THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES THE INTERFEROMETRY OF REVERSED AND NON-REVERSED SPECTRA BY CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON WASHINGTON, 1916 86443 CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No, 249 ., y PRINTED BY J. B. LIPPINCOTT COMPANY AT THE WASHINGTON SQUARE PRESS PHILADELPHIA, U. S. A. 8 V V K SEg pt. /-z CONTENTS. CHAPTER I. The Interferences of Crossed Spectra. PAGE 1. Introductory ........ ..................................................... 2. Coincident spectra with one reversed on a given Fraunhofer line. Figs. I, 2, 3 . 3. The same. Further experiments ........................................... 1 1 4. Coincident spectra with one reversed on a given longitudinal axis. Figs. 4, 5, 6. 12 5. Interference of the corresponding first-order spectra of the grating, in the absence of rotation. Figs. 7, 8, 9, 10 6. Conclusion f ;y CHAPTER II. Further Study of the Interference of Reversed Spectra. "* 7. Apparatus with one grating. Figs. 1 1, 12, 13, a, b 19 8. Observations and experiments with a single grating. Fig. 14 22 9. Inferences. Fig. 15, a, b 24 10. Apparatus with two gratings. Figs. 16, 17, 18 26 11. Experiments continued. New interferometer. Figs. 19, 20 30 12. Experiments continued. Homogeneous light 32 ^ 13. Experiments continued. Contrast of methods 33 14. Experiments continued. Rotation, etc., of grating. Figs. 21, 22 33 v 15. Tentative equations. Figs. 23, 24, 25 36 16. Experiments continued. Analogies. Figs. 26, 27 38 f> 17. Subsidiary diffractions. Figs. 28, 29 43 * 18. Conclusion 45 CHAPTER III. The Interferences of Non-reversed Spectra of Two Gratings. 19. Introduction. Method. Figs. 30, 31 46 20. White light. Colored fringes. Tables i, 2, 3. Figs. 32, 33, 34, 35 47 21. Homogeneous light. Wide slit. Transverse axes coincident. Tables 4, 5. Fig. 36.. 52 22. Homogeneous light. Fine slit. Transverse axes not coincident. Table 6. Fig. 37. 54 xi 1 23. Homogeneous light. Slit and collimator removed. Table 7. Fig. 38 55 ^ 24. Inferences. Figs. 39, 40 56 ^ 25. Rotation of colored fringes. Non-reversed spectra. Figs. 41, 42 58 ^_ 26. Final treatment of reversed spectra. Hypothetical case. Figs. 43, 44, 45, 46 .... 60 ^ 27. Case of reflecting grating. Homogeneous light. Figs. 47, 48 64 J 28. Non-symmetrical positions. Fore-and-aft motion. Fig. 49 67 j CHAPTER IV. The Distance Between Two Parallel Transparent Plates. 5 29. Introductory 69 >j 30. Apparatus. Figs. 50, 51 69 i 31. Equations. Figs. 52, 53 7 * 32. Method 72 v 33. Observations and corrections. Preliminary work. Figs. 54, 55 73 CHAPTER V. Interferometers for Parallel and for Crossed Rays. 34. Introduction. Methods. Figs. 56, 57 78 35. Experiment. Reflecting grating. Parallel rays. Fig. 58 79 36. Experiments. Transmitting grating. Parallel rays 8 1 37. Experiments. Transmitting grating. Crossed rays. Figs. 59, 60, 61, a, b 82 38. The same. The linear phenomenon. Fig. 62 85 39. The same. Inferences. Figs. 63, 64, 65 87 40. Experiments. Reflecting grating. Crossed rays. Figs. 66, 67 88 41 . The same. Compensators 9 l 42. Miscellaneous experiments. Fringes with mercury light 9 1 43. Inferences. Figs. 68, 69 9 2 3 4 CONTENTS. CHAPTER VI. Channeled Spectra Occurring in Connection with the Diffraction of Reflecting Gratings. 44. Introductory 95 45. Apparatus. Fig. 70 95 46. Scattering 95 47. Fringes with white light 96 48. Fringes with sodium light 97 49. Grating on a spectrometer. Fig. 71 98 50. Inferences 100 CHAPTER VII. Prismatic Methods in Reversed and Non-reversed Spectrum Inter ferometry. 51. Purpose 102 52. Method and apparatus. Figs. 72, 73 102 53. The same. Crossed rays 103 54. Another method. Fig. 74 104 55. Methods using prismatic dispersion. Fig. 75 105 56. Methods with paired prisms. Fig. 76 106 CHAPTER VIII. The Linear Type of Displacement Interferometers. 57. Introductory 107 58. Apparatus. Fig. 77 107 59. Film grating. Adjustment. Figs. 78, 79 109 60. Michelson's interferences no 61. Film grating. Another adjustment. Fig. 80 in 62. Equations in CHAPTER IX. The Use of Compensators Bounded by Curved Surfaces. 63. Introduction 113 64. Lens systems 113 65. Effective thickness of the lenticular compensator. Fig. 81 115 66. Observations largely with weak lenses and short interferometer. Figs. 82, 83 1 16 67. Remarks. Fig. 84 1 18 68. Observation with lens systems on both sides. Figs. 85, 86 1 19 69. Telescopic interferences. Figs. 87, 88, 89, 90, 91 120 CHAPTER X. The Dispersion of Air. 70. Introduction. Table 8 124 71. Observations with arc lamp 124 72. Observations with sunlight. Single tube. Table 9 125 73. Two (differential) refraction tubes. Table 10. Fig. 92 127 74. Differential and single refraction tubes. Sunlight. Tables 11,12 129 75. Distortion of glass absent 131 76. Further observations with sunlight. Table 13 131 77. Conclusion 132 CHAPTER XI. The Refraction of Air with Temperature. 78. Apparatus. Fig. 93. Table 14 133 79. Observations 134 80. Computation 135 81. Final experiments at 100. Table 15 136 82. Experiments at red heat 137 83. Further experiments at high temperatures. Fig. 94. Table 16 139 84. Flames 140 85. Conclusion 141 CHAPTER XII. Adiabatic Expansion Observed with the Interferometer. 86. Introductory. Table 17 142 87. Experiments with short, bulky air-chambers 143 88. Effect of strained glass 145 89. Equations 146 90. Experiments with long tubes. Diameter, i inch. Table 18 148 91. The same. Diameter of tube, 2 inches. Table 19 150 92. The same. Diameter of tube, 4 inches. Tables 20, 21. Fig. 95 151 CHAPTER XIII. Miscellaneous Experiments. 93. Effect of ionization on the refraction of a gas 154 94. Mach's interferences. Fig. 96 155 95. A Rowland spectrometer for transmitting and reflecting gratings, plane or concave. Figs. 97, 98, 99 156 PREFACE. The following account of my experiments has been given chronologically. Although many of the anomalous features, in which the interferences of superposed coordinated spectra first presented themselves, were largely removed in the later work, yet the methods used in the several papers, early and later, are throughout different. It therefore seemed justifiable to record them, together with the inferences they at first suggested. The pursuit of the subject as a whole was made both easier and more difficult by the un- avoidable tremors of the laboratory in which I am working; for it is possibly easier to detect an elusive phenomenon if it is in motion among other similar stationary phenomena. But it is certainly difficult, thereafter, to describe it when found. It will be convenient to refer to the cases in which one of the two coincident spectra from the same source is rotated 180 with reference to the other on a transverse axis (i.e., an axis parallel to the Fraunhofer lines), under the term reversed spectra; while the term inverted spectra is at hand for those cases in which one of the paired spectra is turned 180 relative to the other on a longitudinal axis (i.e., an axis parallel to the r-v length of the spectrum). In this book the latter are merely touched upon, briefly, in Chapter I, but they are now being investigated in detail and give promise of many interest- ing results. The chapter contains a full account of what may be seen with a single grating the linear phenomenon, as I have called it, and which, if it stood alone, would be difficult to interpret. In Chapter II, therefore, the interferences of reversed spectra are treated by the aid of two gratings, in virtue of which a multitude of variations are inevitably introduced. The phenomena are thus exhibited in a way leading much more smoothly to their identification. This endeavor is given greater promise in Chapter III, which contains a comparison of the interferences of reversed and non-reversed spectra, the latter produced in a way quite different from those in my earlier work. Nat- urally these in their entirety are even more bewilderingly varied, and become particularly so when, as in Chapter IV, an intermediate reflection of one spectrum is admitted. But with this I was on more familiar ground, as I have hitherto, in these publications, given such investigations particular attention. The flexibility of the new methods is well shown in Chapter V, where separated component beams can with equal facility be made to run in parallel, or across each other at any angle, and perhaps both, with the double result visible in the field of the telescope. In case of crossed rays a remarkable phenomenon is shown, in which very small differences in wave-length imply a remarkably large difference in rotational phase (virtually resolving power) of the two interesting groups of interference fringes due to each wave-length. 5 6 PREFACE. Spectra obtained with two, or at times even with one grating, are often annoyingly furrowed with large transverse fringes. These are investigated in Chapter VI, and referred to diffractions resulting from residual errors in the rulings. In Chapter VII, finally, several examples of new methods of investi- gation are given. They show the important bearing of the diffraction at the slit of the collimator on all these experiments. The cleavage of a field of diffracted rays as an essential preliminary is here put in direct evidence. In Chapters VIII to XIII I have returned to my older experiments with the displacement interferometer. The subjects adduced, like the dispersion of air at low and high temperatures, the adiabatic expansion of air, etc., are pursued less with the object of reaching results of precision than of testing the limits of the displacement method and developing it. My thanks are due to Miss Abbie L. Caldwell for very efficient assistance in preparing the manuscript and drawings for the press. CARL BARUS. BROWN UNIVERSITY, Providence, Rhode Island. CHAPTER I. THE INTERFERENCES OF CROSSED SPECTRA. 1. Introductory. If two component spectra from the same source coincide throughout their extent the elliptic interferences will be spread over the whole surface, provided, of course, the respective glass and air-path differences of the two component rays are not so great as to throw the phenomenon beyond the range of visibility. In the usual method of producing these interferences, where the corresponding reflections and transmissions of the two component rays take place at the same points of the same plane surface, the interference pattern is automatically centered, or nearly so. This is not the case when, as in the following experiments, the interfering beams are separated in some other way; and the problem of centering is often one of the chief difficulties involved; and if the beams are to be treated independ- ently, it is difficult to obviate this annoyance. Suppose, now, that one of the spectra is rotated around an axis normal to both, by a small angle. Will the interferences at once vanish, or is there a limiting angle below which this is not the case? In other words, how far can one trench with light -waves upon the case of musical beats, or of inter- ferences not quite of the same wave-length? Instead of approaching the question in this form, in which it would be exceedingly difficult, experimentally, I have divided it into two component parts. Let one of the spectra be rotated 180 around a longitudinal axis, parallel to the red-violet length of the spectrum and normal to the Fraun- hofer lines. In such a case, interference should be possible only along the infinitely thin longitudinal axis of rotation to which both spectra are sym- metrical, one being the mirror image of the other. One would not expect these interferences to be visible. It is rather surprising, however, that this phenomenon (as I have found) may actually be observed, along a definite longitudinal band in the spectrum, about twice the angular width of the distance between the sodium lines and symmetrical with respect to the axis of rotation. It is independent of the width of the slit, provided this is narrow enough to show the Fraunhofer lines to best advantage. Again, let one spectrum be rotated 180 about a given Fraunhofer line (transverse axis), the nickel or mean D line, for instance. The two coplanar spectra are now mutually reversed, showing the succession red-violet and violet-red, respectively. Interference should take place only along the mean D line and be again inappreciable. Experimentally, I was not at first able to find any interferences for this case in the manner shown below, but this may have been due to inadequacies in the experimental means employed, for the dispersion was insufficient and the reflecting edge of the paired mirrors too rough. Improving the apparatus, I eventually found the phenomenon, 7 8 THE INTERFEROMETRY OF but appearing as a single line, vividly colored above the brightness of the spectrum; or, again, more jet-black than the Fraunhofer lines and located in the position of the coincident wave-lengths of the two superimposed spectra. It is possible, however, as will be shown in 4, to obtain two spectra in such a way that if their longitudinal axes coincide the Fraunhofer lines intersect at a small angle, and vice versa. In such a case, for coincident Fraunhofer lines, interference occurs in a band around these lines and is absent in the rest of the spectrum; whereas, if the longitudinal axes are coincident, the interferences are arranged with reference to these axes. These results seem to bear on the question, but it is difficult to clearly resolve them. The methods used in this paper consist chiefly in bringing the two first- order spectra of a grating, or the second-order spectra or their equivalents, to interfere. In this respect they contain an additional method of inter- ferometry which may be useful, if for any reason it is necessary that the two component beams are not to retrace their paths. 2. Coincident spectra with one reversed on a given Fraunhofer line. In figure i, L is a narrow vertical sheet (subsequently broadened by the dif- fraction of the slit) of white sunlight or arc light from a collimator, G the transparent grating ruled on the side g, from which the first or second order of spectra gM and gN originate. M and N are opaque mirrors mounted adjustably on a firm rail, RR, each of them with three adjustment screws relative to horizontal and vertical axes. M is provided with a slide micrometer (not shown). From M and N the beams pass to the smaller paired mirrors, m and n, which should meet in a fine vertical line at a very obtuse angle. A silvered biprism would here be far preferable, but none having the required angle was available. From n, m, the beams pass into the telescope T. As the spectra are each divergent after issuing from g, they can be made to overlap on leaving n, m, by aid of the adjustment screws on M and N. More- over, as the spectra are mirror images of each other, as suggested in figure i, any spectrum lines (as, for instance, the D) may be put in coincidence on using one of the adjustment screws specified. It is necessary that the telescope T be sufficiently near M in order that the micrometer may be manipulated. REVERSED AND NON-REVERSED SPECTRA. The D lines placed in coincidence are obviously opposites, each line being paired with the mate of the other. A fine wire must be drawn across the slit of the collimator, in order that the vertical coincidence may be tested. One should expect the interferences to appear between the D lines on gradually moving the micrometer mirror M, parallel to itself, into the required posi- tion. As stated above, I did not at first succeed in finding the interferences, but the experiment is a delicate one. In a repetition with first-order spectra, it would be advisable to replace the plane mirrors m, n, by slightly concave mirrors, about 2 meters in focal distance, and to replace the telescope T by a strong eyepiece. This is the method used in the next paragraph, and it was more easily successful. Later I returned to the experiment with the same adjustment, except that the plane mirrors m, n, were placed beyond the grating, with the object of using the equivalent of second-order spectra to get more dispersion. This plan did not fail, and, having once obtained the interferences, the reproduction seemed quite easy, as they remained visible while the micrometer M was moved over about 5 mm. or more, a very important observation. Their appearance with a small telescope was that of a single fine line, alternately flaming yellow (very bright on the yellow background of the surrounding part of the spec- trum) and jet black as compared with the D lines, between which the interferential line was situated, and on an enhanced yellow ground. The flicker is referable to the tremor of the laboratory, which makes it im- possible to keep these interferences quiet. Shutting off the light from either mirror, M or N, naturally quenches the interferences, but leaves the yellow part of the spectrum behind. Obviously, coincidence of the longitudinal axes of the spectra alone is needed. Therefore, upon moving the two double D lines apart, by aid of the adjustment screws on the mirror M and N, symmetrically to the ends of the yellow field in the telescope, the interferences were isolated and located midway between the D doublets of each spectrum, i.e., in the center of the field of the telescope. They could now be observed to better advantage. In the small telescope there is appar- ently but one dark line. If stationary, its ultimate character, when centered, would be surmised to be given by the intersection of a vertical diameter with a series of confocal ellipses, successively bright and dark, as indicated in figure 2. The light and dark parts alternate or flicker. On moving the micrometer, the vertical intersector A takes a more and more lateral position like B, so that the trembling interferences would soon be invisible, as they rapidly become finer and hair-like (not shown). On using higher magnification (larger telescope), two black lines bordering Sb 10 THE INTERFEROMETRY OF a bright line, or a black line between two bright lines, seemed to be visible; but the interferences would have to be stationary to be definitively described, since the width of the pattern is not more than one-third to one-half of the distance between the sodium lines. 1 The interferences, moreover, did not now readily conform to the design B, figure 2, anticipated, but were more of the type C, with long, dark lines slightly oblique to the vertical, and vibrating within a vividly yellow band. Sometimes these were heavier, with two or three faint lines on one side. Further experiment showed that the phenomenon is not influenced by the width of the slit, except that it is clearest and sharpest with the narrowest slit possible and vanishes when the slit is made so wide that the Fraunhofer lines disappear. It may easily be produced by the modified method following, in any wave-length red, yellow, green, etc., with no essential difference except in size. It is present, moreover, in all focal planes, i.e., the ocular of the telescope may be inserted or pulled out to any distance, yet the same phe- nomena persist on the vague, colored background. A number of observations were made to detect the change of the pattern of the interference, between its entrance into the field and its eventual evanescence, in case of the continuous displace- ment of the mirror M over 5 mm. In figure 2 this would be equiva- lent to a passage of B into B' through A, and the fringes for a distant center should therefore o rotate, as they actually do in the experiments of the next para- graph. But in the present case the type C persists; the lines may become longer or all but coalesce and their inclination may change somewhat. They nevertheless remain fine and nearly vertical, until they vanish completely and there is no rotation. Nor could the phenomenon be found again within the length of the given micrometer screw. Hence it is improbable that these interferences conform at once to the ordinary elliptic type for which figure 2 applies, even if the ellipse is considered exceptionally eccentric. The use of two slits, one following the other, does not change the pattern. The modified method of experiment was one of double diffraction. In figure 3, L is the blade of light from the collimator, which passing under the plane mirror, m, penetrates the grating G, whence the diffracted first-order beams reach the opaque mirrors M and N. These return the beams, nearly normally but with an upward slant, so that the color selected intersects the lf The use of the DI D 2 distance of the sodium lines for the measurement of the breadth of the interference phenomenon is a mere matter of convenience in describing it. It will be shown in the next report that the breadth of the strip carrying interference fringes is quite independent of the dispersion of the optic system. REVERSED AND NON-REVERSED SPECTRA. 11 grating at a higher level than L. A second diffraction takes place at about the same angle, 9, to the direct ray t, and the coincident rays now impinge on the mirror m. They are thence reflected into the telescope at T. This method admits of easier adjustment, as everything is controlled by the adjust- ment screws on M and N. Plane mirrors M, N, and m only are needed, the latter being on a horizontal axis to accommodate T. The direct (white) beam is screened off after transmission through the grating, if necessary. But it rarely enters the telescope. 3. The same. Further experiments. In place of the plane mirror, m, a slightly concave mirror (2 meters in focal distance, say) may be used with advantage and the telescope T replaced by a strong eyepiece. In this way I obtained the best results. It is to be noticed that the apparatus (fig. 3) may serve as a spectrometer, provided the wave-length X of one line and the grating space D are known, and the mirror, M, is measurably revolvable about a vertical axis. In this case any unknown wave-length, X', is obtained by rotating M until X' is in coincidence with X. Supposing the X's of the two spectra to have been origi- nally in coincidence and that 6 is the angle of M which now puts X' in coin- cidence with X, it is easily shown that X'-X = X (2 sin 2 0/2 + \/> 2 A 2 -i sin 0) Angles must in such a case be accurately measurable, i.e., to about o.i minute of arc per Angstrom unit, if the grating space .0 = 351 Xio- 6 , as above. Counter-rotation of the mirror N till the X's coincide would double the accu- racy. The usual grating, however, has greater dispersion and would require less precision in 6. Finally, a still simpler and probably more efficient device consists in com- bining the mirror m and the plane grating G, or of proceeding, in other words, on the plan of Rowland's method for concave reflecting gratings. In such a case the light would enter in the direction TG, figure 3, be reflected along CM , back along MG, and then return along GT at a slightly higher or lower level than on entering. The equation just given would still apply, and many interesting modifications are suggested. Experiments of this kind are to be tested. Moreover, in case of the plane-transmitting grating and plane mirror, as above shown, the same simplification is possible if the lens is replaced by the telescope at T. But in this case the spectra are intersected by strong, stationary interferences due to reflections from front and rear faces and consequently not conveniently available. A reflecting grating and telescope would not encounter this annoyance. In general, however, as in the disposition adopted in figure 3, the light enters opposite the observer, and, as the light directly transmitted can be screened off, this is a practical convenience in favor of the transparent grating. The reflected spectra used may be placed at any level by rotating the mirror m on a horizontal axis. On further repeating the work by the use of the concave mirror m, a strong eyepiece at T, figure 3, and using a compensator, I eventually succeeded in 12 THE INTERFEROMETRY OF erecting the interference design C, figure 2. It then took the form given at D, and this seems to furnish the final clue to the subject. In other words, the design consists of a new type of extremely eccentric ellipses, with their long axes parallel to the Fraunhofer lines, each end having the outline of a needle- point, possibly even concave outward. Only one end of the long, closed curves is obtainable. These jet-black lines dance on the highly colored back- ground of less than half the width between the two sodium lines. The inter- ference design would, therefore, be the same (apart from color) as that which would be obtained if the spectrum containing ordinary elliptic interferences were to shrink longitudinally from red to violet, till it occupied less than half the space between the two D lines. In fact, I have at other times obtained just such patterns, with all the colors present, but not in the pure yellow, as in the present case. Vertically, the path-difference is always due to more or less obliquity of the rays passing through the plate of the grating. Horizontally, however, the equivalent path-difference is complicated, in the present case, by the fact that one wave-length of a pair has increased, whereas the other has diminished, while both may pass through the same thickness of glass and air. 4. Coincident spectra with one reversed on a given longitudinal axis. For this experiment it is necessary to reflect the first-order spectra issuing at the grating G, figure 4, from the ruled face g (a narrow, preferably horizontal, blade of white light is here furnished by the collimator L with a horizontal slit, and the rulings of the grating are also horizontal and parallel to it) , twice in succession and preferably from mirrors M and N and m and n, reflecting normally to each other and inclined at an angle of, roughly, 45. Each of the mirrors M and N must be revolvable about a horizontal axis parallel to the slit and furnished with three adjustment screws relatively to axes normal to each other, one of which is horizontal. The mirrors m, n are the silvered faces of a prism right-angled at the edge. It is, moreover, to be placed on the slide of a Fraunhofer micrometer so that the prism may be moved, grad- ually up and down, for the adjustment of distances. On leaving the mirror m, n, the two spectra are carried by nearly horizontal and parallel sheets of divergent rays, which pass outward from the diagram. But it will be seen that one of the two spectra reaching the observer is reversed on the longitudinal axis relatively to the other; i.e., if one is in the position 1 violet, the other will be red j b ttOm j violet, j [ top j The subsequent passage of the rays is shown in figure 5, which is the side elevation and therefore at right angles to the preceding figure. The rays from m and n impinge on a distant, slightly concave mirror K (about 1.74 meters in focal distance), placed somewhat obliquely, so that when the rays come to a focus at F near the micrometer they may just avoid it . The partially overlapping spectra at F are viewed by a strong eyepiece, E. The observer at E can then control the Fraunhofer micrometer by which m, n is raised and lowered, and the three adjustment screws of M. REVERSED AND NON-REVERSED SPECTRA. 13 The adjustment consists in first roughly placing all parts in symmetry with sunlight, until the two spectra appear at E. The lens may be removed. There should be a bright, narrow spectrum band on each side of and near the edge of the prism mn; for it is clear that after passing the lens E, correspond- ing rays from M and N must both enter the pupil of the eye to be seen together. To make the spectrum parallel, the mirror mn is rotated, as a whole, around a vertical axis. The three screws on the mirrors M and N then assist in completing the adjustment; the rotation around the horizontal axis brings the sodium lines in coincidence (both must be clearly seen and sharp and at an appreciable distance apart); that around the oblique axis gives rise to more or less overlapping, as required. The need of a sharp coincidence of the sodium lines is very essential in all these experiments. After proper vertical position of mn has been found by slowly moving the micrometer screw up and down, the fringes appear. They are usually very fine lines, possibly indicating distant centers of the ellipses to which they belong . The appearance is roughly suggested in figure 6 . They are thus totally different from the preceding set, 3. They pass from the type a through b (contraction toward the violet end was not noticed) into the type c, when the mirrors mn move in a given direction. The center of the ellipses is in the vertical through the field of view for the adjustment b, in which case the lines pass from end to end of the spectrum as a narrow band near the longi- tudinal axis of actual coincidence of spectra, symmetrically. The height or breadth of the longitudinal interference band, d in figure 6, is not greater than 1.5 to 2 times the distance apart of the sodium lines at right angles to the band. From this the angular divergence of the breadth of the band may be found, since \ = D sin 6, where X is the wave-length of light, D the grating space, and 6 the angle of diffraction. Hence for the two sodium lines A0 = AX/Z)cos 6 14 THE INTERFEROMETRY OF Since D = 3$iXio- 6 , cos d=.gS6, and AX = 6Xicr 8 ; therefore A0= 1.7X10-" radians. Since the width of the band is about twice this, it will be 68 seconds of arc, or, roughly, about a minute in breadth. Within the strip, when the fringes are horizontal, I counted about five of them, so that their distance apart would be about 14 seconds of arc. It appears, therefore, that rays of a given color, say of the wave-length at D, which leave the grating at a given point and at an angle of about one minute in the plane of the D line, are still in a condition to interfere ; whereas one would anticipate that only those rays which lie in the common longitudinal axis of rotation of the two coincident spectra, symmetrical to this, should be in this condition. Such interference should not be appreciable, since the white rays are independent and apparently come from two different points of the slit. If we consider the angular deviation of pencils of parallel rays crossing the grating to be equivalent to the divergence of their respective optical axes at the collimating lens (about 45 cm. in focal distance), the dis- tance apart of two points of the slit, the rays of which are still able to produce interference, is * = 45XA0 = 45X i. 7X10^ = 7.6Xio- 3 cm. or nearly o.i mm. Hence points of white light in the slit about o.i mm. apart along its length are included in the band of interferences in question, extending in colored light from red to violet. This seemingly anomalous result will be fully interpreted at another opportunity. 5. Interference of the corresponding first-order spectra of the grating, in the absence of rotation. This apparatus seemed to be of special interest, since the rays used do not retrace their path and are thus available for experi- ments in which rays traveling in one direction only, are needed.