind the other, at B. The lever was kept tightly pressed against the specimen by a spring of flat steel G, s i J-U-l i Figure 2. Device with lever magnification and sliding electrical contact for measuring the difference between the linear compressibility of iron and the specimen S. 176 BRIDGMAN. pulling on the upper end of the lever through a link motion L. The nichrome measuring wire C is attached to the upper end of the lever, and slides over a contact at D which is attached to a plate insulated from the rest of the apparatus with a thin mica washer. At E is the potential terminal fixed to the wire. Flexible leads pass from E and D to the two potential terminals of the three terminal plug, and the current terminal is attached at F. The second current connection is simply the case, to which everything is grounded. The details of connecting the lever apparatus to the three terminal plug so that the specimen attached to the plug can be screwed into the pressure cylinder as one self-contained piece offer no difficulties of design, and need not be shown. The apparatus as shown is flexible enough to be adapted to a wide variety of sorts of specimens. Variations in the diameter of the speci- men may be allowed for either by making different steel caps, or preferably simply by making suitable steel collars to slip around the specimen by which it is held securely in the pointed end pieces. Varia- tions in the length of the specimen are allowed for by changing the length of the dummy piece of iron at H. Specimens used with the apparatus as shown must be stiff enough to support the compressive stress due to the spring on the lever. But the compressibility of wires may also be measured by a simple change that will readily sug- gest itself, replacing the specimen S by a slotted frame of iron in which is laid the specimen in the form of wire, attached to the frame by a clamp at the lower end, and at the upper end attached to a clamp, which is in turn attached to the surrounding envelope by a pin. The wire is thus put under tension instead of compression. Measurements may be made on wires of any diameter large enough to stand the necessary tension. One must of course be sure that the elongation of the wire due to the tension of the spring is not large enough com- pared with the relative deformations produced by pressure to intro- duce any such complication as might arise from a change in Young's modulus under pressure. A simple calculation will enable one to stay on the safe side of this requirement. The magnifying power of the lever had to be determined by a special arrangement, it not being possible to measure the dimensions directly accurately enough. An arrangement was made by which a microm- eter screw could advance the short arm of the lever by a known small amount, the micrometer screw being provided with a vernier scale permitting readings to 1/20°. The accuracy obtainable with the special arrangement was considerably better than 1/10%. COMPRESSIBILITY OF METALS. 177 JV 1 For specimens whose compressibility was fairly high compared with that of iron the complications of the lever were dispensed with, and the relative change of length was measured directly by a wire attached directly to the end of the specimen. This is shown in Figure 3. The specimen is compressed against the lower end of the iron container by a spring M and also is pressed against the sliding contact D by a spring N. The final measurements of compressibility were made with one or the other of three different pieces of apparatus; two of these magnified the motion with levers, one having a long envelope for long specimens (maximum capacity 16.5 cm.), and the other a short envelope for pieces of a maximum length of 2.5 cm. The third apparatus was for the direct measurement of the relative change of length, as described in the preceding paragraph. Each form of apparatus was checked by making with it blank runs, using as the specimen a piece of the same iron as that whose absolute linear compressibility was measured with the other apparatus. If the compressi- bility of the envelope (which was made of commercial bessemer steel) was the same as that of pure iron, and if it were perfectly homogeneous, so that it experienced a volume compression without change of figure, then the changes of resistance of the wire under pressure should be such that when the various corrections were applied for the change in specific resistance of the wire under pressure, etc., there should be indicated no out- standing relative motion of specimen and surrounding envelope. Of course for one thing the compressibility of the pure iron and the bessemer steel was probably not exactly the same, so that perfect agreement was not to be expected, but nevertheless the correction so deter- mined due to all these possible sources was very small indeed, and was about that due to the relative com- pression of the iron and the mica washers (which were 0.012 cm. thick) by which the fixed contact D was in- sulated. It is to be noticed that after this correction was applied the results gave accurately the difference fig. 3. Figure 3. Device with sliding electrical contact for the direct measure- ment of the difference of linear compressibility between iron and the specimen S. £ PU=4 178 BRIDGMAN. of compressibility between the specimen and pure iron, independent of any imperfections in the apparatus, which were thereby elimi- nated. Except for the convenience of using small corrections, it would have been quite possible to have made the envelope of some other metal, such as brass. Besides the empirically determined correction just discussed a number of other corrections have to be applied for the change of re- sistance of the nichrome wire under pressure, for the difference of compressibility between nichrome and iron, and for the temperature effects. These corrections are essentially the same as those needed in getting the absolute linear compressibility of iron, and will not be separately discussed; they are small and involve no difficulty. The data requisite for making these corrections were determined by direct experiment. For those substances whose compressibility is near to that of iron the percentage magnitude of the correction on the differ- ence of compressibility may of course be greater than in the experi- ment on the absolute linear compressibility of iron. The percentage magnitude of the corrections is approximately inversely proportional to the difference of compressibility. The total motion of the wire was in all cases small, being seldom more than a small fraction of a millimeter. It will of course be under- stood that in dealing with such small quantities every feature of the design had to be carefully thought out, and that the apparatus as finally used embodied many details whose best form was found only after several trials, but which it will not be profitable to describe here. Many changes were made in several different pieces of apparatus, and readings were made continuously over a period of four months before the design used for the final measurements here recorded was attained. By far the most trouble was found with the lever apparatus. It is sufficient to mention here that the feature requiring the greatest care is the construction of the bearing points and seats of the lever. These must be of the greatest geometrical perfection, and highly polished, and will repay a considerable amount of labor. The radius of the bearing point must not be too small or the point will break, hence the necessity for a high polish. The result of failure to get proper bearing points is irregularity in the readings; the observations skip erratically about, or often show systematic departures from the correct course for considerable successions of readings. The pressure was determined, as in all this work, from the changes in the resistance of a coil of manganin wire, and the corrections have been discussed in detail previously.7 During the course of the work COMPRESSIBILITY OF METALS. 179 the coil was calibrated against the freezing point of mercury at 0° at sufficient intervals to insure the accuracy of the pressure measurements. After the readings had been made of the potentiometer setting as a function of pressure over the entire pressure range, and the various corrections applied to these readings, the points were plotted, potentio- meter setting against pressure. This plot is nearly a straight line. A straight line was now passed by computation through the zero point and the point at the highest pressure, and the intermediate points on the line computed corresponding to the intermediate observed pres- sures. It was necessary to use computation for the intermediate points, as graphical construction would not have been accurate enough. The difference between the linear values and those actually observed was now found by subtraction, and these differences were plotted on an enlarged scale. A smooth curve was now passed through these difference points. This curve could be drawn with sufficient accuracy in most cases free hand. In most cases this curve was symmetrical about the mean pressure (6000 kg.), and could be represented within the limits of error by a parabolic formula. This means that within the limits of error the change of volume is given by a two constant AV formula of the type used for iron above, namely -p~ = — (a + bp)p. The change of volume under 12000 kg. was next computed, using as the most probable value of the change of resistance that obtained from the smoothed curve of departures from linearity. In computing this change of volume all the corrections were applied. The initial rate at which the volume changes with pressure was also computed. This initial rate was the average rate to 12000 corrected by a factor ob- tained from the smoothed difference curve. The correction factor was usually given with sufficient accuracy by doubling the ratio of the deviation at the mean pressure to the mean ordinate, and applying the result as a correction factor to the average rate to 12000. If the deviation from linearity was so great as to make this procedure in- accurate, the change of volume was calculated directly at several intermediate pressures. Having thus obtained the initial rate of change of volume, and the change under 12000, the constants of a formula like that above were at once determined. For several sub- stances, particularly sodium and potassium, the deviation from linearity was so great that it was necessary to calculate from the smoothed curve the changes of volume at even thousand intervals of pressure, and tabulate these, instead of trying to reproduce the results by a single formula. 180 BRIDGMAN. The procedure in making the calculations outlined in the paragraph above was applied independently at the two temperatures 30° and 75°. Departure from linearity was in most cases so small that it was useless to attempt to try to get the temperature coefficient of the departure. Accordingly in the calculations the mean of the departures found independently at the two temperatures was usually used as that most probably accurate. The behavior of the metals under pressure gives in many cases a useful check on their probable homogeneity and therefore on the equality of their compressibility in all directions. If the metal is under no internal strains, the relation between deformation and pressure should be single valued, without hysteresis, and there should be no permanent change of dimensions, even after the first application of pressure. The absence of hysteresis could of course be checked by the readings themselves, but the freedom from set on the initial application of pressure was not so easy to determine, because there were sometimes slight initial irregularities in the apparatus itself which were smoothed out by the initial application of pressure. In all cases the metal was subjected to a preliminary application of pressure over the entire range before readings were begun, and the change of zero produced by this preliminary application was also recorded. This should give an upper limit to any actual change of dimensions pro- duced in the specimen by pressure. If the substance is carefully prepared there should be no change, and tins was in almost every case the fact. It was possible to make castings of lead, for example, that showed no perceptible permanent change of length after the initial application of 12000 kg. In one or two cases, however, there were comparatively large changes, and these were usually accompanied by hysteresis on subsequent applications of pressure, showing internal strains. These cases will be described in detail later. There follows now the detailed presentation of data. First are given those metals crystallizing in the cubic system. There are included here a few metals whose structure has not yet been deter- mined, since the chances are that any metal selected at random is cubic. The metals are arranged in order of compressibility, beginning with tungsten, the least compressible. After this are given data for several metals not crystallizing in the cubic system. The work on these latter metals must be extended. The present data are com- petent to give only an idea of the amount of variation to be expected in the compressibility in different directions. COMPRESSIBILITY OF METALS. 181 Metals Crystallizing in the Cubic System. Tungsten. It is well known that, because of its methods of prepa- ration, this metal may show considerable differences of density, depending on the amount of mechanical working to which it has been subjected, the finer wires having the greater density. It therefore seemed to me desirable to measure the compressibility of two samples differing considerably in the amount of working to which they had been subjected. Through the kindness of the General Electric Com- pany I obtained a number of samples of pure tungsten, and selected two from these best adapted to the measurements. The first was 0.48 cm. in diameter; it had been swaged to these dimensions from the original sintered bar, but had not been drawn through dies. It was mounted as a compression specimen in the lever apparatus, with a length of 8.68 cm. The density at room temperature was found by weighing in air and water to be 19.137. The second specimen was of drawn wire 0.051 cm. in diameter, and was mounted as a tension specimen in the lever apparatus. The length was about 10 cm. The density of this small sample could not be determined with sufficient accuracy by weighing, but tables compiled by the General Electric Company show that the density to be expected on the average for wire of this diameter is 19.48. The difference between the densities of the two samples is thus considerable; it is questionable how much is due to closing of the pores and how much is due to the breaking up of the crystalline structure by working, replacing the crystalline material by amorphous. Regular readings were made on these two samples at 30° and 75°. The readings were very satisfactory, considering the small compressi- bility of this metal. Discarding the 8 most irregular points, the aver- age numerical departure from a smooth curve of the remaining 48 points was 0.28% of the effect produced by the maximum pressure. The accuracy of the readings with the two different samples was essen- tially the same. The departure of the points from linearity, as com- pared with iron, was well marked, and at the mean pressure was about 2.5 times as great as the average error of a single observation. The deviations from linearity determined for each sample independently from the series of readings at the two temperatures differed by not more than 10% from the mean. The direction of this departure from linearity was such as to make the change of compressibility of tungsten with pressure less than that of iron, as would be expected from its 182 BRIDGMAN. smaller absolute compressibility. The final results are expressed in the following formulas, p being in kg/ cm2: AV Swaged rod, At 30° — = - 1(H (2.93 - 1.5 X KHp) p I o AV At 75° — = - 10-7 (2.95 - 1.5 X I0~5p) p 1 o AV Drawn wire, At 30° — = - 10~7 (3.15 - 1.6 X lO^p) p I o AV At 75° — = - 10-7 (3.16 - 1.5 X l(Hp) p. I 0 It is perhaps not to be expected that the drawn wire with the greater density should also have the greater compressibility. This is possibly due in part to the greater amount of the amorphous metal that the drawn wire doubtless contains, since it is usually true that an amor- phous phase is more compressible than the corresponding crystalline one. The only previous determination of the compressibility of tungsten seems to be by Richards.1 His material was in the form of fused buttons, also obtained from the General Electric Company, and had a density at room temperature of 19.231. He finds for the initial com- pressibility 2.7 X 10"7. The value is given to only two significant figures, and I gather from personal conversation with Professor Richards that he regards as possible an error of several units in the last place. Platinum. Runs were made on two specimens, of quite different dimensions, in two different pieces of apparatus. I am indebted to the kindness of Baker and Company for the loan of the two specimens. The material was stated by them to be chemically pure platinum. The first specimen was 2.5 cm. long and 0.S cm. in diameter. It was used as a compression specimen in the lever apparatus for short speci- mens. It was cut from a drawn rod of platinum, and before the meas- urements was annealed for several hours in an electric furnace at a temperature of 800°. The second specimen was of drawn wire, an- nealed, 0.065 cm. in diameter and about 10 cm. long, and was mounted as a tension specimen in the lever apparatus for long specimens. The regular set of readings at two temperatures was made on each specimen. The readings with the shorter specimen we're not COMPRESSIBILITY OF METALS. 183 nearly as regular as those with the longer wire, which indeed is not surprising in view of the smallness of the effect. The average arith- metical departure from a smooth curve of the 28 observed points with the short specimen (no discards) was 2% of the maximum pressure effect, and the departure from linearity, which was perfectly well marked, was 3% at the maximum. The series of readings at the two temperatures gave the same departure from linearity to one significant figure. It is obvious that a high degree of precision cannot be expected for the pressure coefficient of compressibility of this specimen. The average arithmetical departure from the smooth curve of all the read- ings on the wire (no discards) was 0.7% of the maximum pressure effect, but here the deviation from linearity, although perfectly evi- dent, was only two thirds as great as the error of a single reading. The deviation from linearity at the higher temperature was about 50% greater than at the lower. The average value of the compressibility obtained from the wire sample may be expected to be relatively more accurate than that obtained from the massive specimen, but the differences are beyond the errors of the measurements, and point to a real difference between the two specimens. The pressure coefficient of the difference of compressibility between iron and platinum can- not obviously claim any great accuracy, but the actual pressure coeffi- cient, obtained by combining these readings with the absolute values for iron, may be expected to be somewhat more accurate. The results found are reproduced by the following formulas: AV Wire, At 30° — = - 10~7 (3.G0 - 1.8 X 10^p) p At 75° — = - 10-7 (3.64 - 1.8 X l(Hp) p I ii Rod, At 30° — = - 10-7 (3.05 - 0.0 X 10-^p) p I 0 AV At 75° — = - 10-7 (3.09 - 0.0 X I0~bp) p. I o It is of course not likely that the compressibility of the rod actually does not change with the pressure, but at any rate the change is small. The initial compressibility of platinum has been found by Richards to be 3.7 X 10~7. The agreement with the value found above for the wire is well within the limits of error. Richards also obtained his material from Baker and Company. He found for the density at 20° 184 BRIDGMAX. 21.31. The density of the larger of the two pieces measured above was 21.34 at 20°. Evidently the mechanical condition of these two specimens must have been nearly the same. Molybdenum. The treatment of molybdenum was like that of tungsten. Through the kindness of the General Electric Company I obtained a number of pieces in different stages of mechanical working; the compressibility of two of these was measured. One specimen, which had been swaged only but not drawn, was used as a compression specimen in the lever apparatus; its diameter was 0.48 cm. and length about 10 cm. This specimen was in two pieces, held butting end to end in a steel sleeve. It is often possible in this way to fit together several shorter pieces and obtain effectively a longer piece, thus in- creasing the accuracy of the measurements. The density of this speci- men at 20° was found by weighing to be 10.185. The second specimen was in the form of drawn wire 0.051 cm. in diameter and 10 cm. long, mounted as a tension specimen in the lever apparatus. Its density could not be obtained with sufficient accuracy by weighing, but the average density found by the General Electric Company for wire of this diameter is 10.20. The regular series of runs, at 30° and 75°, were made for each speci- men. The accuracy of the measurements was about the same for the two specimens. Discarding 5 points, the average arithmetical de- parture from a smooth curve of the remaining 50 points was 0.36% of the maximum pressure effect, and the departure from linearity (that is, the departure from linearity of the difference of compressibility of iron and molybdenum) was 1.40% of the maximum pressure effect. The departure from linearity of each specimen was sensibly the same at the two temperatures. The final results are : AV — = - 10-7(3.47 - 1.2 p) p I o AT' — = - 10-7 (3.48 - 1.2 p) p » o A T7* — = - 10-7(3.61 - 1.0 p) P » 0 t^L = -10-7(3.62-1.0p)p. > o The compressibility of the drawn wire is seen to be higher than that of the swaged rod in spite of its higher density, as was also the case for tungsten, and probably for the same reason. Swaged rod, At 30° — 1 0 At 75° — » 0 Drawn wire, At 30° ^ » 0 At 75° £ > 0 COMPRESSIBILITY OP METALS. 185 Richards has found for the initial compressibility of molybdenum in the form of fused buttons of density 10.21 the value 4.5 X 10~7. This again is considerably higher than the value found above, but the discrepancy is not more than is to be expected when the small size of Richards' sample is considered, and the fact that one of his two results was 40% higher than the other. Richards' object in his measurement was only to obtain the order of magnitude. Tantalum. This was in the form of a drawn wire 0.062 cm. in diameter and 10 cm. long, mounted as a tension specimen in the lever apparatus. I am indebted for this material to the kindness of the Fansteel Co., of North Chicago. It was stated by them to be of unusually high purity, but I have no analysis. As a partial means of estimating its purity I determined its temperature coefficient of re- sistance between 0° and 100°. The relation between temperature and resistance is sensibly linear over this range, and the average coefficient is 0.00335. This is materially higher than the coefficient of a piece which I had formerly obtained from the General Electric Company, namely 0.00293, and for which I have determined the effect of pressure on electrical resistance. The coefficient is less than that of a sample of Holborn,8 0.00347. Two runs were made on this specimen, as usual, at 30° and 75°. The results at 30° were appreciably more regular than at 75°. At 30° the average arithmetical departure of the observed points from a smooth curve (no discards) was 1.4% of the maximum pressure effect, and at 75° it was 3.2%. The departure from linearity was sensibly the same at the two temperatures, and at its maximum was 6.8% of the pressure effect. It must be remembered that because of the small difference of compressibility between tantalum and iron a large per- centage error in the difference of the two compressibilities need not mean a large error on the absolute compressibility. The final results are given by the formulas : AV At 30° =f = - 10-7 (4.79 - 0.25 X 10"5 p) p I o AV At 75° ^r = - 10-7 (4.92 - 0.25 X 10~5 p) p. I 0 The close approach of the compressibility to constancy with pressure is to be noted. The compressibility of tantalum at 20° over a small pressure range 186 BRIDGMAN. has been found by Richards to be 5.2 X 10~7. His material was from the General Electric Company, and was in the form of fused buttons. Palladium. Runs were made on two different samples, both of which I owe to the kindness of Baker and Company. They were said to be of the highest attainable purity, but I have no analysis. One of these was a massive specimen, 0.75 cm. in diameter and 2.5 cm. long. It was used as a compression specimen in the lever apparatus for short specimens. The second was in the form of wire, 0.062 cm. in diameter, 10 cm. long, and was mounted as a tension specimen in the lever apparatus for long specimens. Each sample was annealed after receiving from Baker, the massive specimen by heating to 800° for two hours in an electric furnace and slowly cooling, and the wire by heating in a Bunsen burner to a bright red. Two regular runs, at 30° and 75°, were made on each specimen. The average arithmetical departure of the readings of the massive specimen from a smooth curve (no discards) was 2.2% of the maxi- mum pressure effect. It was not possible to detect any departure from linearity. For the wire the average arithmetical departure was 3.0% of the maximum pressure effect, and again there was no detecti- ble departure from linearity. The final results found for these two specimens are: AV Massive specimen, At 30° — = - 10~7 (5.19 - 2.1 X KH p) p >o AV At 75° — = - 10-7 (5.11 - 2.0 X KHp) p I o Drawn wire, At 30° ^- = - 10-7 (5.28 - 2.1 X 10"5 p) p Jo At 75° — = - 10-7 (5.31 - 2,1 X H)-5 p) p. Vo It is to be noticed that the temperature coefficient of the massive palladium appears to be negative. This is a very unusual occurrence, and it is to be doubted whether the accuracy of the measurements is so high as to compel the conclusion that this is actually the case. The initial compressibility of palladium at 20° has been found by Richards to be 5.3 X 10~7, agreeing perfectly with the value found above for the wire. Richards' material was also obtained from Baker, and was a massive specimen weighing 94 gm. Its density was 12.14, against 1 1 .97 for the massive specimen above. Nickel. Measurements were made on samples from two sources. COMPRESSIBILITY OF METALS. 187 One was a piece of commercial nickel obtained from the International Nickel Company in the form of a drawn bar 0.75 cm. in diameter and 16 cm. long. The purity was high commercial purity, a little better than 99%. It was annealed by heating for several hours to a bright red after its final machining, and was used as a compression specimen in the lever apparatus for long specimens. The second sample piece of nickel was obtained from the research laboratories of the Leeds and Northrup Company, and was of very unusually high purity. It is the same material as that for which I have already published data for the effect of tension on resistance 9 and pressure on thermal conduc- tivity.10 I have no chemical analysis, but have determined the tem- perature coefficient of resistance between 0° and 100° to have the mean value 0.00634, which is very high. This material was provided in the form of small cast ingots a couple of mm. thick and 5 or 6 cm. long. There were flaws in these castings large enough to be visible to the eye. I tried to get rid of these flaws by forging the castings to considerably smaller dimensions. The forged castings were mounted as compression specimens in the lever apparatus for long specimens, and their compressibility measured, but there was a rather large permanent set on the first application of pressure, and the relation between pressure and deformation showed a rather large amount of hysteresis. In order to get rid of the effect of the flaws, therefore, this forged piece was drawn down to wire of 0.079 cm. diameter, annealed at a bright red after the last drawing, and used as a tension specimen in the lever apparatus for long specimens. The regular series of readings were made on each specimen, at 30° and 75°. The average arithmetical deviation from a smooth curve of the readings on the commercial rod (no discards) was 0.8% of the maximum pressure effect. The corresponding average deviation from a smooth curve of the readings on the wire, making three dis- cards (it is usually true that the tension specimens do not give such regular results as the compression specimens) was 1.1% of the maxi- mum effect. Neither specimen showed any perceptible deviation from linearity, which means of course that the variation of compressibility with pressure is the same for nickel as for iron. The final results are as follows : AV Commercial rod, At 30° 77- = - 10"7 (5.25 - 2.1 X KHp) p I 0 AV At 75° — = - 10-7 (5.28 - 2.1 X 10^p) p Vo 188 BRIDGMAN. Pure drawn wire, At 30° — = - 10"7 (5.29 - 2.1 X lO^p) p AV At 75° — = - 10-7 (5.35 - 2.1 X KHp) p. I 0 A rough value for the average compressibility to 12000 kg. of the forging of pure nickel was 5.16 X 10-7. The values above for the two samples are seen to be nearly the same, and it is therefore probable that the usual impurities, such as cobalt and iron, have little effect on the compressibility. The initial compressibility has been found by Richards to be 4.3 X 10~7. His material was in the form of cubes, in which the pure metal is often supplied by chemical houses. Richards states that the nickel as originally provided was full of flaws into which the mercury was forced by pressure, and that the effect of these flaws was got rid of by heavy forging. My experience with the cast specimen above would strongly indicate (because of the hysteresis shown by this sample) that the forging must have introduced considerable internal strain, which is doubtless responsible for Richards' low value. The effects of hysteresis would be especially pronounced over a small pressure range, and would be in the direction to account for Richards' low value. Cobalt. This I owe to the kindness of Professor C. C. Bidwell of Cornell University, by whom it was in turn obtained from Dr. Herbert T. Kalmus, who had prepared it for the Canadian Government. It was of high purity, and had the following analysis: Fe 0.14, Ni 0.00, S 0.019, Si 0.02, C 0.09, Co 99.73. It was in the form of wire about 0.075 cm. in diameter and 6 cm. long, and was mounted as a tension specimen in the lever apparatus for long specimens. The material had been formed into wire by swaging followed by drawing at a red heat. It was not perfectly straight as supplied. It is rather brittle to bending, but I was able to make it perfectly straight by rolling it between red hot iron plates, thus at the same time annealing it. The two visual runs at 30° and 75° were made. The results were gratifyingly regular, considering the smallness of the effect. The average arithmetical departure from a straight line of the 28 readings (no discards) was 1.3% of the maximum pressure effect; 1.3% on the difference of compressibility means 0.1% on the actual compressibility. It was not possible to detect any departure from linearity. The final results are expressed by the formulas : AV At 30° — - = - 10-7 (5.39 - 2.1 X I0~5p) p y o COMPRESSIBILITY OF METALS. 