* I have tried both the adjustments given in figures 7 and 8, the latter, since the rays are more nearly normally reflected at the mirrors M and N, having some advantages ; but the other succeeds nearly as well. The difficulty encountered is a curious one of adjustment, which was not anticipated. In other words, if the longitudinal axes of two identical spectra are in coincidence, the Fraun- hofer lines are likely to be at a small angle to each other and complete inter- * Cf. Am. Journal of Sci., xxxiv, p. 101, 1912, on the interferometry of an air column carrying electrical current. REVERSED AND NON-REVERSED SPECTRA. 15 ference is therefore impossible. Again, if the spectrum lines are in coincidence, the longitudinal axes usually diverge by a small angle. Furthermore, the interferences are almost always eccentric and the lines hair-like, indicating distant centers. I have not succeeded in making a perfect adjustment, systematically; but the discrepancies indicated are themselves interesting in their bearing on the subject of this paper. In figures 7 and 8, L is a vertical blade of white light from a collimator with fine slit, and G is the grating. The two first-order spectra leaving the ruled face at the line g strike the opaque mirrors M and N, the former on a micrometer moving the mirror parallel to itself. From M and N the rays reach the half-silvered plate of glass HS, where one is transmitted and the other reflected into the telescope T. The coincident rays R are superfluous. After placing the parts and roughly adjusting them for symmetry with sunlight, the finer adjustment may be undertaken. It may be noticed that the two systems M and N, and G as well as HS, can be used for further adjustment separately. All are provided with adjustment screws relatively to rectangular axes. To put the mirrors M and N in parallel and in the vertical plane with the grating G, the half -silvered plate should be removed 10 and replaced by a small white vertical screen of cardboard, placed at right angles to the direction of HS in figure 7 and receiving both spectra. A fine wire is drawn across the slit to locate the longitudinal axis, and an extra lens may be added to the collimator and properly spaced until the doublet insures sharp focussing. Both mirrors, M and N, are now rotated on hori- zontal axes, until the longitudinal black lines in their spectra cease to diverge and coincide accurately. G, M, N, may now be considered in adjustment. On returning the half -silvered plate, HS, it in turn is to be carefully rotated around horizontal and vertical axes, until the horizontal black line in the spectrum and the sodium line (always incidentally present in the arc lamp) both coincide. But, as a rule, it will be found that if the longitudinal axes, ww, figure 9, coincide, the D lines cross each other at a small angle, exagger- ated in the figure. The interferences, when found by moving the micrometer at M, are usually coarse, irregular lines, indicating a center not very distant and located on the level of a band where the D lines cross. On the other hand, if the D lines are brought to coincidence by moving the adjustment screws on M and N (which throws them out of parallel), the longitudinal axes ww, w'w' , figure 10, diverge at a small angle and the inter- ferences are found in a vertical band where the lines ww and w'w' cross. This band is relatively wide, however, as compared with the cases in paragraphs 2 and 3. Nevertheless, I have looked upon these results as additional proof 16 THE INTERFEROMETRY OF of the possibility of interference; for in neither case ought they to occur if the spectra are not quite coincident horizontally and vertically. If they do occur, it would at first sight seem that a certain small latitude of wave-length adjustment is permitted even with light-waves. I was at first inclined to refer the cause of this lack of simultaneous parallel- ism to the grating itself, as it occurred with an Ames grating ruled on glass, with a Michelson reflecting grating, and with a film grating, in about the same measure. But subsequently, on adopting the method of figure 8, the divergence was largely removed and the interferences were now visible throughout the whole of the spectrum. The discrepancy is probably due to insufficient normality of the plate of the grating to the incident white ray, since one of the rays is twice reflected. In any case the adjustment of the coincident sodium lines must be very accurate if the fringes are to be sharp ; certainly as little as half their distance apart will obscure the phenomenon. Though the spectra are bright, the interferences are not as good as with the usual method (paragraph i); i.e., the dark lines are not black. Neither have I found an available or systematic method for centering the fringes, so that the lines obtained are usually delicate. Again, the position of the colli- mator, both as regards slit and lens, is here of very serious importance. Any micrometeric horizontal motion of either, in its own plane, will throw the fringes out. Finally, the whole spectrum travels with the motion of the micrometer mirror M. The apparatus is thus too difficult to adjust for use, to be of practical interest when simpler methods are at hand. The effect of tremors acting prejudicially on so many parts is exaggerated. 6. Conclusion. The phenomena of paragraphs 2,3, and 4, showing definite and characteristic interference in case of two coincident spectra crossed either on a longitudinal or transverse axis, represent the chief import of the present chapter. These results can not be directly due to the diffraction of a slit (regarding the line of coincidence as such), owing to their relatively small magnitudes and their independence of the breadth of the slit. Since there is in each case but a single line of points or axis, the disturbance of which comes from identical sources, we might regard the image of this line in the telescope to be modified by the diffraction of its objective. But if the inter- ferences originated in this way, the Fraunhofer lines of the spectrum should show similar characteristics and the diffraction pattern should differ from those observed. Thus the conclusion is apparently justified that distinct and independent points of the narrow slit whose distance apart on its length is not greater than o.i mm. contribute rays to the field of interference in each of the colors of the spectrum (longitudinal axes coinciding). The phenomenon of inversion is virtually one of homogeneous light, the same type of interference occurring in each color from red to violet. When the fringes are horizontal, homogeneous light and a correspondingly broad slit would replace the spectrum. They belong, moreover, to the elliptic category, being of the same nature, apart from their limitations, as those REVERSED AND NON-REVERSED SPECTRA. 17 used in displacement interferometry. With the exception of the points lying on the longitudinal axis of rotation or of coincidence, all the pairs of points of the two coincident spectra owe the major part of their light to different sources; i.e., the points of the superposed spectra are not colored images of one and the same point in the slit. Again, in case of rotation of one of the coincident spectra around a trans- verse axis (Fraunhofer line), colors which differ in wave-length by about half the distance apart of the two sodium lines seem also to admit of interference. This permissible difference of wave-length is thus relatively about ' or less than o.i per cent. The character of these interferences is distinctive. They are not of the regular elliptic type, but arise and vanish in a succession of nearly vertical (parallel to slit), regularly broken lines. Later observation, however., revealed as their true form a succession of long spindles or needle- shaped designs. The chief peculiarity observed is their almost scintillating mobility, which in the above text has been referred to the inevitable tremors of the laboratory. It is, however, interesting to inquire into the conditions of the possibility of observable beating light-waves. For two waves, very close together, of frequency n and n' and wave-lengths X and X', if V is the velocity of light, the number of beats per second would be Therefore in case of the two sodium lines, for instance, w'-w = 3Xio l X6Xio- 8 /348oXio- 12 =5Xio 11 i.e., about 5 X io 10 beats per o.i second, the physiological interval of nickering. Naturally this seems to be out of all question, even if one is confronting a source which is an approach to a mathematical line. The endeavor will have to be made to produce these interferences under absolutely quiet surroundings. Their appearance is altogether singular and not like the case of paragraph 4, where there is also perceptible tremor, or with the general case of trembling interference patches, with which I am, unfortunately, all too familiar. In this place, however, it is my sole purpose to present, at its face value, an observation which is spatial, independent of time consideration; and the laterally cramped character of the new interference, with its long, hair-like lines thrust into a strip less than half the distance apart of the sodium lines, is the only evidence submitted. If the coincident path of two rays of slightly different wave-lengths, X and X', which interfere, is x, then there are x/\ and x/\', complete waves in the given path, and, in case of original identity in phase, instantaneous reenforcement will occur when In other words, at the wth reenforcement 18 REVERSED AND NON-REVERSED SPECTRA. Hence, since X 2 is very small and x relatively very large, the small value of AX (i.e., the very thin strip of spectrum within which the phenomenon occurs) is apparent. In the above experiments the estimates, in round numbers, were AX = 2.4Xicr 8 , X 2 = 36Xicr 10 . Hence if n=i, so that one reenforcement would have to occur about at each 1.5 mm. along the rays. Nevertheless, the formidable difficulty remains to be investigated, viz, why these nominally beating wave-trains, with an infinitesimal group period (io~ u sec.), could be recognized at all. The characteristic feature of the new phenomenon is this, that apart from intensity it persists, without variation, through a path-difference of over 5 milli- meters; i.e., through 15,000 or 20,000 wave-lengths. It follows, since the optical paths grating-mirror-grating are alone significant, that two individual light-waves of the same ray over 15,000 wave-lengths apart are still appre- ciably identical. Beyond that the waves under consideration no longer correspond in orientation and can not interfere in a way to produce alterna- tions of accentuated brightness and darkness. CHAPTER II. FURTHER STUDY OF THE INTERFERENCE OF REVERSED SPECTRA. 7. Apparatus with one grating. The different methods suggested in para- graph 3 were each tried in succession, but none of them were found equally convenient or efficient in comparison with the method finally used in the preceding paper. To begin with the annoyances encountered in the use of a reflecting grating, it was found that the impinging light from the collimator and the reflected doubly diffracted beam from the grating lie too close together, even if all precautions are taken, to make this method of practical value. The use of Rowland's concave grating without a collimator is out of the question, since the spectra formed on the circular locus of condensation, if reflected back, will again converge into a white image of the slit, colored if part of the spectrum is reflected. The plane-reflecting grating, though not subject to this law, requires a collimator, and, since marked obliquity of rays is excluded, it will hardly be probable that the elusive phenomena can be obtained in this way. A compromise method, in which both the reflecting and the transmitting grating are used, will be described in paragraph 10. Though apparently the best adapted of all the methods used, it has only after difficult and prolonged research led to results. These, however, proved very fruitful in their bearing on the phenomena. For first-order spectra, where there is abundance of light (it is often difficult to exclude all the whitish glare in the field of the telescope completely), the method of figure 1 1 , which shows normal rays only, is still preferable. Here the impinging collimated beam L passes below the opaque mirror m and through the lower half of the grating G. The diffracted pencil is reflected nearly normally but slightly upward, by the mirrors M and N (the former carried on a micrometer slide) , to be again diffracted at the grating and there- fore to impinge as definitely colored light on the lower edge of the concave mirror m (about 1.5 to 2 meters in focal distance), whence it is brought to a focus at F and viewed by the strong eyepiece E. Considerable dispersion and magnification is obtained in this way; indeed, the two D lines stand far apart and the nickel line is distinctly visible between them. There must be a fine hair wire across the slit so that the longitudinal axes of the spectra may be accurately adjusted. The mirror m above the impinging beam must be capable of rotation about a vertical and a horizontal axis in order that the focus F may be appropriately placed between M and N. With G at i meter and m at 2 meters from F, the disposition is good. The micrometer M is easily at hand. Though the direct beam may be screened off, the glare reflected back from the grating and the glare from the objective of the colli- mator are not excluded, as stated. In fact, it was eventually found necessary 19 20 THE INTERFEROMETRY OF to cany this pencil in an opaque tube reaching from the objective of the collimator, as far as the grating. With first-order spectra this method always succeeded satisfactorily, and in case of a ruled grating the phenomenon is exhibited brilliantly, if the paths CM and GN are optically nearly equal. After some experience it is fairly easy to find it. I have not, however, been able to obtain it with a film grating, even after using a variety of excellent samples. This is not remarkable, for the film grating is hardly sufficiently plane to produce clear regular reflection, and the corresponding paths GM and GN would not, therefore, be definite. Second-order spectra are too faint and can not be seen, unless the glare is excluded in the manner stated. All modifications of the method seemed with- out avail, until finally the light was led from the collimator objective C, figure n, to the grating G, in a cylindrical tube, whereupon both the glare from the objective and the rearward reflection from the grating were effec- tively screened off. This tube must, of course, lie below the returning pencil, i.e., it must not (in section) cover more than the lower half of the grating. In this case the second-order spectra, though faint, were seen clearly; but the scintillating interferences could not be observed until the very weak eyepiece, E, was used with the concave mirror m; or a weak telescope with a plane mirror. It was then detected, but showed no essential difference from the case of first-order spectra. The larger dispersion, in other words, was unavailable. The phenomenon was seen most distinctly by drawing out the eyepiece of the telescope, as the light is thereby concentrated, although the Fraunhofer lines vanish. Second-order spectra are therefore not necessarily advantageous. The phenomenon is very hard to find, and the experiments were persisted in only to obtain the result under different conditions. The tube-like light conductor referred to above is, of course, advantageous in case of first-order spectra. If the concave mirror is used, the phenomenon may even be seen brilliantly with the naked eye. An alternative method of half-silvering the ruled face of the grating and then using it as a reflector was tried with success. The beam of parallel rays from the collimator L, figure 12, is transmitted by the grating (ruled, half-silvered face, g toward the mirrors M and N) and the two diffracted beams then returned by the opaque mirrors M and N, to be in turn diffracted by reflection into the telescope T. In fact, this method succeeds with the unsilvered grating; for the rays diffracted, by reflection, from the ruled face (toward the telescope), but not very well. The reflection from the rear face REVERSED AND NON-REVERSED SPECTRA. 21 of the grating is so cut up by the strong, stationary interferences that it is unavailable. The grating plate must, of course, be slightly wedge-shaped, otherwise all the spectra would be superposed. In case the ruled face is half -silvered, however, the stationary interferences are practically absent, while two strong spectra are reflected from the silvered side. The phenome- non may then be produced at all distances of G from M and N (2 meters and less), but best at distances within i meter. It is, however, frequently hard to find unless different distances apart of the mean D lines are tested. This may be due to the fact that the silver film is not quite equally thick. Besides the symmetrical position, gT, figure 12, the two corresponding unsymmetrical positions g'T' were tested with success; and it appeared that while in the case gT the phenomenon is virtually linear, dark or bright, like a Fraunhofer line, a succession of dark lines inclined to the vertical may appear for the unsymmetrical position g' T'. Dark lines are apt to be broadened. Questions relative to the effect of oblique incidence were also tested by aid of the concave-mirror method shown in figure n, the white light from C to G being conducted in an inch tube of pasteboard, immediately under the concave mirror, m. Figure 13,0, shows the general disposition of apparatus. \3 The angle of incidence * is gradually increased, until the return rays from N meet the grating at nearly grazing incidence. No essential difference in the phenomenon was observed, however, except that it was apt to be broader in the non-symmetrical positions and to suggest fine new lines in parallel with the old. In a return to the symmetrical position, sharp lines were especially distinct, usually showing one dark and two bright lines, while two dark and one bright occurred less frequently. It could be seen quite vividly with the naked eye. When the telescope was used and the ocular drawn far forward, the multilinear form was often suggested. On broadening the slit the black lines vanish first and a flickering band remains after the Fraun- hofer lines are gone. Finally, the phenomenon could be seen even when the longitudinal axes of the spectra were not quite coincident, but it rapidly became fainter in intensity. Figure 13, b, suggests a method of using a reflecting grating, either plane or (possibly, if the incident light is parallel) concave, for the production of the phenomenon. G is the grating, receiving the collimated white light, L, which is diffracted toward M and N, thence reflected (at a different elevation) back to G, to be again diffracted towards T, above or below the direct beam, where it is observed. I have not, however, been able to obtain results with these methods owing to subsidiary difficulties. 22 THE INTERFEROMETRY OF 8. Observations and experiments with a single grating. On considering figure ii, it will be seen that the doubly reflected, doubly diffracted rays are also in a condition to interfere. Thus the rays GMGNG and GNGMG have identical path-length, or at least path-difference; but it is improbable that superimposed on the strong spectra this effect could be seen, for the reflec- tion from the ruled face of the grating is very slight and the divergent spectra have weakened seriously. The scintillating interferences, on the other hand, are much brighter than the superposed spectra. Such interfer- ences, also, should be independent of the play of the micrometer M, since the path-difference of these beams is not changed thereby, each being identi- cally lengthened or shortened. Furthermore, the interposition of a thick plate-glass compensator in CM should have no effect. Neither of these infer- ences applies for the phenomenon in question, which persists for a definite displacement of M, only, and the introduction of a compensator requires the usual equivalent displacement of M, within the range of the phenomenon. Finally, the interferences relatively to a phenomenon produced by double diffraction would not be modified. Many experiments were made to ascertain the path-difference within which the phenomenon is visible. This can not be accurately determined, since it is a question of stating when an observation, which is becoming rapidly less distinct, has actually vanished. Moreover, any imperfection of the microm- eter throws out the coincidence of longitudinal spectrum axes, while a read- justment breaks the continuity of the micrometer displacement, or reading. Results were obtained as follows, for example, AN" being the displacement of the mirror M: With telescope ............................ AA r = 0.34, 0.45, 0.41 cm. With concave mirror and lens .............. 0.45, 0.35, 0.41 cm. With concave mirror and adjustment ....... 0.50 to 0.60 cm. The low readings are due to the rnicrometric wabbling of the micrometer slide. Since AAT is the double path-difference, the number of wave-lengths in question may be put i.e., the distances along the ray are 15,000 to 20,000 wave-lengths apart, about as estimated in the above paper. This is the characteristic feature of the phenomenon. Between its extreme ranges of visibility the appearance of the phenomenon scarcely changes. It ceases to be visible rather suddenly; and this is to be expected, since we are dealing directly with two wave-trains displaced rela- tively to each other. It is visible for a wide slit even after the Fraunhofer lines vanish. It disappears by decreasing in width, when the slit is closed. If the ocular of the telescope is drawn out, the phenomenon may even be observed after the Fraunhofer lines have vanished, in the dark, stringy spec- trum of an extremely fine slit. When the longitudinal axis of the spectrum is indicated by a fine wire across the slit, the adjustment consists in bringing REVERSED AND NON-REVERSED SPECTRA. 23 the black longitudinal lines of the two spectra together. The question thus arises how close this coincidence is to be. When the phenomenon is sharp, it has been found possible to displace the two black lines so that a fine, bright strip of spectrum may just be seen between, without quite destroying the interferences. Naturally they are then much weaker. This result is in harmony with the observations made on rotating one spectrum, on a longi- tudinal axis, 1 80 with reference to the other. Since the phenomenon was originally produced with sunlight, it might be supposed that the edges of the Fraunhofer line, under conditions of tremor, would interfere with each other as indicated in figure 14, where A is one and B the other of the two superposed spectra. The change of wave-length is suggested by the slant of lines on the ~ / diagram. In such a case, whereas the ^ r- . y - _^:_ v {L , ,_. i) conditions a and c would show bright $ ^o~ f ==== ^r. i and w, z and ^4, are still in a condition to interfere. The phases i3=z= i, having the same path-difference. At Xo, 0o, the two phases 7XI Ni= i.4X io 6 beats per sec. which would also be inappreciable. REVERSED AND NON-REVERSED SPECTRA. 25 If both the difference of wave-length and wave-velocity are considered, we should have for the first spectrum v and n, and for the second spectrum v and n'. The conditions would be left unchanged, if the second velocity is taken equal to the first and the frequency n'(v'/v) replaced by n'. From this it follows that the number of beats N is nearly If 5X is considered negative, if p.=A B/\ 2 and the multipliers /* and n 2 be neglected, which is the difference of the two cases above computed. As the first is very large compared with the second, the visibility of the phenomenon is not changed. The theory of group waves usually introduces a factor 2. Thus if Xi, vi, n\, be the group wave-length, velocity, and frequency, or, or with the above data results otherwise like the above and without bearing here. There is a possible question whether differences of wave-length due to velocity and not to period can be treated as dispersion. The occurrence of forced vibrations has also been looked to as an explana- tion. Though here again, even if the spectra are almost always of unequal intensity, the reason for the preponderance of one would have to be stated. True, equal mean strength is not equivalent to equal instantaneous strength. In the case of forced vibrations, however, if the harmonic forces of one spec- trum are F=A cospt (forced, T=2ir/p), of the other F=A'cosqt, (free, T=2ir/q) and there is no friction, the resulting harmonic motion will be given by Now if we regard the case of figure 15, on one side of the line of coincidence Xo, q 2 >p 2 ', on the other side, p 2 >q 2 . Hence, whenever a brilliant line flashes out due to coincident phases, there should also be a black line due to opposi- tion; and, in fact, when the phenomenon is produced under conditions of perfect symmetry of the component beams, this seems to be its character; i.e., the enhanced line cuts vertically across the breadth of the spectrum. The case q 2 = p 2 , being of infinitely small breadth, would not be visible. It is not to be overlooked, however, that in certain adjustments, particularly in 26 THE INTERFEROMETRY OF the non-symmetrical case of figure 13, more than two black lines frequently occur. (Cf . 1 5.) These accessory lines are ordinarily very thin and crowded on one side of the phenomenon only. It is thus merely the prevalent occur- rence of paired dark and bright lines that are here brought to mind. Again, the suggestion of many oblique lines has occurred in some of the observations. These would be quite unaccounted for. Finally, many attempts were made to find whether the phenomenon would occur again beyond its normal range of about 2X0.5 cm. of displacement. But, though the micrometer screw actuating the mirror M was effectively 2X3 cm. long, no recurrence could be found. At the ends of its range the phenomenon drops off rather abruptly. None of the inferences put forward adequately account for the phenome- non as seen with a single grating, as a whole. In this dilemma I even went so far as to suppose that a new property of light might be in evidence. One feature, it is true, has been left without comment, and that is the width of the slit-image. If ab, figure 15 b, is the angular width (d&) of this image, the case of figure 15 a should be additionally treated in terms of figure 156. But within the limits of the present method of experiment, with but one grating, this circumstance seems to offer no clue. If, for instance, the spectra actually coincide in color throughout their extent, as in ordi- nary interferences, the interference patterns should be enormous, for the path-difference may be zero. The invariability of the present phenomena as to size within its long range of presence, the occurrence of intensely sharp and bright or dark single lines, with a distance (d&) much less than the distance apart of the DI, D% lines, is in no way suggested by the width of slit-image. Moreover, in spite of its persistence, the interference phenomenon of reversed spectra has the sensitiveness of all interferences. Slight tapping on the massive table throws it out altogether. Clearly, therefore, a modification of method is essential if new light is to be thrown on the phenomenon, and from this viewpoint a separation of the two diffractions seems most promising. 10. Apparatus with two gratings. All the varied experiments described in the preceding paragraph failed to show any essential modification of the linear interference pattern obtained. In a measure this was to be anticipated, inas- much as both diffractions take place at the same grating. It therefore seemed promising to modify this limitation of the experiments, although the difficulty of finding the phenomena would obviously be greatly increased. The separa- tion of the two diffractions, however, seemed to be alone capable of resolving the phenomenon into intelligible parts. In the present method the glass grating G, figure 16, receives the white beam L from the collimator, which is then diffracted to the opaque mirror M REVERSED AND NON-REVERSED SPECTRA. 27 (on a micrometer slide) and N, thence to be reflected to the reflecting grating G', plane or curved. Here the two beams of the identically colored light selected are again diffracted to the telescope or lens at T. Since the gratings G, G', rarely have the same grating constant, their proper position must be found by computation and trial. In my work the distances to the line of mirrors NM were 165 cm. for G and 90 cm. for G' . This method automati- cally excludes the direct beam a and all glare, and gives excellent spectra both in the first and second orders. The use of two gratings, however, introduces the difficulties of adjustment specified, as the two D doublets corresponding to AT and M will not, as a rule, be parallel and normal to the longitudinal axes of the spectrum, unless all cardinal features, like the rulings and their planes, are quite parallel. If the grating is not normal to the impinging beam, the axis of the corresponding spectrum is a curved line. The spectra are, moreover, likely to be unequally intense, a condition not infrequent even in the preceding method. It is possible that this may be due to the grating itself, but probably unequal parts of the corresponding beams are 16 used in the two cases, or the mirrors are unequally good. As a result, in my earlier work I was not able to produce the phenomena with two gratings, after many trials, in spite of the clearness of the overlapping spectra; but the same serious difficulties are encountered whenever interferences are produced from two independent surfaces. Later, having added a number of improvements to facilitate adjustments, I returned to the search again and eventually succeeded. There are essen- tially four operations here in question, supposing the grating G approximately in adjustment. By aid of the three adjustment screws on each of the mirrors M and N, figure 16, the fine wire drawn across the slit may be focussed on the grating, if an extra lens is added to the collimator and the black horizontal shadows of that wire, across the corresponding spectra, placed in coincidence. The grating G' is then to be moved slowly fore and aft, normal to itself, on the slide, so that the position in which the sodium lines are nearly in coinci- dence to an eye placed at the telescope, T, may be found. The grating G' is next to be slowly rotated on a line (parallel to LT) normal to its surface, to the effect that the black axes of both spectra (i.e., the spectra as a whole) may coincide. This must be done accurately, and the last small adjustments may be made at the screws controlling M and N. Finally, the micrometer 28 THE INTERFEROMETRY OF a slide carrying M is to be moved fore and aft until the interferences appear. These operations are difficult even to an experienced observer. The fringes are very susceptible to tremors, and only under quiet surroundings do they appear sharply. At other times they move, as a whole, up and down and intermittently vanish. The fringes so obtained, figure 17, were totally different from the preceding and consisted of short, black, equidistant, nearly horizontal lines across the active yellow strip of spectrum, at the axis of coincidence. The strip was about of the same width as above. Thus the pattern presented the general appearance of a barber's pole in black and yellow, the width being less than the sodium interval, D\, DZ, and the distance apart of fringes usually smaller. They were visually in motion up and down, rarely quiet, no doubt owing to tremor. Since the fringes were nearly horizontal or less than 30 degrees in inclination, it was possible to enlarge the width of the slit without destroying them, as in case of the hair-like vertical fringes in paragraph 2 above. In this way a breadth of strip greater than the distance Di, DZ, could be obtained with sunlight or arc light, though a moderately fine slit was still desirable. - ' _", < c- d 17 f h In general, the characteristics noted above were again observed. Thus on moving the micrometer screw controlling M, the interferences appeared rather abruptly. They vanished in a similar manner, after about 0.4 cm. or more of the micrometer screw had been passed over. In other words, the fringes remain identical for a path-difference of about 2X0.4 cm., or nearly 15,000 wave-lengths. If we call the four D lines available in the two solar spectra D\, DZ, D\, D'z, respectively, a number of curious results were obtained on placing them variously in approximate coincidence. Thus figure 17 a, when each D line of one spectrum coincides with the mate of the other (D it D' 2 ; D'i, D 2 ), equi- distant dots, surrounded apparently by yellow luminous circles, appeared between the two doublets. On widening the slit the dots changed to a grating of nearly horizontal lines covering the strip DI, D 2 , figure 176. The lines in one part of the slit seemed to slope upward and in another to slope downward. With a large telescope the phenomenon was more dim and quiet, apparently. The fringes often lie in more definite focal planes and cease to be visible when the ocular of the telescope is far outward, differing from the case above. The phenomenon of chief interest, however, was observed (figure 17 c) in placing two identical D lines in coincidence (Di; D 2 D' 2 ; D'i). The fringes REVERSED AND NON-REVERSED SPECTRA. 