189 AT' At 75° — = - 10-7 (5.47 - 2.1 X I0~5p) p. I 0 As far as I can find, the compressibility of cobalt has not been pre- viously measured. The value found above, close to that for nickel and iron, is of the order of magnitude that one would expect. Nichrome. The reason for determining the compressibility of this alloy is that the difference of compressibility between it and iron enters as a correction on the observed readings. The correction is very small, and at the maximum amounts to only 0.3 mm. on the slide wire of the potentiometer. The material was obtained from Driver Harris Company, and is of the grade known as Nichrome II. Its composi- tion is Ni 80% and Cr 20%. The sample measured was from the same length as the wires attached to the lever or the specimen, by which the relative changes of length were measured. Its diameter was 0.030 cm. and length about 10 cm. It was mounted as a tension specimen in the lever apparatus for long specimens. Because of the smallness of the correction it was necessary to meas- ure the compressibility only at 30°. Here the mean departure from a smooth curve (no discards) was 1.5% of the maximum pressure effect, which corresponds to 0.09% on the actual compressibility. The deviation from linearity was sensibly not the same as for iron, and at the maximum was 5.5% of the maximum pressure effect. The results are contained in the formulas : AF At 30° — = - 10-7 (5.50 - 1.5 X I0~5p) p. I o Gold. Measurements were made on two samples, both of which I owe to the kindness of Baker and Company. The specimens were stated to be of the highest purity, but I have no analysis. The den- sity was found to be 19.272 at room temperature. Both specimens were in the form of drawn rod 0.75 cm. in diameter, and were annealed at a bright red before the measurements. The first was 2.5 cm. long, and was mounted as a compression specimen in the lever apparatus for short specimens; the second was 12 cm. long, and was mounted as a compression specimen in the lever apparatus for long specimens. The compressibility of gold is exceedingly close to that of iron, and hence accurate results are not to be expected for the difference of com- pressibility. Apart, however, from the smallness of the effect, the apparatus did not function as well as usual during these measurements, there being minor electrical troubles, probably due to short circuits, 190 BRIDGMAN. so that it was necessary to repeat some of the readings. Four sets of readings were made on the shorter sample, two at 30° and two at 75°, and three sets on the longer sample, one at 30° and two at 75°. The readings on the short sample at 75° were never satisfactory, and those on the long sample at 75° left much to be desired. The readings at 30° were fairly satisfactory. At 30° the average arithmetical deviation from a smooth line of the readings on the short sample was 5% of the maximum pressure effect. For the long sample the deviation at 30° averaged 5.9% and at 75° 4.2%. The results are given by the formulas: AV Short sample, At 30° — = - 1(H (5.84 - 2.1 X KHp) p y o Long sample, At 30° -==- = - 1(H (5.77 - 3.1 X 10^p) p I o At 75° -p- = - 10-7 (5.70 - 2.1 X 10^p) p. I o The details of the variation of the coefficients in these formulas probably are not accurate. Because of its greater length the results for the long sample are doubtless to be preferred. Probably the saf- est conclusion to draw from these measurements on gold is merely that at 30° the average compressibility to 12000 is approximately 5.40 X 10-7, and at 75° it is 5.45 X 10"7. The compressibility of gold has been measured both by Richards and A. W. J. Richards finds the initial compressibility at 20° of gold of density 19.24 to be 6.3 X 10"7, and A. W. J. find the compressi- bility to be constant with pressure, and its value over the range of 12000 kg. to be 5.6 X 10-7. It is to be noticed that Richards makes gold more compressible than iron, and A. W. J. less. The measure- ments of this paper would seem to leave no room for doubt that it is actually less than iron. It is to be noticed that A. W. J. used a con- stant value for the compressibility of iron, namely 5.9 X 10-7. If for this is substituted the new value found above for the average com- pressibility over the range of 12000, namely 5.62 X 10"7, A. W. J.'s value becomes 5.3 X 10-7, agreeing within one unit in their last signi- ficant figure with the value found above. It is possible that the high value of Richards is due to the fact that his gold was not annealed after the final drawing; the difference is in the same direction as pro- duced by a similar effect in tungsten and molybdenum. The low value of the compressibility of gold is somewhat surprising COMPRESSIBILITY OF METALS. 191 both when one considers the mechanical softness of the metal, and also that its position in the periodic table below copper and silver suggests a compressibility following the succession of values of these two other metals, which would lead to a figure more than twice as high as the actual one. Copper. Copper from two sources was used. One was commercial drawn rod 0.6 cm. in diameter. Two sets of runs were made with this, the first with the rod in the commercial drawn condition, and the second after annealing. The second grade of material was pure copper from the Bureau of Standards, cut from one of their melting point samples, and had the following analysis: Sb 0.004, As 0.0020, S 0.0026, Cu 99.987. This second sample was measured only in the annealed condition. Both samples were mounted as compression specimens in the lever apparatus for long specimens ; it was necessary to dowel to- gether two pieces of the pure copper in order to get the requisite length, which was 12 cm. In addition to the runs on these long samples, at least four runs were made with preliminary forms of apparatus on shorter samples of the pure copper. The accuracy was not as high as in the final runs, and it is not necessary to describe the details, but within the somewhat wider limits of error, the results of these prelimi- nary measurements agreed with the final results. It was during the preliminary measurements that an attempt was made to find a differ- ence of compressibility of copper in different directions, with negative results. The commercial copper was quite unusual in that it showed evidence of internal strains. The readings both before and after the annealing were affected by considerable hysteresis, and after the annealing the first application of pressure produced a rather large permanent dis- tortion. The average arithmetical deviation from a smooth curve of the results for the unannealed copper (two discards) was 2.6 % of the maximum pressure effect. Almost the entire amount of this deviation is due, not to irregularity of the individual points, but to hysteresis, which at the maximum was 5.5% of the maximum pressure effect. For the commercial annealed copper the average arithmetical deviation from smoothness was 1.6% of the maximum effect, and the greatest width of the hysteresis loop was 5.5%. The loop was of a different shape from that of the unannealed copper. The pure copper from the Bureau of Standards showed no hysteresis; the average arithmetical departure (no discards) from a smooth curve was 0.47%, and the maximum departure from linearity was 1.44%. The results are contained in the following formulas: 192 BRIDGMAN. Commercial drawn rod, unannealed, At 30° ^- = - 10-7 (7.32 - 2.7 X lO-5^) p I o At 75° ^- = - 10-7 (7.39 - 2.7 X 10-*p) p , I o Same commercial rod, annealed, At 30° ^- = - 10-7 (7.29 - 2.7 X 10-5^) p \ i) At 75° % = - 10-7 (7.37 - 2.7 X KHp) p. I o Pure copper, At 30° %- = - 10~7 (7.19 - 2.6 X lO-5^) p I o At 75° %■ = - 10-7 (7.34 - 2.7 X ICHp) p. r> 0 Richards has found the initial compressibility at 20° to be 7.4 X 10-7. A. W. J. could find no departure from linearity. Correcting their results for the new value for iron, their average compressibility to 10000 was 7.1 X 10-7, against 6.84 X 10-7 given by the formula above for pure copper. Uranium. I am indebted for this to the kindness of Dr. A. W. Hull of the General Electric Company. It was furnished in the form of rolled strip 0.05 cm. thick, 0.6 cm. wide, and 5 cm. long. I have no chemical analysis. The temperature coefficient of the electrical resistance between 0° and 100° was 0.0022. Because there are no other measurements at present on the electrical properties of uranium, this does not mean much as to the purity, except that probably the purity was not high, but if in the future the properties of pure uranium are measured, this coefficient should give more definite information. This sample was mounted as a tension specimen in the lever apparatus for long specimens. The usual two runs were made, at 30° and 75°. The average arithmetical departure from a smooth curve of all 28 points (no dis- cards) was 1.0% of the maximum pressure effect. The average deviation at 75° was two or three times as great as at 30°. The maximum departure from linearity was 0.59% of the maximum pres- sure effect. It is therefore evident that the accuracy of the departure COMPRESSIBILITY OF METALS. 193 of the difference of compressibility between uranium and iron is not high. The results are embodied in the formulas: AT' At 30° — = - 10-7 (9.66 - 2.5 X 10"5 p) p W 0 AV At 75° 77- = - 10-7 (9.55 - 2.2 X 10-5 p) p . V 0 It is to be noticed that the temperature coefficient of compressibility seems to be negative. This is unusual, and perhaps is not genuine, but I give the results as they were found because the magnitude of this temperature effect is greater than the probable error. The compressibility of uranium seems not to have been previously measured. The value found is not inconsistent with that which might be expected from its position in the periodic table, but this is always a rather inaccurate way of getting compressibility, and in the case of uranium, because of its position at the end of the list of the elements, it is a particularly poor way of guessing the compressibility. Silver. This I owe to the courtesy of Baker and Co. I have no analysis, but it was said to be of the highest possible purity. It was provided in the form of drawn rod 0.6 cm. in diameter and 12 cm. long. It was annealed at a red heat before the measurements, and mounted as a compression specimen in the lever apparatus for long specimens. The density at 20° was 10.486. I also made measurements on another similar piece of material in the lever apparatus for short specimens. This was done before the final improvements had been made in the apparatus, and the results were rather irregular, but agree with those found for the other specimen within the limits of error. The regular two runs at 30° and 75° were made. The average arithmetical departure from a smooth curve of the 28 readings (no discards) was 0.4S% of the maximum pressure effect. The maximum departure from linearity, which was sensibly the same at the two temperatures, was 2.05%. The final results are given by the formulas ; AV At 30° — = - 10-7 (9.87 - 4.4 X 10~5 p) p I o AT' At 75° -=- = - 10-7 (10.04 - 4.5 X 10-5 p) p. J 0 The initial compressibility found by Richards for silver of density 10.5 was 9.9 X 10-7. A. W. J. found no departure from linearity with 194 BRIDGMAN. pressure; their mean value, corrected for the new value for iron, was 9.4 X 10-7, both results agree very closely with the value above. Aluminum. For these measurements I was fortunate to obtain two samples of exceptionally high purity. One was in the shape of a hard drawn rod and the other a casting, both 1.3 cm. in diameter. The density of both specimens at atmospheric temperature was the same, 2.700. The analysis of both specimens was also the same: Si 0.008, Fe, 0.013, Cu 0.014, Mn Nil, Al (by diff.) 99.965. For measurements these pieces were turned to a diameter of 0.6 cm., and were mounted as compression specimens in the lever apparatus for long specimens. The length of the casting was 7 cm. and that of the drawn piece 12 cm. By way of curiosity I made measurements on the hard drawn piece first before annealing. I expected hysteresis and other evidence of internal stress, but to my surprise the results were perfectly smooth and gave evidence of nothing of the sort. These measurements were made only at one temperature and to a maximum pressure of 10000 instead of 12000, there being some temporary trouble with the pressure apparatus. This hard drawn piece was then annealed at 300° for several hours, and then the regular series of runs at 30° and 75° was made. The average arithmetical departure from a smooth curve of the 28 observed points (one discard) was 0.3% of the maximum pres- sure effect, and the maximum departure from linearity was 0.75%. The departure from linearity seemed to be somewhat greater at 30° than at 75°, but on the other hand the accuracy of the readings at 30° was materially greater than at 75°. The mean of the departures found at the two temperatures was used in the final computations. The cast specimen was used without annealing; it showed no hysteresis or other evidence of internal strain. The average arith- metical departure from a smooth curve of the 28 observed points (one discard) was 0.3%, and the maximum departure from linearity 1.43%. The deviation from linearity was exactly the same at both tempera- tures, and this quantity seemed to be given much more satisfactorily bv the measurements on the castins; than by those on the drawn rod. Besides the runs just described, at least six other complete sets of runs were made on aluminum, mostly with shorter specimens. A number of the preliminary forms of apparatus were tested by measur- ing with them the compressibility of aluminum. These early results were of course less accurate than the final ones, but agreed with them within the wider limits of error. COMPRESSIBILITY OF METALS. 195 The final results are given by the formulas : AV Drawn rod, Hard, At 30° — = - KH (13.40 - 3.5 X 1Q-5 p) p I o AV Drawn rod, annealed, At 30° — = - 10-7 (13.34 - 3.5 X 10-5 p) p V 0 AV At 75° 77- = - 10-7 (13.91 - 3.5 X 1CH p) p 1 o AV Casting At 30° — = - 10~7 (13.43 - 5.0 X 10~5 p) p I ' o AV At 75° — = - 10-7 (13.76 - 5.1 X 10-5p)p. I 0 Some time ago I found the average compressibility to 6000 kg. of a piece of commercial aluminum rod to be 11.7 X 10-7, distinctly lower than the values found above. These early readings were not accurate enough to show a departure from linearity. The specimen showed set on the initial application of pressure, but otherwise there was no evi- dence, such as hysteresis, of internal strain. Richards has found the compressibility of aluminum of unstated analysis of density 2.60 to be 14.1 X 10~7, somewhat higher than the above. A. W. J. for aluminum containing 0.235% Si and 0.016% Fe were not able to detect any departure from linearity over the range of 12000 kg., and give for the average compressibility (corrected by my new value for iron) 12.7 X 10-7, which agrees to the third figure with the value given by the above formula for cast aluminum. Germanium. The crystal structure of this substance has not yet been published; I owe to the kindness of Dr. A. W. Hull of the General Electric Company the information given in personal correspondence that X-ray analysis shows it to be cubic, of the same type of lattice as diamond and silicon. For the sample of germanium I am indebted to the kindness of Professor C. C. Bidwell of Cornell University, who in turn obtained it from Professor L. M. Dennis of the Cornell Chemistry Department. It is the same specimen as that whose electrical properties he has reported in the Physical Review.11 The material is evidently ab- normal in some way, because the electrical resistance behaves normally below 100°, but above 100° decreases instead of increasing with rising temperature. The specimen was in the form of a casting about 2.5 196 BRIDGMAN. cm. long and of square section, 0.5 cm. on the diagonal. Its density at 20° was 5.302. It was mounted as a compression specimen in the lever apparatus for short specimens. The compressibility measure- ments, as well as those on electrical resistance, showed that their is something abnormal about this substance. At 30° the points with increasing pressure lay regularly on a smooth curve, but at the maxi- mum pressure there was an abrupt displacement corresponding to a shortening by an amount 7% of the maximum pressure shortening, and the subsequent points lay on another smooth curve displaced by this amount. The points at 75° showed no irregularity. On taking the apparatus apart, I found that one of the corners of the specimen was broken off. The abrupt displacement was ascribed to the breaking off of this chip, and no more was thought of it. The fractured material is more like a glass in appearance than a metal. Several days later, the specimen meanwhile having rested quietly wrapped in tissue paper, I found that one end of the specimen had spontaneously fractured into a great many small pieces, as badly annealed glass does some- times under internal strain. It suggests itself that there may be another polymorphic modification at high pressures, the transition not being sharp, but viscous, and that it may have been the internal strains produced by the gradual transformation of this new modifica- tion that caused the fracture. Making correction for the displacement from one smooth curve to another, the average arithmetical departure from a smooth curve of the 28 readings at both temperatures (no discards) was 0.33%, and the maximum deviation from linearity was 2.1% of the maximum pressure effect. There was no difference detectible between the deviations from linearity at the two temperatures. The final results are given by the formulas : AT At 30° — = - 10-7 (13.78 - 6.8 X 10-5;;) p I o AV At 75° — = - 10-7 (13.64 - 6.8 X 10~*p) p. I o Again we find a compressibility less at the higher temperature. The variation with temperature seems to be well beyond the experi- mental error, and perhaps is not surprising in view of the abnormal behavior of the electrical resistance of this substance at higher tem- peratures. There seem to be no previous measurements of compressibility. COMPRESSIBILITY OF METALS. 197 Lead. This material was a melting point sample from the Bureau of Standards. The analysis of this specimen had not been completed. A presumably similar piece of purity 99.9948 is reported in detail in a previous paper.12 Two samples were prepared in two different ways. The first was cast in a graphite mold, and chilled by lowering the mold slowly into water while the upper part of the casting was kept hot, in this way ensuring solidification from the bottom up without the formation of flaws. This was seasoned by heating in oil to 200° after easting. Its density at 20° was 11.347. It was machined to 0.75 cm. diameter and 16 cm. long, and was mounted in the apparatus for direct measurement without multiplication. The second piece was cast like the first, then extruded cold from a diameter of 1.5 to 0.75 cm., annealed at 230°, and measured in the same apparatus as the first piece. Its density at 20° was 11.337. Neither specimen showed any perceptible permanent set even after the first application of pres- sure. Regular runs were made on each sample at 30° and 75°. The cast sample gave points whose arithmetical departure from a smooth curve (no discards) was 0.14%, and the maximum departure from linearity averaged 3.1%. The average arithmetical departure from a smooth curve of the points for the extruded sample (no discards) was 0.22%, and the maximum deviation from linearity was 2.2%. It is remark- able that both samples agreed in showing a departure from linearity nearly 50% greater at the lower temperature, but this seemed to me so unlikely, that I have preferred to regard it as due to a chance accu- mulation of error, and have used the mean of the deviations at the two temperatures in computing the final formulas, which are: AV Casting, At 30° — = - 10~7 (23.73 - 17.25 X I0~5p) p I o AV At 75° — = - 10-7 (24.33 - 17.7 X lO"5?) p I o AV Extruded casting At 30° — = - 10~7 (23.05 - 12.3 X 10-5p) p I a AV At 75° — = - 10-7 (23.63 - 12.3 X lO^p) p. I o It is to be noticed that these two samples differ chiefly in their pressure coefficient of compressibility; the average compressibility to 12000 kg. of the two samples is practically the same. 198 BRIDGMAN. The initial compressibility of lead at 20° has been found by Richards to be 22.8 X 10~7, appreciably smaller than the above. A. W. J. gives for the initial compressibility 21.7 X 10~7. They also give a value for the decrease of compressibility with pressure, but they find a much smaller decrease than I do. Their decrease of instantaneous compressibility over a range of 10000 kg. (using my new values for the change of compressibility of iron with pressure) would be 1.2 X 10-7, against a minimum value twice as great from the formulas above. A. W. J.'s average compressibility to 10000 kg. is 21.2 X 10~7 against 21.8 above for either sample. Thallium. This material was prepared electrolytically by me several years ago, and I have already published data for some of its electrical properties. Judging by the temperature coefficient of resistance, its purity is high. Since the previous work the metal has been kept in a sealed glass tube. The metal was now fused under KCN, the fused button was hammered to fit the extrusion block, and it was extruded from a diameter of 1.2 to 0.6 cm. The final length was 13 cm. This was mounted in the apparatus for direct measure- ment without magnification. There was no perceptible set on the first application of pressure. Discarding the three worst points, the average arithmetical devia- tion from a smooth curve of the remaining 27 points was 0.34%, and the maximum deviation from linearity was 4.0% of the maximum pressure effect. The deviation from linearity is unusually large for thallium, and furthermore it is not symmetrical about the mean pressure, so that the results cannot be represented by a too constant formula. I have therefore computed the change of volume at inter- vals of 3000 kg. (in terms of the volume at 30° and atmospheric pressure as unity) and give them in the following table. TABLE I. Change of Volume of Thallium under Pressure. Pressure kg /cm2 AV/V0 30° 75° 3000 -.0097S -.01011 6000 .01872 .01908 9000 .02700 .02724 12000 .03505 .03554 COMPRESSIBILITY OF METALS. 199 Richards gives the initial compressibility of thallium which con- tained a slight amount of lead as 23 X 10-7. The initial compressi- bility may be found graphically from the data of the table just given, or it may be computed. I have passed a three constant power series curve through the points at 3000, 6000, and 9000 kg., and from this find the initial compressibility at 30° to be 34.8, and at 75°, 36.7 X 10-7. This is so much higher than the value of Richards that it seems difficult to ascribe to experimental error or slight impurity. The crystal structure of thallium does not seem to have been determined as yet; the difference in these values of compressibility strongly suggests that the crystal system cannot be cubic. Cerium. This material I owe to the kindness of Dr. A. W. Hull of the General Electric Company. It was provided in the form of a square bar. I extruded it at about 400° to a round section 0.3 cm. in diameter, thereby reducing its section about four fold, and obtaining a piece 4.5 cm. long. This was mounted as a compression specimen in the lever apparatus for long specimens. The purity of this specimen was presumably not high, as its mean temperature coefficient of re- sistance between 0° and 100° was only 0.001, but since the compressi- bility seems never to have been measured, and impurity usually has only an additive effect on compressibility, I thought it of sufficient interest to run through the measurements. The two regular runs were made, at 30° and 75°. Because of trouble with the pressure apparatus the maximum pressure of these runs was only 10000 instead of the usual 12000 kg., but it did not seem worth while to go to the labor of repeating the measurements over the greater range. The average departure from a smooth curve of the 23 observed points (no discards) was 0.47%, and the maximum departure from linearity was 2.2% of the maximum pressure effect. The final results are given by the formulas : „ ATr At 30° — = - 10-7 (35.74 - 19.0 X 10~5 p) p I 0 AV At 75° — = - 10-7 (35.80 - 19.7 X 10-5 p) p. I o This compressibility is not inconsistent with that to be expected for cerium from its position in the periodic table. Calcium. Material from two sources was used. The first was obtained from the General Electric Company and was cut from the same block 1 inch in diameter as I used for previous determinations 200 BRIDGMAN. of the density and thermal expansion.13 This block was perfectly free from any flaws on the exterior, but in cutting it in two preparatory to getting out a smaller specimen, an inclusion of slag was found in the very center. The previous measurement of density must therefore have a small error, and the value for the expansion also an error, but presumably not so large. On making a second cut no new inclusion was found. A clean piece was got from the original cylinder approxi- mately 1.3 cm. in diameter, and this was extruded at a temperature of 450° to 0.6 cm. in diameter. A piece 10.5 cm. long was finally obtained and mounted in the apparatus for direct measurement. Its density at 20° was 1.555 against the previous value 1.556. The second piece I owe to the kindness of Professor C. C. Bidwell. I do not know the origin, but it was stated to be of unusually high purity. This was originally of roughly square section 1.6 cm. in diameter. It was extruded in two steps at 450°, first to a round sec- tion 1.3 cm. in diameter, and then to 0.6 cm. diameter. The final length was the same as that of the other piece, and it was measured in the same apparatus. Its density at 20° was 1.532. Professor F. A. Saunders was kind enough to make a spectroscopic analysis of these two samples for me, with the following results: Calcium from General Electric Co.: Sr, considerable, perhaps 1 to 5% (2% would account for the difference of density), Cu trace, Mg trace, Al trace, Si trace. Calcium from Professor Bidwell; remarkably pure, Mg trace, Cu trace, nothing else. The regular runs at 30° and 75° were made on each specimen. The average arithmetical departure from a smooth curve of the points from the sample from the General Electric Co. (no discards) was 0.14%, and the maximum departure from linearity was 3.55% of the maximum pressure effect. The corresponding values for the sample of Professor Bidwell were (one discard) 0.11% and 3.7%. In spite of the high compressibility of calcium, the departure from linearity was symmetrical about the mean, so that it is possible to represent the results by a two constant formula. The results follow: Calcium from General Electric Co., AV At 30° -=r = - 10-7 (59.46 - 48.8 X 10-5 p) p > o AV At 75° r- = - 10-7 (59.67 - 48.8 X 10-5 p) p. I 0 COMPRESSIBILITY OF METALS. 201 Calcium from Professor Bidwell, AV At 30° -=r = - 10-7 (56.97 - 47.2 X 10"5 p) p "o AV At 75° — = - 10-7 (58.50 - 52.7 X 10-5 p )p. v o The higher temperature coefficient of the second and purer sample is to be noticed ; the pressure coefficient of the two samples is about the same. The initial compressibility of calcium of density 1.54 has been found by Richards to be 56 X 10-7. This agrees with the value above within the limits of error indicated by Richards' two significant figures. Strontium. This material was from the same lot most kindly sup- plied me by Dr. B. L. Glascock as that for which I have previously determined the changes of resistance under pressure.14 The purity is unusually high. The most homogeneous lump of this lot was chosen, and it was extruded in two stages to a final diameter of 0.3 cm. and machined to a length of 1.1 cm. The extrusion was performed at a temperature of 270° and under oil to avoid chemical action. The sample was mounted as a compression specimen in the lever apparatus for long specimens. One preliminary run was made, in which it was not possible to reach more than 10,000 kg. because of trouble with the pressure apparatus, and then two regular runs to 12,000 at 30° and 75°. Within the limits of error the results of the first incomplete run agreed with the final results. The average departure of the 28 observed points of the regular runs from a smooth curve (no discards) was 0.35% and the maximum deviation from linearity was 3.9% of the maximum pres- sure effect. The deviation was the same at the two temperatures. In spite of the high compressibility of strontium the deviation from linearity was symmetrical about the mean pressure, so that it is pos- sible to represent the changes of volume by a two constant formula, which follows: AV At 30° — = - 10-7 (81.87 - 72.5 X I0~5p) p r 0 AV At 75° — = - 10-7 (82.68 - 71.7 X I0~5p) p. V 0 202 BRIDGMAN. There seem to be no previous determinations of the compressibility of strontium. Lithium. This was from Merck, obtained before the war, sealed in glass under oil, and was from the same lot as that used in determining the effect of pressure on resistance. A chemical analysis showed 0.7% Al and a trace of Fe. This was extruded cold to a diameter of 0.6 cm. The original piece was of square section; the extrusion produced little change of section or of length, but only a change in the shape of the section from square to round. The density at 20° was 0.546. The final length was 6.3 cm. and it was mounted as a compression specimen in the apparatus for direct measurement without magnifica- tion. Two regular series of runs at 30° and 75° were made. The average departure of the 29 observed points from a straight line (no discards) was 0.15% and the mean departure from linearity was 4.9% of the maximum pressure effect. The departure from linearity was about 15% higher at the higher temperature than at the lower, a difference much beyond the limits of error. The final results are contained in the formulas : AV At 30° — = - 10-7 (86.92 - 97.5 X l^v) V V 0 AV At 75° -p- = - 10-7 (89.72 - 107.3 X lO^p) p. ''o The initial compressibility at 20° of lithium of density 0.534 was found by Richards to be 88 X 10~7. This agrees with the above with- in the limits of error indicated by Richards' two significant figures. Sodivm . This was from Eimer and Amend, obtained before the war. It was not the identical lot, but was obtained at about the same time and is presumably similar to the lot for which I have already deter- mined the effect of pressure on the electrical conductivity and melting point. No chemical analysis has been made, but the sharpness and high value of the freezing point are evidence of its high purity. The specimen was prepared by cutting from the interior of a large coherent block of sodium a smaller piece, perfectly clean and with no inclusions of any sort, and forming this by cold extrusion and pressing in a special mold into a cylinder 0.9 cm. in diameter and 2.5 cm. long. Tins was mounted as a compression specimen in the apparatus for direct meas- urement without magnification. The two regular runs were made at 30° and 75°. There was no COMPKESSIBILITY OF METALS. 