29 were then seen across the coincident lines, now no longer visible, quite inde- pendent of the absence of light. This would seem to mean that the otherwise quiet ether within the black line is stimulated into vibration by the identical harmonic motions of the bright fields at and beyond the edges of the line (diffraction). The question will presently be broached again in a different way. Here I may note that in the above cases of transverse lines ( 8) it is often possible to observe a very fine parallel yellow line within the coincident DZ, D'z, or Di, D'i, doublets, excited, therefore, in the dark space and splitting the line. The experiments were now repeated with the sodium arc, and these also gave some striking results. Thus in the case of figure 17 d the lines were separated, but the yellow striations seemed to show across the dark space between DZ and D' 2 . When the yellow light was too weak, cross-hatchings were seen only across D'z, as in figure 17 e. Frequently the phenomenon figure 1 7 / occurred on broadening the slit, in which D 2 and D'z interfered, but only D'z was marked. Screening off D 2 (left mirror) at once removed the fringes. I have interpreted this observation as the result of parallax, due to the fact that the lines and the interferences are seen in different focal planes. On the basis of these results one might with some plausibility adduce the following remarks in explanation of the phenomenon: In figure 18 a, let 5i and 5 2 be the overlapping reversed spectra and let the line of symmetry be at Xi, X 2 . Then if identical ether vibrations can react on each other across a narrow ether gap, rays as far as X'i, X' 2 and X"i, X" 2 being of identical source and wave-length, respectively, are still in a condition to interfere. There would then be three groups of interferences, Xi X 2 , X'i X' 2) X% X" 2 . If, figure 1 8 b, all are in phase, we should have a brilliant line; if all are in opposite phases, a dark line on the principle of figure 18 c. Naturally, if wave-trains react on each other across an ether gap, small as compared with the D\, DZ interval, the assumption made above relative to interference of different wave-lengths is superfluous. My misgiving in the matter arises from the misfortune of having taken down the original apparatus, for modification, and having since been unable to reproduce them with anything like the decisiveness with which they were at first apparently observed. I can not now be certain whether what occurred was actually what I seemed to see, or whether the broad illumination of the sodium flash (broad individual lines, DI to D 2 , virtually a continuous spectrum) may not have misled me. The experiments were continued, as follows. 30 THE INTERFEROMETRY OF 11. Experiments continued. New interferometer. At the outset it was necessary to ascertain the reason for the difference of the phenomena, as obtained with one grating in paragraph 8 and with two gratings in para- graph 10. As the probable cause is a lack of parallelism of the rulings in the latter case, it was necessary to remount the second grating G' in the manner shown in figure 19. Here A A is a baseboard, capable of sliding right or left and of rotating on a horizontal axis parallel to the grating. The latter (in a suitable frame) is held at the bottom by the axle, e, normal to the grating and by the two set-screws a and b carried by the standards c and d. Thus the grating could be rotated around an axis normal to its plane. At first a Michelson plane-reflecting grating G' and a telescope were used, as in figure 16; but it was found preferable (fig. 20) to use a Rowland concave reflecting grating G', with the strong lens at T, the grating receiving a beam of parallel rays of light for each color from the collimator and first grating G. In this case, with sufficiently high dispersion, a large, strong field was obtained, in which even the very fine lines of the solar spectrum were quite sharp. Rotating grating G' around a parallel horizontal axis, like A A, figure 19, made little / \ / ^ Hl^ ^ 7 _gie_ c4 ^ c4 19 difference, relatively speaking; but rotation around the axis e, normal to its plane, carried out by actuating a and 6 in opposite directions, made funda- mental differences in the appearance of the phenomenon and eventually suggested a new interferometer for homogeneous light. The adjustments are the same as in case of figure i6,G being the transparent grating, except that G' is now a concave grating and T a strong eyepiece. The distances G'T and GT were of the order of i and 2 meters. On rotating the grating G' on an axis normal to its face, from a position of slight inclination of the rulings toward the left, through the vertical position, to slight inclination to the right, the fringes passed through a great variety of forms, to be described in detail in 13 below. Difference of focal planes between the Fraunhofer lines and the interferences were common, so that effects of parallax were apt to occur. Thus when D z and D' 2 coincide, the ladder-like phenomenon may lie between D' t and D\\ or the ladder may pass obliquely between the D z D l and D\ D' z doublets. The first experiment with the new and powerful apparatus (plane transparent grating 6", grating space 3SiXio- 6 cm., and the concave reflecting grating G', grating space I73XIQ- 6 cm., fig. 20) was made with the object of verifying, if possible, the REVERSED AND NON-REVERSED SPECTRA. 31 reaction of parallel ether wave-trains on each other across a very narrow ether gap. The sodium arc lamp was used as a source of light. The results as a whole were negative, or at least conflicting. Usually when strong inter- ferences were observed for coincident positions of DZ, D'%, for instance, there was no passage of fringes across the dark space when DZ and D'% were slightly separated. At the beginning of the work (possibly as the result of lines broadened by a flash of sodium light) the stretch of interference fringes across the dark space was certain; but such evidence is not quite trustworthy, for a continuous spectrum (i.e., lines broadened by the flash) would necessarily produce the striations. With a very fine slit the coincident D\, D'i or D?, D'% was frequently much broadened by a sort of burr of fringed interferences. When the lines are self -reversed, superposition of Di, D'i, etc., frequently showed vivid interferences across the intensely black middle line. This and the passage of the bright and dark lines across the superposed D\ t D'\ lines of the solar spectrum are thus the only evidence of the reaction of separated light-rays on each other across an ether gap observed in the new experiments, and the above results could not be repeated. On introducing a refined mechanism to establish the sharpest possible coincidence of the DI, D'\ or Dz, D' z lines, it seemed as if these lines could at times be brought to overlap with precision, without the simultaneous appearance of the interferences around then; but on drawing out the ocular of the telescope or the lens the cross-hatching invariably appears. If the coincidence is not quite sharp, the phenomenon is usually very strong in the isolated bright strip. Horizontal fringes are best for the test. An additional series of experiments was made some time later by screening off parts of the concave grating G', in order to locate the seat of the phenom- enon at the grating. Screening the transmitting grating G was without con- sequence; but on reducing the area G' to all but the middle vertical strip about 5 mm. wide, a very marked intensification of the phenomenon followed. Although the spectrum as a whole was darker, the interferences stood out from it, relatively much sharper, stronger, and broader than before. The Fraunhofer lines were still quite clear. Thus the pattern, g, figure 17, was now very common, both with sunlight and with sodium light. For a given slit the phenomenon began with a strong burr c, figure 17, completely oblit- erating and widening the superposed D 2 , D'z lines. When these lines were moved apart, the striations followed them, as in figure 17, h and i, to a limit depending on the width of the slit. A still more interesting pattern is shown in figure 1 7 k, in which the interferences proper are strong and marked between the two Di D'i doublets, but much fainter striations are also evident, reaching obliquely across and obviously with the same period. With this improvement I again tested the ether-gap phenomenon, using the sodium arc, and to my surprise again succeeded. D\ D\ lines of half the breadth of the doublets apart induced strong fringes between them, and the experiments were continued with the same results for a long time. Several days after, however, with another adjustment, it in turn failed. Clearly 32 THE INTERFEROMETRY OF there is some variable element involved that escaped me, and it will hardly be worth while to pursue the question further with the given end in view, without a radical change of method. Screening middle parts of the grating (in relation to 15) did not lead to noteworthy results here, but such experiments will become of critical importance below. A word may be added in relation to Fresnellian interferences in the present work. These would be liable to occur if the observations had been made outside of the principal focus, with the sodium lines blurred. In all the experiments on the excitation of a narrow ether gap, however, the D lines were clearly in sight and sharp, so that the phenomena of non-reversed spectra and homogene- ous light (in the next section) are not here in question. True, such interfer- ences may often be found in the case of reversed spectra, when the sodium lines are purposely blurred, by pushing the ocular toward the front or to the rear. 12. Experiments continued. Homogeneous light. To turn to a second class of experiments : very important results were obtained with homogeneous light (sodium arc) on placing the DiD'i or D 2 Z/ 2 lines in coincidence and then broadening the slit indefinitely or even removing it altogether. A new type of interferences was discovered, linear and parallel in character and inter- secting the whole yellow field. These lines could (as above) be made to pass from a grid of very fine, hair-like, nearly horizontal lines to relatively broad, vertical lines, on changing the orientation of the grating G f , figure 16. Small changes of position of the grating produced a relatively large rotation and enlargement of the lines of the interference pattern. The fringes, when verti- cal and large, are specially interesting. The distances between successive fringes obtained were about the same (accidentally) as the DiD z distance of the sodium lines. They are quiet in the absence of tremor. If DiD'i or D 2 D'z were only present, the field would be an alternation of yellow and black striations; but as both doublets are present, the interferences overlap the flat (non-interfering) yellow field of the lines not in coincidence. The fringes are nevertheless quite distinct. A single homogeneous line (like the green mercury line) would give better results. It is necessary that the line selected (say DiD'i) should coincide horizontally and vertically before the slit is broadened. Otherwise no fringes appear in the yellow ground, or at least not in the principal focal plane. On using a thin mica compensator, it is easy to make these fringes move while the mica film is rotated; and they pass from right to left and then back again from left to right, as the mica vane passes through the normal position of minimum effective thickness. Thus this is a new form of interferometer with homogeneous light. The fringes remain identical in size, from their inception till they vanish, while the microm- eter M, figure 1 6, passes (as above) over about 15,000 wave-lengths. In this respect the new interferometer differs from all other types, the two air-paths, GMG' and GNG', alone being in question. The condition of occurrence will be investigated in paragraph 13. REVERSED AND NON-REVERSED SPECTRA. 33 13. Experiments continued. Contrast of methods. As these fringes were produced with a concave reflecting grating, the question may be put whether they would also appear in case of the plane reflecting grating, G', in the adjust- ment of figure 1 6. The experiment was therefore repeated with a wide slit, or with no slit at all, and there was no essential difference in the two classes of results. On the contrary, when the method of but one grating and sodium light was used (fig. n), the interferometer fringes, in case of a very wide slit or the absence of a slit, could not be produced over the yellow field, as a whole. There appeared, however, an obviously pulsating flicker in parts of the field, on reducing the width of the slit till the sodium lines were each about the width of a DiD 2 space, with either DJD'i or D^D'z superposed. The sharply outlined slit showed an irregular, rhythmic brightening and darkening over certain parts of its length. These broad pulsations were very violent, very much in character with the linear phenomenon above. This behavior is very peculiar; recalling the appearance of a bright yellow ribbon undulating, or flapping fore and aft, so as to darken parts of its length rhythmically. The pulsations, moreover, were quite as active if seen at night, when the tremors of the laboratory were certainly reduced to minimum. Nevertheless, I am now convinced that such tremor only is in question. Regarding the phenomenon as a whole, one may argue that in case of the wide slit and single grating, in which the lines for both diffractions are there- fore rigorously parallel, the interference fringes are on so large a scale as to cover the whole field of view and thus to escape detection; i.e., that a single vague, quivering shadow of a flickering field is all that may be looked for, in the limited field of view of the eyepiece. Returning to the case of two gratings and the wide vertical interference fringes and, in turn, all but closing the slit (vertical interferences and sodium arc light), the pulsating phenomenon simply narrowed in width. The two or three sharp vibrating lines, alternating in black and yellow of the original phenomenon (Chapter I), did not appear. The cause of this is now to be investigated. 14. Experiments continued. Rotation, etc., of grating. The method of two gratings (fig. 16 or 20, plane transmitting and concave reflecting) was first further improved by perfecting the fore-and-aft motion of the grating G' (G r movable in the direction G'T on a slide), as well as the precision of the independent rotation of G' normal to its face ; i.e., around G'T. These adjust- ments led to further elucidation of the phenomenon. To begin with the fore- and-aft motion of the concave grating G' (i.e., displacements in the directions G'T, fig. 20), it was found that the fringes, figure 21, a, b, c, d, e, in any good adjustment, pass from extremely fine, sharp, vertical striations, which gradu- ally thicken and incline to relatively coarse, horizontal lines, finally with further inclination in the same direction into fine vertical lines again, while G' continually moves (through about 5 cm.) on the slide normal to the face 34 THE INTERFEROMETRY OF of the grating. It was not at all difficult to follow the continuous tilt of these lines through the horizontal, occurring on careful and continuous front-and- rear motions of the grating G' through the limiting positions. The fringes usually vanish vertically merely because of their smallness. Again, on rotating the grating G' around an axis normal to its face, the fringes merely vary in size, without changing their inclination. Thus if the horizontal fringes (which were here always closer than the inclined set) are in view, these will pass from extremely small-sized, fine, hair-like striations, through a maximum (which is a mere shadow, as a single fringe probably fills the field) back into the fine lines again. Only a few degrees of rotation of the grating suffice for the complete transformation. The maximum is frequently discernible only in consequence of a flickering field. An oblique set of fringes is equally available, remaining oblique as they grow continually coarser and in turn finer with the continuous rotation of the grating. When the very large horizontal fringes are produced by this method, the change into vertical fringes by fore-and-aft motion of G' is very rapid, so that relatively wide, nearly vertical forms may be obtained. All these effects may be produced by solar or by arc light, around the line of symmetry of the overlapping spectra; ^/ ' ' \ or with sodium light when either DiD\ or DJD\ 21 \ // ^ coincide. / """ \^ The fine vertical or inclined lines appear as & fa Qj di such when the slit is widened, either in case of white or of sodium light. These are the inter- 22 ( */ ferometer fringes seen above (6), coarse or ^ ^ fine. With sodium light any width of slit, or no slit at all, is equally admissible. The same is true for the narrow maxima. Lines nearly horizontal were sometimes obtained, pointing, as a whole, toward a center. Finally (and this is the important result) the extremely large horizontal maxima, when a single fringe fills the field, can not be seen apart from pulsa- tions, in the case of a wide slit. With a very narrow slit, such as is suited for the Fraunhofer lines, these horizontal fringes appear as intensely bright or very dark images of the slit. In other words, the normal phenomenon of overlapping symmetrical spectra as described in Chapter I is merely the vertical strip of an enormous horizontal interference fringe, made sharp and differentiated by its narrowness. This case occurs at once when the rulings of the two gratings G and G' are all but parallel, and hence it is the regular phenomenon when but a single grating is used for the two diffractions, as in figures ii and 12. In later experiments on the effect of the rotation of the grating, G', around a normal axis, the above results were found to be incomplete. If the rotation is sufficient in amount (a few degrees, always very small), it appears that, after enlarging, the fringes also rotate. But the rotation in this case corre- sponds to a vertical maximum, as indicated in figure 22, the vertical set being REVERSED AND NON-REVERSED SPECTRA. 35 the coarsest possible for a given fore-and-aft position of the grating G'. In the figure, the sequence a, b, c, d, e is obtained for a continuous rotation of the grating (in one direction around a normal axis). It now became interesting to ascertain how the vertical set c, figure 22, would behave with the fore-and-aft motion. The experiments showed that there was no further rotation, but that, while G' passes normally to itself over about 1.5 cm. on the slide, the vertical fringes pass from extreme fine- ness at the limit of visibility, through an infinite vertical maximum (a single vague shadow pulsating in the field) , back to extreme fineness again, without any rotation. If the edges of the corresponding yellow strips (superposed DI, D'\ lines) did not quite coincide, the fringes were seen outside of the prin- cipal focal plane, as usual. Probably the vertical and horizontal maxima are identical in occurrence and appear in case of parallelism in the rulings of the two gratings G and G', and the absence of path-difference. Hence if a single grating is used, as in the original method, the interferometer fringes are not obtainable. This is an important and apparently final result, remembering that fore-and-aft motion is probably equivalent to a rotation around a vertical axis, parallel to the grating. With regard to the rotation in case of fore-and-aft motion of G f , it is well to remark that in approaching the position c, figure 24, it is apt to be very rapid as compared with the displacement, precisely as in the case of the picket- fence analogy. Hence the original phenomenon, consisting of single lines, can not be mani- folded by increasing the width of slit. It vanishes for a wide slit into an indiscernible shadow. The phenomenon is a strip cut across an enormous black or bright horizontal fringe, by the occurrence of a narrow slit. More- over, the scintillations variously interpreted above are now seen to be due to tremors, however different from such an effect they at first appear; i.e., the enormously broad, horizontal fringe changes from dark to bright, as a whole, by any half wave-length displacement of any part of the apparatus. It is thus peculiarly sensitive to tremors. On the other hand, oblique or fine vertical fringes are always recognizable for any size of slit. The inquiry is finally pertinent as to why the phenomenon is so remarkably sharpened by a narrow slit ; but this must be left to the following experiments. To be quite sure that the concave grating G' had no fundamental bearing on the phenomenon, I again replaced it by the Michelson plane reflecting grating (fig. 16, G transmitting, G' reflecting). In the same way I was able to rotate the fringes, continuously, through a horizontal maximum of size by fore-and-aft motion of G'. Rotation of G' in its own plane increased or decreased the breadth and distance apart of the fringes through a maximum, coinciding with the parallelism of the rulings of the two gratings. Here I also showed decisively that as the rungs of the interference ladder (fig. 2 1 c) thickened and receded from each other, the design passed, in the transitional case, through the original phenomenon of the single vertical line dark or brilliant yellow, for a slit showing the Fraunhofer lines clearly. The phenome- 36 THE INTERFEROMETRY OF non vanishes with the spectrum lines as the slit is widened, but, on the other hand, persists as far as the interference of light for a narrow slit. Finally, the apparent occurrence of more than one line is referable to the presence of more than one nearly horizontal wide band in the field of the telescope. Thus, for instance, cases between b and c near c and between c and d near c, figure 24, are the ones most liable to occur when both diffractions take place at a single grating. This result will be used in paragraph 15. 15. Tentative equations. In the first place, the actual paths (apart from the theory of diffraction) of the two component rays, on the right and left sides of the line of symmetry, II' Z, figure 23, will be of interest. The compu- tation may be made for the method of two gratings at once, as the result (if the distance apart of the gratings is C = o) includes the method with one grating; i.e., the more complicated figure 23, where G is the transmitting and G' the reflecting grating, resolves itself into a case of figure 24, with but one grating, G. M and M' are the two opaque mirrors, I the normally incident homogeneous ray. Supposing, for simplicity, that the grating planes G and G' are parallel and symmetrically placed relatively to the mirrors M and M' , as in the figure, the ray Y diffracted at the angle 0i is reflected into X at an angle 82-61 and diffracted into Z normally, at an angle 0%, on both sides. Under the condition of symmetry assumed X+Y (X'+Y'}=o, or without path-difference. Let N be the normal from I to M, and n the normal from I' to M, with a similar notation on the other side. Hence if I be given an inclination, di, 61 is incremented by d0\, Y-\-X passes into y\-\-y-\-x, Y'-\-X' into y'\+y'+x r , decremented at an angle dB'i, while both are diffracted into Z'. Since generally sm 0i-sm* = X/.L> cos 0id0i = cos i d8 i for homogeneous light and the same di. Hence ddi = dd'i=dd, say. If 5= 2 0i, and rcosa-+n/ cos (' 2 ~X 2 ) cos (a/2} 2D'(Ncos 2 -X 2 dd= = = i^ 4N tan 4N *N if D is the grating space. To find the part of the spectrum (dX) occupied by a fringe in the case postulated, since sin = \/D, dd d\ ^D cos Z) cos 2 4A/' 86413 38 THE INTERFEROMETRY OF and from the preceding equations, finally, where dK would be the wave-length breadth of the fringe, remembering that the fringes themselves are homogeneous light. In the grating used " 8 cm. X = 6oXio~ 6 cm. w=ioocm. 400 This is but 1/200 of the distance, dX = 6Xicr 8 cm., between the D lines. Hence such fringes would be invisible. Moreover, d6 oc i/N; the fringes, there- fore, should grow markedly in size as AT is made smaller. Experiments were carried out with this consideration in view, by the single-grating and concave- mirror method, N being reduced from nearly 2 meters to 20 cm., without any observable change in the breadth or character of the phenomenon. It showed the same alternation of one black and one or two bright linear fringes, or the reverse, throughout. Hence, it seems improbable that the phenomenon, i.e., the interference fringes, are referable to such a plan of interference as is given in figure 24. Similarly, for the case of two gratings, figure 23, COS Qr/2) (D'2-X 2 ) 2D'(N COS =B siny D' S 7 E sinx B siny 40 THE INTERFEROMETRY OF If E is expressed in terms of D and D', and B in terms of 5 and S', and the first equation is used, then 5 S' sin y D from which, in the fourth equation, E sin

sin v> Again, the angle y' is given from sin y' D or on reduction , _ D sin

KD=D', or 5=5', then D sin (?/a) cos (>/a) tan y' = . v = cos (

/2) so that for an acute angle cos 6 de D ~ cd\ D Thus conditions for practical interferometry would actually appear, the fringes being of DiD 2 width for c = 0.17 cm., provided a wide slit and homo- geneous light of at least Di or D z grade is used. Such an interferometer seems to differ from other forms, inasmuch as the fringes remain of the same size and distribution, from their entrance into the field to their exit; or for a motion of the opaque mirror M of about 6 mm. REVERSED AND NON-REVERSED SPECTRA. 43 To resume the evidence thus far obtained, we may therefore assert that in the case of homogeneous light and a wide slit, or the absence of a slit, the field would either be bright or dark, as a whole. There is a single enormous horizontal fringe in the field. Hence the pronounced flickering with half wave-length displacements of any part of the apparatus. With the slit narrowed until the Fraunhofer lines are seen sharply, the linear phenomenon in question (Chapter I) appears. This may become ladder-like, but it always remains very narrow (^ DiDz) when the rulings of the two gratings are not quite parallel. 17. Subsidiary diffractions. The behavior of the linear phenomena some- times suggests probable relations to the Fresnellian interferences, produced, however, not within the telescope, as in 27, 28, Chapter III (for the inter- ferences are seen together with the Fraunhofer lines in the principal focal plane), but outside of it, at the grating, as suggested by figure 26. If the concave grating G' is screened off, until a width of strip parallel to the rulings and not more than 5 mm. wide is used, the linear phenomenon is much en- hanced, being both broader and stronger, without losing its general character. Here the D lines are still visible. The ladder-like patterns show an equally pronounced coarsening. So far as these phenomena go, it is obvious that the resolving power of the grating must be in question, seeing that the total number of rulings has been greatly reduced. The use of screens with narrower slits carries the process farther; but after the opening is less than 2 mm. in width the available light is insufficient for further observation. If a small lens is used, the phenomena can still be seen over 2 meters beyond the principal focus of the grating. A screen was now made as in figure 28 a, with two slits about 2 mm. wide and 2 mm. apart (6), and placed over the effective part of the grating. The result, after careful trial as to position, was noteworthy. Oblique fringes were widened to many times the DiD 2 space and coarsened, showing a definite grid-like design, as in figure 28 b, whereas, on removing the screen, the original pattern of a regular succession of brilliant dots (fig. 28 c) again appeared. It was with the linear fringes, however, that the evidence obtained was most striking; for these now showed all the Fresnellian interferences (fig. 28 d). On removing the screen, the brilliant linear phenomenon (fig. 28 e), which in all the experiments made had thus far resisted manifolding, appeared at once. The pattern, d, moreover, when viewed with a small lens, within a meter in the direction of the rays, showed very definite enlargement with distance. Though a fine slit was needed, the resolving power of the grating was now too small to show any Fraunhofer lines. Similar results were obtained for a wire i or 2 mm. in diameter. With the screen, figure 28 a, and a bar, 6, i mm. wide, the fine interference grid due to the bar, and the coarse grid due to the spaces (the fine lines being about twice as narrow as the coarse, but all of the same inclination) were often obtained together (fig. 28/). A space i cm. wide intersected by a bar 2 mm. wide gave similar results, fine grids or 44 THE INTERFEROMETRY OF thick lines, according as one or both spaces were used. If either mirror, M or N, is screened, the whole phenomenon vanishes. It follows, then, if rv and v'r', figure 29, represent the two reversed, over- lapping spectra at the grating, b the focus and aa'b the direction of the homo- geneous diffracted rays condensed at 6, that about 0.5 cm. of the spectrum, d'a' and ad on either side of a, is chiefly active in modifying the resulting diffrac- tion pattern. Within this the homogeneous rays, cc' and dd', are capable of interference. Although the wave-fronts entering b are slightly spherical, their radius is about r=i meter, and they may therefore be regarded plane. In such a case the angular width dx of the illuminated strip at b, for a width of screen dd' = i cm., between two extinctions, may be written to X | = 6oX^ = 6>->' 2 (i-cos0) In case of the Wallace grating below =1.75X10-* X Thus if the inclosed angle ii' between the plates is i degree, or 0.0175 radian, D 7?' = 5.3Xio- 7 , about 0.3 per cent of D and equivalent to about 43 lines to the inch. With adequate facilities for measuring i, this method may be useful for comparing gratings, not too different, in terms of a normal or standard, practically, since the finite equations may also be expanded. In a similar way the slight adjustments of the longitudinal axes of the two spectra may be made by rotating one grating around a horizontal axis ; but this correction is less easily specified. Finally, one should bear in mind that with film gratings there is liable to be an angle i-4' between the adjusted plates. Fortunately this has very little bearing on the method below. The range of displacement of grating within which the fringes may be used with an ordinary small telescope extends from contact of the two gratings to a distance of e>= 2 to 3 cm. beyond. In figure 31, which is a plan of the essential planes of the apparatus, G, G' being the ruled faces of the gratings in parallel, 7, 7', 7", three impinging rays of white light diffracted into D, D f , the points a, b, c, a ', a", b are in the same phase, so that the path-difference of the rays from b at g and / is easily computed. If the single ray 7 is diffracted into D and D' or 7 and 7' into D, 7 and I" into D', I' and 7" into D and D', the equations for these fringes should be (if AP is the path-difference), AP = e(i-cos 0)=e(i-Vi-X 2 / 2 ) where D is the grating space, e the distance apart, and X the wave-length. Thus the micrometer value of a fringe for a color X should be, under normal incidence, ~i-cos = I - A / I _xyz? 2 For two colors X and X' n\ = e(i-cos &)=eM (n+n')\' = e(i-cos tf}=eM e REVERSED AND NON-REVERSED SPECTRA. 49 if n' is the number of fringes between X and X'. Thus M'X-MX' (3) n = e- XX' or the number of fringes increases as e is greater. Equation (2) does not, as a rule, reproduce the phenomenon very well. Since the grating space D of the two gratings is rarely quite the same, the air-plate inclosed, in case of apparent coincidence of the sodium lines, is slightly wedge-shaped, as in figure 32. Hence the two diffractions take place at incidences o and a, respectively, and the corresponding angles of diffrac- tion will be and 0'. If we consider the two corresponding rays I and I", diffracted at the first and second face, respectively, and coinciding at c in the latter, the points a, b, and a' (ba r normal to ac), are in the same phase, and we may compute the phase-difference at the coincident points at c. Since the distance be is , _ cos a cos cos oi) 32 the path-difference is whence (4) 33 1 34 e cos a cos0 - r- n - c cos (B-a .. cos 0) Xcos(0-a) cos cos a (i cos 0) which changes into equation (2) when a = o and w=i. Fortunately this correction is, as a rule, small. In case of the Wallace gratings (D = i .75 X icr 4 cm.), for instance, if X= 58.93 X iQ- 8 , then 0= 19 40' or de= i.oi X iQ- 3 ; whereas if a = 5, then te= 1.04X10-*; if a=io, then fo?= 1.07X10-*, etc. If the incidence is at an angle i and the plates are parallel, figure 33, the inquiry leads in the same way to an equation of more serious import. If the gratings G and G' are at a distance e apart and the incident rays are I and /', the points a, b, c are in the same phase. Hence the two rays leaving d and diffracted along D correspond to a path-difference (5) whence cost v(i-cos(0-*)) X cos i i cos (0 t) 50 THE INTERFEROMETRY OF Table i and fig. 35 show the variation of fringes with the angle of incidence t, equation (5). Hence if the angle of incidence is changed from 5 to + 5, de increases to nearly 3 times its first value. This, therefore, accounts for the large discrepancies of 8e found in the successive data below. To secure in- creased sensitiveness and to make the apparatus less sensitive to slight changes of i, this angle should be about 25, in which case 8e is about three wave-lengths per fringe. But normal incidence is frequently more convenient. Finally, in figure 34, if the angle of incidence is i and the two faces G and G' make an angle a with each other and are initially at a distance e apart, changing successively to e' and e", the points o, b, c, being in the same phase, the two rays D and D 1 leaving at d, at an angle Bi, will have a path- difference at d equal to cos cos a whence (6) e : 77; \li ^ua \u tyi cos ^ cos (Q ay _ \cosicos (6 a) cos a cos 6 (i cos (0 &')) TABLE i . Wallace gratings. D= io-*X i .75. i itfXSe f I0 3 X5 + 19040' at -5 0.643 + 15 +10 16.740 4.090 10 -15 0-443 0.321 +5 1.840 -20 0.240 I.OI2 -25 0.185 This equation reproduces the preceding equation (5) if a = o and the origi- nal equation (2) if a = i = o. It shows that a discrepancy or angle between the plates is of minor importance. Hence the change of this angle may be used to bring the sodium lines in coincidence when the gratings differ slightly in their grating constants D. On the other hand, changes of incidence * are of extreme importance. Experiments made with the film grating showed that equation (2) not only fits very badly, but that 8e per fringe is a fluctuating quantity. Table 2 gives some results obtained by measuring the successive values obtained for teXio 8 , corresponding to 10 fringes. Fringes were distinctly seen within 3 cm. of displacement by an ordinary telescope. TABLE 2. Gratings 15,050 lines to inch; computed values io 3 5c=o.94 cm. per fringe. First sample: Second sample: ioe=i.22 cm. 10^=0.98 cm. 1.07 1.16 i. 06 i. ii (normal incidenc (oblique incidence) REVERSED AND NON-REVERSED SPECTRA. 51 TABLE 2. Continued Wallace gratings 14,050 lines to inch; computed value io 3 3e=i.oi cm. per fringe. io 3 5c=i.22 cm. (large fringes); ioSc=i.35 (different *') 1-24 1-32 1-25 1-30 1-23 1.35 1.23 (small fringes) 1.33 1-23 1.33 1.32 1-34 The reason for lack of accord is given in equations (5) and (6) and table i. Any wedge effect of the glass plate is probably negligible. To show that the irregularity of the above results is to be sought in the accidental varia- tions of the angle of incidence i at both gratings, the rough experiments in table 3 suffice. TABLE 3. *, negative (less than 10),] &Xio 3 =o.77 cm. Ocular drawn in, [ .78 focus changing. J .74 *==to; ocular set for } SeX io 3 = 1.18 cm. principal focal plane. Na lines in field and coincident, *, positive (less than lo),l SeX io s = 1.69 cm. ocular drawn out, / 1 .66 Thus, as equation (6) implies, small variations of i produce relatively large variations of de, and if * passes continuously through zero, from negative to positive incidence, de increases continually and may easily be more than doubled. If the phenomenon is in focal planes in front of the principal plane (ocular in), 8e is small, and vice versa. Moreover, this enormous discrepancy is quite as marked for thin glass (2 mm.) as for thick glass plates (8 mm). Again, the rather stiff screw of the micrometer, which twisted the whole apparatus slightly, was sufficient to introduce irregularity. Placing the tele- scope close to the grating or far off made no difference. Hence the position of the optical center of the objective does not affect the result. An additional result was obtained by placing a plate of glass between the two gratings G and G f . The effect was an unexpected enlargement of fringes, increasing with the thickness of the glass plate (0.6 cm. or more). The reason for this is given by equation (3), in paragraph 2, for the number of fringes n' between two colors X and X', XV where M= i-cos0, M'= i-cos0'. Since n' is a number, the glass plate can be effective only in changing 6 and 0'. As both are diminished by refraction, the cosines are increased and i-cos 0, i-cos 0' are both decreased. Hence n' is decreased or the number of fringes is decreased, and their distance apart is thus larger. It is obvious that when the sodium lines are not superposed the fringes can not He at infinity, but are found in a special focal plane, depending on 52 THE INTERFEROMETRY OF the character of coincidence; i.e., whether the rays are convergent or diver- gent. Finally, a slight rotation of the slit around the axis of the collimator rotates the fringes in the opposite direction to the sodium lines, and it is rather surprising that so much rotation of slit (10 or 20) is permissible without fatally blurring the image. The slightest rotation of one grating relatively to the other destroys the fringes. Naturally, the colored fringes vanish when the slit is widened or when it is removed. To give them sharpness, moreover, the beam passing through the grating must be narrow laterally. It is possible to see these colored fringes with the naked eye; but the transverse and longitudinal axes must in this case be slightly thrown out of adjustment, so that the fringes are no longer visible in the telescope. To the eye they form a somewhat fan-shaped set of colored fringes; i.e., narrower below than above. Neither are the lines quite straight. If the collimating lens is removed, a slit about o.i cm. wide across a white flame will also show (to the telescope or to the eye) fine, strong lines rotating in opposite direction to the slit, according as the transverse and longi- tudinal axes are differently placed. As has been already stated, it is with the latter condition that the focal plane in which the fringes lie varies enor- mously. Finally, when the sodium lines are superposed but the longitudinal axis of the spectra not quite so, a second class of fringes appear, which, however, are always more or less blurred. They rotate with great rapidity over 180 when one grating rotates over a small angle relatively to the other and the angle between the longitudinal axes of spectra passes through zero. In the latter position the regular fringes appear in full strength in the principal focus. To see the secondary interferences, the ocular must be drawn inward (toward the grating), and these fringes increase in size with the displacement of the ocular away from its position when regarding the principal focal plane. This secondary set of fringes is always accompanied by another very faint set, nearly normal to them and apparently quivering. The quiver may be due to parallax and the motion of the eye. These are probably the vestiges of the regular set of fringes, out of adjustment. 21. Homogeneous light. Wide slit. Transverse axes coincident. If there is no color-difference, fringes of the same kind will nevertheless be seen in the telescope, on widening the slit indefinitely. Path-difference is here due to differences of obliquity in the interfering rays. As in the preceding case, accurate adjustments of the longitudinal and transverse axes (in case of sodium, DI and D\ or D 2 and D' z coincide horizontally and vertically) of the homogeneous color-field are essential if strong fringes are to appear in the principal focus. These fringes are, as a rule, well marked, and widening the slit merely increases the width of the channeled, homogeneous field of view. If, owing to slight differences of grating space, the sodium lines are not quite superposed automatically, this may be corrected by rotating either grating, or else the apparatus as a whole, until the fringes are strongest. The REVERSED AND NON-REVERSED SPECTRA. 53 fringes may be made to vanish under inverse conditions. Table 4 shows their close relation to the preceding colored set, so far as motion of the micrometer is concerned. TABLE 4. Ives grating. 15,050 lines to inch, computed Se= io- 3 Xo.94 cm. per fringe. lo 3 Se=o.72 cm. (large fringes) 10*86=0.83 cm. (small fringes) .80 77 .80 Wallace grating. Wide slit. Coincident Na lines. Fringes in principal focus, very clear and strong. lo 3 3e= 1.16 cm. io 3 de=l.o6 cm. small f"ij J'JJ fringes Y*J| Wallace grating. Wide slit. Non-coincident Na lines. io 3 5e= 1.34 cm. io 3 Se= 1.38 cm. (small fringes) 1.28 (large fringes) 1.35 The fringes decrease in size as e increases and exhibit the same irregularity of 8e values, due, no doubt, to the same causes (equation 6). Moreover, 8e is here below the normally computed value, supposing the angle i to be negli- gible. In fact, figure 36 shows the optical center of the collimator C; so that 37 36 Ca and Ca' are the axes of parallel pencils, diffracted by the gratings G and G' at the angles 6 for Ca and e' for Ca'. The rays are subsequently condensed at F, the focus of the telescope, L being the principal plane of the objective. The general path-difference is thus, by equation (5), e(icos (0'-H"))/cos i, which distributes the fringes from right to left with variation of *. (small fringes) 54 THE INTERFEROMETRY OF If the grating G' is displaced de parallel to itself, however, the path-difference will again be increased by X whenever _ X cos i 1= i -cos (0'+*) Since i is small, this equation will not differ appreciably from equation (2), with which it coincides for the central fringes. If the sodium lines are not superposed, these fringes may still be seen, but they are not in the principal focal plane and the new focal plane changes con- tinually, as the fringes grow in size. Examples are given in table 4. The large values of be show that i was not actually negligible. Experiments similar to the above, bearing on the reason for the discrepancy (equation 6), were tried with the thin Wallace gratings, and the results are given in table 5. TABLE 5. Thin Wallace grating. * negative (within 10) 5eXio 3 =2.6o cm. 5eXio 3 =2.4O cm. Ocular in, 2.34 2.45 2-57 *=o, normal incidence, 8eX io'= 1.48 cm. SeX io 8 = 1.37 ( (small Ocular set for principal 1.50 1-37 '^fringes) focal plane, 1.37 1.32 [(large 1.19 \fringes) i positive (within 10) SeXio 3 =o.86 cm SeX io*= 0.96 Ocular out, .88 .86 cm 85 .87 .96 fringes) As before, the effect of i passing from negative (through zero) to positive values is enormous, 8e increasing nearly threefold for a change of * estimated as within 20. Here, however, the drawn-out ocular (towards the observer) corresponds to the small values of 8e, whereas above the reverse was the case. This depends upon which of the spaces D or D' is the greater. 22. Homogeneous light. Fine slit. Transverse axes not coincident. To obtain this group of interferences, the two sodium lines from a very fine slit are thrown slightly out of coincidence; i.e., by not more than the DiD 2 dis- tance. In the principal focal plane, therefore, these doublets are seen sharply, while if the ocular is drawn sufficiently forward or rearward, an interesting class of fringes soon appears which resemble Fresnel's fringes for two virtual slits. These fringes may be seen, however, on both sides of the focal plane and in- crease in size with the distance of the plane of observation (focus of ocular) in front or behind the principal focal plane. In figure 37 the two gratings, G and G', are struck by parallel pencils from the collimator at different angles of incidence (o and i ). The two diffracted pencils of parallel rays are caught by the objective L of the telescope and condensed at the principal foci, F and F' t appearing as two bright yellow lines. In front and behind the plane FF', therefore, are two regions of interference, I and /', throughout which the Fresnellian phenomenon may be seen in any plane parallel to FF', observed by the ocular. When the electric arc is used with a very fine REVERSED AND NON-REVERSED SPECTRA. 55 slit, these sodium fringes often appear at the same time as the colored fringes, and, though they are usually of different sizes, their lateral displacement with a change of distance apart of the gratings, 8e, is the same. The fringes in question appear alone when the sodium burner is used. They may then (at times) be observed with the naked eye, with or without a lens, and they fail to appear in the telescope unless the objective is strengthened by an additional lens. They are always vertical, but finer in proportion as the DiD 2 and D'iD'z doublets are moved farther apart. They become infinite in size, but still strong, when the doublets all but coincide, showing a ten- dency to become sinuous or possibly horizontal. Rotation of either grating G around an axis normal to itself and relative to the other produces greatly enhanced rotation of the fringes, as in all the above cases, but they soon become blurred. Only in the case when the horizontal axes of the field coincide (parallel rulings, etc.) do they appear strong. When the angle of incidence (or non- coincidence) is increased for both gratings, the size of the fringes increases; but when the e distance is increased by the micrometer, the fringes are appar- ently constant as to size. However, after displacement of 4 mm. they are liable to become irregular and stringy, though still moving. A fine slit is not essential, particularly when e is small. They vanish gradually when the slit is too wide. If a telescope with a strong objective is used, these fringes may be seen, retaining their constant size long after those of the next paragraph vanish. Examples of data are given in table 6, and 5e is too low in value as compared with the computed datum for i = o. With the Wallace gratings, these fringes were best produced by the aid of the sodium lines, in the ordi- nary electric arc, simultaneously with the colored fringes and for the case of a very fine slit. They were apparent both with an ocular drawn out or drawn in. In the former case several successive groups were observed. Beginning with the sharp sodium lines in principal focus (D z and D' 2 coincident), a slight displacement of the ocular outward showed the first group, this resembling a grid of very fine striations. Further displacement outward produced a second set, equally clear but larger. A third displacement of the ocular outward showed the third set, and these now coincided with, and moved at, the same rate as the colored fringes in the same field. Other groups could not be found. No doubt, for these four successive steps the interference grids of D\ and D\, D 2 and D's are coincident and superposed, until they finally find their place in the colored phenomenon. TABLE 6. Ives grating. Homogeneous light. Fine slit. Sodium lines not coincident. 5eXio 3 =o.87 cm. 23. Homogeneous light. Slit and collimator removed. Fringes similar to those seen with the wide slit above may be observed to better advantage by removing the slit altogether. The sodium flame is then visible as a whole; and if the adjustments are perfected it is intersected with strong, vertical black 56 THE INTERFEROMETRY OF lines, visible to the naked eye or through a lens or a suitably strengthened telescope. They decrease rapidly with increase in e, but vanish to the eye before the preceding set in paragraph 22. The sodium lines need not be in adjustment, but the longitudinal axes of the field must be, as usual. If diffuse white light is present, faint colored fringes may be seen at the same time. If the collimator only partly fills the field of view, these diffuse light fringes and the preceding set may occur together. Both rotate markedly for slight rotation of either grating in its own plane. There seems to be a double peri- odicity in the yellow field, but it is too vague to be discerned. When magni- fied with a lens, they admit of a play of e within about 0.6 cm. from contact. When the sodium lines are not coincident, the focal plane continually changes with e. Otherwise it remains fixed. Some data are given in table 7. TABLE 7. Ives grating. Homogeneous light. Collimator and slit removed. Focus continually changing. [ 5eXio 3 =o.95 cm. Large fringes; ocular out; lens on \ Ocular in, lens on Very small fringes, lens off j Wallace grating. Sodium lines coincident. Principal focal plane i.oi i-<>4 1.02 6eX io*= 1 .08 cm. 1.18 These data are similar to the above and subject to the same discrepancy whenever slight variation of the angle of incidence accidentally occurs. In figure 38 the case of three rays from a given flame-point F is shown corresponding to the ^ equation X cos * ~~ i cos (0-H) when i passes from positive to negative values. If either of the gratings is displaced and if they are parallel, the focal plane will not change; but if G and G r are not parallel, the focal plane differs from the principal plane and now moves with the grating. 24. Inferences. The above data show that the equation underlying all the interferences observed is the same. The interferences themselves may result from different causes, but their variation in consequence of the motion of the grating, 8e, is due to one and the same cause. This is best seen by producing them simultaneously in pairs. As a means of finding an accurate compari- son of the number of lines per centimeter on any grating, in comparison with those on the given grating, the method used in paragraph 20 deserves consideration. REVERSED AND NON-REVERSED SPECTRA. 57 If the fringes are to be used for practical purposes, great care must be taken to keep the angle of incidence of the impinging light constant. This was not done in the present paper, where the purpose is merely an identification of the phenomena. Moreover, a micrometer with the screw running easily is essen- tial, as otherwise the frame is liable to show appreciable twist (change of inci- dence) during displacement of the fringes. The fringes are not of the sensitive type, but they admit of a large range of displacement and are therefore adapted to special purposes. With regard to their bearing on the behavior of reversed spectra, for the interpretation of which the present experiments were undertaken, it is obvious that the interferences with homogeneous light and a wide slit (paragraph 21), or in the absence of a slit (paragraph 23), are of analogous origin in both cases. It makes no difference, therefore, whether one of the spectra is reversed or not, except, perhaps, that in the former case (inversion), the coincidence of longitudinal and transverse axes is a more insistent condition. The colored fringes of paragraph 20 obviously can not be produced with reversed spectra. There remain the fringes with the fine slit and homogeneous light (paragraph 22) ; in other words, the occurrence of a sort of generalized Fresnellian inter- ferences, within the telescope, modified by causes which lie outside of it. Thus DI and D\ or D 2 and D' t may be placed sufficiently close together to pro- duce a region of interference before and behind the principal plane in which the sodium lines are in focus. If the DiD'i lines are o.oi cm. apart and the fringes seen likewise at o.oi cm. apart, their position, measured from the principal plane, will be at or less than 2 cm. The ocular would then have to be displaced forward or rearward by this amount. But there are two sodium doublets, each pair of which is to interfere. Suppose that D 2 and D\ are in coincidence so that the 40 scheme is DI\ DzD'i: D' 2 , as in figure 39, where o is the principal plane of the objective and DI to D' 2 the principal focal plane. We should then have the separate regions of interference I and /' and the combined regions I" and /'". When the breadth of the latter is the whole number of fringes, the two pat- terns clearly merge into a single pattern. The experiments show several of these stages, terminating outermost in the focal plane of the colored fringes under the given conditions. Since the fringes lie on hyperbolic loci the problem 58 THE INTERFEROMETRY OF itself is beyond the present purposes; but it appears that the colored fringes will not appear until the corresponding DI and D 2 lines are shared by the whole of the two continuous spectra. The final question at issue is the bearing of the present Fresnel phenome- non on the reversed spectra. If in figure 40, 5 and s' represent the traces of two reversed spectra in the principal focal plane, superposed throughout their extent (i.e., in longitudinal coincidence), the rays a\a'iB, through the line of symmetry a, a', are at once in a condition to interfere with a given difference of phase; but so are all the symmetrically placed pairs of colors, c, c', b, b', of the two spectra (the distances cc r , bb', being arbitrary), provided the corre- sponding rays meet. As they do not meet in the principal focus, they can interfere only outside of this b and &' at B, c and c' at C, etc. Similar con- ditions must hold at B' and C' within the principal focal plane. The linear interference is thus successively transferred to different pairs of wave-lengths. The phenomena of this paper can not, therefore, be detected in case of reversed spectra, because in the principal focal plane different wave-lengths are every- where superposed, except at the narrow strip aa', which experiment shows to be about one-third of the width of the sodium doublet, in apparent size. Beyond the principal focus the corresponding conditions in turn hold for the rays at B, C, etc., B', C', etc. Hence there can not be any Fresnellian inter- ferences (paragraph 22), for there are not two virtual slits, but only a single one, as it were, and the interferences are laid off in depth along the normal C'C. The phenomenon may, in fact, be detected along this normal for 2 or 3 meters. 25. Rotation of colored fringes. Non-reversed spectra. When the slit is oblique, it effectively reproduces the wide slit, locally, and therefore does not destroy the colored fringes. At every elevation in the field the slit is neces- sarily linear, though not vertical. In figure 41, let the heavy lines, H, denote the colored fringes for a fine vertical slit and white light, showing nearly the same distance apart, throughout. Let the light lines, L, denote the fringes for a wide vertical slit and homogeneous light, X. These fringes are due to the successively increased or decreased obliquity of the rays in the horizontal plane. Now let acb be the image of the oblique slit in homogeneous light. It is thus merely an oblique strip, cut from the area of light lines or striations, as it were, and consists of an alternation of black and bright dot-like vertical elements in correspondence with the original striated field. We may suppose ab to have rotated around c, so that the vertical through c is its position on the colored field (white light and fine vertical slit). A color, X' (near the one X), corresponding to the field of the lines L in case of a wide slit and homogeneous light X', will supply nearly the same grid, so far as the distance apart of fringes is concerned. But the grid is displaced laterally, in consequence of the different angle of diffraction, 6. This is shown by the dotted lines D in figure 41, the effect being as if the slit had been dis- placed laterally. If the wide slit for homogeneous light X' is now narrowed and REVERSED AND NON-REVERSED SPECTRA. 59 inclined as before, an alternation of bright and dark elements will appear in the image of the slit ed, corresponding to X'. If we suppose that for white light and the fine vertical slit the position of the fringe (X') was at c f , we may again regard c' as an axis of rotation. To find the fringes such as ff, it is then only necessary to connect corresponding black elements on ab and ed. Their inclination is thus opposite to ab and ed, or they have rotated in a direction opposite to that of the slit. If, for instance, the slit image ab or ed is gradu- ally moved back to the vertical, the points g and h will move with great rapidity and in both directions toward infinity and the fringes // and /'/' become vertical lines through c and c', respectively. It is interesting to inquire into the frequency of fringes, n, when the angle of diffraction, 6, is changed. From the original equation e=n\/(i-cos 6), since d\/d6=D cos 6, the rate of change dn e i e dd D i-f-cos 6 D+V > 2 -X 2 where e is the distance apart of films and D the grating space. Since cos 6 varies but slowly with 6 and is additionally augmented by i, dn/dd is nearly constant and about equal to e/2D. 41 /^ 42 The fringes and slit images are thus given by the two sides of the parallelo- gram cgc'h for the two colors X and X'. The diagonal cc' represents dd; the diagonal gh has no signification. On the other hand, the normal distances apart, D' and D", of// and/'/ and ab and ed are both important. If D' and D" are the normal distances apart of the fringes and the slit images, respectively, B and B f , the two diagonals of the rectangle cgc'h, modi- fied for convenience in figure 42, >' =>"(cos . In such a case the diffracted rays are no longer parallel. If B' and B" are two angles of diffraction on the right and on the left, sin 0'+sin (0-^)=sin (0+ whence sin B ''-{-sin B' = 2 sin

is very small, are the rays appreciably parallel on rotating G'G" around O normal to the diagram; but this is usually the case, as