203 incident at 30°; the points all lay smoothly and there was no percep- tible permanent deformation after the run. At 75°, however, which is much nearer the melting point of this very soft substance, there was permanent set after the run, and it was obvious from the shape of the curve that the set had taken place during decreasing pressure, at the lower end of the pressure range. This set may easily have been due to viscosity in the transmitting medium. The last points at 75° were therefore discarded. Except for these points the mean departure of all the observed points from a smooth curve was 0.22%, and the maxi- mum deviation from linearity 7.4% of the maximum pressure effect. The deviation from linearity was perceptibly greater at the higher temperature. Furthermore, the deviation is not symmetrical about the mean pressure, and it is not possible to represent the change of volume by a two constant formula in the pressure. In fact the devia- tion from linearity is so marked that I have thought it best to com- pute the changes of volume directly from the original data at every thousand kilogram interval at 30° and 75°. The values so computed are listed in the following table. The change of volume at each TABLE II. Change of Volume of Sodium under Pressure. Pressure AV/Vo kg /cm2 30° 75° 1000 -.0153 -.0166 2000 .0299 .0321 3000 .0437 .0466 4000 .0570 .0602 5000 .0097 .0732 6000 .0819 .0S56 7000 .0937 .0976 8000 . 1050 . 1093 9000 .1159 .1206 10000 . 1263 .1316 11000 . 1365 .1423 12000 . 1465 . 1528 temperature is computed in terms of the volume at atmospheric pressure and 20° (room temperature) as unity. For comparison with the initial compressibility of Richards we may 204 BRIDGMAN. either plot the above results on a large scale and draw the tangent to the curve at the origin, or we may pass a power series curve through the low pressure points. I have passed a three constant curve through the points at 1000, 2000, and 3000 kg. at 30° and obtain the formula: AV — = - 1.562 X lO-5^ + 3 X 10-]V + 1.6 X 10~n ps. > o .- This formula is not adapted to represent the change of volume outside its own range; already at 4000 kg. the change of volume given by the formula is 0.0566 against 0.0570 experimental. The above formula would give for the initial compressibility 156.2 X 10~7. Richards' value for the average compressibility over the range of 500 kg. is 153 X 10~7. He states that over his range the compressibility did not depart sensibly from constancy, but the above formula would show that the average compressibility over the range of 500 kg. differs from its initial value by 1.5 X 10-7. Making this correction, the agree- ment with the result of Richards (now 154.5 X 10~7) becomes closer, and perhaps is as good as could be expected.25 Potassium. This was from the same lot as that for which the effect of pressure on freezing point 15 and electrical conductivity 16 has already been determined. The high value and sharpness of the freezing point are evidence of its high purity. The material was prepared for these measurements by first filtering by forcing in a melted condition through a fine gauze under oil (Nujol); it was then cast under oil into a coherent slug, cleaned by scraping the outside surface, and finally formed by squeezing in a mold into a cylinder 0.9 cm. in diameter and 1.7 cm. long. This was mounted as a com- pression specimen in the apparatus for direct measurement without magnification. A considerably more elaborate series of measurements was made on this than on other materials. I had already found 12 that the be- havior of electrical resistance under pressure was unusual in that above 6000 kg. the temperature coefficient of resistance, which is usually constant over the entire pressure range, begins to decrease rapidly. At the time of making this observation I ventured the guess on theoretical grounds that at high pressures the thermal expansion would show a large rate of decrease. In order to determine the be- havior of the thermal expansion at high pressures the following measurements were made. It is to be noticed that theoretically a measurement of the compressibility at two different temperatures is COMPRESSIBILITY OF METALS. 205 capable of giving information as to the variation of thermal expansion with pressure, but the information so given is not very accurate. In the first place, the compressibility of potassium was measured in the ordinary way by two runs at 30° and 60°. The maximum pressure of these measurements was 10000 kg. instead of the usual 12000; the compressibility of potassium was so high that at 12000 kg. the relative deformation was beyond the range of the apparatus. At 60° the pressure was not allowed to fall below 500 kg., because of the prox- imity of the melting point. After these two series of runs over the pressure range, direct measurements of the thermal expansion were made by changing the temperature from 30° to 60° at constant posi- tion of the piston (constant mean pressure) at a number of different pressures distributed over the range. Two series of these tempera- ture observations were made. The scheme of the method and the manner of computing the results is precisely the same as that which I have previously described in detail in connection with the measure- ment of the properties of liquids under pressure.17 The only difference is that here the dimensions of the specimen are obtained by an electri- cal measurement instead of by measurement of the position of the piston. Is it not worth while again going into the details of the computations, which involve nothing not sufficiently obvious. The average arithmetical departure from a smooth curve of the points of the two pressure runs (no discards) was 0.41% of the maxi- mum pressure effect, and the average change corresponding to the thermal expansion under a 30° change of temperature was about ten times this. The measurements of thermal expansion, being much smaller, were not relatively so consistent among themselves. The first run for the determination of expansion gave scattering points at the high pressures, which, however, were not inconsistent with the later more accurate results. On the second run a better technique was used, and all the points except one lay on a smooth curve, this one lying off by some 25% of the observed effect. It is a question whether weight is to be attached to this discordant point. It has seemed to me that a certain amount of weight is to be attached to this point, as thus a curve is obtained without sudden changes of curvature, and the results tabulated have been thus computed. If no weight whatever is attached to this point, then the thermal expansion in the extreme case remains constant over the range from 6000 to 12000 kg., instead of dropping from 83 to 50, as shown. The curvature of the graph of volume against pressure is so great that, as also in the case of sodium, it was necessary to compute the 206 BRIDGMAN. changes of volume directly from the original data at intervals of 1000 kg. In the following table is given the changes of volume at even 1000 intervals at the mean temperature of 45°, obtained by taking TABLE III. Volume Changes of Potassium under Changes of Pressure and Temperature. Pressure AV/F, kg /cm2 at 45° 1000 - . 0330 2000 .0615 3000 .0867 4000 . 1095 5000 .1306 6000 .1504 7000 . 1694 8000 .1877 9000 . 2053 10000 .2222 11000 . 2385 12000 . 2544 Mean Thermal Expansion between 30° and 60° 0 .000211 2000 148 4000 105 6000 083 8000 073 10000 060 12000 050 the mean of the two series of measurements at 30° and 60°, and there is also shown the mean thermal expansion over this range, obtained by dividing the smoothed observed expansion over the range from 30° to 60° by 30. The tabulated changes of volume and the expansion are those of a mass of potassium that occupies 1 c.c. at 20° at atmospheric COMPRESSIBILITY OF METALS. 207 pressure. The initial point of the thermal expansion curve was taken from the work of others; these measurements cannot give it. To obtain the initial compressibility I have proceeded as in the case of sodium, passing a three constant power series in the pressure through the three lowest points. The formula so found is : AV =y- = - 3.565 X 10-5p + 2.85 X 10-9 f - 2 X KHV- J o The formula is not good outside its range; at 4000 kg. it gives for AV 0.1098 against 0.1091 experimental, and at 10000 kg. 0.27 against 0.22 experimental. The formula shows that the initial compressi- bility at 45° is 356.5 X 10-7, and the average compressibility over the range of 500 kg. 342.3 X 10-7. The above values for the dependence of thermal expansion on pressure show that the temperature correc- tion on this compressibility for reducing from 45° to 20° is 12.5 X 10~7, making the average compressibility over the first 500 kg. at 20° 329.8 X 10-7. The value given by Richards is 311 X 10-7. Perhaps this is as good as could be expected in view of the experimental error and the range of the extrapolation above. Especial attention is to be paid to the behavior of the thermal ex- pansion under pressure, which is that anticipated theoretically. There can be no question of the large decrease; at the minimum, which is improbable, it has decreased by a factor of 3, and more probably by a factor of 5. The compressibility in the same range has decreased by a factor of 2. This reverses the initial behavior, for at low pressures the relative decrease of compressibility with pressure is about 50% greater than that of thermal expansion. The difference of behavior of metallic potassium and an ordinary liquid is of interest. Some time ago I measured the compressibility and expansion under pressure of a number of organic liquids.13 The behavior of all these liquids is on the average roughly the same. The total change of volume under 12000 kg. of these liquids is about the same as for solid potassium, being 0.292 for the liquids against 0.254 for potassium. But the compressibility of the liquids shows a change which is rela- tively high, and is also much higher than the change of expansion. In the range of 12000 kg. the average compressibility of the liquids has changed by a factor of 15.2, whereas the expansion has changed by a factor of 4.66; potassium on the other hand has a considerably larger relative change in the expansion than in the compressibility. The significance of the behavior of potassium I believe to be that the outer shells of the atoms are being pushed tightly into permanent contact, 208 BRIDGMAN. that the thermal agitation consists merely of a swinging back and forth within the atom of the heavy nucleus, and that the restoring force on the nucleus is proportional to a linear function of its displacement. This completes the list of those substances which are known or pre- sumed to crystallize in the cubic system, and for which therefore the assumption is justified that the linear compressibility is the same in all directions. Of the above list all except three have been definitely proved to be cubic. These three are Sr, Ur, and Tl. Sr resembles Ca chemically and is probably cubic. With regard to Ur there is not much basis for estimate. We have seen above that it is highly probable that Tl is not cubic because of the discrepancy between my value of the compressibility calculated above on the basis of uniform compressibility in all directions and that of Richards. Metals Crystallizing not ix the Cubic System. Xi"> system except the cubic enjoys in general the property of having the same compressibility in all directions under hydrostatic pressure. It is, however, possible that there should be such relations between the elastic constants as to give this property to particular metals crystal- lizing in other systems, or it may be that the difference of compressi- bility in different directions is so small that experimentally it is not of importance. Xow a study of crystal models makes it seem highly probable that in the hexagonal close packed arrangement of spheres the compressibility is the same in every direction. In fact the hexag- onal and the cubic close packed arrangement of spheres are the same as far as the adjacent layers of atoms are concerned, so that any de- parture of the elastic properties of the hexagonal from that of the cubic close packed arrangements can be due only to forces between atoms separated from each other by at least one intervening atom, and these forces are presumably weak. Xow of the metals studied here, magnesium crystallizes in the arrangement of hexagonally close packed spheres. (Of the metals listed above as cubic it will be found that Hull has evidence that two, cobalt and cerium, may also crystal- lize in the hexagonal system, but since the hexagonal arrangement in which these crystallize is the close packed arrangement of spheres, the compressibility would be expected to be the same in all directions, so that the possibility of the existence of two forms introduces no compli- cations in the case of these two metals. I It is probable therefore that the method of linear compressibility COMPRESSIBILITY OF METALS. 209 gives at once without question the cubic compressibility of magnesium, and the data are accordingly given here. Magnesium. This material I owe to the kindness of Professor C. C. Bidwell, who stated that it was of unusually high purity, but the exact chemical analysis was not known. Professor F. A. Saunders was kind enough to make a spectroscopic analysis for me with the following results: Fe, none, Zn, perhaps none, Ca small, Cu small, Al faintest trace, Sr none, Cd faintest trace. The material was originally provided in the form of a square bar, about 1.3 cm. on a side and 1 cm. long. I extruded this in two stages at 500° to a piece of round section 0.3 cm. in diameter. It was straight- ened after extruding by rolling between iron plates heated to a dull red, and then annealed by heating for several hours to 300°, sealed in a glass tube to protect from oxidation. The final specimen, which was 8 cm. long, was placed in a steel sleeve in order to adapt it to the apparatus, and mounted as a compression specimen in the apparatus for direct measurement without magnification. The regular series of runs at 30° and 75° was made. The average arithmetical deviation from a smooth curve of the 29 observed points (one discard) was 0.49% and the maximum deviation from linearity was 3.8% of the maximum pressure effect. The maximum deviation from linearity was the same at the two temperatures. Within the limits of error the deviation from linearity is symmetrical about the mean pressure so that it was possible to represent the results by a two constant formula. In addition to the measurements on this pure sample, in the early stages of the development work several series of readings were made on an extruded sample of commercial magnesium of unknown purity obtained from Eimer and Amend. These results agreed within the limits of error, two or three per cent, with those found for the pure sample. This makes it probable that small impurities do not exert an important effect on the compressibility, and also lends much plausibility to the contention that the compressibility of this material is the same in different directions. The final results are given by the formulas: At 30° — = - 10-7 (29.60 - 20.3 X 10"5 p) p I o At 75° — = - 10-7 (29.97 - 18.0 X 10- p) p. ^0 210 BRIDGMAN. Richards gives for the average compressibility to 500 kg. (no dis- tinction possible between the initial and the average compressibility) the value 29 X 10-7, pressure being expressed in megabars. The cor- rection for converting to kilograms would reduce this figure by about 2%. The formulas above give for the average compressibility to 500 kg. at 20° the value 29.4 X 10~7. Considering the number of signi- ficant figures given by Richards, the agreement is within the possible experimental error, and thus we again have presumptive evidence of the equal compressibility of magnesium in all directions. Magnesium is the only one of the non-cubic metals that may be expected to have equal compressibility in all directions. The other metals of this group measured were Bi, Sb, Cd, Sn, Zn, and Te. A complete description of their behavior would demand a determination of the compressibility in different directions specified with respect to the crystalline axes. This is much beyond the scope of the present work. For some of these metals the most that can be obtained from these measurements is some idea of the magnitude of the variation with direction that may be expected. This is given roughly by the variation in the numbers obtained for the compressibility with differ- ent methods of preparing the sample. The first strong evidence of this effect was found with bismuth. Bismuth. The material used for these measurements was com- mercial electrolytic bismuth, obtained through the courtesy of the U. S. Lead Refinery, Inc. The properties of this bismuth have already been discussed in considerable detail 18; the only measurable impurity is 0.03% of silver. Measurements were made on four samples. Two of these were castings, one cast in an iron mold, and the other in a thin walled graphite mold, chilled by slowly lowering into water. The other two specimens were extruded from the casting, the reduction in diameter being 2 to 1. Measurements on two of these samples, the casting in the iron mold, and one of the extruded pieces, were made with the preliminary apparatus, before the greatest accuracy was obtained. The initial compressibility at 30° of the extruded piece was 31. S X 10-7, and that of the casting 25.6 X 10~7. The measure- ments of the casting were repeated to be sure that there was no error. Although these measurements were not as accurate as those finally made, still the accuracy was certainly one or two per cent, and the difference between these two samples was far beyond any possibility of experimental error. Measurements on the two other samples, the casting made in graphite (density at 20° 9.803) and one of the extruded pieces (density COMPRESSIBILITY OF METALS. 211 at 20° 9.797) were made with the finally perfected apparatus, that for direct measurement without magnification. The pieces were mounted as compression specimens, and their length was 16.3 cm. The mean deviation from a smooth curve of the 28 readings on the extruded specimen (no discards) was 0.32% of the maximum pressure effect, and the maximum departure from linearity was 3.5%. The corresponding figures for the casting (no discards) were 0.24% for the deviation from a smooth curve and 3.07% for the maximum departure from linearity. The final results are given by the formulas : AV Extruded cylinder, At 30° tt = - 10~7 (35.35 - 28.0 X 10~5p) p I ' o AV At 75° r- = - 10-7 (35.94 - 28.7 X I0~5p) p I o AV Cast cylinder, At 30° — = - 10~7 (22.02 - 9.0 X lO^p) p AV — = _ io-7 (22.11 - 9.0 X l0-*p) p. y o At 75 o These results were calculated on the assumption of equal compressi- bility in all directions. One third the difference between the results is a minimum measure of the difference of the linear compressibility in different directions in the crystal. It is to be noticed that both the pressure coefficient and the temperature coefficients are greater in that direction in which the linear compressibility is the greater. Richards and A. W. J. have both measured the cubic compressibility, that is, the mean of the linear compressibility in all directions. Richards finds for the initial value of bismuth of density 9.80, 29 X 10~7. A. W. J. find the same initial value, and the average compressi- bility to 10000, corrected by my new value for iron, has dropped by 2.3 X 10-7 below its initial value. It is seen that both the initial value and the pressure coefficient of A. W. J. are included in the range of values above. It should be possible to calculate the difference of the linear com- pressibility in different directions from the elastic constants of bismuth, but these do not seem to have been determined with sufficient accuracy. Tin. The material was a melting point sample from the Bureau of Standards, and had the following analysis: Pb 0.007, Cu 0.003, Fe 0.002, Sb not detected, As and S trace, Sn (by diff.) 99.988. It was investigated in the form of a simple casting, cast in graphite and 212 BRIDGMAN. annealed at a temperature of 150° for several hours after casting, of density 7.296 at 20°, or in the form of an extruded easting, the extru- sion reducing the diameter from 1.2 cm. to 0.0 cm. The extrusion was performed at a temperature of about 125°, and after extrusion the piece was seasoned at 150° for several hours. The density of this piece at 20° was 7.302. Two runs were made on the extruded pieces; one was with a pre- liminary form of apparatus and the results were not as accurate as those finally obtained. The preliminary run gave for the average compressibility to 12000 kg. at 30° 16.2 X 10~7. The other run on the extruded piece was on a sample about 16 cm. long, mounted as a com- pression piece in the apparatus for direct measurement without mag- nification. The average arithmetical deviation from a smooth curve of the 40 readings (three discards) was 0.23°^ and the maximum devia- tion from linearity was 2.0°^ of the maximum pressure effect. The runs on the simple casting were also made in the final apparatus for direct measurement, and the specimen was also 16 cm. long. The mean arithmetical deviation from a smooth curve of the 28 readings (no discards) was 0.25ro and the maximum deviation from linearity was L.09* , of the maximum pressure effect. The values of cubic compressibility given by these two samples, assuming in the computa- tion equal compressibility in all directions, was: AT' Extruded rod, At 30° — = - 1(H (19.53 - 9.6 X 10-* p) p ' o AT At 75° — = - 10-7 (20.11 - 9.9 X 10-*p) p ' 0 Cast rod, At 30° A I* To 10-7 (17.01 - 5.17 X lO-^p) P AT" At 75° — = - 10-7 (17.37 - 5.75 X 10^) p. I 0 The difference between extreme values is not as large as in the case of bismuth. Again the direction in which the linear compressibility is the greatest is the direction in which the pressure and the tempera- ture coefficients of compressibility are the greatest. The crystal structure of ordinary tin determined by X-ray methods is side-face- centered tetragonal. The initial compressibility at 20° of tin of density 7.29 is given by Richards as 19 X 1Q-7. A. W. J. give 1S.5 for the initial and 17.0 X COMPRESSIBILITY OF METALS. 213 1(H (corrected for the new value of iron) as the average to 10000. The initial values of these two observers fall within the extremes above, but the decrease of compressibility with pressure found by A. W. J. is somewhat greater than the larger of the two above values. Antimony. Two different specimens from two different sources were used. One was so-called chemically pure antimony from the J. T. Baker Chemical Co. It was cast in a graphite mold, the mold being preheated to above the melting point of antimony, and chilled by slowly lowering into water. The second specimen was antimony from Kahlbaum, his "K" grade. I have no analysis, but antimony from this source is known to be of high purity. This also was cast, but by pouring into a groove machined in a massive iron bar, the bar being cold. The manner of chilling the two castings was therefore entirely different, and the crystalline orientation in the two castings should be different. The casting of Baker's antimony, which was 14.6 cm. long, was mounted as a compression specimen in the appara- tus for direct measurement without magnification. Its density at 20° was 6.678. The other casting, which was given its final shape by grinding, was 2.3 cm. long, and was mounted as a compression speci- men in the lever apparatus for short specimens. The regular series of runs at 30° and 75° was made on each specimen. The average arithmetical departure from a smooth curve of the read- ings on Baker's antimony (one discard) was 0.48%, and the maximum departure from linearity was 1.67% of the maximum pressure effect. For the casting of Kahlbaum's antimony the average arithmetical deviation from a smooth curve (four discards) was 0.68%, and the maximum deviation from linearity was 2.7%. The results, computed on the assumption of equal compressibility in all directions, are: Baker's antimony, slowly cooled in graphite, VA At 30° — = - 10-7 (14.69 - 6.2 X 10-5p) p I i) AV At 75° — = - 10-7 (14.80 - 6.3 X 10~5 p) p . V 0 Kahlbaum's antimony, rapidly chilled in iron, AV At 30° — = - 10-7 (20.41 - 12.9 X 10~5 y) p V o AV At 75° -77- = - 10-7 (20.50 - 12.9 X 10-5 p) p . V o 214 BRIDGMAN. The range of values is nearly as great as for bismuth. Here the temperature coefficient of compressibility is greater for that direction in which the linear compressibility is less, although the pressure coefficient of compressibility is less in this direction, as is to be expected. For the initial value of the compressibility of antimony of density 6.71 Richards gives 24 X 10~7. If there is no error, the significance of this result, when compared with that above, is merely that the two methods of casting adopted above did not bring out the extreme possible difference in the orientation of the crystalline grains. Cadmium. Runs were made on three different samples. The first two were made from chemically pure cadmium from Eimer and Amend. The first was extruded at a temperature slightly below the melting point from a diameter of 1.2 to 0.6 cm., and was annealed at 230° for ten or fifteen minutes. Its density as 20° was 8.652. The second came from the same source and was cast in a graphite mold, preheated above the melting point, and the casting was cooled by lowering the mold slowly into water, and annealed after casting, as above. The density at 20° was 8.644. These two pieces were ma- chined to a length of 16 cm. and mounted as compression specimens in the apparatus for direct measurement without magnification. The source of the other specimen was Kahlbaum, his "K" grade. This is known to have only a few hundredths of per cent of impurity. This was cast in a special manner. The metal was melted in a tube of pyrex glass of diameter about 0.6 cm. in a cylindrical electric furnace mounted with its axis vertical. After temperature equilibrium was attained the glass tube was lowered slowly through the bottom of the furnace into the air of the room, thus allowing the metal in the lower part of the tube to solidify. The lowering was accomplished by clock- work, and was at the rate of about 12 cm. per hour. It will be seen that the conditions were highly favorable to a similar orientation of the crystals through the entire mass of the casting, except perhaps at the lower end where solidification began. It seems likely that in the long narrow tube, even if crystallization may originally begin about several nuclei, that eventually one surface of advance of the crystal, that on which growth is most rapid, will predominate over the others, and that after this, crystallization will consist merely in a single sur- face sweeping along the tube at a speed to keep pace with the lowering of the tube. This presumption as to the uniformity of crystal struc- ture throughout the casting should be verifiable by X-ray analysis, or perhaps by a study of the etch figures; I have not yet made such an analysis. In the case of zinc and tellurium, which were cast in the COMPRESSIBILITY OF METALS. 215 same way, the appearance of the casting indicated almost unescapably the similarity of the crystalline orientation throughout the mass. At any rate, this method of casting seems to offer important possibilities, and has at least brought out great differences in compressibility. A casting made in this way I shall call a "unicrystalline" casting. Regular runs were made on the first two samples of cadmium at 30° and 75°; on the third a run only at 30° was made. The results with these castings of cadmium had as a rule more error than the other compressibility measurements; the irregularity is probably a real phenomenon, and may be due to a slipping of the crystalline grains under pressure. In one case the permanent deformation after the initial application of pressure was considerably greater than for most other metals. For the simple casting of Eimer and Amend cadmium the average arithmetical departure from a smooth curve of the 27 observed points (no discards) was 0.29% and the maximum deviation from linearity was 2.28% of the maximum pressure effect. For the extruded casting of Eimer and Amend the arithmetical departure from smoothness (five discards) was 0.44% and the maximum deviation from linearity 2.33%. The unicrystalline casting of Kahlbaum's cadmium at 30° (one discard) gave the figures respectively 1.8% and 2.03%. The results calculated from these different specimens, assuming equal compressibility in all directions, are: Eimer and Amend, simple casting, AV At 30° -- = - 10-7 (19.54 - 10.7 X 10"5 p) p V 0 AV At 75° — = - 10-7 (20.19 - 11.0 X 1Q-5 p) p . I o Eimer and Amend, extruded casting, FA At 30° — = - 10-7 (14.17 - 7.5 X 10-*p) p V AV At 75° — = - 10-7 (14.53 - 7.7 X 10-5 p) p . > 0 Kahlbaum's, unicrystalline casting, AV At 30° — = - 10-7 (8.57 - 3.8 X 10"5 p) p . Y0 216 BRIDGMAN. The range in values is very striking, being almost two and a half fold. The difference between the figures for the ordinary and the unicrystal- line easting tends to be the extreme. This is as one would expect, because the manner of chilling would tend to make that surface which grows along the axis of the unicrystalline cylinder grow at right angles to the exterior curved surface in the simple casting, advancing along the radius. The initial compressibility at 20° of cadmium of density 8.60 was found by Richards to be 21 X 1(H. A. W. J. found for the initial compressibility 22.2 X 10-7, and for the average compressibility to 10000, 19.2 X 10~7. This pressure coefficient of compressibility is at the extreme edge of the values found above, but the initial compressi- bility of these two observers is outside the range above, which would mean a greater variation of compressibility with direction than is sug- gested by my measurements. Zinc. All the specimens were made from melting point samples of the Bureau of Standards of the following analysis: Pb 0.0026, Cd 0.0022, Fe 0.0034, Cu (not detected), Zn (by diff .) 