v = o, x' = o; if and T' 3 are also practically parallel. Hence in the case of symmetry and coincidence of MI and Ni the points 8', ft', 7, 8', a', and e are in the same phase (diffrac- tion). In other words, there is no path-difference between Y-\-X and Y'-\-X', whether the angle of incidence is zero or not (Yi+Xi and Y'i+X'i). The whole field in the telescope must therefore show the same illumination (homo- geneous light, wide slit) between the maximum brightness and complete darkness. Interference fringes can occur only when the opaque mirror, MI, is displaced parallel to itself out of the symmetrical position. If MI and Ni are symmetrical, as in figure 47, the displacement of G', fore and aft, parallel to itself, is without influence. This reduces the whole discussion to the normal displacements of the sys- tem G', Mi, NI, given in figure 48. Let the mirror Mi be displaced over a normal distance e m to the position M 3) Ni remaining in place. Then the image of G' will be at G\, at a perpendicular distance, e, from its original posi- 66 THE INTERFEROMETRY OF tion G'i. The path-difference so introduced, since a and b (ab normal to the ray V 2 impinging on M s at c and reflected to &) are in the same phase, is 26 m COS 5/2 and the displacement per fringe 1 will be 5* m = X 2 COS 5/2 which is nearly equal to X/2, as in most interferometers, remembering that e m and de m refer to the displacement of the mirror M\. Two interfering rays will be coincident at b. In the next place e and de may be reduced from the corresponding displace- ments e m and 8e m of the mirror M\. In figure 48 the figure Jdbe is approxi- mately a parallelogram with the acute angles 5/2. Hence, since 2 = ( 5+00/2 2e m cos enters equation (5) again, but only to a small extent. CHAPTER IV. THE DISTANCE BETWEEN TWO PARALLEL TRANSPARENT PLATES. 29. Introductory. The problem of finding the distance separating two parallel glass disks, as well as their degree of parallelism, is frequently one of practical importance. Thus, in my work on the repulsion of two such disks, it would enter fundamentally, and it has long been my intention to repeat that work with two half-silvered glass disks, for comparison with the case of metallic disks. It has since occurred to me that the method devised by my son, Mr. Maxwell Barus, and myself* would probably be ideal for the purpose, both for very small distances (within o.i mm.) as well as for distances ten or more times larger. This method admits of use of the film grating, and there are three types of interferences of successive orders of fineness, the first virtu- ally involving the colors of thin plates (resolved spectroscopically), the other two being dependent on diffraction. To measure the thickness of the air-space it would be necessary to count the number of fringes between two definite Fraunhofer lines only, supposing the constants of the grating to be given. 30. Apparatus. The apparatus has been designed for transmitted light, in preference, though the case of reflection is also available. fc 3 aj= 1 6 m 2 ) n\ = 2en cos B' m (3 ) n\ = 2efjL (cos r cos ff'm) where X is the wave-length of the color used, n the order of the interference, e the thickness of the sheet to be measured, and /* index of refraction, if i is the angle of incidence of the white light on the grating, r the angle of refrac- tion in the plate (/*), and 0' m the angle of diffraction of the wth order of spec- tra therein. If the sheet is an air-space, these equations become simplified, since M = i and r is replaced by i, d' m by 8 m , the angle of diffraction in air. Thus, since positive values are in question, n\ = 2e cos B m ( 6 ) n\ = 2) 2 (9) n\ = 2e(i cos m ) = 2e(i A/! if D is the grating space, and the interference in question is due to the grating spectrum of the wth order. The meaning of the equations (7), (8), and (9) is given in figure 52. The case of equation (7) may be seen in the direct white ray, figure 52 a, provided the light of the focussed slit-image is resolved by direct-vision spectroscope. For this purpose Mr. Ives's grating with attached direct grating prism may conveniently be placed in front of the telescope T, figure 50, focussed on the slit. After adjustment these fringes appear strong. Of course, H and G a b c \\ \\ OS \\ \V V \ & 52 must be parallel and all but touch. Under the same conditions the fringes may be seen laterally in any order of spectrum, as in figure 52 b. Figure 52 c illustrates equation (8) and figure 52 b equation (9). Figure 53, finally, illustrates the general case of incidence, i. The first and second orders of spectra are alone intense enough to pro- duce marked effects. In case oii Q, a double diffraction of the first order, 6' reinforces a single diffraction of the second order, 2 , since \/D = (sin 0'-sin 0) =sin 2 /2, (sin e'-\/D) = (2\/D)/2 or sin e' = z\/D Probably for this reason they are visible. The general case, equations (4), (5), and (6), is illustrated in figure 53, the rays /, I', and I" being incident, R reflected, and D diffracted. The retardations are ef and df, respectively. If the diffractions differ by a whole number of wave-lengths the total diffrac- tion is obtained. One would be tempted to resolve the case by aid of a wave- front ab, in which case the equations would be different; but they do not reproduce the phenomenon. 72 THE INTERFEROMETRY OF 32. Method. Suppose, now, two Fraunhofer lines, X and X' of the spectrum, are selected as the rays between which interference fringes are to be counted. Then, in case of equation (7), if n' is the number of interference rings between X and X', ( I0 ) n\ (n) n (12) 2 e In order to measure e, therefore, it is necessary to count the number of fringes n' between X and X', and e varies directly as n'. If the mean D and magnesium b lines be taken as limiting the range, io 6 X = 58.93 cm., io'X' = 5i.75 cm., Ci=io- 4 X4-2S; then n'= i io 3 e = 0.21 cm. = 10 = 2.1 = 100 = 21 etc. As it will not be convenient to count more than 100 lines ordinarily, the method is thus limited to air-spaces below 0.2 mm. and becomes more avail- able as the film is thinner. Of course, in case of plates which contain specks of dust or lint, or are not optically flat on their surfaces, it is extremely diffi- cult to get e down below 0.002 cm., so that ten fringes between D and b would require very careful preparation. If equation (8) is taken, X is to be increased to L = X/ cos m = X/Vi - (m\/DY where m is the order of the grating spectrum, whose rays interfere. Thus equations (n) and (12) now become, since nL = (n+n')L' = 2e (13) n = n'L/(L-L'} (14) 2e = n'LL'/(L-L')=C 2 n' If first order of diffractions are in question, ra = i, io 6 L = 59.11, io 6 L' = 52.33, '2=10^X4-20. Thus for = i io 3 e= 0.21 cm. 10 02.1 100 021. scarcely differing from the preceding case, so that one would not know in which series one is working. If the diffractions occur in the second order, m = 2, 6 2 .56 io 6 L' 2 = 54.15 C" 2 = 10^X4.03 thus again differing but slightly from the above. If we inquire into the condition of coincidence and opposition of these fringes, the following results appear: Let the spectrum distance between the G and 6 line be taken as unity, and let there be m and n, first-order fringes in this distance. Then in ~ is the difference of distance per fringe. Let REVERSED AND NON-REVERSED SPECTRA. 73 x be the number of long fringes, to restore the original coincident phase; i.e., let x longer fringes gain one long fringe on the x shorter fringes. Then that is, x fringes constitute a new period. From the above data It follows that the length of coincident strips is subject to or C- where C is the new constant. This would place the fringes beyond the coarse group below, but naturally C is enormously dependent on small errors in C\ and C 2 . - Finally, if equation (9) be taken, the X is to be increased to M = X/(i cos m )=X/(i V i in order that equations similar to the above may apply. Thus n'M MM' , '' M'-M ' M'-M In the diffractions of the first order of spectra m=i and io 3 M = 4.i5o, io 3 M' = 4-747 "'3 = 0.0330 These are the coarse order of fringes, so that n= i 20 = 0.033 10 0.33 100 3.3 , etc. Fringes are thus still strongly available, even if the distance apart of the plates is over 2 cm. If the diffractions are of a second order of spectra, m = 2, io 3 M=i.oi6 ioW=i.i6s C'"s=io-*X7.85 These fringes are therefore of intermediate order, since n= i 20 = 0.0078 cm. 10 0.0785 100 0.785 They would be enhanced, since they cooperate with the double diffractions of the first order. 33. Observations and corrections. Preliminary work. The following work was done merely with a view to testing the equations and with no attempt at accuracy. The grating was left unsilvered, so that the ruled sur- 74 THE INTERFEROMETRY OF faces confronted the half-silvered surface on ordinary plate glass. Conse- quently, the fine fringes were observed by transmitted light behind, and the medium and coarse fringes by reflected light in front. The micrometer was a good instrument for general purposes, but hardly equal to the present work, where the slightest rocking of the slide introduces annoyances. To count the number of fringes between D and b, since the fringes were not generally seen in the principal focal plane of the telescope, it was considered sufficient to rotate the cross-hair into an oblique position, until its ends ter- minated in the D and b lines, respectively, and then to count the number of fringes on running the eye down the wire from end to end. When there are many fringes, 25 to 50, the eye is apt to tire before reaching its destination, so that several counts must be made and the mean taken. / -01 The results are given as a whole in figure 54, where the distance between plates, measured in centimeters on the micrometer, beginning at an approxi- mate zero, is laid off horizontally and the number of fringes vertically, in case of each of the three series. The computed line e = Cn'/z is drawn in full and the observations laid off with regard to it. The zeros do not quite correspond, as very small distances here are significant. With the fine fringes I did not spend much time, as they are virtually colors of thin plates seen by diffrac- tion. The chief difficulty with these small distances is that the plates touch and a complete readjustment is necessary. After touching, the micrometer acts like a forcing screw and its reading is too low. This is the meaning of the data in the curves a and a', the latter with its horizontal scale magnified ten times. The object of this series is chiefly to locate the position of the line in relation to the other lines. REVERSED AND NON-REVERSED SPECTRA. 75 The observations for medium and fine fringes were made together, so that a single micrometer reading suffices. Beginning with very small distances apart, called zero, this was rapidly increased to nearly i cm. The fine fringes soon vanished, later the medium fringes vanished, finally (when e is several centimeters) the coarse fringes also vanish. The three together, therefore, cover with accuracy a relatively enormous range of displacement for measure- ments of this kind. The curves b and c show that the observations are not completely repro- duced by the line. Mean lines drawn through the observations indicate that the zeros do not correspond sufficiently for the two lines b and c to locate the common zero. This is inevitable, since the micrometer begins to count at a small distance as specified, which is otherwise arbitrary. In fact, it should be noticed, as an accessory property of this interferometry, that the two lines for finding the zero determine the absolute reading of the micrometer, mutu- ally, and" these readings are here 0.22 mm. too large. But even if the zeros were horizontally to coincide, the observations would not adequately conform to the computed lines. All that can be affirmed is that the angle between the observed and computed loci is about the same. The main reason for the divergence is referable to the fact that the air-space is not quite plane parallel, but slightly wedge-shaped, so that the effect of the angle of the wedge is superposed on the interferences. Any slight unsteadi- ness of the micrometer slide, for instance, would already introduce the wedge discrepancy, without necessarily interfering with the sharpness of visibility, while any attempt to readjust would destroy the continuity of measurement. There will also be many secondary reasons for divergence, as, for instance, the three separate focal planes in which the fringes lie and the fact that the glass plates which limit the air-space are themselves wedge-shaped; other, but fainter, fringes are marching through the spectrum, such cases as coincidence and opposition, for instance, as were CA pointed out above, etc. But the ade- quate reason for the discrepancies in this paper is the incidental change of the angle of incidence, i. If the film is wedge-shaped, very little disturbance results; but the cor- rection to the second order of small quantities is unfortunately somewhat cumbersome. Let the edge of the wedge of air be vertical and subtend a small angle, , figure 55, between the two faces A and B. Let I and /' be the two corresponding rays incident at the angle i at the first face and at the angle i-\-V at the second face, n and ri being the normals. Let e and e' be the con- secutive thicknesses of the air-plate, taken normal to B for convenience. Then the I rays R' will issue at the A face, after reflection, at an angle i+z very small, this becomes If in the first equation i is replaced by the angle of diffraction 6, the equation for the diffracted fringes, as far as

at the mirror M must be moved in compensation. Thus w 4 ^ w this is another method of displacement interferometry and ///, ^ \\ the usual equation suffices. ^ m, t in! y.' The following rough experiments were made: Placing , ~ the strong fringes in the center of the field (slit image), ^ the reading of the micrometer was taken. Then a thick glass plate, = 0.71 cm., was inserted in one beam, nearly normally, and the micrometer displace- ment, A/V, was found when the fringes were brought back to the center of the field again. The results were (for instance) 0.393 cm. The displacement equation is (p being the index of refraction of the plate) where the correction for dispersion may be put 2.B/X 2 = o.o26. Hence M = 1.50, 1.52, as was anticipated. On using white light, where there is but a single strand, a cross-hair, and greater care as to the normality of the plate compensator, etc., there is no reason why results of precision should not be obtained. 38. The same. The linear phenomenon. The occurrence of the linear phenomenon reciprocally with the fringes for homogeneous light is interesting. It usually appears when there is a flash of the arc lamp, i.e., a displacement of the crater, introducing white light into the sodium arc. It is thus undoubt- edly due to the reversed spectra for white light and may, in fact, be produced by using the white arc or sunlight in place of the sodium arc. When the mirror M is displaced on the micrometer parallel to itself, the linear pattern moves through the wide-slit image from right to left; or the reverse. It does so also when either mirror, M or N, is slightly rotated on a vertical axis. The change in appearance during this transfer is very striking. In the middle, between the extreme right and left positions, the linear phenomenon is excep- tionally strong and fairly tumbling in its mobility. Toward the right or left from the center it becomes gradually less intense, and on one side merges into the homogeneous striations which then appear. On the other side it seems merely to vanish. Doubtless the linear phenomenon is found, as usual, at the line of symmetry of two reversed spectra; but, as both spectra are shrunk to very small lateral dimensions, many colors probably adequately coincide. In an achromatic reproduction of the slit all colors will coincide. It is thus not necessary that the edges of the slit images should be super- posed to produce the linear phenomenon. What is still more curious is the 86 THE INTERFEROMETRY OF result that not even the longitudinal axes of the spectra need be quite in coin- cidence, though, of course, the phenomenon appears most intensely for the case of precise superposition. The angle of admissible separation of longitu- dinal axes is, however, much larger here than in the usual cases above, so that one of the longitudinal guide-lines of the two spectra may be appreciably above the other. The last result and the fact that the linear phenomenon appears here with an indefinitely wide slit are new features. The cause of the latter has just been referred to the exceptionally reduced width of spectrum resulting from the double diffraction. If the dispersion were quite reduced to zero, all colors in a definite narrow, transverse strip of the white slit image would be in a condi- tion to interfere. This strip contains the superposed images of an indefinitely fine slit. The slit in any other position, right or left, would have two non- coincident images. Hence, when one wide-slit image moves over the other, there is also a shift of the linear phenomena. To produce the linear phenomenon with sunlight is difficult. The inter- ferences should first be produced with the sodium arc, strongly, and the arc thereafter replaced with sunlight entering the slit at the same angle. Further- more, the pencil leaving the collimator should be a narrow, vertical blade of light, and at the mirrors, M and N, red and green light should be screened off, retaining only a narrow strip of yellow light for each. Finally, to avoid glare, the slit is not to be too broad nor too narrow to cut off the yellow field of the telescope. Under these circumstances of completed adjustment, the linear phenomenon usually appears strongly. Its form may be greatly modified by rotation of either mirror, M or N, micrometrically, around the vertical axis, as already suggested. The types are given in figure 62, quite fine, nearly vertical lines, q, changing to moving, coarser forms, m, and these into the tumbling variety, t, very coarse and nearly horizontal. The latter change by rotation and dimi- nution into m' and q', while N is being continually rotated over a very small angle, sliding one slit image continuously over the other. In the condition t, the fringes rotate with astonishing rapidity, and this rotation is nearly 180; i.e., if the angle between m and m' is a, the angle of rotation has been 180- a, so that between q and q' there is about 180 of rotation. At the stage /, with fine micrometric adjustment, the fringes may be made quite horizontal, and they are then relatively large and square, or at times shaped like blunt arrow- heads. This rapid rotation of fringes near t accounts for their turbulence, since tremors have the effect not merely of raising and lowering them, but also of producing the rotary motion in question. They may also be rotated, of course, on slightly tilting the grating about an axis normal to its face. Rotating the latter on an axis parallel to its face places the phenomenon in different parts of the superposed yellow field. Since a preponderance of yellow homogeneous light is present in the whole of the superposed wide-slit images in the telescope, it is not difficult to suggest the cause for the variations of the interference pattern when one image passes REVERSED AND NON-REVERSED SPECTRA. 87 horizontally over the other. The forms, t, correspond to minimum path- difference, remembering that in accordance with figure 59 all rays pass the plate of the grating twice. Further experiments were made with sunlight to detect the changes which befall the phenomena in different focal planes. The ocular of the telescope was gradually drawn out from an inner extreme position to an outer extreme position, through the normal position for principal focal plane. In this case a variation of form corresponding closely to figure 62 was also observed. The characteristic feature, however, was the prevalence of arrow-head or caret- shaped lines, both in the case of the extremely fine striations and of the coarser nodules. In the former case these roof -like designs were closely packed from end to end of the phenomenon and usually pointed upward. They recall the top edges of extremely eccentric ellipses in displacement interferometry, and in view of their lateral motion with the micrometer M and the decreased dis- persion due to double diffraction, their origin may be similar. 39. The same. Inferences. When the pencils, Mm and Nn, figure 59, are parallel and sodium light is used, the whole field is uniformly striated, whether the striations are made fine or coarse. I have found it impossible, on placing plate compensators (0.5 to 1.5 cm.) in both beams and rotating these to any degree whatever, to produce any suggestion of a secondary periodicity in the field. The fringes for a thick compensator, slightly wedge-shaped, merely become a little finer. Films of mica are liable to blur the field. In general, moreover, reflections would be relatively weak and thus inappreciable. They would require a separate adjustment for coincidence and not appear with the principal phenomenon. Hence the strands of interferences obtained in case of crossed rays are in a measure unique. The second periodicity is not stationary, but a part of the phenomenon. The glass plate of the grating produces an effect in virtue of its thickness, precisely as in the case of the dis- placement interferometry of my earlier papers. Experiments made with polarized light proved to be entirely negative. The phenomenon appears between a polarizer and an analyzer so long as sufficient light is present to ex- M hibit it. Observation with a ty 71 nicol, in the absence of the polar- izer, showed nothing but the ob- - / vious effect of reflection. & a 63 The occurrence of these par- allel strands for crossed rays and homogeneous light is thus diffi- cult to explain. I have tried a great variety of methods of superposing special interferences, etc., to produce the nodules with parallel rays, mM and nN, or to break them with crossed rays, mN and nM, without avail. There is no focal plane effect, nor any polarization effect. It is therefore necessary to confront the case at its face value, as in figure 63. Here S gg THE INTERFEROMETRY OF and 5' are the traces of two longitudinally coincident reversed spectra, drawn apart for distinction, the region of the D lines only being used. The light is homogeneous to this extent and the slit wide, so that there is oblique inci- dence. Then every point of 5 should (on adjustment) interfere with every point of 5', the result showing a uniformly striated field in the telescope. This is emphatically the case for the parallel rays, mM, nN; but with the crossed rays, mN, nM, the interference is confined to the rays in the equi- distant positions, n, in figure 63, and midway between them the field is a neutral yellow. In other words, between the rays n the rays are displaced, as shown by the arrows, recalling the arrangement of nodes in acoustics. Corresponding rays a and a' (for instance) do not coincide and hence can not interfere, the region aa r remaining neutral. In figure 64 the rays crossing at c (fig. 57) have been shown for three nodes and the transverse arrows indi- cate the directions in which the rays have been urged laterally. Naturally, I am merely stating the case as immediately suggested by the results. One may argue that there may be a secondary periodicity in the grating. But why 65 does it not appear at all in the case of parallel pencils, when it is so obtrusive in the case of crossed pencils of rays? Again, the interferences are unquestion- ably due to Di and L> 2 light, simultaneously. If the grids for these two wave- lengths should be at a slightly different angle to each other, their superposition would give something like the observed phenomenon, apart from details. Thus in figure 65 the two grids due to D\ and D 2 , intersecting at a small angle, may be interpreted as appearing strand or cord like at N, and neutral at / and I'. With white light the linear phenomenon would eventually become achromatic. But, again, why should lines so close together as DI and D 2 show any appre- ciable difference of angle or rotational phase-difference in their interference pattern? Intersecting grids, moreover, can be produced by other methods and nearly always betray their origin. The final inference is that suggested by figures 63 and 64, that homogeneous rays on crossing (here in a medium of plate glass) may exert a lateral influence on each other, to the effect that identical rays emerging from the crossing are arranged in equidistant nodal planes according to figure 63. 40. Experiments. Reflecting grating. Crossed rays. In the preceding experiments the remarkable phenomenon of double interferences was ob- tained with glass-plate apparatus. It is improbable that any secondary inter- ference can have been produced by the presence of reflected light, since the reflected pencils will be weak as compared with the primary pencils and REVERSED AND NON-REVERSED SPECTRA. 89 differently situated. It is nevertheless necessary to forestall all misgivings by avoiding glass plates altogether and adapting the methods of figure 57, where reflecting surfaces (front faces) only are present, to the experiment for crossed rays mN and nM. In the apparatus as finally perfected, G, figure 57, was a Michelson plate grating and G' a Rowland concave grating, each with about the same grat- ing constant. A strong lens was placed at T for observation at the focus of the concave mirror of G'. The latter was capable of fore-and-aft motion, of rotation about a vertical axis in its own plane and about an axis normal to that plane ; G was capable of rotation about a horizontal axis parallel to its plane. Thus the possibility of fore-and-aft motion and the three cardinal rotations for the gratings, together with a micrometric fore-and-aft motion of M , was at hand, as well as the rotation of M and N about horizontal and vertical axes. The interferences were found after establishing the coincidence of the yellow homogeneous fields, in the manner described in the preceding paragraph. The fringes were at first small and apparently single, but they could be enlarged at pleasure and the two definite systems separated by fore-and-aft motion of G'. They occupied only a part of the wide yellow slit image, the sodium arc being used. On actuating the micrometer at M there was dis- placement of the interference pattern as a whole, so that the conditions of displacement interferometry are here also implied, though the equations are liable to be different. On rotating M, micrometrically, about a vertical axis, the structure of the interference reticulations changed and was at times reduced to a single set. Whenever the arc flashed, or when white light was used, the linear phenom- enon appeared alone, either cross-hatched or longitudinal, depending upon the character of the reticulated pattern for homogeneous light. With sun- light, even after narrowing the blade from the collimator and screening off red and green light, the phenomenon was faint and hard to find, unless it was produced alternately with sodium arc. With the arc freshly charged with sodium, but a single set of interferences or else the linear phenomenon appears, since the broadened sodium lines are equivalent to a continuous spectrum in this region. Not until the excess of sodium has all been evaporated and the sodium lines are normal does the true reticulation show itself. It is interesting to describe two cases of this double- interference pattern, obtained by gradual and successive fore-and-aft motion of the grating G', between limits, while the edges of the two wide-slit images, respectively horizontal and vertical, are kept in contact throughout. Suppose the original fine fringes to be nearly vertical; then the apparently simple fringes, a, figure 66 (their appearance, however, would lead one to sus- pect their simplicity), change to the cord-like strands b, appearing like helices of a very large pitch. Both interference fringes are still nearly parallel, and they cover the whole wide-slit image uniformly. These eventually pass into the square or rectangular reticulation, c, with both systems equally strong. Probably intermediate forms have here been skipped. The system, e, occurs 90 THE INTERFEROMETRY OF very soon afterward, in which the difference in size of fringes has become enormous. Following e, the procession is reversed in g, h. Both systems (a and systems, say) have passed through maxima, but not at the same time, or not for the same fore-and-aft adjustment. Both sys- tems have rotated, the rotation being very rapid near the maximum. The reticulations quiver and look precisely like capillary waves in a rectangular trough of mer- cury, except that they are usu- ally at an angle to the bound- ing edges of the superposed slit images. In this quivering system of two identically strong fringes it is difficult to make out the rotations, but after consider- able revision the sequence in figure 67 was definitely ascer- tained. Beginning with the extreme fore-and-aft position of G', and moving it successively forward in steps of i or 2 millimeters, the apparently single grid, i changes to 2, where the two systems a and /3 can be disentangled, a expanding and rotating more rapidly, so that 3 and 4 follow. Here a is horizontal and probably of maxi- mum size, /3 is still nearly vertical and but slightly expanded. Therefore, while the a-effect wanes the /3-effect waxes, and the squared or orthogonal type, 5, is produced. The lines are here equally strong and it is the symmetrical figure of the series. Thereafter in 5, 6, 7, 8, and 9 the chief expansion and rotation is transferred to the /3 system, with which the a system has changed functions. Hence both systems rotate nearly 180 in the same direction and pass through maximum size; but the maximum is retarded in rotational phase for one as compared with the other. Rotation and growth are accelerated near the maximum. The total displacement of the grating G' between the cases i and g (fig. 67) was about 2 cm. ; but this depends upon the obliquity of the grating and incidental conditions, as explained above. Suppose, in the second place, that the original fringes, i, figure 67, were nearly horizontal; in such a case the evolution is much the same, but the sym- metrical form number 5 becomes smaller and more and more flatly rhomboidal horizontally. Probably the scheme of rotation is the same, but is much harder to ascertain in view of the flat forms. On the other hand, the field now abounds in vertical strands of interferences, like those of the preceding paragraph, and nodules are often in evidence, as before. If the original lines are quite vertical, they do not seem to rotate with fore- and-aft motion of G', but form intersecting, vertical, apparently simple sys- tems throughout the motion. Slight departure from the vertical produces rhomboids very long vertically and often very coarse. REVERSED AND NON-REVERSED SPECTRA. 91 41. The same. Compensators. A compensator of ordinary plate glass, at the intersection c, figure 57, produces no effect, if symmetrical to both beams. If not symmetrical, the interferences are displaced to right or left in the field of the telescope, as in any case of displacement interferometry, depending on which component beam receives the longer glass-path. Thus this adjustment corresponds to the grating in the preceding paragraph, the difference being that in the latter case the same ruling is used for both diffractions. Hence the interference figures obtained are simpler, showing vertical strands only. In the present case strands occur in all directions. The maxima for oblique positions of the glass plate were not found with reflecting gratings. If the compensator is within i inch in thickness, its introduction occasions no difficulty. The interference pattern may be changed, but it remains the same during the rotation of the compensator; but if the latter is thicker than 2 inches, the figure is usually so small as to be found with difficulty, unless the grating G' is brought forward, to allow for the mutually inward refraction of the rays. If this is done, the same figure may be reproduced. On advancing the grating, plate compensators much over 3 inches thick were tested without the slightest annoyance. Lenticular compensators require special adjust- ment and are very difficult of use. The effects of rotating the grating about the three cardinal axes have all been considered above. In the present instance two sets of fringes are sym- metrically rotated, subject to the same conditions. Rotation of G' around a horizontal axis requires an elevation or depression of the arc lamp, if the fringes are to remain in the field. Rotation around a vertical axis separates the slit images, and a readjustment for superposition is necessary. Results so obtained are therefore complicated and were not studied. 42. Miscellaneous experiments. Fringes with mercury light. A few random experiments made with the sodium arc, in the presence and absence of the magnetic field, showed no results; nor was this to be expected, as a reasonably strong field would blow out the arc. Again, the insertion of a glass compensator, 0.7 cm. thick, in one of the component beams, developed no maximum on rotating the compensator about a vertical axis. Thus with reflect- ing gratings the peculiar behavior of the transmitting grating, showing a maxi- mum on either side of a symmetrical minimum (36, 37), is not reproduced. The effect of rotating the first reflecting grating G on a vertical axis is only to throw the sodium light out of one side or the other of the (superposed) slit images. No available means of enlarging the fringes indefinitely was found. It is probable that this would require fine adjustment for symmetry. The field of interference, as a whole, is within a spot-like area which may be moved up and down, or right and left, by the vertical and horizontal adjust- ment screws on the mirror M. Coincidence at the two sides of the slit favor different interferences. The case is always as if , at a single point of the field only, there were actual coincidence, and that the interference pattern is grouped closely around it. 92 THE INTERFEROMETRY OF With the use of an ordinary glass mercury lamp (27 storage cells, 5 amperes) the fringes are found with difficulty when the beam at the first grating is wide. On using a vertical blade of light the definition was improved. The fringes are faint, very susceptible to motion, and at times even absent. They occur, however, as a single set, as was anticipated, showing that the above duplicated fringes are actually due to the two sodium lines. The mercury fringes are easily rotated and pass through a horizontal maximum with fore- and-aft motion. Rotating G about a normal axis may further increase this maximum size to a limit at which the fringes appear irregular or sinuous. A displacement of the mirror M over 0.7 cm. was easily permissible, without destroying the fringes. They occur, as above stated, within a certain adjusted spot area of the field of view. An attempt was again made to detect a Zeeman effect by placing the poles of an electromagnet on the two sides of the lamp; but here again no difference was discernible on opening and closing the electric circuit. The field, however, for incidental reasons, could not be made strong enough for a critical experiment. 43. Inferences. After these experiments (made with the apparatus figure 57, free from glass plates and depending on reflections only) the cause of the phenomenon is no longer obscure. Obviously one of the paired grids in figure 66 or 67 belongs to each sodium line. The retardation of one phenomenon, rotationally, as compared with the other, is due to the difference in wave- length between Di and D z . The phase-difference between numbers 4 and 6 (fig. 67) is thus equivalent to 6 Angstrom units. If the displacement of G' is about 0.3 cm., there should be about 0.5 mm. displacement, fore and aft, for i Angstrom unit. If the grating, G', is on a micrometer, this should be a fairly sensitive method of detecting small differences of wave-length, or give evidence of doublets lying close together. The sensitiveness clearly increases with the length of path of the component rays and may thus be increased. With this definite understanding of the phenomenon, it is desirable to deduce the equations, which in the occurrence of parallel rays would not differ essen- tially from those of Chapter II or III. It is useful, however, to treat the new case of crossed rays. In figure 69 the angles of diffraction are 0i and 2 , if the incidence of light, L, is normal at G and at an angle iz at G' t G and G' being parallel. The mirrors are set symmetrically at angles ffi and 0- 2 to the normal in question, and the diffracted rays are reflected at angles 0:1/2 and 2 /2, respectively. The reflected rays cross the normal at an angle . Then sin 0i = \/D i sin 2 = \/D 2 - sin i t where DI and D 2 are the grating constants. From the figure l/2 = 0i+ intersection, and consequently be separately diffracted into T\ and T\. Hence DI and D 2 are differently circumstanced in relation to the fore- and-aft motion, and the rotation produced will thus be advanced in one case, as compared with the other, for the reason discussed in Chapter III, paragraph 26. It is also clear that the difference of phase in the two rotations will be greater, as the total path of rays between G and G' is greater, so that the large distances used in the present experiments (nearly 3 meters) account for the astonishing sensitiveness of the phase of rotation to the wave-length difference. In fact, D 2 will be in the same phase as DI, if the grating is moved forward from G' to g', figure 68, since in both cases the rays intersect in the normal. Hence if R is the total path GmNG\ and if the angle of dispersion between DI and D 2 is dd, 2 the angle of diffraction at G', and h the displace- ment from G' to g', D the grating space, 69 = hsin 2 and the resolving power , fl fcx *'*DR d\ Dh X R cos 2 RI/ Z)2_x 2 In the given adjustment, roughly, dQ = 3.'jXio~ 4 R = 3oocm. 2 = 2o c whence - _ 0.34 94 REVERSED AND NON-REVERSED SPECTRA. As the resolving power is, roughly, h/R, and if h = 0.003 cm. is still appreciable, 3Xio 2 i.e., lines i/ioo of the distance apart of the sodium lines should be rotationally separated. Again, the displacement, fore and aft, between like rotational phases of Di and Dz should be about 3 mm., and this agrees fairly well with the order of values found. The case of the transmitting grating (fig. 59) is thus also elucidated, though it is not clear to me why the duplication of fringes is so efficiently concealed in the nodular forms observed. The reason for the minimum of size, for the symmetrical position i = o, and the two maxima for oblique positions of the grating (^=20 about), suggests an explanation similar to that given in. Chapter II. In other words, in the oblique position the short path-length is compensated by the increased thickness resulting from the greater obliquity of grating, whereas the long path-rays traverse the plate of the grating more nearly normally. In this way the path-difference is reduced as compared with the symmetrical position, and the fringes are therefore larger. The oblique grating acts as a compensator in both of the component beams, and the fringes may be visible, even if in the original position (fig. 59) they are all but invisible. If, however, the apparatus (fig. 57) is used with a plate-glass com- pensator symmetrical at c, there are no maxima or minima for any obliquity. Hence the tentative explanation for the case of figure 59 is not warranted. The fore-and-aft motion of the plate grating (fig. 59) produces no effect, since the rays are reflected back so as to retrace their paths. They are also reflected between parallel mirrors N, m and n, M. Thus the path-difference is not modified. The result is merely a decrease of the distance M, N, and a corresponding increase of m, n, and vice versa. The marked effects produced by rotating the transmitting grating around a normal axis, finally, follow the explanations given for the rotation of fringes of non-reversed spectra in Chapter III, paragraphs 25 and 26. In conclusion, an interesting application of the apparatus (fig. 56) or the other similar types may be suggested. By half-silvering the mirrors and pro- viding a similar opaque set beyond them, there should be no difficulty (in the case of homogeneous light) of bringing the interferences due to crossed rays, c, and to parallel rays, a'b', into the field of the telescope together. Strictly homogeneous light (mercury arc) would be needed to obviate the duplication of the sodium arc. In such a case, therefore, the parallel fringes could be used after the manner of a vernier on the crossed fringes, with a view to a repetition of the experiment of Michelson and Morley, if this experiment had not been so thoroughly carried out by the original investigators. However, the plan would be to rotate the apparatus, as a whole, so that the two crossed rays would be alternately in and at right angles to the earth's motion, whereas the two parallel rays would preserve the same relation to that motion. Naturally, the parallel and crossed paths would in such a case have to be lengthened by multiple reflections. CHAPTER VI. CHANNELED SPECTRA OCCURRING IN CONNECTION WITH THE DIFFRAC- TIONS OF REFLECTING GRATINGS. 44. Introductory. Throughout the preceding work I had noticed that the spectrum due to either of the component beams, after successive reflection from two reflecting gratings, was often regularly furrowed by transverse black bands, before the two spectra were brought to interfere. As these fringes are stationary, they do not modify the phenomenon investigated; but questions now arise as to whence these reflected fringes of a single beam come. They are not strong, as a rule, and I was therefore inclined to attribute them to some imperfection of the silvering of the opaque mirrors, but this proved not to be the case, so that it seemed worth while to examine them by special experiments. 45. Apparatus. The apparatus for this purpose, as one beam only is wanted, is quite simple. In figure 70, L is a vertical blade of parallel rays of white light from a collimator and slit. These rays impinge on the plane grat- ing G, whence the orders 1,2, 3 , etc. , of spectra are reflected . Either of these pencils maybe received by the second grat- ing G', plane or concave, from which spectra of any order are available. If o denotes the reflected pencils, the groups from two gratings may be dis- tinguished as (3, i), (3, o), (3. -i), (3. -2), etc., as in JQ the figure. Any of these two different pencils is to be examined at T by a lens or telescope, for instance, and the latter (with strengthened objective where needed) is more convenient, even when the concave grating is used. A wide slit 5, revolvable about G, is often useful for screening off spectra or parts of spectra. In some experi- ments the grating G' may be replaced by an opaque mirror. The gratings are provided with the usual adjustments for parallelism of rulings and slit. G' and T must be capable of considerable right-and-left motion, and G, in particular, of controllable fore-and-aft motion. 46. Scattering. An interesting result of this work is the evidence and spectroscopic quality of scattered rays, incidentally encountered. For instance in figure 70, if the slit 5 is narrow, it cuts off all the rays but the orange yellow of the third order, and the reflected spectra (3, i), (3, -i), etc., will largely 95 96 THE INTERFEROMETRY OF consist of orange-yellow light. Associated with each of these reddish-yellow patches, however, are vividly violet-blue patches, each separated from the reddish yellow by an almost total absence of green, relatively speaking. If the light is very intense, the connecting part of the spectrum also appears, but it is always far less vivid than the ends of the spectrum in question. Inasmuch as all violet radiation proper has been screened off at 5, it is obvious that violet light must have been scattered in all directions from G, a part of which, therefore, passes the slit and is resolved by the second grating G'. Moreover, as the scattering lines of the grating are equidistant, the scattered light has a regular wave-front. (Cf. Cam. Inst. Wash. Pub. No. 229, 1915, pp. 100102.) The correlative experiment of detecting the reddish light transmitted after scattering was also tested. For this purpose the reflecting grating G may be replaced by a transmitting grating, slit 5 placed beyond, and the light then analyzed by a second grating G' behind the slit and diffracting toward it on one side. But no results of value were obtained. 47. Fringes with white light. The experiments with the apparatus (fig. 70) were commenced with sunlight and (what is essential) a fine slit. Fringes are found in all combinations of doubly diffracted pencils (3, +i), (3, i), (3, -2), etc.; (2, i), (2, -i), etc.; (i, -i), (i, i), etc., but none in the reflected pencils (3, o), (2, o), (i, o), etc., as a rule. Whether the grating G' be concave or plane, it is best to use a telescope at T, because (when provided at the objective with an auxiliary concave or a convex lens) it more easily offers a wide range of observation along its axis than an ocular. The latter must be wide and has to be shifted bodily; but both methods were used. A concave grating at G and plane grating at G' gave no results. The concave grating is usually more free from channeled spectra. Of the great variety of fringes obtained, I shall give only two typical examples. The second order of spectra for G (plane) was separated from the others by the slit S and diffracted into G' (fig. 70). The successive fringes appear as the ocular is drawn outward from the principal focus. Combination 2, G'o : Only a good sodium doublet, which became washed on drawing out the ocular of T, was obtained; no fringes appeared. Combination 2, G' i: Just outward from the principal focus a large, coarse, irregular set of fringes appeared; next (ocular farther out) a large regular set, somewhat diffuse, possibly double and superposed; then a finer, half-size, very regular set, possibly decreasing. After this the mottled sur- faces of the gratings were successively in focus. A weak spectacle lens was now added to the objective of T, whereupon very large regular fringes were seen when the ocular was far out. Combination 2, G- 2: The ocular moving outward from the principal focus, the fringes seen in succession were as follows: large, regular, vague; half-size sharp; surfaces vertically striated; (lens on) fine regular set in red; doubled regular set in green. REVERSED AND NON-REVERSED SPECTRA. 97 Combination 2, G 3 : Fine set just before the surfaces appeared, which were delicately striated; fine regular set; coarse set, both close to surface; (with lens on) fine regular set; doubled, strong regular set. Different distances between G and G' had very little influence on the size of the phenomena. A few examples may be given, which are observed when the ocular is moved outward. 6*3, Gi : Distance 10 cm. Fringes, faint regular; strong irregular; faint regular; flat field; surfaces visible; faint regular. Distance 25 cm. Strong irregular; faint regular; small regular; large (double) irregular; lines slit into fine fringes; large faint regular. Distance 46 cm. Large strong, with two absorption bands; fine regular; double-sized faint ; surfaces with fine striations ; alternations of fine and coarse lines; faint, regular, large, etc. Fringes of different color are often in different focal planes. When a lens is used with the concave grating, observations must sometimes be made 2 meters off to get the large regular fringes. Red fringes may be narrower than the corresponding violet set. If the grating G is moved fore and aft, parallel to itself, the fringes are shifted across the stationary sodium line, as in displacement interferometry. Whereas in the positive combination (3, i), (3, 2), etc., the spectra widen, they tend to close up for the negative combinations (3, i), (3, 2), etc. With two identical plate gratings they may image the white slit. But this seems to have little effect on the fringes seen as a whole when the ocular is out of focus. When white light is used and the grating G' replaced by an opaque mirror, or in case of combinations which involve direct reflection (2, o; 3, o; etc.) at G', there seem to be no fringes. 48. Fringes with sodium light. While there is some difficulty in obtaining the fringes with white light, fringes with homogeneous light are obtained at once, provided the light is sufficiently intense. A sodium arc lamp, or a mercury lamp, with a fine slit, must therefore be used. In this case, moreover, the grating G' may often be replaced by an opaque mirror, or the fringes of the order 2 G'o, GzGo, etc., may be produced with entire success. On moving G fore and aft, they again travel across the sodium line. Often, in fact, two sets of fringes seem to be shifted. A few examples again may be given of the great variety in this display while the ocular is being drawn out : Gi,G' 2~. Sodium lines D\D Z single size; large strong fringes, lines split. Gi, G' i : Closed spectrum; striations continuous. Gi, G'o: Reflection; D\Dz single size; surfaces of gratings finely striated. Gi, G'i: DJDz double size; strong grid seen very near the surface of G'. Gi, G'i : DiDz treble size, out of reach. G2, G'o: Reflection; no fringes. 98 THE INTERFEROMETRY OF G2,G'i: Distance 12 cm. Coarse irregular; with lens fine regular set, near and beyond the surfaces. 2 , G'i: Distance 45 cm. Surfaces with doubled fine striations; with lens finally strong and regular. 2, G'i: Distance 60 cm. Regular faint; irregular double, very strong; surfaces striated; with lens strong double irregular; finally regular small. G$, G'I: Regular; regular line split; irregular coarse; surfaces finely striated, G coarser; fringes grow continually larger without vanishing. On moving G fore and aft, two grids seem to travel through each other in opposite directions. This probably accounts for the occurrence of irregular fringes. The size of fringes seems to be a minimum for a conjugate focus near the surfaces. The whole phenomenon is continuous. Irregular fringes, probably superpositions, become regular in other focal planes. 3, G'z: About the same; minimum size at the surfaces, increasing about three times as the ocular is drawn either way. 3, G'o; also 3, mirror: About the same results, only brighter and better. Hence in case of large dispersion two gratings are not needed. The two sodium lines, when the ocular is drawn out of focus, multiply themselves at regular intervals, so that the grids are sometimes distinct, sometimes partially superposed. Thus the classic diffraction phenomena of a slit suggest them- selves as the starting-point for an explanation of the present phenomena as a whole. 3. G'o, produced alternately with sodium light and sunlight, showed the same sequence of fringes (the large ones with a tendency to split) in the former case, while nothing appeared in the case of white light. 49. Grating on a spectrometer. It seemed necessary, therefore, to con- sider the diffraction of a fine slit, when seen in the telescope, somewhat in detail. In Chapter III the production of beautiful Fresnellian interferences from two identical slit images and homogeneous light was demonstrated; but an equally clear manifestation of the diffraction of a slit image, when the ocular is out of focus, does not seem to occur. The broad image of the slit out of focus shows a stringy structure only, but no separation is easily obtain- able. Fringes, as such, are quite absent when the ocular is drawn out. The light of the sodium arc was now passed through a very fine slit and collimator and reflected from a plate grating. The above intermittently regular and irregular fringes were strikingly obtained with the ocular out of focus. As this is successively more and more drawn out, fine lines become coarser, and then seem to subdivide, giving the structure a fluted appear- ance, frequently regular. There is, in other words, a double periodicity. In REVERSED AND NON-REVERSED SPECTRA. 99 the case of highly diffracting grating (D== lo^Xiys), the results appear best in the second order. The same beautifully duplicated fringes were obtained with a transmitting film grating of about the same dispersion, particularly well in the first order. The sodium flame gives too little light for the present purposes, but the phenomenon is seen. Believing that some irregularity might be introduced by the double-sodium line, I installed a mercury lamp for comparison. In the first experiment a film grating (D= 173 X io~*) was used, the ocular traveling outward from the principal focus. Both the green and the double yellow mercury lines enlarged and showed fringes of increasing size and number together. The green field had a darker band, the yellow a bright band in the middle. As the fringes enlarged, each split up into secondary fringes, 4 or 5 eventually, and this again occurred for both green and yellow fields. Rotating the grating around a vertical axis seemed to shift the primary fringes laterally over the stationary secondary fringes. A concave lens for positions anterior to the principal focus and a convex lens for posterior positions (toward the eye) were successively added to increase the range of observa- tion. On both sides of the principal focal plane (fig. 71) fringes occur, which enlarge with the dis- tance x from that plane. As they enlarge, each fringe splits up into secondary fringes, which in turn enlarge. Sometimes the arrangement is irregular. Green and yellow fields may overlap, but they do not do so conformably. The undeviated ray, however fine the slit may be, merely shows a stringy field, sometimes suggesting structure, but never showing clear-cut fringes. The same kind of results were obtained with a reflecting grating of about the same dispersive power. In the second order the fringes were particularly clear and regular. Primary fringes, finally, carried three to four secondary fringes each. Next, a ruled transmitting grating of less dispersive power (grating constant 352 X icr 6 cm.) was adjusted for mercury light. Here in the undeviated ray and in the first order no clearly separated fringes were obtained. In the second and third orders, however, they were very perfect, and followed the above rules, showing sharp secondary fringes. It follows, therefore, that a certain degree of dispersion is needed to resolve the fringes, which is inadequate in amount in the order zero, in this case, and scarcely so in the first order. In the higher orders the conditions are met. Using a very fine slit, however, I later just succeeded in separating the fringes in the first order. Finally, I returned to the endeavor of detecting diffraction fringes in the undeviated image, using a micrometer slit, a good achromatic lens (or no lens), and a distant (2 meters), moderately strong telescope. In this case separated and distinct diffraction fringes, white throughout, were undoubtedly obtained. They moved with the eye so as rarely to be stationary and in the same direc- 100 THE INTERFEROMETRY OF tion if the ocular is drawn out, or the reverse if it is thrust in. On close examination two sets, in different focal planes, seemed to be present, one stationary and the other moving as described, and accounting for the observed pronounced parallax. Suggestions of movable fringes accompanying the stationary are also present when the latter are produced by the grating. In this case the stationary fringes are strong; in the case of simple diffraction the movable fringes are more prominent. 50. Inferences. There can be no doubt that the great variety of chan- neled spectra obtained, when white light is successively diffracted by two gratings, is referable to the fringes obtained in the diffraction of homogeneous light, observed outside the principal focal plane, on a spectrometer. In other words, if light of a given pure color (sodium, mercury) is used, a single grating suffices. Each line of the spectrum is resolved into well-defined groups of fringes, if it is observed either in front of or behind the principal focal plane. The arrangement of fringes varies in marked degree with the distance of the plane observed from the latter (x, fig. 71). If reflecting gratings are used, there is no other possible source of interferences ; but reflecting and transmit- ting gratings show the phenomenon equally well. After finding how easily the Fresnellian interferences of two virtual slits could be reproduced in the telescope (Chapter III) and observed on either side of (before or behind) the sharp images, it seemed reasonable to suppose that the diffraction of a slit could also be produced and exhibited in this way; but the availability of this anticipation is attended with much greater diffi- culty. The image of a very distant slit does indeed show separated diffraction fringes on either side of the principal focal plane in the observing telescope. But they move right and left with the eye, in the same direction if the ocular is drawn outward from the principal focal plane, and in the direction opposite to the eye if the ocular is thrust in. Hence, in this respect, the fringes do not at once recall the phenomena under consideration. Usually the blurred image, out of focus, is stringy, without definite structure. It is resolved in a single focal plane only. To obtain sharp stationary fringes from an image of the slit, this image must be produced by the diffraction of a grating having a dispersing power above a certain minimum. Thus in a grating of about 7,000 lines to the inch the un- deviated slit image and the image of the first order are not clearly resolved, unless the slit is very fine. In the second and higher orders, however, the res- olution is very pronounced and the fringes stationary. The resolution of fringes is equally manifest in front of or behind the prin- cipal focal plane, so that if a weak convex lens is added to the objective of the telescope, the succession of fringes is found with an outgoing ocular; if a weak concave lens is added to the objective, the succession is found with an ingoing ocular, starting in each case near the principal focus. As the fringes increase in size they in turn subdivide, sometimes irregularly, as if each fringe were a new slit image, capable of undergoing secondary diffraction. Beyond these secondary fringes no further resolution was detected. REVERSED AND NON-REVERSED SPECTRA. 101 Returning to the work with two successive gratings and white light, the channeled spectra obtained are too complicated for concise description. A very interesting result, however, is the passage of the fringes across the sta- tionary sodium line, when the first grating G is moved fore and aft in a direc- tion normal to its plane. The region of the D line is thus alternately dark and bright. The direction of these rays remains unaltered while the illumined strip is shifted horizontally across the ruled space (fig. 70) of the second grating. Usually it is difficult to see the D line in the focal plane of the fringes. When homogeneous light is used this fiducial mark is necessarily absent and the cross-hairs of the ocular must be supposed to replace it. The shift of the fringes is then equally obvious, and sometimes (sodium light) different groups seem to travel in opposite directions while the grating G moves in one direction. In case of homogeneous light and two gratings, moreover, the fringes seem to be of minimum size in the conjugate focal plane of the gratings. They increase in size and in turn split up in focal planes before and behind this. An insight into these occurrences was finally obtained in observation with homogeneous light in the spectrometer by shifting the grating (transmitting) in its own plane, right and left. The fringes in such a case move bodily across the field of the telescope, new groups entering on one side for those which leave on the other. These fringes, even if quite distinct, are differently arranged in coarse and fine series and are frequently accompanied by dark or bright bands. This probably also accounts for the effect of the fore-and-aft motion of the grating, mentioned above. Moreover, it would be interesting to search for repetitions of given groups of fringes while the grating is being shifted parallel to itself, from end to end, as this might indicate the residual imperfections of the screw with which the grating was ruled. If the ocular is drawn and set outward from the principal focal plane (at which the slit image is quite sharp) into a different position, the fringes move in a direction opposite to the grating. If the ocular is set inward from the principal focal plane, they move in the same direction as the grating. This would not be unexpected ; but secondary fringes or something else in the field seem to remain stationary. Successive fields may be quite different as to arrangement of fine and coarse lines, but all plane gratings exhibit the same phenomena. Thus it is obvious that the fringes of the present paper result from a residual irregularity in the rulings of the grating. Micrometrically, the successive strips of a slit image, however fine, are of unequal intensity. Between these there is diffraction, as may be tested by examining the clear glass at the edge of the ruled space. To attempt a theory of these phenomena seems premature ; but it is obvious that in the otherwise indistinguishable images of a slit in homogeneous light, however sharp or however narrow, the nature of its origin still persists and may be detected by observations outside of the principal focal plane. A fine slit is in all cases presupposed, and all the phenomena vanish for a wide slit. On the other hand, the width of the pencils of parallel rays may be far greater than is necessary to show the strong Fraunhofer lines, if indeed there is any limitation to this width. CHAPTER VII. PRISMATIC LONG-DISTANCE METHODS IN REVERSED AND NON-REVERSED SPECTRUM INTERFEROMETRY. 51. Purpose. It is preliminarily the object of the present paper to examine a variety of new methods for the production of interferences with spectra, with a view to the selection of as simple a design as possible for practical pur- poses. Some interesting differences appear in the results, so that the sim- plicity of construction does not necessarily recommend the apparatus for use. In the second place, the endeavor will be made to assemble appurtenances in such a way that the extremely mobile phenomena may be under control, even in a moderately agitated laboratory. In case of the early interferometer experiments, the interferences disappeared on merely touching the apparatus, and are rarely or never at rest; whereas it is, of course, necessary that they should remain visible while the micrometer is being moved. These experi- ments are now nearly completed, but will preferably be described in a succeed- ing report. 52. Methods and apparatus. Some prismatic methods wefe tested in the earlier volume, but not developed; for the plan of using a transmitting grating twice, or two gratings in succession, seemed to contain greater promise. The prism method is, however, more sim- ple than any of the others and there- fore deserving of special study. In figure 72 the large right-angled prism P, with its faces silvered, re- ceives the pencil of parallel white rays, L, on its orthogonal faces and reflects them to the plane opaque _ o mirrors n and m. From here the rays are further reflected, either nearly in parallel, as in the figure, or crossed, as at c, c', to the remote opaque mirrors N and M, which in turn re- flect them to the plane or concave grating G. If the rays converge at the appropriate angle of diffraction, 0, a ~ selected color will be diffracted in the direction of the normal to G in each case. If the two paths are nearly equal, these rays will therefore interfere in the axis GT and the results may be observed by a telescope or a lens at T. In my apparatus the distances mM and nN were of the order of 2 meters. In consequence of the three successive reflections, it is somewhat difficult to 102 c/lf REVERSED AND NON-REVERSED SPECTRA. 103 obtain spectrum lines normal to the axis of the spectrum, so that if the latter are superposed the lines will be at an angle. But if this is small, it does not seriously interfere with the occurrence of fringes, as they extend from top to bottom of the spectrum. The appearance in general is of the linear character heretofore described. They pass symmetrically from extreme fineness, through a maximum size, to fineness again, with the fore-and-aft motion of the grating G, and they usually rotate near the maximum. If the mirror M is displaced nearly in a direction normal to itself, on a micrometer, the fringes undergo the same evolution, and in this respect differ from the case where the primary differentiator, P, was also a grating. In this case the displacement of M showed no discernible modifications of the size or character of the fringe pattern. The fringes merely moved. In figure 72 the effect of moving G or M fore and aft is similar, since it throws the point of convergence of the rays NG and MG in front of or behind the grating. The result is therefore different when white light impinges on G from what it is when the light is already nearly homogeneous. The limit of visibility is also inferior to the double-grating method heretofore used, for the fringes passed between the limits of visibility through the maxi- mum size, for a displacement of M of only about 3 mm. Smaller ranges may occur. On limiting the incident beam at L to a breadth of about 0.5 cm., the fringes became much broader and relatively intense. There is, of course, an abundance of light, so that the screening of the incident beam is not disadvantageous. In this case, when the fore-and-aft position (illuminated strips on the grating coincide, as in figure 72) and the position of the grating relative to its normal axis were carefully adjusted, large arrow-headed fringes, as in figure 73, were obtained, usually less closely packed vertically. Apart from tremors, these move slowly up and down (breathing), as a result, no doubt, of changes of temperature in the air- paths. A mica film inserted into one beam and slowly rotated produced similar motion, besides introducing its own grid of vertical and parallel fringes. The reason for the occurrence of these arrows is not quite clear to me, though they are associated with horizontal fringes and homogeneous light, the doubly inflected forms belonging to inclined fringes and homogeneous light. In the endeavor to reproduce these fringes with the sodium arc, I failed after long trials. The reason may be sought in the nicker of the arc, whereby the beam passes from one side to the other of the edge of the prism P, but it is probably due to the inadmissibility of a wide slit. 53. The same. Crossed rays. The present method, using four mirrors, has, nevertheless, the advantage of admitting the use of either parallel or crossed rays. Inasmuch as these rays are white until they leave the grating, the method is interesting. On being tested it showed the same peculiarities as the preceding. The crossed rays (cc f , figure 72) are more nearly normal to the mirrors M and N; nevertheless the range within which the interfer- 104 THE INTERFEROMETRY OF ences are visible is not above 2 mm. of displacement of M. The fringes may, as usual, be made as large as possible, by first superposing the two illuminated strips on the grating G (by fore-and-aft motion) and then rotating the grating on an axis normal to its face until the best conditions appear. Both spectra are very bright, but liable to be in different focal planes from inadequate planeness of the reflecting system. If work of precision is aimed at, this con- dition is of foremost importance. 54. Another method. If the opportunity of using crossed pencils of white light is to be dispensed with, the prism method may be simplified, as shown in figure 74. Here P is a prism with silvered sides and a prism angle of less than 30. It receives horizontal white rays L from a collimator, which, after reflection from the opaque mir- rors M and N, impinge on the grating G, plane or concave, and are observed at T by a telescope or lens. If