99.992. More elabo- rate measurements were made on this than on any other metal, and the results show a greater effect of crystal structure. In general the results were much more irregular than for any other metal; there were more likely to be permanent distortions after the first application of pressure, there was quite often hysteresis, and the relation between pressure and deformation did not depart regularly from linearity, but on several occasions curves were found with distinct points of inflec- tion. (The total departure from linearity of these abnormal cases was never large.) It will not pay to describe in detail all the anomal- ous effects found, but I will give merely the average compressibility between 0 and 12000 kg. at 30°. The effect of temperature was studied in only one or two cases. With the preliminary apparatus measurements were made on three different samples. The first of these, a casting, had an average com- pressibility between 0 and 12000 kg. of 24 X 10"7, the second, an extruded casting, had a mean compressibility of 12.8 X 10-7, and the mean compressibility of the third, also an extruded casting, was 10.7 X 10~7. Measurements with the final form of apparatus and another simple casting gave for the mean compressibility 9.04 X 10-7, and on an extruded casting 8.58 X 10-7. The density at 20° of the simple casting was 7.129, and of the extruded casting 7.134. Finally a unicrystalline casting was made by the procedure described under cadmium, and measurements made of the compressibility of COMPRESSIBILITY OF METALS. 217 this in three mutual perpendicular directions. This was accomplished as follows. First the casting in its original form, 16 cm. long, was measured in the regular way as a compression specimen. From this casting were then cut four cubes 0.6 cm. on a side, marking the orienta- tion with respect to the original casting, and these cubes were piled on top of each other, and the linear compressibility measured along the axis of the pile for two different ways of piling the cubes together, corresponding to two directions at right angles to the axis of the original casting. These measurements were made in the lever appara- tus for long specimens. The mechanical properties of this casting were quite unusual in that it was very flexible, so that great difficulty was experienced during the machining of the cubes to prevent the casting from bending. The bending was not accompanied by the characteristic noise of the bending of an ordinary casting. It seems not unlikely that the stiffness of the ordinary zinc casting is due to a dovetailing together of crystal grains in different orientations, which cannot be deformed as a single homogeneous mass because of the great difference of the elastic constants of the grains in different directions. Dr. G. L. Clark has been so kind as to make an X-ray analysis of this specimen of zinc for me, and finds, within the possible errors of the method, that the crystalline orientation is indeed the same throughout the specimen, and furthermore that the direction of the hexagonal axis is along the axis of the original casting within a few degrees. The results found for the cubic compressibility of the unicrystalline casting in three mutually perpendicular directions, (that is, the linear compressibility in these directions multiplied by three) were : along the axis of the original casting 4.98 X 10~7, one of the perpendicular direc- tions, 15.9 X 10-7, and the other perpendicular direction 21.4 X 10-7. The average of these three is 14.1 X 10-7, which is approximately the average cubical compressibility, and would be the result of a single measurement by the method of Richards or A. W. J. The initial value of Richards is 17 X 10-7. The initial value of A. W. J. is 17.1 X KH, and their average value to 12000, corrected according to the new values for iron, is 15.1 X 10-7. The agreement of this last figure with the average above is as close as could be expected when one considers the possible error of my measurements on four small cubes piled together, and constitutes very strong evidence as to the reality of the large dif- ferences of linear compressibility found above in different directions. Tellurium. This material I owe to the kindness of the Raritan Copper Works, from whom I obtained it a number of years ago. I 218 BRIDGMAN. have no chemical analysis. This was prepared as a unicrystalline casting in the manner already described for cadmium. The crystal- line figures were very prominent in this casting, and show the orien- tation of the crystal to be the same throughout the entire casting. The appearance of the casting is somewhat tree-like, the axis of the tree is inclined at an angle of about 15° to the axis of the casting. The specimen was mounted as a compression specimen (length 16 cm.) in the apparatus for direct measurement. Only a few measure- ments were made on it, because it was at once evident that the behav- ior is such as to be worthy of much more detailed study, which must be put off until a systematic investigation can be made of the variation of compressibility with direction of a large number of crystals. The measurements, as far as they went, were perfectly regular. There was very little permanent set on the first application of pressure, and the relation between distortion and pressure was sensibly linear. The startling result found for this substance was that its compressi- bility along the axis of the casting is negative; that is, that it lengthens along the axis when subjected to a uniform hydrostatic pressure all over. Its average linear compressibility in this direction was -2.36 X 10-7. Paradoxical as this behavior appears, there is nothing inherently impossible about it, and something analogous has been observed by Voigt in the negative Poisson's ratio in certain directions in pyrites. (This remark is merely by way of analogy; the linear compressi- bility of pyrites is not negative in any direction.) Discussion of Results. Although the experiments just described give the compressibility of some substances not measured before, and also I believe give the compressibilities with greater accuracy than hitherto attained, the principle interest in the measurements is in the variations of com- pressibility with temperature and pressure. The variation of thermal expansion with pressure is also given by the experiments, because of course it is a mathematical identity that the pressure coefficient of thermal expansion is the same as the temperature coefficient of compressibility. The results of these experiments are collected in Table IV. In the first column is given the name and description of the substance, in the second column the initial compressibility at 30° multiplied by 107, COMPRESSIBILITY OF METALS. 219 TABLE IV. Collected Results. Substance xo at 30° XlO' XlO6 \xdpJo / lc1«\ XlO \atdp / o -1 f l da\ Xo \a dp J c IK-) \X2 dP/O Al, hard drawn 13.40 .52 3.90 drawn and annealed 13.34 .50 P76 13^2 3.96 cast 13.43 .74 1.02 7.6 5.70 Sb, Kahlbaum's cast 20.41 1.26 0.61 3.0 6.2 Baker's cast 14.69 .84 0.75 5.13 5.74 Bi, cast 22.02 .82 0.50 2.27 3.92 extruded 35.35 1.58 3.3 9.30 4.50 Cd, Eimer and Amend extruded 14.17 1.06 1.08 7.64 7.48 Eimer and Amend cast 19.54 1.10 1.S9 9.67 5.60 Kahlbaum's uni- crystalline Ca, G. E. Co. 8.57 59 . 46 .88 1.64 ".62 L04 10.32 2.76 Prof. Bidwell's 56.97 1.66 4.5 7.90 2.90 Ce, extruded 35.74 1.06 2.98 Co, wire 5.39 .78 '48 8.93 14.28 Cu, Bureau of Standards 7.19 .72 .67 9.35 10.06 commercial, unan- nealed 7.32 .74 .31 4.23 10.06 commercial, annealed 7.29 .74 .36 4.94 10.06 Ge, casting 13. 7S .98 .... 7.16 Au, short specimen long specimen Fe, pure 5.78 5.77 5.87 .74 1.08 .70 '37 6 '30 18.66 12.16 Pb, cast 23.73 1.46 1.51 6.35 6.14 extruded 23.05 1.04 1.46 6.32 4.64 Li, extruded 86.92 2.24 3.3 3.8 2.58 Mg, extruded 29.60 1.36 1.05 3.54 4.64 Mo, drawn wire 3.61 .58 2 5.54 16.12 swaged rod 3.47 .72 2 5.76 20.8 Nichrome, wire 5.50 .54 9.94 Ni, commercial rod 5.25 .80 'l6 3~04 15.2 pure wire 5.29 .80 .32 6.05 15.0 Pd, drawn rod 5.19 .80 . . • • 15.54 drawn wire 5.28 .80 AS 3.41 15.02 Pt, drawn rod 3.05 .0 .33 10.8 .0 drawn wire 3.60 . 1.00 .33 9.2 28.0 K, 356.5 23. 20. 5.62 9.76 Ag, drawn rod 9.87 .90 .66 6.68 9.04 Na, 156.2 7.8 .37 4.43 4.92 Sr, extruded 81.87 1.78 2.16 Ta, wire 4.79 .10 1.26 2.63 2.18 Sn, extruded 19.53 .98 1.92 9.85 5.04 cast 17.01 .60 1.19 7.00 3.58 W, drawn wire 3.15 1.02 2 1.24 32.2 swaged rod Te, unicrystalline casting Tl, extruded 2.93 -7.1 34.2 1.02 3'36 A P53 13.7 4.48 35^0 L94 Ur, rolled sheet 9.66 .52 5.36 220 BRIDGMAN. (this is merelv the coefficient "a" of the formula — = — ap + op2, > o where p is expressed in kg/ cm2). In the third column is given the initial proportional change of compressibility with pressure, or ) . Here — is 26 of the above formula for change of volume, Xdp/o dp and Xo is the "a" of the formula. In the fourth column is given the /l daN initial proportional change of thermal expansion with pressure /da\ In calculating the figures of this column, the derivative I — ) , is the same as I — ) or — , was computed from the detailed formulas \dt/o ot already given for the compressibility at 30° and 75° merely by sub- tracting the value of "a" at 30° from the value at 75° and dividing by 45. The value of a0 1 took in most cases from the table in the paper of Richards 1 summarizing his compressibility data. In the fifth column is given the proportional change of expansion of the fourth column divided by the compressibility "a." Finally in the sixth column is given the proportional change of compressibility divided by the compressibility. An examination of the details of the measure- ments will show that less accuracy is to be attached to quantities involving changes with temperature than with pressure. In every case the compressibility decreases with increasing pressure. This is entirely what we would expect, but I do not know that there is any necessity here, so that a case of increasing compressibility with pressure would violate any of the laws of physics. In fact it is not at all inconceivable that this may be found to be actually the case for some exceptional substances. For instance, there are a number of cases known in which the polymorphic form with the smaller volume has the larger compressibility, and it is conceivable that a change similar to one which in the case of polymorphism takes place abruptly might in some cases be brought about gradually by increasing pressure, so that the form stable at the higher pressures and therefore with the smaller volume should be the more compressible. In fact for potas- 1 fdv\ , ,. , . sium the proportional compressibility,-! — I , does increase slightly at the higher pressures. The order of magnitude of the change of compressibility with pressure, when pressure is expressed as here in terms of kilograms per We may write— — as -• ( - — ). Now the dimensions of 1/x are COMPRESSIBILITY OF METALS. 221 square centimeter, is seen to be 10~5. The dimensions of this number are those of the reciprocal of a pressure. The thermal expansion also decreases numerically with increasing pressure in nearly all the cases above, but there are at least two sub- stances in the list above which give a negative change of thermal expansion with pressure that seems to be beyond the experimental error. These are germanium and uranium. Unfortunately the ex- pansion at atmospheric pressure of these substances does not seem to have been determined, so that the proportional change could not be computed, and blanks were accordingly left in the third column of Table IV. The other blanks in this column are also due to missing data, either my own, or that of expansion at atmospheric pressure. Again the order of magnitude of the change of expansion with pressure is 10-5. The physical significance of the fifth and sixth columns is easy to see. 1 *x as 1 (I d* X2 dp ' ' X \X dP those of a pressure, and numerically 1/x is the pressure required to 1 3x . halve the volume. Hence — — is the proportional change of com- X2 dp pressibility under a pressure that would halve the volume. Similarly llda. ., . . — is the proportional change of thermal expansion under a pressure which would halve the volume. Both of these quantities are dimensionless, and the most immediately interesting thing about them is their order of magnitude, which is that of a small number. That this would be found to be the case was anticipated some years ago by the theory of the solid state of Griineisen.19 Griineisen's theory went farther, and gave an exact expression for this number in terms of the exponents in the assumed law of attraction and repulsion between the atoms. This exact expression does not seem, however, to fit at all well, and it is most doubtful whether the details of Griinei- sen's theory can be maintained. The fact that these two ratios are of the order of magnitude of a small number would probably be given by a number of simple theories, simply because they are dimensionless. The relative magnitude of the changes of compressibility and ex- pansion with pressure is of some interest. The considerations which I have already mentioned in the case of potassium would lead one to expect that at very high pressures the thermal expansion becomes vanishingly small. But if one plots the above values, ruling out those metals which do not crystallize in the cubic system, it will be found 222 BRIDGMAN. that there are only three, Al, Pb, and Li, for which the thermal ex- pansion initially decreases more rapidly than the compressibility. Even potassium is in the list of those whose compressibility at first decreases more rapidly than the expansion. But at high pressures we have seen that this state of affairs is reversed in the case of potassium. Computation will show that this is also the case for sodium, the ex- pansion at 12000 kg. having dropped to about one third of its initial value, while the compressibility has dropped to two thirds. This may well be the behavior of all metals at sufficiently high pressures. Theoretical Considerations. By far the most successful theoretical attempt to account numeri- cally for the compressibility of solid substances is that which Born 3 has developed and applied to crystals of the type of NaCl and also to CaF2 and ZnS. The fundamental thesis of this theory is that the solid is maintained by electrostatic forces, there being in the position of the center of each chlorine atom, in NaCl for example, a single elementary negative charge, and at the center of the Xa atom a single elementary positive charge. In addition to the forces between the charged ions, which act on each other according to the inverse square law, there are also mutual repulsions between adjacent atoms due to the electrons in the outer shells of the atoms. Equilibrium is due to a balance between the attractive and repulsive forces. It should be possible to calculate the repulsive forces as well as the attractive forces if the distribution of the electrons within the atom is known. From the assumption that the electrons are in cubical array in the interior of the atom, as they are supposed to be from other considera- tions, Born has deduced that the potential of the repulsive forces is as the inverse ninth power of the distance between atomic centers, and has furthermore shown that the inverse ninth power gives numerically very approximately the compressibility of crystals of the type of NaCl. Naturally the first inquiry of an attempt to extend this theory to include metals is whether the fundamental thesis still holds, namely that the forces are essentially electrostatic in nature and are due to single elementary charges or small integral multiples of them situated at the centers of the atoms. A dimensional argument as to the order of magnitude of the quantities involved suggests that the same fundamental thesis does indeed hold. A quantity of the dimensions of compressibility (M_1 LT-2) is to be built up from the electronic COMPRESSIBILITY OF METALS. 223 charge e (dimensions of e2 are MLT-2) and 8, the distance of separation of atomic centers (L). The required combination is at once found to be 8i/c-. The very fact that it is possible to build up a combination of these two quantities of the right dimensions is presumptive evidence of the correctness of our general considerations, because in general it would require three (instead of two) quantities to give in combination the dimensions of any one arbitrarily given quantity. This dimen- sional argument suggests, therefore, that compressibility should be of the order of magnitude of 8i/e2. Now assuming simple cubic structure, and expressing 5 in terms of the quantities in terms of which it is de- termined in practise, that is in terms of atmoic weight, density, and the mass of the hydrogen atom, we find that the compressibility is of (Mass of H atom) t ;/At.Wt.\f the order of magnitude of : X ( v^ r- ) , or sub- e2 \Density/ stituting numerical values for e and the mass of the hydrogen atom, . /At. Wt.\! Compressibility is of order of 8.6 X 10_H I — — I . Here of course, compressibility is in absolute units, that is, the unit of pressure is 1 dyne/cm2. In table V is given the compressibility in absolute units divided by /At. Wt.\t 10"14 ( — '- — 7— ) , for most of the metals measured above. It is seen \Density/ that the numbers are of the order of magnitude of 8.6, as our argument suggests. Although the argument is crude, it seems to me that the result is of great significance, because it suggests that in the metal, as well as in salts, the atoms become charged by losing valence electrons, and the metallic structure is held together by the forces between positively charged residues of the atoms and the lost electrons. It is furthermore to be presumed that the lost electrons take some definite position in the crystal lattice, as for instance, the measurements of Hull 20 make so probable in the case of calcium. To make the next step in refinement in the computation, we must take account of the exact arrangement of the atoms in the lattice, and must also know the law of repulsion. As far as the arrangement of the atoms in the crystal structure goes, the results of X-ray analysis indicate very definitely in at least some cases what we must expect. Thus Hull 20 has found in the case of metallic calcium that the Ca atoms have exactly the same arrangement as the Ca atoms in CaF2. 224 BRIDGMAN. TABLE V. Comparison of Compressibility with the Order of Magnitude Sug- gested by Dimensional Analysis. Metal Compress /At Wt.\| 10-14 ) \ Density j Li 27 . 5 Na 23.0 K 22.1 Ca 7.43 Sr 7.30 Mr 8.8 Zn 7.8 Cd 6.7 Cu Ag 4.4 Au 2.6 Ge 4.4 Sn 4.6 Pb 1 8 Sb 5.05 Bi 5.10 Mo 1.72 W 1.4s u 3.26 Fe 4.4 Xi 4.3 Co 4.1 Pd 2.7 Ft 1.9 The inference is strongly suggested that in metallic Ca the electrons take the place of the F atoms in CaF2, and if this is the case, it should be possible to take over with slight modification the considerations COMPRESSIBILITY OF METALS. 225 which Born has applied to calculating the compressibility of CaF2. There is also another group of metals about which we may make very plausible assumptions about the positions of the electrons in the lattice. The Na atoms by themselves or the CI atoms by themselves form in NaCl a face centered cubic lattice. Hence it is natural to expect that the complete crystal structure (meaning by this the arrangement of ions and free electrons) of those metals which are univalent, and therefore readily lose a single electron, and whose atomic nuclei have been proved to crystallize face centered cubic, should be exactly like that of NaCl. Several of the metals of the above list are in this class: Cu, Ag, Au, and probably Pb. In addition to the crystal structure, we must also know the law of repulsion. This is a much more uncertain matter. An uncritical extension of the argument given by Born would indicate a repulsion whose potential is as the inverse fifth power of the distance of separa- tion of ions and electrons, since his first term in his fundamental de- velopment for the potential of a neutral cube and a single point charge is the inverse fifth. A more critical examination of this discussion will raise several doubts, however, and in fact it is very questionable to me how much significance is to be given to the inverse ninth power which Born deduces for NaCl. It is in the first place disconcerting to discover that the inverse fifth power term for a compound lattice of electrons and ions has the wrong sign, so that instead of a repulsion we have an attraction. However, in computing what the action is between a neutral cube and an electron there are several matters with regard to which we are in serious doubt and which may greatly affect the result. Thus the statement that the force between an ion and an external electron is of the wrong sign is based on the assumption that the electron is situated on one of the three axes of the cube perpendicu- lar to the faces at their mid-points. This corresponds to the arrange- ment assumed by Born for the cubical atoms in NaCl. But it is evi- dent that if the electron approaches the neutral cube along a diagonal, the force will be of repulsion instead of attraction. In crystals not of the simple type of NaCl it is not at all obvious what the arrangement of the cubical atoms in the crystal lattice is, for the symmetry of the crystal does not seem to be consistent with the cubical symmetry of the atom. Thus in the case of CaF2, the Ca atom is surrounded by 12 other atoms symmetrically situated, so that the symmetry of the dodecahedron is indicated. Even if the orientation of the atoms with respect to the electron is known and is that assumed by Born, still the inverse fifth power must 226 BRIDGMAN. be open to grave question. Born developed the potential of the cube at external points in a power series, of which the first term was the inverse fifth, the development being good at points whose distance from the center of the cube is large compared with the semi-diagonal, or in other words, the dimensions of the atom are assumed small in comparison with their distances apart. Now this is almost certainly not the case, but a number of lines of argument indicate that the atoms pretty completely fill the total space, and that their external shells are nearly in contact. Hence in most of the cases in practise we can certainly say that more terms than the first of the series are neces- sary to give an adequate representation, and that probably under many actual conditions the series is actually divergent. There is another important consideration with regard to the series development. In computing the compressibility it is necessary to differentiate the expression for the potential. Now although the numerical magnitude of the potential itself may perhaps be given with sufficient approximation by the first term of the series, it is quite another matter to expect the derivative to be given by the derivative of the first term. Apart from these considerations, which after all are concerned with the method of computing the results, there is still another considera- tion which reaches deeper, and involves the physical picture back of the computations. This has been suggested by Schottky 21 in a recent paper. He finds quite general theorems for electromagnetic systems like the system of ions and electrons in a crystal which connect the changes of internal kinetic and potential energy of the system with the external forces acting on the system. In particular, for a system under external hydrostatic pressure, Schottky obtains the result, dl= - dU+ 3d (pV) dE = 2dU - U (pV), where L is the internal kinetic energy, E the internal electro-magnetic energy, and U the total energy of the system. The dashes indicate average values over a time sufficiently long for the average values to be constant. If in particular we now subject the system to an iso- thermal change of pressure, we have by thermodynamics dU = COMPRESSIBILITY OF METALS. 227 Substituting above, d~L = }3v + t Now in a solid under normal conditions we may suppose the kinetic energy of motion of centers of mass of the atoms to be a function of temperature only, so that at constant temperature this part of L is constant. But the equation above gives at moderate pressures an increase of L with pressure. As Schottky points out, this means an increase of internal kinetic energy of rotation of the electrons in the atoms about the nuclei, which again means a decrease of atomic radius. In other words, the atoms shrink in size as pressure increases. This reminds one strongly of the compressible atom of Richards.22 What is more, the change of size is very considerable, so that when a substance is compressed the force between the atoms changes for two reasons, both of the same order of magnitude, one reason being the change in the distance between atomic centers, and the other the change in the dimensions of the atom. This will evidently introduce another term into Born's equation, and will essentially modify his results. The modified analysis is perhaps worth giving. While we are giving the analysis, we may as well carry the work one step further than Born has in his published papers, because we can thereby get the variation of compressibility with pressure. If the atom is deformable, the coefficient of the repulsive force can no longer be regarded as a constant, but will depend on the distance of separation of the atoms. Born's analysis for the forces exerted by a cube suggests that this coefficient is proportional to the fourth power of the atomic radius. We assume as the starting point, as does Born, that the potential energy per unit cell of the lattice is of the form a b $ = — - -4- — 8 is the lattice constant, and may be computed in terms of the lattice structure and its elementary charges. The coefficient b should be directly computable in terms of the details of the structure, but it must also satisfy a condition imposed by the stability of the lattice, n should also be computable in terms of the details of the structure, but again may also be found in another way, namely, from the compressi- bility. Thus we have here the possibility of two checks, one on n and 228 BRIDGMAN. one on b. Born has availed himself only of the check on n given by compressibility measurements. All the following considerations take no account of temperature; they hold rigorously at absolute zero, but approximately at higher temperatures, because compressibility does not greatly depend on temperature. Given a structure with a potential, like the above, and subjected to external pressure, then a necessary condition of equilibrium in general is that $ + p83 shall be a minimum for changes of 5, or d$ M + ^ = °' In particular, when there is no external pressure, and 8 = 50, d$\ m\ =0- The compressibility we define in accordance with the experimental usage above as J_ dV _ 35* d8_ <503 dp d03 dp ' d8 We can get — by differentiating the equation of stability, which is, solved for p, 1 ^ V ~ ~ 362 db " Hence dp 2 d

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 ' 2a n(n+l)b 2n db lffl dS2 ~~ ~ 53 + 8n+2 ~ 8^dS + 8"dJ2 COMPRESSIBILITY OF METALS. 229 With regard to b as a function of 8, we shall content ourselves with the next order of approximation beyond assuming that it is constant, and put b = a + j8S. Substituting this value back, d& a n(a+P&) 0 db ' ' 82 ~ 8n+] 5" d2<$> 2a ra(re+l) 2re 8n+2 5' and the special value of 37 at 5 = 80 gives do a_ refa+ffio) 0 502 o0n+1 + o0n ' Now substitute these values for the derivatives of <£, and we get dp 4a re(re+3) (ra+2)(re-l) rf5 355 35n+4 a — 35n+3 A 352 X~ 503 355 w(w+3) (n+2)(n-l)fl- 4a — — rr-; — a — - .„ » p sn-l s;n-2 In this equation replace a in terms of 80 by the stability relation, put 8 = 5o + A50, where A<50 is small, so that and r — < n - _1}4io 1 1 5n-] ' ' 50n-] 1 - (re - 1) — do 1 1 gn-2 "- bn-2 1 oo . A50 AF replace — by 3 — , and substitute back, obtaining 00 v 0 230 BRIDGMAN. - 95o4- x= (n- l)(a- — 1+1 7 + 2w+l a(n + 3) - 2 P oo a — 0 50n-2 The initial compressibility is therefore, -9 So4 Xo / i8 (to- 1)1 a- — F0 This agrees with Bom's expression on putting j3 = 0. Now apply this formula to numerical computation. What we are interested in is the order of magnitude of /3, to see whether it checks with what might be expected by Schottky's theorem. We apply the formula to calcium. For this metal the evidence is particularly strong that we are dealing with a space lattice of doubly charged positive ions and electrons, the ions occupying the position of Ca in CaF2, and the electrons the position of F. Born has worked out the analysis for this type of lattice, so that we may apply his results directly. We have numerically a Xo 38.7e2 = 8.78 X 10-18, 5.56 X 10-8, - 5.7 X 10-12. Let us assume at first that /3 = 0. Then solving for to, we find to = 2.72, which is much lower than the 5 suggested by Born's analysis for neutral cubes and electrons. Let us now assume n = 5, and see what this gives for j8. = 8.5 X 10-40 Now let us see what Schottky's theorem would lead us to expect for the order of magnitude of /3. We have COMPRESSIBILITY OF METALS. 231 0 = db db dp d8 dp d5 dS dp- 5x 3 ' p = 3 db X$dp Hence Now the analysis of Born (equation 4, page 235, Verh. D. Phys. Ges. 20, 1918) shows that b is of the form b = Const e2 r4, where r is the semidiagonal of the neutral cubic atom, and the con- stant, which varies with the type of lattice, will be of the order of magnitude of 10. (For instance, the coefficient of e2 in the above expression for a is 38.7, for NaCl it is 13.94). Hence as a rough approximation put b = 10 c2 r\ db dr and — = AOe2)-3— . dp dp dr Now Schottky's theorem gives — . At ordinary temperatures, at low pressures, his formula gives very closely j- = 3F, dp and hence, applied to an elementary cell of the lattice, dL dL dr dp dr dp' Now the variable part of L we take as the contribution of the 8 outer electrons rotating about a nuclear charge of 8 at distance r. (This assumes most of the deformation of the atom under pressure is con- fined to the outer shell). The kinetic energy of these electrons is one half their potential energy, or 232 and finally, BRIDGMAN. L = 32 e2 r ' dL 64 e1 dr r-i > dr 353r2 dp ' 64 e2 ' db 15 s, , dp -8«V, 45 5V 0 = " 8~T To find r, we assume that one side of the cubic atom is f <5, 80 or r = — p = 0.577 5o. V3 Hence, substituting numerical values 0 = + 1.04 X 10-40. Now this is of the same sign, and of the same order of magnitude as the value required experimentally to make a plausible change in the value of n. When we are dealing with exponents as high as 40, it would seem that an agreement of order of magnitudes is of considerable significance. Hence we conclude that the Schottky effect must cer- tainly be expected to play an important part (a conclusion also stated by Schottky, although without numerical computation). Considera- tion of the effect makes the value of n required to give the right compressibility larger than it otherwise would be. The conclusion to be drawn from all these considerations I believe to be that we are not yet in a position to attach much significance to the precise value of n necessary to give the best value of compressi- bility. An interesting suggestion as to the compressibility of solids at extremely high pressures is contained in Schottky 's equation COMPRESSIBILITY OF METALS. 233 The term r I — ) is small compared with 3v, and since it becomes in- \or/p creasingly smaller at higher pressures, we will neglect it. We have seen that at low pressures ( — ) is positive. But the term p ( — j is negative, so that there is here the possibility of a reversal of sign of — ) at sufficiently high pressures. Now this is an exceedingly un- dp/T likely state of affairs, because it would mean that at low pressures the atoms shrink with increasing pressure, but that at sufficiently high pressures they begin to expand again. Hence if we assume that at the utmost ( — ) can only become zero, we find that the upper \dp/T limit of p - ( — ) is 3/4 (numerically). If this constitutes any real V \Op/T restriction, the place to look for it is in the compressible substances. Of the metals above, potassium is the most compressible. At 12000 1 /dv\ kg., the maximum pressure of these experiments,^-!— 1 has the value 0.19, and thus is still safely below the upper limit. However, 1 /dv\ . , . . if we plot p - I — ) against p, we obtain approximately a straight line, V \up/T so that at some pressure below 50,000 kg. there must be an essential change in the character of the relation between pressure and volume. In this connection the possibility of a new polymorphic modification of potassium at high pressures must be kept in mind ; it is natural to expect one in analogy with the new form of caesium which I have re- cently found. A new polymorphic form of potassium at very high pressures also seemed indicated to me a number of years ago 23 by the character of the curve giving change of volume on melting against pressure. 1 /dv\ The extreme value of p - I t" ) for potassium is somewhat higher than the value for the most compressible of the organic liquids in- vestigated at high pressures, ether, which has the value 0.17 at 12000 kg. Change oj Compressibility with Pressure. The theory of Born as extended above gives the compressibility as a function of pressure (or 234 BRIDGMAN. change of volume). Neglecting at first the Schottky effect, or putting /3 = 0, we have 95o4 x = - a(n— 1) 10 + n AV 1 + ^t; It is most interesting to notice that the lattice constants are not con- tained explicitly in the term giving the variation of compressibility with change of volume, but only the exponent n. 1 d53 . If we had defined the compressibility as — — instead of as we AV 7 + ?i _,. did above, then the factor of -==- would have become — - — . This V o o agrees with a formula which" Born has been so kind as to communicate to me in a personal letter. AV At high pressures -zzr is negative, so that the formula predicts a numerical decrease of compressibility at high pressures, as is most natural. Furthermore, the compressibility decreases more rapidly than the volume itself, and by a factor of the order of a small integer. Now this factor has already been tabulated in Table IV in the column under 2b/ a2. We see that this quantity is indeed of the predicted order of magnitude, so that Born's theory is suggestive also with regard to the pressure change of compressibility, as well as its absolute magnitude. But the exact value cannot be given by his expression, which demands values of n ranging from negative numbers for Ca and Sr to positive values as high as 95 for W. It is not surprising that the theory does not give precise results here. If we had to object to the single differentiation of the first term of a power series involved in computing the compressibility, still more must we object to the double differentiation involved in the computation of the change of compressibility with pressure. It is difficult to know exactly how much significance should be attached to the prediction of a correct order of magnitude for the change of compressibility with pressure. Griineisen's theory of the solid -state, which rested on a physically very different picture from Born's, also gave the right order of magnitude. It is perhaps interesting to consider whether the Schottky term gives plausible results here. If we assume as given by experiment both the compressibility and its variation with pressure, then we have two equations from which we may find both n and /3. COMPRESSIBILITY OF METALS. 235 AV . , „ We may write the factor of ~rr in the form a(n - 1) (n - 2) 8 + 2,i + -> ^ L x ado* Now apply this again to calcium. We have the two equations: 95o4 (n - 1) a - /3 5.7 X 10 -12 (), n_2 a(n - 1) (n - 2) 8 + 2n+ ^ Lx ado4 2 9 with the numerical values of a, 5o, and x already used. We notice that /3 does not enter the second of the two equations above, which is a quadratic in n. Solving this for n, we find n = 7.34. Now for /3 we have the equation j8 - 50n"2 a + 95o4 X (n - 1). Substituting numerical values, we find (3 = 2.03 X 10-56. The result found for n does not appear unreasonable, but the value for |3 appears to be of the wrong order of magnitude. It would seem, therefore, that the assumption of an inverse ninth power is so poor an approximation when computation involving two differentiations is involved that the inclusion of the Schottky term (which there is every reason to believe actually exists) cannot help matters. It is to be noticed that if conversely we regard as sufficiently ap- proximate the formula above for the initial compressibility with the Schottky term included, the value of n is very closely determined, if we know merely the order of magnitude of (3, by the requirement that /8/5on_2 be of the same order as " a." If instead now of assuming a specific form for the repulsive potential, we substitute an arbitrary function, a certain amount of information 236 BRIDGMAN. as to its behavior can be found in terms of the density, the compressi- bility, and the change of compressibility with pressure. Let us put in general: As before, the general condition of equilibrium is $ + p 83 is to be a minimum with respect to changes of 8. This condition gives: = - — - L P ' 354 352' If do is the value of 8 when p = 0, a = -8o2f(8o), and if we furthermore write (5/5o)3 = V/Vo, we have the equation /,(5)=//(5o)(yV-3^5o2(^. Here V and p are measured quantities, so that if /'(50) can be found, we have here the means of getting/' (5) at any volume (or 5), and hence f(8) itself, except for the constant of integration. The most informa- tion would be given by the most compressible metals. It is unfortu- nate that sodium and potassium, the most compressible of the metals above, do not crystallize face centered cubic, so that it is not easy to guess what the complete crystal structure of ions and electrons is, and hence not possible to compute the "a" or the/'(5o). For less compressible metals, another way of treating the above general equation gives more information. For these metals, the compressibility is given with the accuracy of the experiment by a linear function of the pressure, X = Xo(l + ap) where a is an experimental constant, and is negative for all the metals studied above. Now the above equation for 4> can be treated in the general case exactly as we did in the special cases above. Writing again X ~ V dp ' we may get x by differentiating the equation of equilibrium, and keeping only small quantities of the first order, we may express x as a COMPRESSIBILITY OF METALS. 237 function of p. It is not worth while reproducing here the algebraic details, but it will be readily found that X = 95n 10a 3V - /"(*>) ( 14 / 10a 1+ 7fXo+ Xo2 I — + f"(tny 2750 ; V This formula may be checked in special cases by substituting the particular values of / used above. In this expression for x> the term outside the square bracket is xo- The factor of p inside the bracket is the empirical constant a. A third relation is given by the stability relations under zero pres- sure. These three relations may be solved for the three derivatives of giving r <*) - - * + Xo 10a 35n3 /"'(*>) = 10a So3 5o ado -42 — 4-27-1 Xo Xo" I am much indebted to Mr. J. C. Slater for pointing out to me an error in the original form of my formula for the third derivative. These formulas may now be applied to numerical computation for those metals above which crystallize face-centered cubic, for in these cases we know the probable complete crystal structure, and can com- pute the "a." Given the structure, the only additional information needed to compute "a" is the magnitude of the charge on the ion. I have assumed that this charge is three electrons for Al, two for Ca, Fe, Co, Ni, Pd, and Pt, and one for Cu, Ag, Au, Ce, and Pb. With these assumptions the results tabulated in Table VI may be found. In the first place we notice that the sign of any given derivative is the same for all metals, and that the signs are alternately negative and positive for the first, second, and third derivatives. This is the general character to be expected for a force increasing more and more rapidly as we approach the center of the atom (in particular, a po- tential as 1/5" gives this relative arrangement of the derivatives). In general the derivatives are numerically larger for the more in- 238 BRIDGMAN. TABLE VI. The First Three Derivatives of the Repulsive Potential. Metal -/'(5o) /"(5o) -/"'(So) Al 17.3 X 10"3 1.70 X 106 .78 X 1014 Ca 4.1 .33 .11 Fe 9.7 1.44 4.9 Co 10.0 1.52 6.6 Ni 10.1 1.54 7.1 Pd 8.1 1.34 7.0 Pt 8.2 1.65 19.2 Cu 2.4 .68 3.4 Ag 1.9 .52 2.1 Au 1.9 .78 7.8 Ce 1.2 .21 .12 Pb 1.3 .27 .55 compressible metals. This is what we would expect, the atoms of the less compressible metals being less deformable. The order of magnitude of these derivatives now gives a little further information. We would expect that if / is one of the more ordinary mathematical functions the natural sphere of variability of / would have the radius 8, so that the proportional change of / and its various derivatives in a distance 8 would be of the order of small numbers. This is true for the first derivative, as we see from the table, for 8of'(80)/f(8o) is of the order of a small number. We can, if we like, reverse this principle, and get the order of magnitude of the unknown / by putting 5o/'(50)//(5o) equal to a small number. We shall find /(<5o) of the order of 10~]0. This is what we would expect on other grounds, for it is also the order of a/8o. The same is true of the second derivative, for 8of"(8o)/f"(80) is also of the order of a small number. The fact that/'(50), which measures roughly the force of repulsion between adjacent atoms, changes only a few fold in the range of 8o, has an important bearing on our picture of the nature of the atomic boun- daries. The meaning is that the atoms do not behave, when pushed into contact, as if they had rigid boundaries, like two bricks, for example. If this were the case, the relative increase of /'(<50) would be very much more rapid. In my previous work on polymorphic changes under pressure, 24 I found it useful to think of the atoms as more or COMPRESSIBILITY OF METALS. 239 less rigid bodies with definite shapes. We now see that this point of view is of restricted usefulness, and in particular is not suited to account for the behavior of the compressibility. We have up to the present followed Born in entirely disregarding the effect of temperature. This may now be taken into consideration in the following way in the domain in which the temperature is high enough for the classical dynamics to apply, so that we may ascribe to each atom a kinetic energy equal to (3/2) kt. The condition of equilibrium may be conveniently expressed in terms of the " total heat" per unit cell of the lattice, and we will restrict the discussion to substances for which the complete lattice structure of both ions and electrons is like NaCl. For the total heat, H, we have in general the expression II — E -\- yV. Here V is to be taken as the volume of the cell of the lattice or 83. E, the energy of the cell of the lattice, is made up of two parts, a potential part, which we write as before as — - + /(<5), plus the kinetic part, due to temperature agitation, which o is 6kt., since there are in the cell 4 atoms, each with energy (3/2)kt. (we suppose that the kinetic energy of the electrons may be disre- garded) . Hence . E= - ^ +/(5) + 6kt. The equilibrium condition is that the total heat is constant for changes at constant entropy, and constant pressure. Hence we have d (E + p 53) = 0 where we are to change 8 at constant pressure and entropy. This gives !+/W + 6«(f)j+3^ = 0. To evaluate ( — ) it is convenient to make connection with the ordi- \d0/a nary formulas of thermodynamics for ( — ) , where v is the volume of material which under standard conditions occupies 1 c.c. We have 240 BRIDGMAN. Now = N8* and /dv\ /dvy Hence substituting back, we have a /'W = " ^2 " ^ At zero pressure 18 KT do /'(S0) ------ So2 We have to compare this with the value /'(So) = — 77 , which we pre- 00 viously found, neglecting the temperature effect. For the case of copper, I have substituted numerical values into the second member of the right hand side of the last equation, obtaining 4 X 10-5, com- pared with the value 2.4 X 10-3 for a/So2. Hence the effect of the temperature term is not important. It is also probable that the new term will not greatly affect the values that we have found for /"(So) and /"'(So), or alter the conclusions that we have drawn from these numerical values, because in the case of copper, the numerical value of the new term which stands besides p in the next to the last equation above is 1 X 1010, and is nearly con- stant, whereas p varies through a range of 1.2 X 1010. COMPRESSIBILITY OF METALS. 241 Summary. A new method has been developed by which the linear compressi- bility of metals or other solid substances with small compressibility may be measured to high pressures. The method is accurate enough to give change of compressibility with pressure and temperature. This paper contains the results of measurements by this method on 30 metals over a pressure range of 12000 kg/cm2, and at 30° and 75°. The majority of these metals crystallize in the cubic system, or else in the hexagonal close packed arrangement of spheres, and for them the change of volume may be obtained immediately from the change of linear dimensions, because the compressibility is the same in all direc- tions. Formulas are given for the change of volume as a function of pressure and temperature over the range of the measurements. In general the compressibility decreases with increasing pressure and increases with increasing temperature. The order of magnitude of the change of compressibility or thermal expansion with pressure is the same for all metals. The compressibility changes under pressure by a fraction which is a small number (varying from 2 to 30) times the proportional change of volume under the same pressure, and similarly the proportional change of thermal expansion under pressure is of the order of a small number times the corresponding proportional change of volume. Six of the thirty substances do not crystallize cubic or in the hex- agonal close packed arrangement of spheres, and for these it was established that there are large differences of compressibility in differ- ent directions; for one substance, tellurium, it was found that in one direction the linear compressibility is even negative. In the theoretical discussion it was shown that it is very probable that the forces resisting compression in a metal are the same in nature as those in a salt of the type of NaCl, that is, the metal may be re- garded as a lattice of ions and electrons acting on each other by electro- static forces due to one or more single elementary charges. In addi- tion there is a force of repulsion. Criticism is made of the details of Born's proof of the inverse ninth power law for the potential of the repulsion, and the conclusion drawn that we have not at present a detailed enough knowledge of the structure of the atom to determine the law of repulsion so accurately as to allow us to differentiate the formulas once and twice, as is necessary in computing the compressi- 242 BRIDGMAN. bility and its change with pressure. It is shown by numerical calcu- lation that the effect discussed by Schottky due to the deformation of the atoms under pressure is of such an order of magnitude that it must be taken into account. Finally, if we do not assume the specific form of the repulsive poten- tial, but replace it by an arbitrary function, it is shown that the first, second and third derivatives may be computed numerically from experimental data now available for a number of the metals of this investigation. From these numerical values the conclusion is drawn that the boundaries of the atoms cannot be as definite as appeared not unlikely from a study of polymorphic changes under pressure. I am indebted to my assistant, Mr. I. M. Kerney, for help in making the greater number of the readings. The Jefferson Physical, Laboratory, Harvard University, Cambridge, Mass. 1 T. W. Richards, Numerous papers, summarized in Jour. Amer. Chem. Soc. 37, 1643, 1915. 2 L. H. Adams, E. D. Williamson and John Johnston, Jour. Amer. Chem. Soc. 41, 1, 1919. 3 M. Born, Numerous papers, in particular Ann. Phys. 61, 87, 1919. 4 P. W. Bridgman, Proc. Amer. Acad. 44, 266, 1909. 5 P. W. Bridgman, Proc. Amer. Acad. 47, 366, 1911. 6 P. W. Bridgman, Proc. Amer. Acad. 52, 609, 1917. 7 P. W. Bridgman, Proc. Amer. Acad. 47, 335, 1911. 8 L. Holborn, Ann. Phys. 59, 145, 1919. 9 P. W. Bridgman, Proc. Amer. Acad. 57, 52, 1922. 10 P. W. Bridgman, Proc. Amer. Acad. 57, 110, 1922. 11 C. C. Bidwell, Phys. Rev. 19, 447, 1922. 12 Reference 10, page 101. 13 P. W. Bridgman, Proc. Amer. Acad. 56, 93, 1921. 14 Eeference 13, page 96. 15 P. W. Bridgman, Phys. Rev. 3, 154, 1914. 16 Reference 13, page 82. 17 P. W. Bridgman, Proc. Amer. Acad. 49, 1, 1913. 13 Reference 10, page 114. 19 E. Griineisen, Ann. Phys. 39, 257, 1912. 20 A. W. Hull, Phys. Rev. 17, 42, 1921. 21 W. Schottky, Phys. Zs. 21, 232, 1920. 22 T. W. Richards, Jour. Amer. Chem. Soc. 36, 2417, 1914. 23 Reference 15, page 157. 24 p. W. Bridgman, Proc. Amer. Acad. 52, 180, 1916. 25 On reading the galley proofs Professor Richards has been kind enough to call my attention to his estimates of the decrease of compressibility with pres- sure of the alkali metals on page 24 of Publication No. 76 of the Carnegie Institute of Washington. He finds the average compressibility of sodium between 300 and 500 kg. to be 2/150 less than between 100 and 300, and that of potassium 12/300 less for the same intervals of pressure. VOLUME 57. 1. Kent, Norton A. and Taylor, Ltjcien B. — The Grid Structure in Echelon Spectrum Lines, pp. 1-18.' December, 1921. $.75. 2. Lotka, Alfred J. — The General Conditions of Validity of the Principle of Le Chatelier. pp. 19-37. January, 1922. 8.75. 3. Bridgman, P. W. — The Effect of Tension on the Electrical Besistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Loins. — Notes on the Early Evolution of the Beflector. pp. 67-74. February, 1922. $.50. 5. Bridgman, P. W. — The Effect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Bridgman, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. $1.25. 7. Pierce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Befraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjacent Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Franklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. $.65. 10. Bennitt, Budolf. — Additions to the Hydroid Fauna of the Bermudas, pp. 239-259. May, 1922. $.65. 11. Brues, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonididre. pp, 261- 288. 2 pis. May, 1922. $.85. 12. Thaxter, Boland. — A Bevision of the Endogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda. pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. B. — Atmospheric Attenuation of Ultra- Violet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Bomberg, Arnold. — The Batio of the Calorie at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Bobert H. — Studies on Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389-422. 2 pis. November, 1922. $1.65. 17. Thaxter, Boland. — Note on Two Bemarkable Ascomycetes. pp. 423-436. 2 pis. September, 1922. $1.35. 18. Becords of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members; Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 3 of cover.) PUBLICATIONS OP THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Series, Vols. 1-4; New Series, Vols. 1-13. 16 volumes, $10 each. Half volumes, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. Vol. 11. Part 1. Centennial Celebration. 1880. pp. 1-104. 1882. $2.00. Part 2. No. 1. Agassiz, A. — The Tortugas and Florida Reefs, pp. 105-134. 12 pis. June, 1885. (Author's copies, June, 1883.) $3.00. Part 3. Nos. 2-3. Searle, A. — The Apparent Position of the Zodiacal Light, pp. 135-157. and Chandler, S. C. — On the Square Bar Micrometer, pp. 158-178. October, 1885. $1.00. Part 4. No. 4. Pickering, E. C. — Stellar Photography, pp. 179-226. 2 pis. March, 1886. $1.00. Part 4. No. 5. Rogers, W. A., and Winlock, Anna. — A Catalogue of 130 Polar Stars for the Epoch of 1875.0, resulting from the available Observations made between 1860 and 1885, and reduced to the System of the Catalogue of Publication XIV of the Astrono- mische Gesellschaft. pp. 227-300. June, 1886. 75c. Part 5. No. 6. Langley, S. P., Young, C. A., and Pickering, E. C— Pritchard's Wedge Photometer, pp. 301-324. November, 1886. 25c. Part 6. No. 7. Wyman. M. — Memoir of Daniel Treadwell. pp. 325-523. October. 1887. $2.00. Vol. 12. 1. Sawyer, E. F. — Catalogue of the Magnitudes of Southern Stars from 0° to — 30° Declination, to the Magnitude 7.0 inclusive, pp. 1-100. May, 1892. $1.50. 2. Rowland, H. A. — On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-186. December, 1896. $2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniaceae. pp. 187- 430. 26 pis. December, 1896. $6.00. 4. Lowell, P. — New observations of the Planet Mercury, pp. 431-466. 8 pis June, 1898. $1.25. 5. Sedgwick, W. T., and Winslow, C. E. A. — (I.) Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II.) Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. 13. 1. Curtiss, D. R. — Binary Families in a Triply connected Begion with Especia Reference to Hypergeometric Families, pp. 1-60. January, 1904. $1.00. 2. Tonks, O. S — Brygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T. — The Spectrum of Hydrogen in the Begion of Extremely Short Wave-Length, pp. 121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H. — Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv. November, 1906. $1.10. 5. Trowbridge, J. — High Electro-motive Force, pp. 181-215. pis. xxv-xxvii. May, 1907. 75c. 6. Thaxter, B. — Contribution toward a Monograph of the Laboulbeniaceae. Part II. pp. 217-469. pis. xxviii-lxxi. June, 1908. $7.00. Vol. 14. 1. Lowell, Percival — The Origin of the Planets, pp. 1-16. pis. i-iv. June, 1913. 60c. 2. Fernald, W. E., Southard, E. E., and Taft, A. E.— Waverley Besearches in the Pathology of the Feeble-Minded. (Besearch Series, Cases I to X.) pp. 17-128. 20 pis. May, 1918. $6.00. 3. Fernald, W. E., Southard, E. E., Canavan, M. M., Baeder, O. J. and Taft, A. E. — Waverley Besearches in the Pathology of the Feeble-Minded. (Research Series, Cases XI to XX.) pp. 129-207. 32 dIs. December, 1921. $6.50. PROCEEDINGS. Vols. 1-56, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. The individual articles may be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. $5.00. Complete sets of the Life and Works of Rumford. 5 vols., $25.00; to Fellows, $5.00. For sale at the Library of The American Academy of Arts and Sciences. 28 Newbury Street, Boston, Massachusetts. 58-6 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 6.— February, 1923. (CONTRIBUTION FROM THE T. JEFFERSON COOLIDGE, JR. CHEMICAL LABORATORY OF HARVARD UNIVERSITY.) A REVISION OF THE ATOMIC WEIGHT OF SILICON. THE ANALYSIS OF SILICON TETRACHLORIDE AND TETRABROMIDE. By Gregory P. Baxter, Philip F. Weatherill and Edward W. Scripture, Jr. (Continued frotn page 3 of cover.) VOLUME 58. 1. Ames, A. Jr., Proctor, C. A., and Ames, Blanche. — Vision and the Technique of Art. pp. 1-47. 28 pis. In press. 2. Bjrkhoff, George D. and Langer, Rudolph E. — The Boundary Problems Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49- 128. In press. 3. Vatnio, Edward A. — Lichenes in Insula Trinidad a Professore R. Thaxter Collecti. pp. 129-147. January, 1923. $1.00. 4. BRrDGMAN, P. W. — The Effect of Pressure on the Electrical Resistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. S.75. 5. Bridgman, P. W.— The Compressibility of Thirty Metals as a Function of Pressure and Temperature, pp. 163-242. January, 1923. $1.70. 6. Baxter, Gregory P., Weathertll, Philip F. and Scripture, Edward W., Jr. — A Revision of the Atomic Weight of Silicon. The Analysis of Silicon Tetrachloride and Tetrabromide. pp. 243-268. February, 1923. S.75. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 6.— February, 1923. (CONTRIBUTION FROM THE T. JEFFERSON COOLIDGE, JR. CHEMICAL LABORATORY OF HARVARD UNIVERSITY.) A REVISION OF THE ATOMIC WEIGHT OF SILICON. THE ANALYSIS OF SILICON TETRACHLORIDE AND TETRABROMIDE. By Gregory P. Baxter, Philip F. Weatherill and Edward W. Scripture, Jr. BOTANICAL HAW* (CONTRIBUTION FROM THE T. JEFFERSON CO0LIDGE, JR. CHEMICAL LABORATORY OF HARVARD UNIVERSITY.) A REVISION OF THE ATOMIC WEIGHT OF SILICON. THE ANALYSIS OF SILICON TETRACHLORIDE AND TETRABROMIDE. By Gregory P. Baxter, Philip F. Weatherill and Edward W. Scripture, Jr. Received November 10, 1922. Presented February 14, 1923. Introduction. At the present time the atomic weight of silicon depends largely upon the analysis of the chloride and bromide of this element. Pelouze 1 and Dumas 2 compared the tetrachloride with silver, while Schiel 3 determined the silver chloride obtained from the same sub- stance. Thorpe and Young 4 converted the tetrabromide to silica, and Becker and Meyer 5 used a similar method with the tetrachloride. Two determinations of the density of silicon tetranuoride have been made by Jaquerod and Tourpaian 6 and by Germann and Booth.7 The results of all these determinations have been collected and re- calculated by F. W. Clarke 8 and are summarized in the following table. Atomic Weight of Silicon Pelouze SiCl4 4 Ag 28.37 Dumas SiCb 4 Ag 28.08 Schiel SiCl4 4 AgCl 27.95 Thorpe and Young SiBr4 Si02 28.38 Becker and Meyer SiCb Si02 28.23 Jaquerod and Tourpaian Density SiF4 28.50 Germann and Booth Density SiF4 28.31 9 1 Pelouze, Compt rend., 20, 1047 (1845). 2 Dumas, Ann. Chem. Pharm., 113. 31 (1860). 3 Schiel, Ann. Chem. Pharm., 120, 94 (1861). 4 Thorpe and Young, J. Chem. Soc, 61, 576 (1887). 5 Becker and Meyer, Z. anorg. Chem., 43, 251 (1905); 47, 45 (1905). 6 Jaquerod and Tourpaian, J. chim. phys., 11, 28, 269 (1913). 7 Germann and Booth, J. Phys. Chem., 21, 81 (1917). 8 Clarke, Memoirs of the Nat. Acad. Sci., Vol. XVI, No. 3. The Constants c of Nature. Part V. A Recalculation of the Atomic Weights. 4th Ed., p. 222 (1920). See also the discussion by Brauner in Abegg's Handb. d. anorg. ;: Chem., Vol. Ill, Pt. 2, 284 (1909). 9 Calculated by Clarke. QQ 246 * BAXTER, WEATHERILL AND SCRIPTURE. From these figures by the method of least squares Clarke computes the mean, 28.25, but calls attention to the uncertain nature of this value and to the desirability of new determinations of the constant in question. In spite of the small percentage of silicon in the chloride and bromide, 16.5 and 8.6 respectively, so that the experimental error is magnified six and twelve times in calculating the atomic weight of silicon from the results of the analyses, on the whole these two substances seem to offer marked advantages for the purpose over any other compounds of silicon. Chief of these are the ease both of preparation and purifica- tion by distillation, and of analysis by comparison with silver. A brief account of our preliminary experiments with silicon tetrachlo- ride was published not long ago.10 These experiments point to the value 28.11 for the atomic weight of silicon. Our subsequent experi- ments with both the chloride and bromide confirm the conclusion that silicon possesses an atomic weight lower than has been supposed, and indicate a value 28.06, which is even lower than the result of the pre- liminary experiments. Purification of Materials. Water. Ordinary distilled water was twice redistilled, once from alkaline permanganate and once from very dilute sulfuric acid. The condensers were block tin tubes, fitted to Pyrex flasks with constricted necks, which served as stills. The connection was made by a water seal, no cork or rubber being used. The water was collected in Jena flasks, generally just previous to use. Nitric Acid. Concentrated C. P. nitric acid was distilled through a quartz condenser, the first two-thirds as well as the last tenth being rejected. Sodium Hydroxide. The best commercial material was dissolved and the greater part of the carbonate was precipitated with barium hydroxide. The supernatant solution was centrifuged until clear and then evaporated to crystallization in a platinum dish. Centrifugal drainage of the crystals in platinum Gooch crucibles n was followed by rinsing with a small quantity of water and a second drainage. Recrystallization in the same way was continued until the mother 10 Baxter, Weatherill and Holmes, Jour. Am. Chem. Soc, 42, 1194 (1920). 