/2, or (1) cos 0/2 =M sin r Again, (2) sin i' = n cos (0/2 -f-r) when *' is the angle of emergence. Hence sin i' = cos 0/2 (\/M 2 cos 2 0/2 sin 106 REVERSED AND NON-REVERSED SPECTRA. -i 0- Thus if /*= i.55> tnensin *' = and ^' = 28.4. Now, since i+8 = 6, the angle 6 will obviously have to be in the second order of the spectra of the grating G. Although the two spectra obtained in this way were highly dispersed and very brilliant, the interference phenomenon itself was not much superior to the case where reflection from the (silvered) faces of the prism was employed. The fringes disappeared, in fact, for a displacement of i or 2 mm. of the mirror M, showing the usual inflation of form just before vanishing. The details also were of the same nature, the large arrow-shaped forms being obtained when illuminated strips on the grating were superposed and the latter slightly rotated until the maximal conditions appeared. To increase the range, the angle 8 must be reduced, as far as practicable. This is possible in the present method, since the points of intersection at a and G may be made to all but coincide. Reflection from the mirrors M and N would then be normal. To attain this end it will be necessary either to have the grating constant or the prism angle predetermined, or to use rays of suitable divergence at L. 56. Methods with paired prisms. White light (fig. 76, L) from a collimator is reflected in turn from the silvered sides of the sharp prism P, from the opaque mirrors M and N, and from the silvered blunt prism P', as shown by the component beams abc and a'b'c' . Thereafter the white beams are diffracted by an Ives film grating G, with attached prism p, and observed in a telescope at T. Interference, therefore, takes place in the focal plane of the telescope and would not (for the case in fig. 76) occur in its absence. Very interesting results were obtained with this apparatus. The spectra are non-reversed or else (if slit and grating are rotated 90) inverted. The work, however, is still in progress and will be described elsewhere. I will merely add, in this place, that the work with prisms is important, inasmuch as it shows the essen- tial part played by the diffraction of the slit of the collimator, in its bearing on the phenomena of the present report. It is the function of the prism P to cleave the diffracted field which leaves the collimator. For this reason pencils identical in source are found on both sides of P. The experiments thus fur- nish the final link in the theory of the phenomena. Furthermore, as the above results already show, the range of displace- ment of either opaque mirror (M, N) within which interference fringes are visible, increases in marked degree with the dispersion to which the white ray is subjected on separation and before the resulting partial rays reach their final recombination. These ranges increase from a fraction of a millimeter to almost a centimeter, while the width of the strip of spectrum carrying the interference fringes, caet. par., remains the same. This also has a fun- damental bearing on the phenomenon and is under investigation. The ques- tion at issue is whether increase of range of displacement results simply from the geometry of the optic system, or whether wave-trains are actually uniform throughout greater lengths, in proportion as they have been more highly dispersed. CHAPTER VIII. THE LINEAR TYPE OF DISPLACEMENT INTERFEROMETERS. 57. Introductory. This apparatus will be referred to in various places in this book and presents certain interesting features. The incidence of the grating is normal (I = R = o), and both component rays in their vertical pro- jection lie strictly in the same plane. To make the horizontal projection also collinear is not quite possible in practice, because the direct or unreflected rays and the corresponding spectra would overlap with the spectra of the interferometer. As the former are much more intense, the interference pat- terns would scarcely be visible in the combination. To avoid this, the rays diverge slightly (a few degrees, depending on the distance between grating and opaque mirrors) in a vertical plane. But this is of no consequence, as the horizontal projections only are used in the measurements. One may note, in passing, that this avoidance of coincidence with undesirable spectra secured by tilting the grating and the corresponding opaque mirror in the same direction is, in general, one of the essentials of the adjustments. The advantage of the linear displacement interferometer is this: that it can be built on a rail and mounted along a wall or a pier. If the rail is tubular, a current of water may be passed through it from the middle toward both ends, to insure constancy of temperature. 58. Apparatus. The apparatus was constructed as follows and gave good results at once, showing strong interferences. The ellipses were, in fact, oblate in the red, circular in the yellow, and prolate in the blues, but clear throughout. 77 Light enters from an arc lamp, A, or Nernst burner, or the sun, at the slit S, and is collimated by the lens L. Then the parallel rays pass the grating G with its ruled side toward L. From the grating the reflected beam returns to the opaque mirror N, and is then reflected into the auxiliary or adjustment telescope, T. The component beam transmitted at G is reflected from the opaque mirror M, returned to the ruled side of G, and thus also reflected into T coincidently with the other beam. 107 108 THE INTERFEROMETRY OF Figure 77 shows that the entering undivided beam LG passes just above the mirror M, and is reflected just below this from the top of N. Similarly, the reunited beam GT passes just above M, but is reflected from the top of M , the object being to make the vertical angle at G as small as possible. The mirrors M and N and the grating G are on adjustable bases, a, a', a", each controlled by three leveling screws on a plane-dot-slot arrangement in the tablets m, m', m", the axis of rotation being horizontal and normal to the diagram. The tablets, furthermore, may be revolved and raised or lowered by the rods n, n', n", which are attached by ordinary clamps to the large, tubular, horizontal rail, RR, in question, admitting of a circuit of water. The latter is secured to the pier. The angles of inclination of the figure are much exaggerated, since the dis- tance MG=GN (nearly) is from one-half to several meters in extent. The mirror M is on a Fraunhofer micrometer suggested at m. The bases, a, a', a", are drawn to the tablets, m, m f , m", by firm springs, preferably run- ning into the tubes below them. The axis of the adjustment telescope, T, lies in the plane of the figure and serves the purpose of bringing the direct slit images into horizontal and vertical coincidence. When this is done it may be removed, if desirable, as the ray GT is not thereafter used. T should not be attached to the rail, but placed on an independent table, or standard, so as not to be an integrant part of the interferometer. The telescope, T (not shown), for the observation of the interferences, should be independently mounted on the same table. This tele- scope lies outside of the diagram, to the right or the left of it, to catch either of the two diffraction spectra selected. It will be seen that these lie quite above the direct diffraction spectra of the ray LGM. Otherwise, as this is much more intense, it would completely wipe out the interference spectra and their combination. The latter, when seen alone, are very brilliant, black and colored patterns, running through the spectrum when the micrometer, m, is manipulated. If the distance GN is large and the grating G, as usual, slightly wedge-shaped, the superfluous rear reflection from G may be blotted out at N by a small screen. It is easily recognized, as it is brown from scattered light. The installation is simple. The parts being adjusted nearly symmetrically, the undivided ray from a wide slit is brought to the top of M by raising or lowering the lamp. This should first be done roughly with the lens and slit removed. N has at the same time been placed just below the beam, and this passes through the middle part of G. The latter is then inclined by the adjust- ment screws until the component beam GN strikes the top of N, symmetri- cally. Next N is inclined and rotated (vertical axis) until the reflected beam enters the telescope, T. Finally, M is inclined and rotated (vertical axis) until the reflected rays MG and GT also enter the telescope, the final sharp adjustment being made with a narrow slit and the eye at the telescope. The mirror M must also have a fine vertical adjustment (not shown). If the REVERSED AND NON-REVERSED SPECTRA. 109 distances NG (face toward the light) and MG are equal, the interferences are then easily found by moving the mirror M on the micrometer toward the grating. As compared with the other non-linear interferometers used under like conditions, the present instrument, even when mounted on a K-inch gas- pipe, RR, showed itself remarkably steady, so that rings could be observed in spite of the tremors of the hill on which the laboratory is built. 59. Film=grating adjustment. Michelson's interferometer. If the grat- ing G is a film grating, like those in the market, with 14,000 lines to the inch, it should be mounted smoothly on the unruled side, on a thick glass plate, with Canada balsam, and without a cover plate for the ruled side. It is to be adjusted with the glass side toward the source of light, so that the reflection taken may be from this side only (see r, fig. 78). In the telescope, T, directed toward the reflected beams, two slits (one for each component beam) only appear, as the glass plate does not reflect on the side covered by the grating (g in fig. 78). The slits placed in coincidence will then show the elliptic inter- Ic/Jt ferences in the diffracted beam D at the proper distances. With so large a dispersion as the above, the ellipses are usually too large. They should then be reduced in size by a compensator placed in the beam on the ruled side of the grating; or, preferably, the grating may be mounted on a plate of glass fully i cm. (or more) thick, as in figure 78. This thick plate has the additional advantage of eliminating the stationary interferences due to the front and rear faces of the grating. In case of thin glass plates (2 or 3 mm.), these stationary interferences are very strong, coarse, vertical lines and exceed- ingly annoying. If the film grating is carefully mounted in this way, it is nearly as good as a ruled grating. There is, however, one insuperable objection, inasmuch as the ruled face, though it does not reflect sharply, does diffract, and this more strongly than the other. Thus there are always 3 superposed spectra in the telescope, the third coming from the film side only, whereas the other two are produced by the rays coming coincidently from r on the unruled side of the grating. Hence the velvety blackness of the interferences in case of the ruled gratings can not be reproduced by the film grating, since the interfer- ences are spread out on a colored ground. They are, however, quite strong enough for all practical purposes, and the lines are sharply and symmetrically HQ THE INTERFEROMETRY OF traced. A vertical wire 2 or 3 mm. thick, placed symmetrically in front of the objective of the telescope, makes the interference relatively strong and sharp, by blotting out the third spectrum partially; but it at the same time diminishes the light available. A wide slit in front of the objective subserves the same purposes better. If the distance apart of the mirrors M and N and the grating G is large, it is best to dispense with the rail RR altogether, and to mount the mirrors and grating directly on the pier or wall. This has the additional advantage of a large free space between M and G or G and N, so that spacious apparatus like a fog-chamber may be independently mounted there. This was the case in the optic experiments on the thermal coefficients of the refraction of air, etc., below, where the distance between MG and GN was nearly 2 meters. In such a case, moreover, in addition to the usual three adjustment screws of the mirror M at the micrometer, it is desirable to have two others bearing on the rigid parts of the support, so that the final adjustment may be made elastically. By devising a tetrahedral plan of bracing M, G, N, independent of each other, using short rods and clamping all parts on relatively short stems, I eventually obtained a mounting which was almost free from tremors, even amid the dis- turbances of the surrounding laboratory. In figure 79 one of these mount- ings is suggested : a and b are y^-inch gas-pipes (about a foot long) , sunk into the wall of the pier at their rear ends ; cd is a cross-rod of same size and material, damped in place, and supporting the grating (or a micrometer) G. The screw h abutting against the wall gives the horizontal elastic adjustment. The braces e and /, which may be adjusted by rotation (screw) abutting in g at the wall, give the grating vertical elastic adjustment. Thus h, e, and /, rotate G around vertical and horizontal axes, respectively. 60. Michelson's interferences. If the collimator, SL, is removed and replaced by a strong sodium flame provided with a condenser, Michelson's interferences will appear at T when the instrument is in adjustment. It is rather surprising that, even in case of a film grating adjusted as above, they are well-defined circles covering the whole field of the telescope. If the col- limator SL is retained and the sodium light introduced from the side by aid of a reflecting mirror, placed between the grating G and the collimating lens L, both interferences may be observed at the same time in corresponding telescopes. The mirror introducing the homogeneous light should in such a case be provided with a clear space (silver removed), through which the white beam, SL, may pass without obstruction. In a vertical plane the interferences have the same size and character at the sodium line. Horizontally the spec- trum interferences vary with the dispersion. If an apparatus constructed of gas-pipe is employed, however, it is far too frail for the practical use of the Michelson interferences. Vibrations within the apparatus are excited on merely touching it. For the purpose of displace- ment interferometry, however, such an apparatus is quite adequate; for the measurements are taken when the tremors have vanished. REVERSED AND NON-REVERSED SPECTRA. Ill 61. Film grating. Another adjustment. The supernumerary spectra may be gotten rid of altogether by using the method shown in figure 80. Here the impinging vertical sheet of white light, L, from the collimator, falls upon the clear or unruled part p of the plate of the grating, the film extending out as far as shown at G. If M and N are the opaque mirrors, the reflected rays a and b passing G are additionally reflected into a' and b', and thence, after leaving the grating, into c and d. As both of the latter pass through the film, both produce spectra; but b' and e may be blotted out by a screen at the mirror N. This leaves only d beyond the grating. Again, the transmitted ray from L, after reflection at M, is again reflected into c and d', which is made coinci- dent with d . But c, being reflected from the unruled side, has no spectrum. Thus the spectra due to the two rays d alone interfere. Had the grating been reversed, caet. par., then the ray c would have pro- duced the strongest spectrum, and superposed on the other two it would have greatly diminished the clearness. In the telescope, whereas the ray a' prolonged is white, the ray d' from M and reflected from the film is strongly azure blue, due to regularly scattered light. This blue image is apt to be less sharp, unless very flat parts of the film are found. The two spectra, however, are good and the interferences satisfactory. The sodium line is sufficiently indicated, though, like the blue image, not quite sharp. This method of using the unruled edge of the plate of the grating for reflec- tion is, of course, equally applicable and advantageous in the case of the ruled grating. Only the two interfering spectra and no diffused light are present in the field of the telescope, and if sunlight is used the Fraunhofer lines are beautifully sharp. 62. Equations. The equations for N e /e, for normal incidence I=R = o, takes its simplest form as (i) Nc/e = M - Un/d\ = A +35/X 2 , nearly where N c is the coordinate of the center of a given ellipse on the micrometer M, for the thickness of glass grating e, index of refraction n, and color of wave- length X. Hence if two different wave-lengths, X and X', are in question (5 refers to differences), w 8N, being the displacement of the micrometer to pass the center of ellipses from line X to line X'. If M=A+/X 2 and Xd/i/dX= -2#/X 2 , then (3) tf.- 3 112 REVERSED AND NON-REVERSED SPECTRA. from which B may be obtained without further measurements. If greater approximation is necessary, so that two constants, B and C, enter the disper- sion equation, (4) W < = 3 so that observations at three spectrum lines, X, X', X", would be necessary. The amount of displacement corresponding to the thickness e of glass is, at a given spectrum line X, where 2J5/X 2 is constant for all values of e, or It is therefore not possible to obviate the term in B, determined as shown, if n is to be measured. If equal distances are cut off at M and N, the interference pattern, of course, remains stationary in the spectrum. It is interesting to inquire to what degree this may be guaranteed. Equation (3) is available for the purpose, and, since X and X' are nearly the same, X' X = SX and -,, ,->2<4^>0o^_ 9.2 X io- u /(59) 2 X io-"+4.5 (59) 3 Xio- !. I2 = 7-7Xio- 5 cm. The datum is of the same order as above, so that the sensitiveness changes but very little for different angles of incidence. Thus there is no disadvan- tage in using 7 = o. CHAPTER IX. THE USE OF COMPENSATORS, BOUNDED BY CURVED SURFACES, IN DISPLACEMENT INTERFEROMETRY. 63. Introduction. The method of increasing the sensitiveness of the dis- placement interferometer by increasing the dispersion of the grating readily suggests itself, but unfortunately the interference pattern loses sharpness in the same ratio and ultimately becomes too diffuse for practical purposes. Similar sensitiveness is secured when the air-paths and the glass-paths of the component beams of light are respectively identical, with the same inadequacy in the huge mobile figures, for the purpose of adjustment. In fact, if for sim- plicity we consider the incidence normal (I = R = o, linear interferometer), the sensitiveness becomes de/dn = \*/[*eD cos 0. ((n+ 2 b/\*)-N))] where 6 is the angle of diffraction for the wave-length \, e the thickness of the plate of the grating, n its index of refraction, D the grating space, n the order of the fringe, and b, N, constants. Hence, other things being equal, dQ/dn increases as D and e grow smaller, where e = o is obtained by a compensator counteracting the thickness of the plate of the grating. It occurred to me that the difficulty of diffuse interference patterns might be overcome, in part, by the use of compensators with curved faces, when the case would become similar to the conversion of the usual interference colors of thin plates into Newton's rings. Naturally a cylindric lens with its elements normal to the slit is chiefly in question, though an ordinary lens also presents cases of interest, chiefly because of the easy conversion of elliptic into hyper- bolic patterns, and the lens is more easily obtained. Other methods were tried. For instance, on using a Fresnel biprism with its blunt edge normal to the slit, two sets of interference patterns, one above the other in the spectrum, are obtained. When the blunt edge is parallel to the slit, either side of the prism gives its own interferences, but they can not be made clearly visible at the same time. A doubly reflecting plate or a thin sheet of mica covering one half of the beam will produce two intersecting patterns, but these also are of little use for measurement. 64. Lens systems. If but a single compensator is to be used, i.e., compen- sation in one of the component beams only, the lens in question must be of very small focal power; otherwise the adjustment will be impossible, as the two direct images of the slit will be in very different focal planes. Moreover, the focal power should be variable. All this makes it necessary to use a doublet, preferably consisting of lenses of the same focal power, respectively convex and concave. If these lenses are themselves weak, say i meter in focal distance, both slit images may easily be seen in the telescope and be 113 114 THE INTERFEROMETRY OF sufficiently sharp for adjustment. If the lens first struck by light is convex and the second concave, their focal distances /i and/ 2 , respectively, and their distances apart D, the focal power i/F of the combination used is (i) D/M-D/J* since /i =/ 2 =/. The position of the equivalent lens is d = DF/fi =/ 2 =/. D, d are both measured from the second or concave lens to the convex lens, and D would always be smaller than /. If the lens system is reversed, F remains the same as before for the same D, the system being again convex, but d is reversed. The equivalent lens again lies toward the convex side of the system. In other words, the equivalent lens generally lies on the same side of the doublet as the convex lens. In the actual experiment, however, the rays go through the lens system twice. In this case it is perhaps best to compute the distances directly. Of the two adjustments, the one with the concave lens toward the grating and the convex lens toward the mirror has much the greater range of focus relative to the displacement D. Supposing the mirror appreciably in contact with a convex lens, therefore, if b is its principal focal distance measured from the concave lens, b-\-D = M its principal focal distance from the convex lens or mirror, where fi is the (numerical) focal distance of the concave and f t that of the convex lens. If we now write (3) b equation (2) is easily converted into (4) -- = - 2B~f 2 so that the usual value of the principal focal distance has been halved rela- tively to the new position of the equivalent lens. If, as in the present case, f /2_ 2 D(f+D) The following table shows roughly the corresponding values of D and M in centimeters : D M=C+D 2B d 2 2450 2500 49 5 950 IOOO 47 10 455 500 45 15 292 333 4i 20 25 212 165 250 200 38 35 REVERSED AND NON-REVERSED SPECTRA. 115 As 6 is smaller than B by equation (3), the equivalent lens is on the side of the convex lens and at a distance behind the mirror, or B-b=f(f+2D)/2(f+D) behind the concave lens. If the system is reversed, /i and f z are to be replaced by /i and ft, whereas D remains positive. Thus the equations become successively i = !/(/!->) - 2 // 2 [ T If /i =/*=/, then 2D J-D Hence the equivalent lens has the same focal distance as before, but it is now placed in front of the system, at a greater distance than it was formerly behind it. Measured from the mirror (mirror distances, M) the data (in millimeters) are roughly as follows : D 2B B- M 2 2500 -5i 5 1000 -52 10 500 -54 15 333 -56 20 250 -57 25 200 -58 The total displacement of the equivalent lens on reversal is about i meter, falling off to 96 cm. in the extreme case. The image is larger if the convex lens is nearer the grating and the concave lens nearer the mirror. 65. Effective thickness of the lenticular compensator. The compensator with curved faces may change the interference pattern in two ways; viz, by changing the angle of incidence and refraction of the rays at the grating, and by changing the path-difference of successive rays passing through it. Both conditions are virtually the same, or at least occur simultaneously. If there is but one compensator, as above, the two effects must be small, since the rays reflected from each of the opaque mirrors, M and AT, of the interferometer, must eventually enter the telescope, to unite in two nearly identical images of the slit. It was rather unexpected to observe that the interferences are still obtained, even when the two slit images are quite appreciably different in size, but they are then confined to a single plane, as will be shown in 69. 116 THE INTERFEROMETRY OF Since the beam of light coining out of the collimator and traversing the grating is a vertical ribbon of light, several centimeters high vertically, but very thin in comparison (a few millimeters) horizontally, it is relative to the vertical plane that the marked effect must be expected. In figure 81, G is the grating, cc the principal plane of the concave, CD that of the convex lens, M the opaque mirror. If the beam consists merely of the axial pencil c, the distorting effect due to the introduction of the lens doublet is slight for any value of their distance apart, D. The two lenses are practically equivalent to a plate. If a broad beam dd is in question and the rays retrace their path, the same is still true. But if, on changing D, the rays do not retrace their path, so that the equivalent lens is convergent or divergent, then the rays after leaving M re-impinge on the grating at different angles than before and the interference pattern is correspondingly changed, principally in its vertical relations. Thus it is the lens system which changes the obliquity of rays lying in a vertical plane and passing through the grating, to the effect that the axial rays may represent a case of either maximum or minimum path-difference. The latter will be the case when the divergent pencil which usually traverses the grating becomes convergent in consequence of a sufficiently large value of the D of the lens system. 81 83 Jib CV 66. Observations largely with weak lenses and short interferometer. The film grating used (Wallace, 14,500 lines to the inch) was cemented with Canada balsam to a thick piece of plate glass, so that the total thickness of plate at the grating was 1,734 cm. This introduces a large excess of path in one of the component beams; but it is generally necessary, if the stationary interferences, due to the reflection at the two faces of the plate of the grating, are to be obvi- ated and if the ellipses produced are to be reasonably large for adjustment (cf . 69). The lens doublet was to be added on the same side as the glass speci- fied, so that the excess of glass thickness on one side was further increased by about 0.19 cm., on the average. Under these circumstances the ellipses were strong, but (in view of the large dispersion) with inconveniently long horizon- On inserting the doublet (convex and concave lens, each i meter in focal istance) with its concave lens at the mirror and gradually increasing the distance D by moving the convex lens toward the grating, a series of forms REVERSED AND NON-REVERSED SPECTRA. 