11 Baxter, Jour. Am. Chem. Soc, 30, 286 (1908). ATOMIC WEIGHT OF SILICON. 247 liquor was free from chloride. This required from three to six crystal- lizations according to the quality of the original material. Two other specimens of hydroxide were made by combining barium hydroxide and sodium carbonate, after both substances had been crystallized until free from chloride. The resulting solution was then evaporated to crystallization, and the product was recrystallized. Bromine. This material had been purified by Mr. A. F. Scott according to the method which has been frequently used at Harvard.12 Crude bromine was first distilled from aqueous potassium bromide in order to remove chlorine. Next one-fourth of the product was con- verted to potassium bromide by means of recrystallized potassium oxalate, and the remaining three-fourths of the bromine was distilled from solution in this potassium bromide. All the product was then converted to potassium bromide by means of potassium oxalate, and the potassium bromide was fused in a platinum dish with enough recrystallized potassium permanganate to oxidize all organic matter. In order to obtain bromine the purified potassium bromide was dis- solved in a solution of pure sulfuric acid. The excess of potassium permanganate contained by the salt caused a small quantity of bromine to be liberated. This bromine was removed by distillation since it might have contained a trace of iodine. Enough pure per- manganate to liberate the greater portion of the bromine was then added, and the bromine was distilled into a receiver cooled with ice. In this last step the bromine received a third distillation from a bromide. The product was separated from the water, dried with resublimed phosphorus pentoxide and once redistilled. Silver. This substance was prepared by standard methods. These consisted in brief of the following processes: double precipitation as chloride, followed by reduction with alkaline sugar solution, fusion of the metal on charcoal, solution and reprecipitation with ammonium formate, fusion on pure lime, electrolytic transport, fusion on pure lime in hydrogen, etching, drying in a vacuum at 500°. Recent work by Baxter and Parsons 13 and Baxter 14 as well as earlier work have shown that these processes yield a product of adequate purity. Nitrogen. This gas was prepared by the Wanklyn process. Air was charged with ammonia and passed over hot copper catalyst. The 12 Baxter, Moore and Boylston, These Proceedings, 47, 585 (1912); Jour. Amer. Chem. Soc, 34, 1644 (1912); Baxter and Grover, Jour. Amer. Chem. Soc, 37, 1029 (1915). 13 Baxter and Parsons, Jour. Amer. Chem. Soc, 44, 577 (1922). 14 Baxter, Ibid., 44, 591 (1922). 248 BAXTER, WEATHERILL AND SCRIPTURE. excess of ammonia was removed by wash bottles containing dilute sulfuric acid. Hydrogen resulting from catalytic decomposition of the ammonia was next burned with hot copper oxide, and the gas was then purified and dried by silver nitrate solution, sodium hydroxide, concentrated sulfuric acid and phosphorus pentoxide. Finally last traces of oxygen were absorbed by hot copper oxide. The apparatus, which was constructed entirely of glass, is described in detail by Baxter and Grover.15 The Preparation of Silicon Tetrachloride. The silicon tetrachloride was prepared by the action of chlorine on silicon, and was purified by distillation in exhausted glass vessels without exposure to air or moisture. Moisture was particularly to be avoided because of the decomposition products resulting from hy- drolysis, i.e., silicic acid, silicon oxychlorides and hydrochloric acid. Because in preparing the chloride it was impossible entirely to avoid exposure to moist air, the process of purification was adapted for the removal of these impurities. In addition to the above impurities, the following were to be expected and provided for: silicon hexachloride and octachloride, titanium tetrachloride and carbon tetrachloride, the last two resulting from impurities of carbon and titanium in the silicon. Fortunately the boiling points of all the impurities are sufficiently far removed from that of silicon tetrachloride to lead to the expectation that they would be readily removed by fractional distillation. So far as can be told this proved to be the case. In the following table are given the boiling points of the impurities in question. bpt. bpt. SiCl4 58° HC1 -83° Si2Cl6 146-148° CC1« 76.5° Si3Cl8 200-215° TiCl< 136° Si2OCl6 137-138° Several specimens of the tetrachloride were prepared, differing chiefly in the method and extent of the fractionation. One lot was purchased from the General Electric Company. This had been made from silicon and chlorine, and had been rectified. Our own preparations were made as follows : Dry chlorine was passed over silicon in a hard glass tube heated to redness, and the product was condensed. The chlorine was drawn from a large tank which served for all the prepara- 15 Baxter and Grover, Jour. Amer. Chem. Soc, 37, 1037 (1915). ATOMIC WEIGHT OF SILICON. 249 tions, and was purified and dried by passing through two towers filled with beads wet with water, then three similar towers containing con- centrated sulfuric acid, and finally through a tube containing resub- limed phosphorus pentoxide. The silicon, in the form of a coarse powder, was contained in a hard glass tube which could be electrically heated and which was inclined toward the large glass bulb cooled with ice which served as condenser. Connection with both the chlorine purifier and the condenser was made with dry ground joints, and all other connections in the apparatus were made with glass seals, except Figure 1. that between the chlorine tank and the purifying apparatus, which consisted of a very short rubber connector. Since the reaction is highly exothermic, comparatively little heat was required. The material obtained from the General Electric Company was a clear colorless liquid. The initial product of our own process was a dark colored liquid containing both dissolved and suspended ferric chloride, titanium chloride, chlorine, hydrochloric acid and probably some or all of the other impurities mentioned above. Sample I was a mixture of General Electric Company product with some of our own. Sample II consisted entirely of General Electric Company material. Samples III and IV were two separate prepa- rations of our own. The fractional distillation of the first three samples was effected in apparatus of the general type shown in Figure 1. This apparatus 250 BAXTER, WEATHERILL AND SCRIPTURE. resembles closely that used by Baxter, Moore and Boylston 16 for the preparation of phosphorus tribromide, by Baxter and Moore 17 for the preparation of phosphorus trichloride and by Baxter and Stark- weather 18 for the preparation of tin tetrachloride. Two bulbs, A and B, of about 300 cc. capacity serve as stills or receivers. The separatory funnel D is employed during the introduction of the mate- rial. The flask C of one liter or more capacity is an expansion reser- voir for removing permanent gases. The small bulbs a, b, c, etc. are fractionating bulbs. The various parts of the apparatus may be dis- connected by sealing the capillaries F, G, J, K, etc. By means of the special joint L, the final receiver B may subsequently be connected with another similar system without exposure to air or moisture. The special joint L is shown on a large scale in Figure 1. The sealed- in capillary P is closed at the end and is scratched with a file at several points to facilitate breaking. To break the joint the closed tube of glass weighted with mercury Q is allowed to strike the capillary with some force. Bruner and Bekier,19 Briscoe and Little 20 and Baxter and Starkweather 18 used a similar device, while Richards and Craig 21 employed a magnetic hammer instead of one operated by gravity. First the stopcock E is made tight by rings of grease on the ends of the cock and by pouring a small amount of mercury into the separatory funnel D. The whole apparatus is next exhausted through // and the capillary G is sealed. Next D is filled with silicon tetrachloride and the mercury and nearly all the tetrachloride are admitted to A. Again a small amount of mercury is poured into D to seal the stop- cock and as soon as possible the capillary F is sealed. By sealing off the globe C at ./ a large part of the permanent gases originally dis- solved in the liquid chloride can be removed. The more volatile fractions of material are condensed in some of the small bulbs, a, h, c, by means of liquid air or carbon dioxide-alcohol mixture, and are sealed off. Then the main portion of material is distilled into B which is sealed off at K, leaving the least volatile fraction in A. Subsequently, if desired, the residue in A can be distilled into d, c,f, for examination. 16 Baxter, Moore and Boylston, These Proceedings, 47 590 (1912); Jour. Amer. Chem. Soc, 34, 263 (1912); Z. anorg. Chem., 74, 365 (1912). 17 Baxter and Moore, Orig. Com. Eighth Internal. Cong. Appl. Chem., Vol. II, 21 (1912); Jour. Amer. Chem. Soc., 34, 1644 (1912); Z. anorg. Chem., 80, 189 (1913). 18 Baxter and Starkweather, Jour. Amer. Chem. Soc, 42, 907 (1920). 19 Bruner and Bekier, Z. f. Elektrochem, 18, 369 (1912). 20 Briscoe and Little, Jour. Chem. Soc, 105, 1321 (1914). 21 Richards and Craig, Jour. Amer. Chem. Soc, 41, 131 (1919) ATOMIC WEIGHT OF SILICON'. 251 A new distillation system can now be sealed on at M and the process repeated. Sample I. All the details of the fractionation of Sample I are shown in Figure 2. The chloride was freed from suspended material by filtration through glass wool and was admitted to the exhausted bulb A together with several cubic centimeters of mercury. After standing for a week with occasional shaking, in order to remove the excess of chlorine as mercurous chloride, three small fractions, a, b, c, were removed and all but about one fifth the original chloride was distilled into B. From B about four fifths of the product was dis- W f D\ c ha 21 20 2 1 SiCI4 II Figure 2. tilled into C, after sealing off the liter reservoir E, and finally all the material was collected in fourteen small bulbs, 1-14. The appearance of a bulb sealed off for analysis is shown at g (Fig. 1). In the figures, special joints are represented as at j, expansion reservoirs as at E, more volatile fractions to the right of the center, a, b, c, less volatile fractions to the left of the center, d, e. The volumes of the fractions are indicated roughly by the size of the circles. Fractions 3, 6, 9 and 12 were analyzed. Sample II. Figure 2 shows diagrammatically the fractionation of Sample II. After standing over mercury for some time, three small light fractions, a, b, c, were removed with liquid air and the greater 252 BAXTER, WEATHERILL AND SCRIPTURE. part of the remainder was distilled into B leaving a residue d of about one-sixth. In spite of the use of the expansion reservoir F, distillation refused to begin when the bulbs, a, b, c, were cooled with carbon dioxide-alcohol mixture. Since, however, liquid air was effective in starting the distillation, and since thereafter carbon dioxide-alcohol refrigerant proved sufficient, it seems likely that hydrochloric acid gas was responsible for the delay. In the next two steps the expansion reservoir G and three light fractions, e, f, g, were removed, and two residues // and j were rejected when the bulk of the material was dis- tilled successively into C and D. From D all the remainder was col- lected in twenty one small bulbs, except for a residue of obviously less volatile oily material, amounting to a few tenths of a cubic centimeter. Of the final fractions, Xos. 1, 7, 13 and 20 were analyzed. In order to discover whether the non-volatile residue was diminish- ing in quantity with successive distillations, residue j was transferred to an exhausted bulb without exposure to air and fractionally distilled. Here the last cubic centimeter, as in the case of the residue in D, con- sisted of a colorless oily liquid, which could be distilled by warming but which was obviously less volatile than the tetrachloride. In the preparation of Sample III it was found that a part of this material is titanium tetrachloride. The remainder is apparently higher chlorides of silicon. Sample III. In preparing Sample III, the product of the reaction of chlorine on silicon was collected in the bulb A (Fig. 3) which was then connected with an efficient water pump. The liquid boiled vigorously under these conditions, so that a large part of the excess of chlorine and probably much of the hydrochloric acid were removed at the outset. Furthermore the boiling of the tetrachloride must have flushed out most of the air originally in the bulb. After removal of the light fractions, a, b, c, with liquid air, the bulk of the material was distilled into bulb B, containing mercury, and allowed to stand for four months. In each of the fourteen distillations of the main portion, a residue of several cubic centimeters was left in the still, and usually was collected as two or more fractions. Furthermore nine more volatile fractions were rejected in the first four distillations. The final product was collected in twenty one small bulbs, of which Nos. 1, 13, 16 and 21 were analyzed. From the point of view of quantity of material bulb 14 represents about the middle of this series. In the first seven distillations the oily, less volatile residue observed in the preparation of Sample II appeared, in gradually decreasing proportions. Some of this material was found by qualitative testing ATOMIC WEIGHT OF SILICON. 253 with hydrogen peroxide to contain titanium. The quantity of ti- tanium contained in fraction j was very small, however, and none could be found in fraction k. Furthermore the amount found in fractions d and e seemed far too small to account for the quantity of less volatile residue, so that we are inclined to believe that a con- siderable portion of the residue consisted of higher chlorides of silicon. Sample IV. Still a fourth sample of tetrachloride was purified as SiCI4 III 21 20 19 Figure 3. 3 2 I described under the preparation of Sample II. Owing to the discovery of the less volatile residue in Sample II, the fractions originally ob- tained in this way were not analyzed. After the preparation of Sample III, however, Mr. A. F. Scott, who was engaged in the Coolidge Memorial Laboratory in the purification of boron trichloride by fractional distillation, found that a Hempel column could be con- veniently employed in a distillation at low temperatures, provided the 254 BAXTER, WEATHERILL AND SCRIPTURE. column was chilled to a suitable temperature intermediate between that of the still and the condenser. Since it was a simple matter to apply this principle to the distillation of silicon tetrachloride, all of the fractions resulting from the first distillation, without regard to purity, together with a considerable number of rejected fractions obtained in 6 6 6 (57jS (§Wf^)W^) (J7M) C 67tt> 6 G (57jf) <8><3>/^)a>(DO(vv}0 Figure 4. the purification of Sample III, were combined in a large bottle by breaking the bulbs with a heavy glass rod, and the mixture was sub- jected to a new fractionation, as indicated in Figure 4. The material was poured into bulb .4 through a glass funnel and the bulb, while immersed in hot water, was exhausted with an efficient ATOMIC WEIGHT OF SILICON. 255 water pump, through a drying tube containing sodium hydroxide. The tetrachloride was allowed to boil vigorously for a short time, until several cubic centimeters had evaporated, together, we hoped, with much of the hydrochloric acid formed by exposure to the moisture of the air. Repeated distillation was then carried out with apparatus similar to that indicated in Figure 5. The Hempel column was cooled with ice and water, while the receiver was chilled with carbon dioxide- alcohol mixture. A continuous reflux action was evident in the Hempel column throughout the distillation. In each distillation a few cubic centimeters of residue were rejected. Furthermore, frequently at the end of a distillation the temperatures of still and receiver were re- Figure 5. reversed so that the most volatile material was largely condensed in the still. Sometimes the same effect was produced by inserting an additional large bulb in the train and condensing the most volatile material in this bulb before the next distillation. After six distillations, one of the capillary connections in the ap- paratus was accidentally broken so that moist air was admitted. With as little exposure to the air as possible the chloride was intro- duced into bulb // and seven more distillations were carried out in the same way. In the last distillation the two most volatile fractions (1 and 2, Fig. 4) were collected for analysis, the main bulk of material was distilled into bulb 0 from bulb A, and the residue was collected in bulbs 14 and 15 for analysis. Since the possibility of the existence of constant boiling distillates is always to be feared in fractional distillation, an effort was made to avoid or detect the effect of such a mixture by further distillation at a different pressure. To do this the still was immersed in warm water, the Hempel column was kept at about 5° and the receiver was sur- 256 BAXTER, WEATHERILL AND SCRIPTURE. rounded with ice and water. In the seventh distillation at the higher pressure the two most volatile fractions, 3 and 4, were collected for analysis, as well as the two least volatile fractions, 12 and 13 (Fig. 4). Unfortunately at this point the apparatus was again opened to the air by the breakage of a capillary. Although the quantity of material was by this time much diminished, about 30 cc, the remainder was introduced into bulb V and the bulb thoroughly flushed out with tetrachloride vapor. Further fractionation at the lower pressure, using carbon dioxide-alcohol as refrigerant, yielded three residual fractions, 9, 10 and 11, while the remainder was collected in the four fractions, 5, 6, 7 and 8. Sample I was purified by Dr. E. O. Holmes, Jr., Samples II and III by Dr. Weatherill, and Sample IV by Mr. Scripture. The Analysis of Silicon Tetrachloride. After being weighed the bulb containing silicon tetrachloride was broken under an excess of sodium hydroxide and the glass was col- lected and weighed. The solution was diluted to considerable volume and made acid with nitric acid, and then was precipitated with a solu- tion of a weighed very nearly equivalent amount of pure silver. The point of exact equivalence between chloride and silver was then found with the assistance of a nephelometer. The bulb of material selected for analysis was soaked first in cleaning solution, then for several days in pure water. Sometimes the capil- lary was slightly scratched with a file. Next the bulb, suspended in a platinum wire basket, was weighed under water of known temperature. After being dried with a clean, nearly lintless cloth, the bulb was al- lowed to remain at least over night in a desiccator containing fused potassium hydroxide. Weighing by substitution followed, and the temperature, pressure, and humidity of the balance case were observed at the same time. From the loss in weight under water the volume of the bulb was computed, and then from the atmospheric conditions the buoyant effect of the air on the bulb. In weighing the fractions of Sample IV, the atmospheric density was found from the weight of a sealed, standardized globe.22 A solution of about fifty per cent excess of the quantity of pure sodium hydroxide (purified as described on page 247) necessary to react completely with the tetrachloride was filtered into a heavy walled 2 22 Baxter, Jour. Amer. Chem. Soc, 43, 1317 (1921). ATOMIC WEIGHT OF SILICON. 257 liter conical Jena flask, which was provided with a particularly well ground glass stopper. The volume of the sodium hydroxide solution varied from 200 to 400 c.c. according to the weight of the tetrachloride sample. The bulb was introduced and after thoroughly wetting the walls and stopper of the flask with the alkali, the bulb was broken by shaking the flask. The fog produced on breaking of the bulb appar- ently disappeared after fifteen or twenty minutes, but the flask was not opened for several hours, in order to make certain .that no silicon tetrachloride or hydrochloric acid vapor was lost. The solution was always perfectly clear, although in a few instances silicic acid separated in the end of the capillary of the bulb. This difficulty remedied itself in some instances, for on standing the silicic acid dissolved in the alkaline solution. In others, this process was assisted by breaking the capillary with a blunt rod. In no case was an analysis continued until all traces of such a deposit had disappeared, and two experiments were abandoned because of undissolved silicic acid in the capillary. Next the contents of the flask were diluted to 700-800 cc. and were filtered through a quantitative filter into a glass-stoppered precipitat- ing flask or bottle. The glass fragments were washed by decantation several times and were collected on the filter, and finally the filter paper was charred and the residue burned in a platinum crucible at as low a temperature as possible so as to avoid the danger of volatilizing alkali from the glass. In order to find out whether it is possible to wash the filter free from sodium silicate and silicic acid, several blank experiments were carried out in which a similar alkaline solution of silicon tetrachloride was filtered in the same way through a quantitative filter, and after thorough washing with water the filter was burned. In still other experiments a weighed empty bulb similar to those used in collecting the silicon tetrachloride for analysis, but open to the air, was broken in the silicate solution and the glass fragments were collected and determined as above. In all these experiments the ash of the paper was found to be slightly in excess of the weight to be expected. Excess mg. Without glass bulb 0.21 0.15 0.1G With glass bulb 0.14 0.35 Average 0.20 258 BAXTER, WEATHERILL AND SCRIPTURE. In experiments where the foregoing method of determining the glass was employed (Series I and II) a negative correction of 0.20 mg. has therefore been applied. Since we found that silicate solutions, of the concentration obtained in these analyses, when acidified with one per cent nitric acid remain clear indefinitely, the experiment was tried of washing the glass frag- ments and the filter paper with nitric acid of this concentration. Somewhat to our surprise not only did the excess in weight of the ash disappear, but a slight deficiency was found. Since this deficiency was practically the same whether or not a glass bulb was involved, it is apparent that some of the mineral constituents of the filter are ex- tracted by either the alkaline silicate or the nitric acid. The following table gives the results obtained in blank experiments in which one per cent nitric acid was used as washing liquid. Without glass bulb Deficiency mg. 0.08 0.08 0.12 0.01 0.09 0.13 0.01 0.02 0.07 With glass bulb Average The method of filtering the sodium hydroxide solution of the silicon chloride through the filter and then washing the glass and filter with one per cent nitric acid was followed in all the analyses of Series III, and in Analyses 21 to 25 of Series V. In these experiments an average positive correction of 0.07 mg. was applied to the weight of the glass before subtracting the weight of the filter ash, 0.11 mg. In all the analvses of Series IV and Analvses 26, 27 and 28 of Series V, the glass was washed with one per cent nitric acid, but a different variety of filter was employed. Blank experiments with these filters gave the following results: Excess mg. Without glass bulb 0 . 05 0.03 ATOMIC WEIGHT OF SILICON. 259 Excess mg. With glass bulb 0.04 0.04 -0.04 -0.04 0.09 0.08 Average 0 . 03 In these experiments' a negative correction of 0.03 mg. was applied to the weight of the glass. The filtrate from the glass was then made acid by adding, in the form of 1 normal solution, a quantity of nitric acid equivalent to the sodium hydroxide used. Since a fifty per cent excess of sodium hydroxide was used, and since four ninths was changed to sodium chloride, after the addition of the nitric acid the total acidity was not far from 0.1 normal. This could do no harm, however, for Honig- _schmid 23 has found that even three normal nitric acid liberates no hydrochloric acid or chlorine from a similar quantity of potassium chloride. From the corrected weight of silicon tetrachloride the weight of silver necessary to precipitate the chloride was computed. This quantity was weighed out, chiefly in the form of a very few large buttons, the final adjustment being made with small electrolytic crystals. After careful solution of the silver in chloride-free nitric acid and elimination of nitrous acid, in a flask provided with a spray trap in the form of a column of bulbs ground into the neck, the solution was diluted to tenth normal concentration or less, and then was added slowly with constant agitation to the chloride solution contained in a glass stoppered flask or bottle. The mixture was allowed to stand for several days with occasional shaking, before testing for excess of chloride or silver in a nephelometer. If an excess of either was found, the deficiency of the other was made up by adding hundredth normal solution until the endpoint had been reached. Even then the solutions were allowed to stand for some weeks longer with occasional shaking in order to allow included or occluded material to be extracted from the precipitate. Only slight changes were ever produced by this standing. The manipulations of precipitation and testing of the solutions were 23 Honigschmid, Z. Elektrochem., 26, 403 (1920); Ber. d. d. chem. Gesell., 54, B, 1873 (1921). 260 BAXTER, WEATHERILL AND SCRIPTURE. always carried out in ruby light. In using the nephelometer all the precautions noted by Richards and 'Wells 24 were observed, such as preparing the comparison tubes under as nearly as possible identical conditions of temperature, concentration, and time, allowing the tubes to come to constant ratio by standing for an hour or more, and taking the average of several readings of each ratio. The Preparation of Silicon Tetrabromide. To prepare silicon tetrabromide, dry nitrogen saturated with pure bromine at a temperature not far below the boiling point of bromine was passed over silicon at red heat in an all-glass apparatus very similar to that used for preparing the tetrachloride. Since the layer of silicon was long, very little bromine passed through the reaction tube into the receiver. About six hundred grams were secured in the single preparation which was made. The bromide was then subjected to fractional distillation for re- moval of the impurities which experience with the chloride led us to expect. Since the boiling point of the tetrabromide, 153°, is far higher than that of the chloride, the distillations were carried out by heating the still instead of by cooling the receiver. Furthermore, Hempel columns were employed in most of the distillations. The boiling points of the probable volatile impurities to be expected are as follows : bpt. bpt. SiBr4 153° HBr -69° Si2Br6 240° CBr4 190° TiBr< 230° Chlorobromides were not to be feared, since the bromine used was chlorine-free. The general type of apparatus used in the distillations is shown in Figure 5. The material in the bull) A, which has been filled from a previous distillation and sealed off at A', is warmed in a water bath and is distilled through the Hempel column D into the bull) B, which is cooled with ice, leaving a residue of a few cubic centimeters in A. The bulb A is then separated by sealing the capillary G. B terminates in a special joint J through which it may be attached to another similar system. The residues left in A and B may then be distilled into the small bulbs a and b by inclining the bulbs to a suitable angle. The complete course of the tetrabromide distillation is shown in Figure 6. 24 Richards and Wells, Amer. Chem. Jour., 31, 235 (1904); 35, 510 (1906). ATOMIC WEIGHT OF SILICON. 261 After the material had been collected in bulb A, a considerable amount of mercury was added and, with the bulb and contents nearly at 100°, the bulb was exhausted through a drying tube by a very efficient water pump. The liquid boiled vigorously so that a large part of the excess 6 6 6 6 0" 6 6 H Si Br. OOO (1)0(3)0 h 0050 000 0000 K Figure 6. of bromine evaporated, together with tetrabromide vapor. The air must have been thoroughly flushed out by this boiling. Probably a large portion of hydrobromic acid formed by hydrolysis was removed at the same time. The bulb A was sealed off while the pump was operating. Upon shaking the bulb A, after cooling, the mercury 262 BAXTER, WEATHERILL AXD SCRIPTURE. combined with the greater part of the bromine. Bulb B, also con- taining mercury, was then attached and exhausted, and the material poured from A to B, where it was again shaken with fresh mercury. Connection was now made with the system C, Y, a, b, First the two liter expansion flask Y was sealed off. Two small fractions a and b were next removed by cooling with liquid air while B was warmed, and all but about 6 cc. was distilled into C and sealed off. The material in a and b was colored yellow, owing doubtless to free bromine. The main fraction in C also was slightly colored, as well as the residue in B. Another distillation into D which also contained mercury followed, and then a series of five distillations into /. In all these distillations a slight cloudiness persisted in the main body of material. Since we feared that this might be a bromide of mercury vaporizing with the silicon tetrabromide, several pieces of bright copper wire were placed in /. These pieces of wire were slightly discolored during the distillation from /, but since a fractionating column, as indicated at c, was used for the first time in the next distillation, it is uncertain whether the copper or the column was responsible for the great im- provement in the appearance of the distillate; for from this point on it remained perfectly clear and colorless. After distillation into the bulb L, the accidental breaking of a capillary admitted air and consequently a small amount of moisture to the tetrabromide. With as little exposure to air as possible the material was poured from L into a suitably prepared bulb M and again exhausted as before. After distillation into N, but before seal- ing off the connection to M, N was warmed and 31 was cooled so that several cubic centimeters were distilled back into .1/. This procedure is indicated in Figure 6 as if a light fraction had actually been collected from X. A similar procedure was followed in the next distillation. The contents of the bulb d, which contained the heavy fraction remaining after the distillation from 0, were tested for titanium, with a negative result. Although a few pieces of copper wire were used in 0, to remove a trace of bromine which was formed apparently by the exposure to the air in L and 31, the first of the four light fractions removed from P had a slight yellow color; the remaining three were colorless. After about half the material had been distilled from S to T, four small samples were collected as indicated at h, the remainder was distilled into T, leaving a residue which was collected in three portions, indicated at k. The samples analyzed were, 1, the least volatile of three light fractions taken from R, 2, the second most volatile fraction from T, ATOMIC WEIGHT OF SILICON. 263 3 and 4, middle fractions from S, 5 and 6, the least volatile material from S, 7, the residue left from the distillation from R and 8, a similar residue from (}. The Analysis of Silicon Tetrabromide. The analysis of silicon tetrabromide was carried out almost exactly as described in the case of the tetrachloride. The glass of the bulb was washed with one per cent nitric acid and allowance made for re- duction of the weight of the filter ash as described on page 258. When the end point of the comparison had been reached, the portions tested with silver and bromide solutions remained almost perfectly clear. This indicates the absence of appreciable amounts of chloride, and is to be expected in view of the pains taken to remove chlorine from the bromine. Weighings were made on a No. 10 Troemner balance, sensitive to 0.02 mg. with a load of fifty grams. The beam was graduated to 0.05 milligram and the 5 mg. rider was used to determine all quantities larger than this, interpolation from the zero points being employed for amounts less than 0.05 milligram. The weights were of gold plated brass, except the fractional weights, and were compared by the Richards 25 method. All weighings were by substitution. In the case of the bulb and the silver, the weights were substituted for the object weighed. In the case of the glass, the crucible was substituted for a similar counterpoise. Impure radium bromide was kept in the balance case to prevent electrostatic effects. Vacuum corrections were applied as follows: Vacuum Correction Weights Density per gram Weights 8.3 Ag 10.49 -0.000031 glass 2 5 +0.000335 air 0.001 293 at0° and 760 mm. Humidity was read on an accurate hair hygrometer. The analyses of Series I, II, III, and analyses 21, 22, 23, 24 and 25 of Series V were performed by Dr. Weatherill. All analyses of Series IV and analyses 26, 27 and 28 of Series V were performed by Mr. Scripture. 25 T. W. Richards, Jour. Amer. Chem. Soc, 22, 144 (1900). 264 BAXTER, WEATHERILL AND SCRIPTURE. m CO uo_ o o cm ^h t-h o t^ o ^# o o ci c ^ o t-H t-H ^t1 a; lo "f ci co t>. •- «j E nddhhh fflOOl'fffl 00 iO ffl CO N N to O CO (O O N O 55 E5 8 ^ O h t-h t-h O O O t-H O o t-h o o o oooocoooo O Sd> — *s •« rs co oo go co co co co co oo oo ooooccocoo cocoocgocococogoco •<:iC0 CI CI CM (M CM CM * — (HI CO ^ G© CI LQ t^ CO CO t-h r- ■* O O O H IMONSNN-t-flM <- LI !D H f 1 H GO CO CO O 00 COrHtOCOH MOOHOOOI- I t-h rt oJI oo r^ oo oo oo t^- r» i^ go i-- r~ o t^ co i— ^t~r^i>r^.i>.i--t-r- Q .5"* co co co co co co co co co co rccocococo MCOMCOCOMCOCOCO "ti • • Ol Ol Ol OlOi OOlQOlOi 0100J03 03 01C101010C71010105 «Srr CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO COCOCOCOCOCOCOCOCO CO ooooo ooooo ooooo ooooooooo 60 T3«! g co ■* i-i i> ClOMH ~*S E iC CO t-h CO - yO 3 = 5> O GO CI LO u ■*HCOM HI > CO LO CM t> ^ O 5 o (MHMH CO i-l M h Ol lO to OOOlO<« TtlOiCO-*lCM'MLO-HCt) LONNN'S 'fOCBffi'S O ><0 LO CO "* lO CM !>• 6C t-h lo t-h co £L a oo o lo >. coo'1,iooilo'*lo cs comocoJ ■* cm ^* co r? LONOcocirooooo *■ Tf N -t Cl tS LOMiHN S OCOOOOiOCOCOCOS; . p • • • • C >^ -*iMOiO<; NOhO<| COCOCONOCOiOCD■ 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. 2. Lotka, Alfred J. — The General Conditions of Validity of the Principle of Le Chatelier. pp. 19-37. January, 1922. $.75. 3. Bridgman, P. W. — The Effect of Tension on the Electrical Besistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Louis. — Notes on the Early Evolution of the Beflector. pp. 67-74. February, 1922. $.50. 5. Bridgman, P. W. — The EfTect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Bridgman, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. $1.25. 7. Pierce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Befraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjacent Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Franklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. $.65. 10. Bennitt, Budolp. — Additions to the Hydroid Fauna of the Bermudas, pp. 239-259. May, 1922. $.65. 11. Brues, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonididre. pp, 261- 288. 2 pis. May, 1922. $.85. 12. Thaxter, Boland. — A Bevision of the Endogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda. pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. B. — Atmospheric Attenuation of Ultra- Violet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Bomberg, Arnold.— The Batio of the Calorie at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Bobert H. — Studies on Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389-422. 2 pis. November, 1922. $1.65. 17. Thaxter, Boland. — Note on Two Bemarkable Ascomycetes. pp. 423-436. 2 pis. September, 1922. $1.35. 18. Becords of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members; Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 2 of cover.) PUBLICATIONS OP THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Series, Vols. 1-4; New Series, Vols. 1-13. 16 volumes, $10 each. Half volumes, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. Vol. 11. Part 1. Centennial Celebration. 1880. pp. 1-104. 1882. $2.00. Part 2. No. 1. Agassiz, A. — The Tortugas and Florida Reefs, pp. 105-134. 12 pis. June, 1885. (Author's copies, June, 1883.) $3.00. Part 3. Nos. 2-3. Searle, A. — The Apparent Position of the Zodiacal Light, pp. 135-157, and Chandler, S. C. — On the Square Bar Micrometer, pp. 158-178. October, 1885. $1.00. Part 4. No. 4. Pickering, E. C— Stellar Photography, pp. 179-226. 2 pis. March, 1886. $1.00. Part 4. No. 5. Rogers, W. A., and Winlock, Anna. — A Catalogue of 130 Polar Stars for the Epoch of 1875.0, resulting from the available Observations made between 1860 and 1S85, and reduced to the System of the Catalogue of Publication XIV of the Astrono- mische Gesellschaft. pp. 227-300. June, 1886. 75c. Part 5. No. 6. Langley, S. P., Young, C. A., and Pickering, E. C. — Pritchard's Wedge Photometer, pp. 301-324. November, 1886. 25c. Part 6. No. 7. Wyman. M. — Memoir of Daniel Treadwell. pp. 325-523. October. 1887. $2.00. Vol. 12. 1. Sawyer, E. F. — Catalogue of the Magnitudes of Southern Stars from 0° to — 30° Declination, to the Magnitude 7.0 inclusive, pp. 1-100. May, 1892. $1.50. 2. Rowland, H. A. — On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-186. December, 1896. $2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniaceae. pp. 187- 430. 26 pis. December, 1896. $6.00. 4. Lowell, P. — New observations of the Planet Mercury, pp. 431-466. 8 pis June, 1898. $1.25. 5. Sedgwick, W. T., and Winslow, C. E. A. — (I.) Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II.) Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. IS. 1. Curtiss, D. R. — Binary Families in a Triply connected Region with Especia Reference to Hypergeometric Families, pp. 1-60. January, 1904. $1.00. 2. Tonks, O. S. — Brygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T. — The Spectrum of Hydrogen in the Region of Extremely Short Wave-Length, pp. 121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H. — Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv. November, 1906. $1.10. 5. Trowbridge, J.— High Electro-motive Force, pp. 181-215. pis. xxv-xxvii. May, 1907. 75c. 6. Thaxter, R. — Contribution toward a Monograph of the Laboulbeniaceae. Part II. pp. 217-469. pis. xxviii-lxxi. June, 1908. $7.00. Vol. 14. 1. Lowell, Percival.— The Origin of the Planets, pp. 1-16. pis. i-iv. June, 1913. 60c. 2. Fernald, W. E., Southard, E. E., and Taft, A. E. — Waverley Researches in the Pathology of the Feeble-Mi nded. (Research Series, Cases I to X.) pp. 17-128. 20 pis. May, 1918. $6.00. 3. Fernald, W. E., Southard, E. E., Canavan, M. M., Raeder. O. J. and Taft, A. E. — Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases XI to XX.) pp. 129-207. 32 pis. December, 1921. $6.50. PROCEEDINGS. Vols. 1-56, $5 each. Discount to booksellers 25%; to Fellows 50%, or for whole sets 60%. The individual articles may be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. S5.00. Complete sets of the Life and Works of Rumford. 5 vols., $25.00; to Fellows, $5.00. For sale at the Library of The American Academy of Arts and Sciences. 28 Newbury Street, Boston, Massachusetts. 58-7 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 7.— March, 1923. CONTRIBUTION FROM THE OSBORN BOTANICAL LABORATORY. THE CHILEAN SPECIES OF METZGERIA. By Alexander W. Evans. ( Continued, from page 3 of cover. ) VOLUME 58. 1. Ames, A. Jr., Proctor, C. A., and Ames, Blanche. — Vision and the Technique of Art. pp. 1-47 28 pis. February, 1923. $3.75. 2. Birkhoff, George D. and Langer, Rudolph E. — The Boundary Problems Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49- 128. In press. 3. Vainio, Edward A. — Lichenes in Insula Trinidad a Professore R. Thaxter Collecti. pp. 129-147. January, 1923. $1.00. 4. Bhidgman, P. W. — The Effect of Pressure on the Electrical Resistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. $.75. 5. Bridgman, P. W. — The Compressibility of Thirty Metals as a Function of Pressure and Temperature, pp. 163-242. January, 1923. $1.70. 6. Baxter, Gregory P., Weatherill, Philip F. and Scripture, Edward W., Jr. — A Revision of the Atomic Weight of Silicon. The Analysis of Silicon Tetrachloride and Tetrabromide. pp. 243-268. February, 1923. $.75. 7. Evans, Alexander W . — The Chilean Species of Metzgeria. pp. 269—324. March, 1923. $1.25. Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 7.— March, 1923. 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. f:N ON DOUBLE POLYADICS, WITH APPLICATION TO THE LINEAR MATRIX EQUATION. By Frank L. Hitchcock. DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY. TABLE OF CONTENTS. Page. 1. Gibbs' Concept of dyadics and polyadics 355 2. Polyadics as vectors in space of higher dimensions 357 3. The fundamental identity of dot multiplication 358 4. Double polyadics 360 5. The Hamilton-Cayley equation for double polyadics 362 6. The scalars as sums of cubic determinants 366 7. Scalars in combination with the idemfactor 367 8. Scalars as square determinants whose elements are polyadics . . 368 9. An invariant property of the scalars 372 10. Forms which show the polyadic character of the scalars .... 374 11. Invariants regarded as products 378 12. Star products 379 13. Scalars formed by star multiplication 382 14. Identities with the star idemfactor 383 15. Illustrations of various types of multiplication 384 16. Transformation from matrices to double dyadics 385 17. Transformation back to matrices 386 18. Character of the coefficients as algebraic polynomials 390 19. The equation of extent unity and order two 391 20. The equation of extent two and order two 393 1. Gibbs' Concept of Dyadics and Polyadics. A system of Vector Algebra in AT dimensions following Gibbs is based on a set of A7 unit vectors d, e2, • • • , e# such that the dot products e»- ejt are unity w7hen subscripts are equal, otherwise zero. Any other vector a may be expressed in terms of the fundamental units, a = fljei + 02e2 + • • • + cin^n- (1) Two vectors a and b written together with no dot or other sign between them constitute a dyad ab called the indeterminate product of a into b. A dyad e^eA; will be called a fundamental dyad or dyad unit. If we have any dyad ab the first vector a is called the antecedent, the second the consequent. Both a and b will be called factors of the dyad. A dyad is zero when and only wrhen all products of scalar elements ciibk are zero, that is, when one of the factors vanishes in all its ele- ments. 356 HITCHCOCK. A dyadic is a sum of dyads, or, with no increase in generality, a sum of dyads each multiplied by a scalar factor. The elements of a dyadic are the sums Zctibk of corresponding products taken from each dyad term, where, for an element Aik of a dyadic A the subscripts i and k are constant and the summation is over the various dyad terms. Thus if A = aibi + a>b2 + • • • + aAbA (2) we have Aik = aubik + 02ib2k + • • • + (ihibhk . (3) It is frequently of use to write the dyad ab as symbolical of the dyadic and dibk as symbolical of the element Aik, omitting the subscript h and the sign of summation with respect to h.1 A dyadic may be written in terms of the dyad units efik thus A = 2Aik9ifik (4) where both subscripts run from 1 to Ar. In a similar manner several vectors abc • • • g written with no sign between them constitute a polyad. When we wish to indicate that the polyad is of order A, that is, it is the indeterminate product of K vectors, we may call it a A-ad. The vectors a, b, c, etc., in order, will be called the first, second, third, etc. factors of the polyad. A polyadic is a sum of polyads, all of the same order. The polyads epeqer- ■ es to K factors will be called the fundamental polyads of order K or fundamental A-ads. It is evident that any polyadic may be expanded in terms of the fundamental polyads of its own order. The elements of a polyadic are the scalar coefficients in this expansion. Thus if we have K subscripts p, q,- • • , s, all of which run from 1 to X, we may write a A-adic A as It is thus apparent that in general a Zv-adic depends on XK scalar elements. These may be primarily regarded as forming a A-dimen- sional block, and most conveniently written by means of adjacent squares when special values have to be assigned. For example if X = 2 and A = 4 we should have the scheme of elements ^4im, -bir: Ami, Aims Amu Ann Al221) Ai22S A2UI, A 2112 -42121, -42122 • I22IIJ -4.2212 A232I) A2222 ON DOUBLE POLYADK S — THE LINEAR MATRIX EQUATION. 357 as one possible way of representing a four-dimensional assemblage of elements on a flat surface. A polyadic is defined to be zero when and only when all its elements are zero. Since our fundamental conception of a polyadic is as a sum of polyads rather than as a mere aggregation of scalars, the equation A = 0 is to be thought of as equivalent to NK equations of the form aipbiq- -(jis + (hjhq- -gis + + ahpbhq- -ghs = 0 . (7) It is frequently useful to indicate such a set of equations symbolically as ab • • g = 0, where, as before, the polyad is written symbolically for the polyadic. 2. POLYADICS AS VECTORS IN SPACE OF HlGHER DIMENSIONS. Investigations on polyadics may be distinguished according to whether the A -dimensional character is important or not. If we agree on some definite order among the fundamental polyads we may write Er= epeq- es (8) and let r run from 1 to NK while each of the K subscripts on the right runs from 1 to N. With Ar= Apq . . s we shall then have A = ^1E1+^2E2+-..+^nEn (9) where, for convenience, n has been put for NK. The polyadic A thus takes the form of a vector in space of n dimen- sions. This concept is justified if we introduce the multiple dot product defined by (eae6- -eff): (epeg--es) = (ea«ep) (e6-e5)- • -(e^e,,) (10) where the colon in every case indicates A'-tuple dot product. In words Definition. The K -tuple dot product of two A-ads is the product of dot products of corresponding factors. The /v-tuple dot product of two A'-adics is the sum of A'-tuple dot products term by term. By virtue of the distributive law of multipli- cation the relation (10) is symbolical of the iv-tuple dot multiplication of two A-adics. When K = 2 we have Gibbs' double dot multiplica- tion. By (8) and (10) it is evident that E, : Ek is unity when i = k, other- wise zero. Thus the fundamental polyads behave with respect to multiple dot product as do unit vectors ei, e2, • • • e„ in space of n 358 HITCHCOCK. dimensions with respect to ordinary dot product. Therefore all properties of such vectors which pertain to dot multiplication alone will go over into properties of polyadics and multiple dot product. To say the same thing in another way, the fundamental polyads epeg- -es form a normal orthogonal system with respect to multiple dot multiplication, in the sense that the product of two unlike polyads of order K is zero while the product of a polyad by itself is unity. The notation E, for the polyad is convenient but not essential. The analogy with er, while of much value in a formal sense, is temporary in character, to be employed or laid aside according as we wish to forget or to emphasize the K -tuple nature of the polyads. 3. The Fundamental Identity of Dot Multiplication. Let there now be a set of NK or n polyadics of order K as Ai, A2, • • • An, arbitrarily chosen. Any one of these, as Ai, may be expanded in the manner of (9), thus Ai = flijEi + «l2E2 H h AinEn. (11) Let there also be another polyadic which we may on occasion call An+i but which it will be more convenient at present to call M and to expand in the form M = J/xEi + il/2E, H h M„E„. (12) If we now take the multiple dot product M:A,, remembering the principle of orthogonality pointed out in the last article, we shall have M : Ai = Mian + M2aa -\ 1- Mnain. (13) There are n equations of this form, one for each of the polyadics Ar • -An. Together with (12) we thus have n + 1 equations linear in the scalars M\---Mn. It is true that (12) is a /v-adic equation and is itself equivalent to n scalar equations. But since all the equations (13) are merely scalar equations, it is not hard to see that the determinant M , E,, E2, •••, E„ M:A], an, ar;,---, r/,„ M:A2, an, (h-i,- • ■ , a2n (14) M:A„, a, a, an2,''', ann must vanish; because when polyadics have to be multiplied by scalars only, all the laws of ordinary algebra are obeyed. ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 359 We have now to develop this determinant by the elements of the first column, yielding the identity C0M + CM : Ai + C2M : A2 -\ hCjtt : An = 0 (15) where the scalar Co and the polyadics Ci • • • C„ are the cofactors of the elements into which they are multiplied. Evidently Co is the determi- nant of the n2 elements 0;jt. To exhibit the character of Ci, Co, etc., let Cik denote the cofactor of au- from the n-rowed determinant Co. On developing (14) it appears that Ci = - (caEx + ci2E2 -\ f- cinEn) (16) whence the Ct- are polyadics whose elements are the negatives of the n — 1 rowed cofactors from Co. So far the n + 1 polyadics M, Ai, A2, • • •, A,, have been quite arbi- trary. If, however, the At- are linearly independent, their determinant Co is not zero and we may introduce the important new set of polyadics A'i, A'2 • • • , A'„ defined by or by (16) C0A't=-Ci (17) A\- = i (caEi + c,2E2 + • • • + CinE.). (18) Co The set A'i is said to be reciprocal to the set A;. By multiplying the expansions (11) and (18), remembering that Cik is the cofactor of ot;: from the determinant Co, we have the relation A'i'.Ak = 1 when subscripts are equal, otherwise zero, which we may also express by saying that a set of n polyadics and its reciprocal set with respect to multiple dot product are bi-orthogonal. The d may now be eliminated by the aid of (17); and the funda- mental identity (15), (on the hypothesis that Co does not vanish), rearranged in the form M = (A'iA, + A'2A2 H 1- A'nAn) : M. (19) The expression in parentheses will be denoted by /; it is an idem factor in the sense that its A'-tuple dot product into an arbitrary Jv-adic leaves that /v-adic unaltered; and it is the most important special case of a double polyadic, — a dyadic whose antecedents and consequents are polyadics. Let the expression in parentheses be now transformed by putting in the values of all the polyadics A and A' from the expansions (11) 360 HITCHCOCK. and (18). Collecting the scalar coefficient of E^E* we find it to be — TZcikCLik summed on i, which is unity. Again, the coefficient of Co E,Ea- when j and A- are unequal is —^CijOik summed on i, which is Co zero. Hence I = ZE^E* summed on k, and this latter expression for the idemfactor is symmetrical in the sense that it is unaltered by interchanging antecedents and consequents. It follows that the reciprocal relation between two sets of polyadics is a mutual one, and we have I = ZE,E, = 2A',A, = 2A,A'A- (20) and also M = / : M = M : I (21) or the idemfactor may be used either as prefactor or as postfactor. To illustrate, if we take K = 1 and A* = 3, our polyadics reduce to vectors in ordinary space; and if our unit vectors be the usual i, j, and k the idemfactor becomes the usual idemfactor of Gibbs, namely ii + jj + kk. If K = 2 and N = 3 with the same notation, our polyadics are ordinary dyadics; the idemfactor (20) is a double dyadic, having nine terms obtained by doubling the nine fundamental dyads, viz. I = iiii + jjjj + kkkk + ijij + jiji + jkjk + kjkj + kiki + ikik (22) and it may easily be verified that the double dot product of any of the nine fundamental dyads ii, ij, etc., either by or into this expression gives the dyad unchanged; whence the same is true for any dyadic. 4. Double Polyadics. Two polyadics A and M written together with no sign between them will constitute a double poly ad AM. A sum of double poly ads is a double polyadic, assuming always that the terms of the sum are of like character. In the present investigation it will always be assumed that the factors A and M of all double polyads are polyadics of the same order K. \Yhen desirable, all the antecedents A may be expanded in terms of the fundamental A'-ads E„ and the consequents collected so that any double polyadic

»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 be expanded in the form (23). The scalar elements of i and i, tp2, 1))s = (n - 2) (( B, (51) where the law of expansion is defined to be as follows: the leading term is written [BnB22- ■ -B„„] and signifies the determinant in the ordinary sense whose elements are the scalar elements of the polyadics which enter: namely, if we expand as B,:y = 6iABi + bifA -\ \- 6«/nBf (52) the leading term in expanding (51) is the determinant of ordinary character Olllj O112, &221> "222> ' , b-22n Until} Onnli " " "j ^nnn (53) the other terms in the expansion of (51) are of similar character, and are formed by writing out the determinant (51) as if the elements were scalars, enclosing each term in square brackets to signify the determi- nant formed from it as in (53). The signs will occur as in ordinary determinants if we adopt the rule that, in every term, the order of the polyadics shall be the order of the rows in the original determinant. The meaning of the determinant (51) is thus defined without ambiguity. We note that by this rule the order of rows in (53) corresponds to the order of rows in (51), in the sense that each row of (53) consists of the elements of a single polyadic from the corresponding row of (51). It is then apparent that (51) is the sum of n! ordinary determinants, hence the sum of in!)2 terms on complete expansion. The order of the rows is non-signant, in the sense that, if two rows in (51) be inter- changed, the value of the expression is quite unchanged; for on ex- 370 HITCHCOCK. panding, first, into a sum of ordinary determinants or square bracket expressions, the sign before every square bracket is changed because the signs behave as if (51) were an ordinary determinant; but, since the development follows the rows, the sign of every determinant (53) will have been changed due to the interchange of rows; thus each square bracket has its sign changed, we may say, both from within and from without. Hence the whole expression is unaltered. On the other hand, an interchange of two columns in (51) changes the sign before every square bracket but leaves the brackets themselves un- affected, for the development is according to rows; hence the sign of the whole is changed. It is now not hard to see that the polyadic determinant (51) is the same as ((2ii + 6222J] — [&121I + &mj, &2ui + &212J] ON DOUBLE POLYADICS THE LINEAR MATRIX EQUATION. 371 which is the same as the difference of two determinants (01110222 — OU2^22l) — ("121^212 — "122021l) and this, again, agrees with the cubic determinant formed with the element of the two dyadics as layers, namely biixy "121 "112, "122 0211j '^221 "212) "222 A number of results follow immediately. We might have formed the left side of (54) by taking the multiple dot product of Ei, E2, etc. into each of the arbitrary double polyadics, as Ei: 01, E2: 01, etc. in the first row, and similarly for the other rows. Instead of (51) we should have had the determinant of the M;/ by developing after the manner (23) instead of (27). The rest of our reasoning would have been un- changed, however, hence the value of the expression would have re- mained the same. The change would in fact be equivalent to chang- ing rows into columns and columns into rows in all the non-signant layers of the cubic determinant, which, since the second and third indices are signant, leaves the result unchanged. This is illustrated by the example just given, where the last index refers to the rows in- stead of to the columns as in (41). If we let 0' stand for 2CMA when

" ' ', • E2, • * » En Ei , Eo , ' > En Ei E, = in - p)!(( : Eo, E E

, • ••, '; 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))s = + •••+ (-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 (-+ AhSBhS (117) As remarked earlier, the symbolic process is valid by virtue of the distributive character of all the steps involved. Thus for m% we may put, in abbreviated language, 2ma= ({#, ' 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. VOLUME 57. 1. Kent, Norton A. and Taylor, Lucien B. — The Grid Structure in Echelon Spectrum Lines, pp. 1-18. December, 1921. $.75. 2. Lotka. Alfred J. — The General Conditions of Validity of the Principle of Le Chatelier. pp. 19-37- January, 1922. $.75. 3. Bridgman, P. W. — The Effect of Tension on the Electrical Besistance of Certain Ab- normal Metals, pp. 39-66. April, 1922. $1.00. 4. Bell, Louis. — Notes on the Early Evolution of the Reflector, pp. 67-74. February, 1922. $.50. 5. Bridgman, P. W. — The Effect of Pressure on the Thermal Conductivity of Metals, pp. 75-127. April, 1922. $1.25. 6. Bridgman, P. W. — The Failure of Ohm's Law in Gold and Silver at High Current Densities. pp. 129-172. April, 1922. $1.25. 7. Pierce, George W. — A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Befraction, and Inverse Hyperbolic Functions of a Complex Variable, pp 173-191. April, 1922. $1.25. 8. Pierce, George W. — Artificial Electric Lines with Mutual Inductance between Adjacent Series Elements, pp. 193-212. May, 1922. $1.25. 9. Barker, Franklin D. — The Parasitic Worms of the Animals of Bermuda. I. Trema- todes. pp. 213-237. 3 pis. May, 1922. $.65. 10. Bennitt, Budolf. — Additions to the Hydroid Fauna of the Bermudas, pp. 239-259. May, 1922. $.65. 11. Brues, Charles T. — Some Hymenopterous Parasites of Lignicolous Itonidida?. pp, 261- 288. 2 pis. May, 1922. $.85. 12. Thaxter, Boland. — A Revision of the Endogoneae. pp. 289-350. 4 pis. June, 1922. $1.25. 13. Clark, H. L. — The Echinoderms of the Challenger Bank, Bermuda, pp. 351-362. 1 pi. June, 1922. $.50. 14. Schaeffer, E. R. — Atmospheric Attenuation of Ultra-Violet Light, pp. 363-374. 1 pi. June, 1922. $.65. 15. Romberg, Arnold.— The Ratio of the Calorie" at 73° to that at 20°. pp 375-387. June, 1922. $.65. 16. Bowen, Robert H. — Studies on Insect Spermatogenesis. IV. The Phenomenon of Polymegaly in the Sperm Cells of the Family Pentatomidae. pp. 389-422. 2 pis. November, 1922. $1.65. 17 Thaxter, Roland. — Note on Two Remarkable Ascomycetes. pp. 423-436. 2 pis. September. 1922. $1.35. 18. Records of Meetings; Biographical Notices; Officers and Committees; List of Fellows and Foreign Honorary Members- Statutes and Standing Votes, etc. pp. 437-567. November, 1922. $.75. (Continued on page 2 of cover.) PUBLICATIONS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. MEMOIRS. Old Sekies, Vols. 1-4; New Series, Vols. 1-14. 16 volumes, $10 each. Half volumes, $5 each. Discount to booksellers and libraries 25%; to Fellows 50%, or for whole sets 60%. Vol. 12. 1. Sawyer, E. F. — Catalogue of the Magnitudes of Southern Stars from 0° to — 30° Declination, to the Magnitude 7.0 inclusive, pp. 1-100. May, 1892. $1.50. 2. Rowland, H. A.— On a Table of Standard Wave Lengths of the Spectral Lines, pp. 101-1S6. December, 1S96. $2.00. 3. Thaxter, R. — Contribution towards a Monograph of the Laboulbeniaceae. pp. 1S7- 430. 26 pis. December, 1S96. $6.00. 4. Lowell, P. — New observations of the Planet Mercury, pp. 431-466. S pis. June, 189S. $1.25. 5. Sedgwick, W. T., and Winslow, C. E. A. — (1.) Experiments on the Effect of Freezing and other low Temperatures upon the Viability of the Bacillus of Typhoid Fever, with Considerations regarding Ice as a Vehicle of Infectious Disease. (II.) Statistical Studies on the Seasonal Prevalence of Typhoid Fever in various Countries and its Relation to Seasonal Temperature, pp. 467-579. 8 pis. August, 1902. $2.50. Vol. 13. 1. Curtiss, D. R. — Binary Families in a Triply connected Region with Especial Reference to Hypergeometric Families, pp. 1-60. January, 1904. $1.00. 2. Tonks, O. S. — Brygos: his Characteristics, pp. 61-119. 2 pis. November, 1904. $1.50. 3. Lyman, T. — The Spectrum of Hydrogen in the Region of Extremely Short Wave- Length, pp. 1121-148. pis. iii-viii. February, 1906. 75c. 4. Pickering, W. H.— Lunar and Hawaiian Physical Features Compared, pp. 149-179. pis. ix-xxiv. November, 1906. $1.10. 5. Trowbridge, J. — High Electro-motive Force, pp. 181-215. pis. xxv-xxvii. May, 1907. 75c. 6. Thaxter, R. — Contribution toward a Monograph of the Laboulbeniaceae. Part II. pp. 217-469. pis. xxviii-lxxi. June, 1908. $7.00. Vol. 14. 1. Lowell, Percival. — The Origin of the Planets, pp. 1-16. pis. i-iv. June, 1913. 60c. 2. Fernald, W. E., Southard, E. E., and Taf t, A. E.— Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases 1 to X.) pp. 17-12S. 20 pis. May, 1918. $6.00. 3. Fernald, W. E., Southard, E. E., Canavan, M. M., Raeder, O. J. and Taf t, A. E. — Waverley Researches in the Pathology of the Feeble-Minded. (Research Series, Cases XI to XX.) pp. 129-207. 32 pis. December, 1921. $6.50.j 4. Pickering, Edward C, Parkhurst, J. A., Aitken, Robert G., Curtis, Heber D., Mitchell, S. A., Alden, Harold L., Simpson, T. McN., Reed, F. W. — Photometric Magnitudes of Faint Standard Stars, measured at Harvard, Yerkes, Lick and McCormick Observatories. pp. 209- . April, 1923. $0.00. PROCEEDINGS. Vols. 1-57, $5 each. Discount to booksellers and Libraries 25%; to Fellows 50%, or for whole sets 60%. The individual articles may be obtained separately. A price list of recent articles is printed on the inside pages of the cover of the Proceedings. Complete Works of Count Rumford. 4 vols., $5.00 each. Memoir of Sir Benjamin Thompson, Count Rumford, with Notices of his Daughter. By George E. Ellis. $5.00. Complete sets of the Life and Works of Rumford. 5 vols., $25.00; to Fellows, $5.00. For sale at the Library of The American Academy of Arts and Sciences, 28 Newbury Street, Boston, Massachusetts. 58-11 Proceedings of the American Academy of Arts and Sciences. Vol. 58. No. 11.— May, 1923. IDENTITIES SATISFIED BY ALGEBRAIC POINT FUNCTIONS IN N-SPACE. By Frank L. Hitchcock. ( Continued from page 3 of cover. ) VOLUME 58. 1. Ames, A. Jb., Proctor, C. A., and Ames, Blanche. — Vision and the Technique of Art. pp. 1-47. 28 pis. February, 1923. $3.75. 2. Birkhoff, George D. and Langer, Rudolph E. — The Boundary Problems and Develop- ments Associated with a System of Ordinary Linear Differential Equations of the First Order, pp. 49-128. April, 1923. S3. 15. 3. Vadjio, Edward A. — Lichenes in Insula Trinidad a Professore R. Thaxter Collecti. pp. 129-147. January, 1923. $1.00. 4. Biudghan, P. W. — The Effect of Pressure on the Electrical Resistance of Cobalt, Alumi- num, Nickel, Uranium, and Caesium, pp. 149-161. January, 1923. $.75. 6. Bridgman, P. W. — The Compressibility of Thirty Metals as a Function of Pressure and Temperature, pp. 163-242. January, 1923. $1.70. 6. Baxter, Gregory P., Weatherill, Philip F. and Scripture, Edward W., Jr. — A Revision of the Atomic Weight of Silicon. The Analysis of Silicon Tetrachloride and Tetrabromide. pp. 243-268. February. 1923. $.75. 7. Evans, Alexander W . — The Chilean Species of Metzgeria. pp. 269—324. March, 1923 $1.25. 8. Brues, Charles T. — Some New Fossil Parasitic Hymenoptera from Baltic Amber. pp. 325-346. March, 1923. $.65. 9. Keistnelly, A. E. — Text of the Charter of the Academie Royale de Belgique, Trans- lated from the Original in the Archives of the Academie at Brussels, pp. 347—351. April, 1923. $.40. 10. Hitchcock, Frank L. — On Double Polyadics, with Application to the Linear Matrix Equation, pp. 353-395. May, 1923. £1.15. 11. Hitchcock, Frank L. — Identities Satisfied by Algebraic Point Functions in N-Space. pp. 397-421. May, 1923. S.S5. Proceedings of the American Academy of Arts and Sciences. 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\,xr • • a„. If we expand the value of the polynomial at each of these points in the form (2) we shall have n equations of the form Ffa) = AiaaK + A2ailK-)ai2 + ^au*-1^ H + AnaiNK, (3) 400 HITCHCOCK. which, together with (2), constitutes a set of n + 1 linear homogen- eous equations in the n -f- 1 quantities A\, A»,- • • An and unity. It is accordingly evident that the determinant F(x), xj F(ai), , • • • e#, and add the results, we shall have the vector identity F(x)C0 + F(a1)6'1 + F(a,)C2 + • • • + F(a„)C„ = 0. (9) It is obvious that the left member of this identity may, if we wish, be written in the form of the determinant (4), the elements of the first column being now vectors. Just as with the scalar identity (5), we may regard the vector iden- tity (9) in two ways, — either as the statement of the linear relation connecting the values of the vector polynomial F(x) at n + 1 arbi- trary points; or as the expansion of this polynomial in terms of its values at n points arbitrary except that C0 does not vanish. The vector function F(x) depends on AT scalar polynomials, each contain- ing n arbitrary numerical coefficients. Whenever these coefficients need to be separately considered they may be designated according to the matrix An , Ai\ , -^31 , ' , An\ A\i , ^22 , Am , ' > Ani An , -^23 , - 1 33 , ' > A„3 A\n , AoN , A3ff , •••, Anff Thus the set of numbers An, like the set a^, contains Nn elements. We may now state the following simple but fundamental theorem: Theorem I. The values taken on by an arbitrary vector polynomial at n + i arbitrary points arc linearly related. The multipliers Co, Ci, C2, • • • Cn are functions of the arbitrary points, but are independent of the coefficients An of the terms of the polynomial. 402 HITCHCOCK. 2. Geometric Meaning of the Coefficients. To take a specific example, let K = 1 and N = 3. The vector determinant of the form (4) becomes P(x) , Xi , x2 , x3 P(a0 , an, ai2, ai3 F(a2) , a2i, a22, (ha F(a3) , o3i, 032, a33 The coefficients Co, Ci, C2, and C3 become determinants of the com- ponents of three vectors. If we denote these determinants by (123), (.r23), (zl3), and (.rl2), the identity (9) becomes F(x) (123) - P(ai) (a-23) + F(a2) (a-13) - F(a3) (xl2) - 0, (10) a relation very familiar to students of vector analysis, F(x) being now a linear vector function. When K = 1 a similar relation holds for any value of N, whence clearly the vanishing of any C means coplan- arity of the vectors which enter into it. Next take K = 2 and N = 3. The vector determinant of the form (4) becomes (11) F(a6) , 0612 , «61«62 , «6lfl63 , «622 , «62«63 , «632 In this case the vanishing of one of the coefficients C means that the six vectors which enter into it lie on a quadric cone, (which may, how- ever, be a degenerate cone). For the equation d = 0 is homogeneous of the second degree in six vectors, and holds true when any two of them coincide, because two rows of the above determinant become equal. Therefore if, as usual, we regard x as a variable vector, the equation d = 0, for values of the subscript other than zero, is the equation of the cone determined by the five fixed vectors which enter into it. Similar reasoning applies for any values of AT and K. F(x) , •Ti2 , XiX-2 , X1X3 , •1*2 , x&z , a-32 F(ai) , Oil2, 011&12 > cindiz , Ol22, "12«13 , Oi32 F(a2) , «212, 021022 , 021023 , ChJ , O22O23 > 0232 3. The Coefficients as Eliminants of K-adics. The coefficients C may be interpreted in another way, so as to con- nect them with the theory of matrices and allied operators. We take, ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 403 as before, a set of n vectors ai, a2, • • • a„. Consider for a moment the case K = 2. A symmetrical dyadic may be written as a sum of terms of the form aa where a is a vector. Let there be a set of constants C\, Co, ■ • • cn not all zero. We define the equation ciaiai + c2a2a2 + • • • + cnana„ = 0 (12) to be equivalent to the set of n scalar equations obtained by selecting pairs of corresponding components in all possible ways: — Ciauda + c2a2ia2j + • • • + cnanianj = 0 (13) where the subscripts i and j may have any values from 1 to N. These equations are linear and homogeneous in the c's; hence the determi- nant whose elements are audhj must vanish if (12) holds true. A glance at (11) shows that this determinant is the same as Co, the minor of F(x),— by interchanging rows and columns. Therefore the neces- sary and sufficient condition for the existence of a dyadic equation of the form (12) is that the n vectors which enter in that equation should lie on a quadric cone. A symmetrical /v-ad should properly be written aaa • • • to K factors (14) but, when no ambiguity is brought about, we may write it as aK. We in general define the A'-adic equation da/+ c2a2* + • • • + cnB,nK = 0 (1|5) to be equivalent to the system of n equations 1f[chahiahi • • • to K factors] = 0 (16) h where in each equation h runs from 1 to n. Passage from one equation of the system to another is by varying each of the second subscripts from 1 to N. By inspection of (4) it is evident that the eliminant of the set of n equations is the same as Co. We thus have Theorem II. A set of n symmetrical K-ads are linearly related if and only if their vector elements lie on a hypercone of order K. 4. Special Forms of the General Identity: Method of Standard Sets. By assigning particular values to the vectors ai, a2, • • • a„ and by selecting particular forms of polynomials F(a), a great number of 404 HITCHCOCK. identities may be obtained from the scalar formula (5) as well as from the. vector formula (9). An elementary method is, evidently, to assign numerical values to the elements an, with the sole restriction that Co shall not vanish. The coefficients C\, C2, • • • Cn are polynomials of degree K. By (5), an arbitrary polynomial F(x) can be expanded in terms of these n polynomials, the coefficients F(&i) being found by direct substitution. It is evident, therefore, that &, Co, ■ • ■ Cn are linearly independent. It is equally evident that not every linearly independent set of n polynomials homogeneous of degree K in N variables can be taken as these nC's; for the totality of all vectors where two or more C"s vanish is comprised by the set of vectors &i, a2, • • • a„. We may embody this distinction in the following definition: Definition I. A set of polynomials Pi, P2, • • -Pn, homogeneous of degree K in N variables, such that Pf(ay) vanishes when i is different from, j but not when i equals j for each of the points ai, a2, • • • an, will be called a standard set. By use of standard sets of polynomials, many of the identities of elementary algebra may be made to appear as special cases of (5). To take the simplest of illustrations, let N = K = 2 so that n = 3, with variables x, y. Let the matrix an be 1 1 0 0 1 1 t(4) becomes F(x, y), *2, xy, Vi F(\, 0), 1, 0, 0 F(h i), 1, 1, 1 m i), 0, 0, 1 By easy calculation C0 = 1,CX= xy — x2, C2 = — xy, C3 =xy —y2. Choosing F(x, y) = (.r + y) (x - y) yields F(l, 0) = 1, F(l, 1) = 0, F(Q, 1) = — 1. The identity (5) appears as (x+y) (x-y) + (xy - .r) + 0 + (-1) (xy - if) = 0 To take a less familiar illustration, let F(x) be a quadratic vector function in ordinary space, Ar = 3, A' = 2. A simple way to build a standard set of polynomials is to choose four vectors bi, b2, b3, b4 of which no three are coplanar, and form six products of linear factors X'bi-byx which are the required polynomials. The six fixed points or ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 405 vectors a^ are the six intersections of four planes, and the vector identity (9) becomes, by putting F(a*) = F(b; X b,) --- ir, lp?> F(x) = x«(bif23b4 + bif34b2 + bif24b3 + b2fi4b3 + b2fi3b4 + b3fi2b4) *x (17) where, if F(x) is an arbitrary quadratic vector function, the six vectors f are arbitrary. In both the above illustrations we have K — 2, and note that the polynomials of the standard set are formed by selecting pairs of linear polynomial from a group of K -{- N — 1 possibilities. This pro- cedure is applicable in all cases. For the number of ways in which K factors can be selected from a group of K + A7 — 1 factors with no repetitions is the same as the number of ways in which A' variables can be selected from N variables allowing repetitions, that is n ways by definition of n. Each linear factor is of the form b;*x where b^x = buXi + bi2.v» + • ■ ■ + biN.vN (18) and there are AT + K — 1 vectors bt so chosen that no N of them are linearly related. We thus arrive at the following: Definition II. A standard set of polynomials which has been formed by taking products of K linear polynomials selected from N + K — 1 such polynomials without repetition will be called a factored set. The vectors ai, a2 • • • a„ may always be chosen so that Pi(a) = 1. We define the vector [b;,bgbr- • • to N — 1 factors] to be the vector whose scalar components are the respective cofactors of the elements of the first row from the determinant bn , bii , biz , ' > biN bpi, bPi, bpz , ' > bpN bgl, bg2, bqZ, " » bqN brl , br2 , brZ , " » brN to AT rows (19) We then define one of the vectors a, say a4, as follows ; let mi denote the vector [bib2b3 ■ ■ • b^i] and write a4 = mi (bjv'mibivr+i'mr • •biv+A'-i'inOiv (20) The other vectors a are built up in a similar manner: each a is a function of all the b's; the denominator is the Kih root of the product 406 HITCHCOCK. of K determinants of the form b*m, that is of the form (19); any selection of A7 — 1 of the b's determines one of the n vectors mi, m2 • • • m„ which forms the numerator of the corresponding a. All the remaining b's occur explicitly in the denominator. The polynomials Pi, Pi,- ■ -Pn may now be taken of the form Ph = (bp'x) (bg«x) (br«x)- • -to K factors. (21) The a of corresponding subscript is that a which contains precisely the same choice of b's in its denominator; for example, if ai be defined by (20) then Pi must be taken as Pi = bN -xbjv+i -x • • • bN+K^ -x (22) By inspection of (20) and (22) it is evident that Pi(aO = 1, and similarly for any other distribution of b's between numerator and, denominator. That is P^(a^) = 1 for all values of h, as was to be shown. It is clear that a set of polynomials defined by (21) will fulfill the conditions for a standard set provided no group of A7 vectors chosen from the b's are linearly related. For the a's may be taken as in (20) and no denominator can vanish. At least one of the b's which occur in m,- must occur in Pi when i is different from j. Hence Pi(a,-) = 0. Definition III. A factored set of polynomials, together with a set of points ai, a>, • • • a?l such that P;(a/) vanishes when i is different from j but equals unity when i equals j, will be called a normal reference system. Any polynomial Pi of a normal reference system agrees with the coefficient d of (5) except for a constant factor, since Pj and C,- both vanish at every a except a,. By putting a^ for x in (5) we have P(ai)C0 + P(ai)Ci(ai) = 0; (23) but F is an arbitrary polynomial, hence P(at) is in general not zero. Therefore we have identically Co + Ci(&i) = 0 (24) and since P;(aj) = 1 it follows that Pi=-% (25) By substituting d = — CoPi in the vector identity (9) we arrive at the following special case of theorem I, — ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 407 Theorem III. An arbitrary vector function homogeneous of degree K in N variables may be written as the sum ofh terms F(x) = Z[fA(b,-x) (bg'x)- • -to A' factors] (26) h where the vectors iu f2, • • ■ fn are arbitrary; in each term occur K linear -polynomials selected from N+ K — 1 linear -polynomials; these linear polynomials are under the sole restriction that no N of them are linearly related. It follows from the form of (26) that, if a& be of the form (20), F(aA) = U (27) It is apparent that expansions like (26) will be of advantage in investigations where symmetry of form is to be sought. As a simple illustration suppose A = 2. We have N + A — 1 = N + 1 and may take as our linear polynomials the N variables Xi, x2, • • • xn together with their sum Xi + x* + • • • + .r#. We see at once that any quadratic vector function can be written in the form TXiiX&j + 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 2cw+ir ' ■ » &w+t, hut at no others of subscript less than n-\- 1 . The polynomial Lj(x) is the determinant of the coefficients of the N vectors x, y, a,, and the N — 3 adjoined vectors, any or all of which might coincide with an equal number of the original set ai---an, according to the value of t. Thus Li(x) is a linear function of a,, and Ly(x) is the same linear function of a, except perhaps in sign. If we write L;(x) = Z/(a<) we may write (37/)) in the form c1L(a1)P1(x) + • • • + cuX(au.)P«,(x) = 0 (37,) It suffices to show that Z(a»_m) = 0 contrary to hypothesis. Now Ci must be a polynomial of degree K — 1 in a*. For all the terms of the supposed identity (37,) are of degree K in a; with excep- tion of the iih term; is independent of a;; and L(s,i) is linear in a*. Furthermore Ci is of degree K — 1 in each of the a's from a„_OT+i to a„ inclusive. For (37,), being an identity, will subsist if we make a» coincide with a„_m+3; Pi will be unaltered since it is independent of a;; all other P's vanish; L(a„_m+y) by hypothesis does not vanish; hence c4 must vanish when at coincides with a„_m+J; that is, Cj is a polynomial of degree K — 1 in aj vanishing at the m = n(N, K — 1) — 1 points an-m+i" • • a„, and is therefore a polynomial of degree K — 1 in each of these points. We may accordingly identify the c's of (37,) with the polynomials Em(a,i) of (32) except, in each case, for a constant multiplier (a8), (z29)£2(a9) (.r37)£3(a7) , (.r38)£3(a8), (.r39)E3(a9) (38) That this determinant vanishes when x coincides with ai is evident, for the elements of the first row all vanish; similarly the second and third rows vanish when x coincides with a2 and a3 respectively. The columns vanish when x coincides with a;, a8, a9. Since as has been shown the determinant must also vanish when x coincides with a-i, aj, or a6, we are presented with three new identities of the form (417)JB1(a7), (418)£1(a8), (419)£1(a9) (427)£2(a7), (42S)£2(a8), (429)£2(a9) (437)£3(a7), (438)£3(a8), (439)£3(a9) - 0 (39) the other two having 5 and 6 in place of 4. In fact if we multiply the elements of the first row of (39) by the three-row determinant (561), those of the second row by (562), of the third by (563), and add, the sum of the elements of each column is zero ; for (561) (41.T)£i(x) + (562) (42a-)£2(x) + (563) (43.r)£3(x) = 0 (40) is an identity of the type (33), a special case of (35). It holds there- fore when x is replaced by a-, a8, or ag. Determinants of similar form to (38) may evidently be written down at once for any value of K when AT = 3. The E's will in all cases denote polynomials of degree K — 1; vanishing at n(N, K — 1) — 1 points. When N is greater than 3 we have always q > K; hence if we select from the equations of form (37) a set of q equations which are inde- pendent we may write the C's proportional to a set of determinants having linear elements; these determinants must accordingly be reduci- ble polynomials and must possess a common factor of degree q — K, the other factor being one of the C's. Not every set of q equations of the type of (37) will be independent. For example let AT = 4, K = 2, so that n — 10, m = 3, q = 4, and Q = S[2, 5] = 10. If we choose our four vectors br'-b^ to be [xa;a5] where i = 1, 2, 3, 4, the coefficients of C/ in our four equations will be (xibj) (pqrj) where p, q, and r are different from i and from each other and are less than five ; while j has any value from six to ten inclusive. The four polynomials in a,- are linearly related; for we may write as a special case of (29) 414 HITCHCOCK. a,(1234) - ai(j234) + a20'134) - a30124) + a4(jl23) = 0 (41) and by multiplying by [xa5a;] (x5jl) (J234) - (x5j2) 0134) + (.r5j3) (J124) - (.r5j4) (J123) = 0 (42) which, being an identity, holds when j runs from 6 to 10. Hence when a5 is common to all four b's the four equations are linearly related. When the four b's do not contain an a in common the four equa- tions are independent. For suppose the b's to be [xaia5], [xa2a5], [xa3a5], and [xaia2]. The four corresponding equations are 2(.rl5j) (234j)C'j = 0 where j runs from 6 to 10, and three other equations of similar form. The determinant of the coefficients of the first four columns is (*156) (2346), (.rl57) (2347), (.rl58) (2348), (a-159) (2349) (a-256) (1346), (z257) (1347), (z258) (1348), (.r259) (1349) (a-356) (1246), (.r357) (1247), (.r358) (1248), (a-359) (1249) (a-126) (3456), (.rl27) (3457), (.rl28) (3458), (.rl29) (3459) (43) In order that this determinant might vanish identically it would be necessary and sufficient that four numbers C\, c2, c3, c4 exist, independ- ent of a, and not all zero, such that Cl(xl5j) (234j) + c2(x25j) (134;) + c3C.r35j) (124;) + Ci(xl2j) (345j) = 0 (44) for the vectors a6, a7, as, a9, represented by a,-, are arbitrary. If we let &]■ = x + a5 this equation reduced to its last term, namely c4(.rl25) (345a:) = 0 whence c4 = 0, for the vectors x, ai • ■ • a5 may have any values what- ever. By letting a, be ai -f- a^, a2 + a4, and a3 + a4 we see that ci, c2, and c3 must all vanish, contrary to hypothesis. Hence (43) does not vanish identically. We may proceed similarly whenever the four b's do not have an a in common. This determinant must accordingly contain as a factor Cw, a quad- ric through the nine points ai,- • • a9. That it vanishes at all these points can be verified by inspection : if x = a4 the last two rows are equal numerically and opposite in sign; if x equals any other a all the elements of a row or of a column vanish. This determinant is of the fourth degree in x and in ar • ■ a5. It is of the second degree in &$■ ■ • a9. Since the factor C\q is a quadric in (46) ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 415 all ten vectors the other factor must be a quadric in x and in ai • • • a5 and be independent of a6- • a9. If we wish to find this latter factor it is allowable to assign to a6- • a9 any values we please. If we take a6 = a3 + a4, a7 = a2 + a4, and a8 = a2 + a3 (45) the determinant (43) becomes 0 0 0 (.rl59) (2349) 0 , (a-254) (1342), (z253) (1342), (a-259) (1349) (a-354) (1243), 0 , (a-352) (1243), (a-359) (1249) 0 , (a-124) (3452), (.rl23) (3452), (*129) (3459) which is equal to the product of (x354) (1243) (.rl59) (2349) times the determinant (x254) (1342), (a-253) (1342) (zl24) (3452), (*123) (3452) which in turn is (1342) (3452) [(a-254) (zl23) - (xl24) (.r253)]; (48) but from (41), by letting a, = x and multiplying by [xaoa5] - (z251) (a-234) - (x253) (xl24) + (*254) (.rl23) = 0 (49) that is, the quantity in brackets in (48) is equal to (.r251) (.r234). Collecting results, we see that the determinant (43), by virtue of the substitutions (45), become the product of eight determinants: (.r354) (a-251) (a-234) (zl59) (1243) (1342) (3452) (2349) (50) It is evident by inspection of this product that the adventitious factor of the determinant (43) must be (.t354) (a-251) (1234) (51) for in no other way can we pick from (50) a factor quadratic in ar • • a5 and x. The algebraic sign is, however, undetermined. Determinants similar in form to (43) can evidently be written down for any values of K and N. The second factor of each element will in every case be a polynomial of degree K — 1 in the variables which enter into it. The first factor is linear in x and in N — 1 other points. We have therefore the theorem : Theorem V. Given a 'polynomial homogeneous of degree K in N variables, and n — 1 points at which this polynomial vanishes: it is in general possible to write down a determinant of order q whose elements 41G HITCHCOCK. are linear in x and are rational functions, of these points and which shall contain the polynomial as a factor. One further illustration may be given of selecting the b's so that the determinant shall not vanish identically, and of factoring the determi- nant. Take N = 5, K — 2, and let the seven b's be [3145], [3356], [3346], [a:567], [3467], [3457], and [3456]. The elements of the first row of our determinant will be (3145/) (2367 'j) where j = 8, 9, • • •, 14; and similarly for the other rows. This determinant does not vanish identically. For if we make the substitutions as = ai + a2, ag = a2 + a3, aio = ai + a3, an = ai + a7, ai2 = a2 + a7, ai3 = a3 + a7, and an = y, an arbitrary vector, the determinant becomes 3 (31452) (12367); 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; Pi ; (33562) (33561) (12473); (12473); (33462) (33461) (12573); (12573); P2 0 P3 0 Pa P6 ; P5 ; Pi ; P9 o o o o o o 0 0 0 ; 0 ; P8 (35671) (.r5672) (.t5673) (12347); (12347); (12347); Pm (34671) 0r4672) (.x4673) (12357); (12357); (12357); Pu (34571) (34572) (34573) (12367); (12367); (12367); P12 0 ; 0 ; 0 ; P0 where Pi, • • • P12 denote elements which are of no consequence, but Po = (3456#) (1237?/). It is evident that this determinant factors into a number of linear determinants together with the two-row determinant (33562); (33561) (33462); (33461) and the three-row determinant (35671) (34671) (34571 ) (.r5672); (35673) (34672); (.r4673) (34572) ; (34573) (53) 3 For compactness in printing this determinant, the two factors of each ele- ment are set one above the other. Thus the element in the first row and first column is (31452) (12367) and similarly for the other factored elements. ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 417 The two-row determinant does not vanish identically since it does not vanish when x = a2 + a4. Nor does the three-row determinant vanish identically, for if x = a4 + ai it becomes (45671) times the two-row determinant (14672); (14673) (14572); (14573) which, in turn, does not vanish if we let ai = a2 + a5. Hence our original determinant does not vanish identically. To complete the factorization, consider the minor of the leading element in (53), namely (*4672) (o;4573) - (.r4572) (.r4673); (54) and let x be expanded in terms of the five vectors a2, &s, a4, a5, and a;, — x(23457) - a2(z3457) + a3(.r2457) - a4(a-2357) + a5(.r2347) - a7(.r2345) = 0 (55) and multiplying by [.r467] - (z4672) (a-3457) + (.r4673) (a!2457) + (.r4675) (.r2347) = 0 (56) whence it is evident that (54) contains the factor (.r4567). In a similar manner it may be shown that the other minors of the elements of the first row in (53) contain the same factor. Thus the determinant contains this factor. In the same way we may show that the minors of any other row contain a common factor, hence the determinant is the product of three linear factors. Again, the same process shows that (52) is a product of two linear factors. The factorization is therefore complete, and the adventitious factor of our original determi- nant is here, as in the former case, a mere product of determinants linear in the points which enter into them. 7. Rule for Constructing these Determinants. It remains to indicate how, in general, the b's may be chosen so that the determinant contemplated in theorem V shall not vanish identically. We have seen that the order q of this determinant is n[(N — 1), K] — N + 2. Let q' be the number analogous to q but associated with polynomials of degree less by one, that is q' = n[N - 1, K - 1] -N + 2 (57) The elements of the determinant are of the form b •a,jE(a,j). A neces- sary condition that the determinant shall not vanish identically is 418 HITCHCOCK. that not more than q' + N — 2 of the E's shall be polynomials of the same standard set. For when several E's belong to the same standard set they are functions of a set of m + 1 of the a's not including a,. Since any element of the determinant contains every one of a set of m + N — 2 of the a's (exclusive of a/), and also contains x, it follows that the corresponding b's will possess N — 3 of the a's in common. That is, they have in common A7 — 2 arbitrary vectors not belonging to the reference system of the E's which occur in the same elements. Now by theorem IV, if q' + 1 of the E's belong to a standard set they satisfy an identity 2L(ay)£(a,-) = 0, but the L's are functions of those a's only which occur in the E's. By writing for the E's an identity of the form (32) it is evident that even when the L's contain N — 2 arbitrary vectors we shall have q' + 1 + (N — 2) of the E's (which now correspond to the C's of (32)), connected by a relation 2LE = 0. Hence only q' + N - 2 of the E's can belong to the same standard set, as was to be shown. We are how in a position to prove that the b's may in general be selected by the following rule : Rule for choice of b's. Select any m + N — 2 from the given set of n — \ vectors ar • -a„_i. Choose q' + N — 2 polynomials Ei, Ei, • • • out of the standard set of degree K — 1 based on m + 1 of these vectors, which we may number from 1 to m + 1. Make a second selection of m + 1 a's by omitting ai from the first selection and adjoining am+2, choose q' -\- N — 3 polynomials of the new standard set based on this second selection of a's. If we have not yet q polynomials, make a third selection of m + 1 a's by omitting am+2 from the second selection and adjoining a^+z', choose q' + AT — 3 polynomials from a third standard set based on this third selection of a's. If we have not yet q polynomials, proceed in a similar manner to form new groups until q polynomials have been obtained: every selec- tion of a's consists of ao, a3, • • -a^i together with one other. Each of the b's contains every a not in its own ' E ' but included in the original choice of m + N — 2 of the a's. We have to prove that the determinant whose elements are b*a; E(&j) does not vanish identically, and that the rule is always possible. We note that, by the rule, ai occurs only in the E's numbered from 2 to q' + N — 2 inclusive. Hence ai occurs in all the b's except those so numbered. If the determinant vanishes identically a set of numbers cr • -cq exists such that Scb»a,-jE(a,-) = 0, the c's being inde- pendent of a; and not all zero. Now let a, = x + g&i where g is ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 419 an arbitrary scalar. All terms vanish except those from 2 to qf-\- N— 2. The equation takes the form 2c»bi»ai.Ei(x + g&i) = 0 where i runs from 2 to q' + N - 2. Consider the quantity Ei (x + jrai). It is a polynomial in g whose absolute term is Ei(x). Since the above equation is to hold for all values of a,- it must hold for all values of g. Hence the sum of the absolute terms must vanish, that is 2cibi*8,iEi(x) = 0 where i runs through the q' + N — 3 values above noted. But this is impossible unless the c's which have these subscripts are all zero : for the equation now has the form 2L(x)£(x) = 0, the only a common to L and E is ai, whence not fewer than q' + N — 2 E's satisfy such a relation, by the corollary to theorem IV. Similarly we note that am+2 occurs only in the second group of E's, and prove in the same way that all c's corresponding to this group must be zero, and so on, till finally only C\ remains. Hence C\ is also zero. This is contrary to hypothesis that the c's be not zero. Hence the determinant does not vanish identically. To show, finally, that the rule is always possible, we note, first, that the rule observes the plan of grouping the E's so that no more than q' + N — 2 belong to a standard set, E\ being common to all the groups. The first group requires m + 1 of the a's. Each new group requires one additional a. We have m + N — 2 a's available for selection. It therefore suffices to show that the number of groups is not greater than N — 2. We have, in other words to prove the inequality q ^ (A - 2) (