117 was obtained which passed from the initial horizontally long ellipse, through circles, vertically long ellipses, vertical lines, into hyperbolic forms of increas- ing eccentricity, as recorded in figure 82. On reversing the system, keeping the convex lens fixed near the mirror and increasing the distance D by moving the other lens toward the grating, the original ellipse usually flattened out further, as shown in figure 83. Moving the lenses sideways parallel to themselves had no definite effect; moving them fore and aft together (D constant) produced results similar to the above. The vertical lines of figure 82 are liable to be sinuous or to resemble the grain of wood around a knot. In case of figure 82, as the equivalent lens lies in front of the mirror, the rays reaching the grating are thus necessarily converg- ing. In figure 83 the equivalent lens lies behind the mirror, so that the rays at the grating are more convergent. Both positions furnish essentially convergent rays. If corresponding to figure 82, the convex lens is kept fixed near the grating and the concave lens gradually moved up to it, the order of forms is reversed, but not quite completely. They usually terminate in long, vertical ellipses, before reaching which the wood-grained forms are sometimes passed. The same is similarly true for the case of figure 83. With cylindrical lenses (respectively convex and concave, each i meter in focal distance) very little effect was observed when the axes of the cylinders were parallel to the slit. With the axes perpendicular to the slit, the effects of spherical lenses were virtually reproduced, except that the central fields partook of a more rectangular character. To carry out the purposes of the present paper with strong lenses, respec- tively convex and concave, the vertical sheet of light from the slit must be diverged into a wedge by the concave lens and then collimated by the convex lens. The mirror, normal to the rays, reflects them, so that they retrace their path and become a sheet of light before the final reflection and diffraction at the grating. The following experiments were made with strong lenses: At first lenses of double the preceding focal power, /= 50 cm., were tried, but with no essential difference in the results. Thereupon strong lenses of focal distances /!= 73 cm. and / 2 = 13.1 cm. were used together, the convex lens being, as usual, near the mirror. For .0 = 7.5 CEa -i about, these gave fairly clear images of the slit and it was easy to find the ellipses, which were now very eccentric, almost spindle-shaped in form. They could be obtained strong and clear without difficulty, and the nearly horizontal lines filled the whole spectrum. Reversal of lenses practically failed to give results, the rays after reflection being too divergent. On the large interferometer, where the distances between mirror and grating are nearly 2 meters, adjustment was more difficult and the result (if parallel rays are retained) less satisfactory, because the slit images are not in focus at the same time. This is particularly the case when the convex lens is nearest the mirror and the concave lens toward the grating. Thus when/= 100 cm. and D = 1 5 cm., the modified slit image may be twice as large as the other and 118 THE INTERFEROMETRY OF the interferences in the principal focal plane of the telescope are only just seen. At D = 5 cm., however, the results are acceptable. When the concave lens is nearest to the mirror and the convex lens toward the grating, the modified slit image is smaller than the other. Adjustment is then easier and the usual elliptic and hyperbolic forms may be observed without trouble. In both cases the flickering of the arc lamp used passes the rays through different parts of the lenses relatively to the center, and the adjustment is thus easily destroyed. If the spectra from M and N, however, are observed, not in the principal focal plane but in advance of it (toward the eye), interferences of great interest will be observed, to be discussed in 69. 67. Remarks. A few explanatory observations may here be inserted. The occurrence of the elliptic or oval and the hyperbolic type of fringes may be most easily exhibited by laying off the order of the fringe in terms of the dis- tance (in arbitrary units) above and below the center of the image of the slit. If we call the latter y and consider the allied colors of thin plates, for instance, n = 2efi cos r/X or more generally n = (en/\)f (y, r) (where e is the thickness of the plate, p. its index of refraction, X the wave- length of light in case of a dark locus of the order n) is to be expressed in terms of y, which itself determines e cos r, r being the angle of refraction in the plate of the grating. The phenomenon will thus be coarser for red light than for violet light, since n decreases when X increases, and any two curves, r and v, figure 84, may be assumed as the loci of the equation in question. If, now, horizontal lines be drawn for n= i, 2, 3, etc., they will determine the number of dark bands in the spectrum for any value of y. If the central ray is also a line of symmetry and intersects the grating nor- mally, it must correspond to a maximum or a minimum of n. These conditions are shown in the diagram at M, where the maximum number of bands occurs, and at m, where the reverse is true. The question is thus referred to two sets of loci, rr' and vo' t or r'r" and v'v*, etc. In the former case e cos r varies with y in the same sense as /z/X; in the latter in the opposite sense and is preponder- ating in amount. Both may vary at the same rates in the transitional case, in which, therefore, the two curves r and v are at the same distance apart for all values of y. Suppose, furthermore, that the same phenomenon is exhibited in terms of wave-length X, as in the lower part of the diagram, the spectrum being now equally wide for all values of y, while at any given y the upper diagram still shows the number of dark points (bands) between r and v. If now, we suppose that under any conditions these dark points are grouped symmetrically with reference to any given color (which is probable, for a maximum or a minimum of any value of y will be so for all values), and that the successive dark points have been connected by a curve, the interference pattern will be of the elliptic type in case of aa' t a"a" f , and of the hyperbolic in the case of a' a". The other features of the phenomenon are secondary and therefore left out REVERSED AND NON-REVERSED SPECTRA. 119 of the diagram. Thus, for instance, the distance apart of the bands shrinks from red to violet, and the ovals, etc., are only appreciably symmetric, because they occupy so small a part of the spectrum. The horizontal distribution of dark bands around the center is determined by variations e cos r and is not linear. Whether the long axes of the ellipses are horizontal or vertical depends upon the slope of the lines r and v. Maxima and minima will not, as a rule, occur close together, though in certain wood-grain-shaped patterns this seems to be the case. In conclusion, therefore, the main feature in modifying the type of inter- ference pattern is the varying thickness of the compensator. For oval types the preponderating lens is convex; for the hyperbolic type it is concave. Neither of these lenses is here appreciably affected in modifying the horizontal distribution of path-difference, because the dispersion of the grating requires a horizontally parallel system of rays. 68. Observation with lens systems on both sides. The method shown in plan in figure 85 (L and U convex lenses, G grating, M and N mirrors, telescope at T) was tested. The outcome can not at once be foreseen, since the focal distances for different colors is different and since slight displacements of either lens must greatly modify the interference pattern. The latter, however, as obtained in every case, proved to be exceedingly fine lines, tipping in the 120 THE INTERFEROMETRY OF usual way with the motion of the micrometer and indicating a center of ellipses very distant in the field of the spectrum. In other words, the interference pattern is no longer automatically centered and is therefore useless. A modification of this plan is the method shown in figure 86 (horizontal section), where B is the beam from the collimator, L, L',L",L"', four condens- ing lenses of the same power (/= 50 cm.), G the grating, Mand N opaque plane mirrors, T the telescope. In all the above cases the horizontal rays from the collimator traverse the grating in parallel and eventually condense to a single point in the field of the telescope. The same is true of all rays having the same angle of altitude. These rays, therefore, act as a whole, since they pass through the plate of the grating at the same angle of incidence. On the other hand, relative to a vertical plane, the rays traverse the grating at different angles, each angle corresponding to a horizontal strip of the spectrum. It is by the easy modification of this obliquity that the curved compensator becomes effec- tive. In figure 86 the rays are also oblique relative to a horizontal plane ; but the result, unfortunately, is not available, since each of these oblique rays must have its own complete spectrum. Consequently the diffracted pencil will con- sist of an infinite number of overlapping spectra, the extreme cases lying within the same angle a shown in the figure. A large telescopic objective would then reunite these spectra into a white image of the slit, while a small objective will show colored slit images, passing from impure red to impure violet. Naturally, the interferences will also overlap, and therefore vanish. 69. Telescopic interferences. If interference patterns of small angular extent are to be obtained, it is essential that the rate at which obliquity increases from ray to ray be made as large as practicable. Probably, therefore, REVERSED AND NON-REVERSED SPECTRA. 121 an opportunity for realizing these conditions will be found within the telescope ; i.e., after the rays pass the objective. The endeavor would therefore be directed to bringing two spectra, focussed in two planes, one of which is behind the other and consequently of different sizes, both vertically and horizontally, to eventual interference. The experiment was made on the long interferometer (fig. 87), the distances between mirror M and grating G and from the latter to the mirror N being nearly 2 meters each. C is the lenticular compensator, consisting of two lenses, respectively concave and convex, each having the same focal distance, /= == 50 cm. The distances apart, D, of the lenses may be varied. The glass plate C', which is revolvable about the vertical, is thick enough to exactly counterbal- ance, if necessary, the thickness of the glass plate of the grating and of the lens system C. A sharp wedge sliding transversely may also be used. It is best to replace C' by two plates of glass, one thick and the other thin, so that the lat- ter may be removed. The telescope directed along the axis R will therefore, in general, see two white slit images, A and A' (fig.88), not both in focus at once, A' coming from cA- $0? (,' Jc' 88 M being larger, A from N (parallel rays) smaller. The focal plane of A' will be towards the grating as compared with A, and A' is larger than A, in proportion as the distance apart of the lenses C is larger. Similarly, the two spectra are observed along the diffraction axis, D, not in focus at once and of different areas. To obtain the interferences the slit image A must be placed anywhere within A', and they will occur at the top of the spectrum if a and a' are vertically in coincidence ; in the middle if b and b' coincide, etc. The plane of the new interferences is no longer the principal focal plane, con- taining the Fraunhofer lines, but lies in front of it; i.e., towards the eye of the observer and away from the grating. This distance, measured along D for the given small telescope used, was fully i cm. The focal planes of the two spectra are usually not so far apart. A' corresponds to a virtual object behind the observer. If the vertical plane in which the interferences lie be taken as the image, the object would be situated about 3 meters beyond the objective of the telescope used. This would place it 30 cm. in front of the mirror M or N, where there is but a single beam in each case. In fact, the telescope may be brought quite 122 THE INTERFEROMETRY OF up to the grating. Hence interference is produced in the telescope itself, where rays are relatively very divergent, a condition which accounts for the smallness of the interference pattern. This understanding of the case is tentatively shown in figure 89, where is the objective of the telescope, M the larger image from the mirror with the lens compensator, and AT the image from the other mirror (parallel rays) . If the corresponding rays be drawn through the extrem- ity of M and N, their fields of interference, F and F r , would begin in the plane I and /'. For axial rays it would be at i. Thus the locus as a whole would not be a plane, and this seems to be the case. If the telescope moves toward the grating, II' moves toward the right in the figure, as though the virtual object beyond the grating were fixed in position. At all events, the problem is to find the interference diagram of two symmetrical plane parallel spectra, of different areas and placed at definite distances apart. The appearance of the fringes is indicated in figure 90, where 5 is the height of the spectrum, usually quite out of focus. There are many more lines than could be drawn in the sketch. The ends a and a' seem to surround small ellipses, but these are not quite closed on the outer edge. The center of symmetry is at C. The demarcations are stronger and broader vertically if the distance apart of the lenses C (fig. 87) is small; fainter, but nevertheless clear and nar- rower, if this distance is large. Horizontally the fine lines thread the spectrum. The best results were obtained when the lenses C are less than i cm. apart, the middle band being about half as high as the spectrum. Two contiguous lenses gave a design which nearly filled the spectrum vertically. For practical pur- poses the lens compensator C is to be attached to the mirror M, just in front of and moving with it . It makes little difference here whether the concave lens or the convex lens of the doublet C is foremost. If the micrometer M is moved, or if the telescope is slid to the right or left, or forward, so as to take in other parts of the spectrum, the nearly closed lines at a and a' become finer and finer crescent-shaped lines, always open outward, till they pass beyond the range of vision. The whole phenomenon remains on the same level of the spectrum. On moving the telescope forward as far as G (fig. 87), the ocular has to be drawn outward (towards the eye) till it is fully 2 cm. beyond the position of the principal focal plane. The whole spectrum is now seen with the interferences from red to violet (no ellipses), but Q having the same relative position as before. The central " ' horizontal band measures about one-fifth the height of the spectrum, while the fine parallel horizontal lines extend to the upper and lower edges. The REVERSED AND NON-REVERSED SPECTRA. 123 appearance is now curiously like a blunt wedge (fig. 91), with a band at b nearest the eye, and the lines dd extending quite to the rear. This impression is probably an illusion, due to the shading; the lines grow finer and are more crowded toward the bottom and top of the spectrum. The illusion of a reentrant wedge is not possible. To use this interference pattern for measurement, the cross-hair is supposed to pass through the region c (fig. 90) symmetrically. Very slight motion of the micrometer mirror M then throws c either to the right or the left of the cross- hair. In this case the lens doublet, C, is attached to the mirror and moves with it, as stated. To obtain the extreme of sensitiveness, the path-difference of NG and GM must be all but zero; i.e., the grating plate G and the lens doublet C (fig. 87) must be all but compensated for equal air-distances by the compen- sator C. In this case of full compensation, the interference pattern, in the absence o a doublet C, would be enormous and diffuse, seen preferably in the principal plane of the telescope, but useless for measurement. The introduc- tion of a lenticular compensator, balanced by a compensator in GN, transforms the huge pattern into the small interference fringes in question, with the advan- tage that the high mobility of the coarse design has been retained. In other words, an index suitable for adjustment has been found, compatible with extreme sensitiveness. In fact, it is difficult to place the micrometer mirror M so that the region c (fig. 90) is exactly bisected. As the plane in which these interferences are seen most distinctly is i cm. or more anterior to the principal focal plane, the Fraunhofer lines are unfortunately blurred and a cross-hair is needed as a line of reference. I may in conclusion refer to a similar series of experiments now in prog- ress, in which the compensators placed in the M and N pencils (fig. 87, C, C'), instead of being of different shapes as above, are plates of different kinds of glass (crown and flint, for instance). Here the successive differ- ences of dispersive power, from wave-length to wave-length, produce effects closely resembling those discussed, with the advantage that difficulties inherent in the curved system are avoided. CHAPTER X. THE DISPERSION OF AIR. 70. Introduction. In view of the long-armed interferometer available, it seemed interesting to test the refraction of air at different wave-lengths, X. An iron tube of inch gas-pipe, 138 cm. long, was therefore placed in one or the other of the component beams. The tube was closed at both ends by glass plates, about one-eighth of an inch thick, kept in place with resinous cement. A lateral tube communicated with an air-pump and drying train, so that the tube could be alternately exhausted and refilled with air. By using sun- light, the different lines of the spectrum were obtained with sufficient clear- ness, and the method consisted in finding the reading of the micrometer for successive Fraunhofer lines, both for the case of a plenum of air and for a vacuum. If AN is the (monochromatic) displacement of micrometer corre- sponding to the latter difference of pressure, M being the index of refraction of air, e the thickness, (i) AAf> To determine MX, we must know d^/ d\. It has been omitted above, because it enters differentially and because of its small value. It appears as a con- stant decrement of AA/x, as X is constant and dMx/ d\ is negative. In the present case, where M is actually to be measured, dnj d\ enters directly and is essen- tial; but it follows from any two experiments when M is found for different colors. TABLE 8. Values of B. Inch iron gas-pipe, 138.0 cm. long. D line. t Bar P 1&&N 22.0 76.20\ 75-o 38.0 20 / 22.3 75.831 74-5 37-9 37-7 20 / 37-6 37-9 37-7 37-6 37-6 37-7 37-5 37-7 Mean. . . . 37.69 71. Observations with arc lamp l n table 8 results are given as obtained with the electric arc, in which the sodium line usually appears with sufficient distinctness in the spectrum to be available as a line of reference for measure- 124 REVERSED AND NON-REVERSED SPECTRA. 125 ment. Disregarding earlier results, the following are mean values of the ten independent data for A/V (each comprising a reading for vacuum and for plenum) : t = 22. 3 = 74.5011. Thus where ^ refers to normal pressure and absolute temperature (r). If /KO is given for the D line, d/i/ aX is determinable. It will be sufficient for the present purposes to put t^A-{-B/\ z , or X.aju/aX = zB/X 2 5 referring to r and p. Mascart's * value for ^ i (agreeing with Fabry's) is icr e X292.7, whence =icr 14 Xi.34 at r and p If the value B be computed from Mascart's observations between C and , D and F, respectively, so that the mean value io 14 S=i.6s m ay be taken. Since the last decimals of /z are in question, it will not be correct to more than 5 to 10 per cent. The value found above (io 14 B = i.34) is therefore somewhat too small. True, since from equation (3) (4) an error of icr 4 cm. in AN is an error of 0.13 X io~ 14 or 10 per cent in B, Very close agreement can not therefore be expected in either result. One is tempted to refer the present low value of B to flexure of the glass end plates of the tube, which, when the tube is exhausted, become slightly saucer-shaped and introduce a sharp concentric wedge of glass into the component beam, whereby the interference pattern is changed, probably in the direction of smaller values, as found. But the direct experiments below do not show this. In any case, the measurement of B lies at the limits of the method. An advan- tage may possibly be secured by using two identical tubes, one in each com- ponent beam, the tubes to be exhausted alternately. The sensitiveness would then be doubled. 72. Observations with sunlight. Single tube. These observations are given in table 9, the exhaustion throughout being 75 cm. and the temperature about 1 6. In the first set sunlight was used without a condensing lens; in * See excellent summary in Landolt and Boernstein's Tables, 1905, p. 214. 126 THE INTERFEROMETRY OF the second set the sun was focussed with a weak lens (0.5 meter in focus) at the point formerly occupied by the electric arc. The spectrum (particularly in the second case) was brilliant and the lines clear. The focus of sunlight is to be placed just outside the focus of the collimator lens, in order that a nearly linear pencil may be available to penetrate the long refraction tube twice. The distance of the collimator lens to the grating was about 2 meters. The spectrum is then a bright band in the telescope, the width being limited by the height of the ruled part of the grating. The strip of white light on the grating should not be more than a few millimeters wide. It must therefore be nar- rowed by an opaque screen (wide slit of the given width) in the path of the beam (see fig. 92 below). TABLE 9. Dispersion of air. Tube /= 138.0 cm. Bar. 77.25 cm. at 19.5. =75.0 cm Line. Temp. io*AN io+ 14 5 C D b 16.0 38.5] 39- 394J i-5 C D b 1 6.0 38.5] 38.9 39-3J 1.4 Improved seeing, weak condensing lens. C D 16.4 38.81 39-2J I-5I C D b F 16.4 38.9] 39-1 ?39-3 40. 1 J 1.65 1.4 C F 16.4 38.81 40. 1 / 1.61 The equation for B in this case, if the symbol 8 refers to differences for two given values of X, is P 273 8 L V if the value of B is to hold for normal conditions. The data are shown in table 9, series i and 2 being obtained without con- densing lens. These are inferior, as regards definition of lines, to the subse- quent set, in which condensed sunlight was used. In all cases there is some- times an irregularity (marked by ? in table 9) in which the observation is obviously discordant, but the reason could not be found. Possibly values of r and p taken were not the actual values. The data for B X io 14 given in the table are mean values. Some of these are low. Later values, where the F line is included, come out larger, the range being from 1.3 to 1.8 or 1.5, on average. It is desirable to use the whole of the available range of the spectrum (sufficiently luminous from C to F) to obtain an acceptable value of the coefficient B and additionally to improve the method by using two REVERSED AND NON-REVERSED SPECTRA. 127 identical tubes, alternately exhausted as suggested above. The attempted B measurement is at the limits of the method, as has already been instanced in the discussion of errors in the preceding paragraph, and it is not to be con- cluded that data which happen to agree with Mascart's result from a correct application of the present method. In fact, there is no reason for excluding the exceptional values, and the present results are to be regarded as preliminary. 73. Two (differential) refraction tubes. In the following experiments two identical iron tubes (138 cm. long, of inch gas-pipe) were installed, one being placed in each of the component beams of light, which subsequently interfered, and the tubes were exhausted alternately. There are apparently three advan- tages in this arrangement. In the first place, the sensitiveness is doubled; in the second, the flexure of glass plate should be the same at each tube, in each experiment, and thus fail to disturb the interference pattern. Furthermore, by using the tubes in parallel (*.*., exhausting both at the same time), any irregularity of flexure effect, etc., should be determinable, as the air in both tubes will be identically circumstanced. Finally, the air being inclosed in a thick metallic envelope at both beams is not subject to incidental disturb- ances. An unexpected difficulty, however, was encountered ; for there is reflec- tion of direct spectra from the eight glass surfaces, and this must be specially met. The direct spectrum is easily eliminated by inclining the grating until the reflected interference spectra are at a different level; but reflections of this spectrum are not so easily dealt with. Fortunately they are weak. Even so, they are very annoying, as they overlap the interference pattern and dull it. They could be eliminated by attaching the glass plates obliquely to the axis of the pipes, but this remedy was not thought of at the outset. Figure 92 is a diagram of the disposition of the parts of the apparatus. L is the beam of white sunlight from the collimator limited laterally by the wide slit (1 inch) 5. G is the grating, T and T' the two refraction tubes, M (micrometer) and N the opaque mirrors, R the refracted and D the dif- fracted (spectrum) beam of light. C is virtually a four-way stopcock (or two 3 -way glass stopcocks) leading respectively to the exhaust pump E and dry 128 THE INTERFEROMETRY OF air supply A, from the tubulures e and e' of both refraction tubes T and T. These are therefore alternately exhausted. Preliminary results are given in table 10, the arc lamp with its sodium line being used in the absence of sunlight. It will be seen that AN, apart from temperature (which is here higher than above), has been doubled. The deflections were symmetrical within o.isX icr 3 cm. TABLE 10. Dispersion of air. Differential tubes, each 138 cm. long. D line in electric arc. Barometer. Temp. lo'Atf (*>-l)Xio -BoXio 14 75.93 cm., 22 76.02 cm., 19.5 21-5 18.0 754 75-3 g 291.6 291.8 1-47 1.49 (Single lube) 23.0 37^8 293-6 i. 80 The values of B found in the first two series of this table, if the standard value of HQ i =0.0002927 is assumed, is somewhat small, but as near to the true values as may be expected. Again, if jE?Xio 14 =i.6s is assumed, the Aio i values given in the table are similarly small, being 0.3 per cent short of standard values. A single-tube experiment made for comparison (series 3), similarly, came out too large in each case. It follows from this that p and t observations are not sufficiently guaranteed. It is hardly probable, however, that with a micrometer reading to iQ- 3 cm. and estimated to lo" 4 cm. (vernier) the precision can be much enhanced ; for since X'AAT 76 T dJ5o = _VAA[^-r_ dp 2 e 273 p* or at the C line and D lines, respectively, 8p = i cm. 8B - dT 2 e 273 p fo.2Xio- 14 1o.2Xio- 14 Now, unless the measurement can be made in terms of rings, it is difficult to detect a few millimeters of pressure difference by displacement only. The interesting question now occurs whether the two tubes, if identical for a plenum of air, remain identical (no shift of the interference pattern) throughout all the stages of identical exhaustion. On trial, nothing could be detected, the fringes remaining stationary during the whole period of exhaus- tion, or during the influx of air following a high vacuum. Hence there is no perceptible difference effect of flexure of the glass ends, and the ultimate question of accuracy depends on the measurement of r and p. To eliminate the possible effect of flexure, an air column of negligible length, in which the glass effect only is present, will have to be tested; otherwise there is no possi- bility of separating the air and glass effect. REVERSED AND NON-REVERSED SPECTRA. 129 74. Differential and single refraction tubes. Sunlight. The direct experi- ments for the coefficient B were now resumed and conducted with sunlight, with the results given in table 1 1 . The first two series were made with the two identical tubes specified, exhausted alternately, one tube containing a plenum of air, while the other was nearly empty in each experiment. The C and F lines alone were used for measurement. In spite of the large displace- ment (AAT=o.o76 to 0.078 cm.), the results were not as satisfactory as was expected, owing to the fact that sharpness of vision is made difficult by the stray reflected spectra to which reference has already been made. But the data for B obtained with one exception (No. 2 in the first series) are consistent and reasonably good. In every other respect the work was satisfactory and could have been improved by using oblique cover-glasses. The values of B obtained are therefore disconcerting. TABLE ir. Dispersion of air. First and second series, Differential Tubes, each 138.0 cm. long. C and F lines. Third and fourth series, Single Tube, good adjustment. Barom. p Line i&AN io u B Temp. Barom. p Line lo'AiV io u B Temp. cm. 77-05\ 74-0 22 / 74.0 74.0 74.0 C F C F C F C F cm. 76.4 78.0 76.1 78.2 76.3 78.0 76.2 78 o I.O 1-3 i.i i.i C 17.4 17.4 17.4 17.4 cm. 77.22} 74.0 21 / 74.0 74.0 74.0 C F C F C F C F cm. 38.4 39-2 38.2 39-3 38.3 39-3 38.2 39-O 0.99 i. ii i.i 8 i. 06 C 18.0 1 8.0 1 8.0 18.0 76.27\ 74.0 23 / 74.0 74.0 C F C F C F 75-8 774 75-4 77.2 75-4 77 * I.O I.I I.I 21.4 21.4 21.4 77.22! 76.0 21 / 76.0 76.O C F C F C F 39-3 40.1 39-2 40.0 39-0 40.0 0.99 99 99 18.1 18.1 18.2 76.0 C F 39-2 40 o 99 18.2 I then went back to the single-tube experiments (in series 3 and 4), and these are the smoothest results obtained. The C and F lines were used as before. In the last series, for instance, the micrometer reading is the same to io~* cm. throughout. In spite of this satisfactory behavior, the value of B obtained is again of the same low order, all the data, both for double and single tubes, being consistent throughout in this respect. Series 3 and 4 agree, although p is changed from 74 cm. to 76 cm. of mercury. In table 12 I have summarized the data in comparison with the standard results, computing /J.Q i and B , for each of the cases, reducing all values to oC. and 76 cm. of mercury. The difference of ju<> i for the F and C lines, which is 3.2 X icr 6 for the standard data, is but 2.3 X icr 6 on the average in the present results. Similarly, the mean values of the latter are io-*X4-i and io~'X3.2 larger for the C and F lines, respectively, than the standard results. 130 THE INTERFEROMETRY OF These conditions are particularly puzzling, since in 73, with the use of the arc lamp, both results were nearly normal. I therefore endeavored to detect the causes for this difference of behavior. TABLE 12. Summary of Table n. "St." refers to standard data. A= (MO i). Line, etc. XX io 6 St. io'A Series. Ser. (i)to( 4 ) -BoXio" (MO- i) St. (I) io A (2) IOM (3) loM (4) IOM SoXio 14 C. -BoX io"P. c 65-63 48'.6i 291.8 295.0 295.0 -3-2 297-3 -3-3 2.3 296.0 i 296.3 -4.2 I -4.5 298.0 ; 298.7 -3-3 I -3-7 2.3 2.4 296.5 -4-7 298.6 -3-6 2.1 1.18 1.20 1.28 1. 08 1.85 2.37 2.22 2-55 1-33 1.56 1.70 1.50 10* Diff . from St. p ioDiff. from St. lo'Diff. FandC. 3-2 The standard of length was first compared with a normal meter, showing that the i = 138.0 cm. for the M tube should be replaced by 137.75 cm. and for the N tube by 137.59 cm- As this correction affects all the results, /* i and B, in the same ratio, it contributes nothing to modify the discrepancy in question. The correctness of the micrometer screw was assumed, as it was of good manufacture. The test next was made to see if the lines taken as C and F actually had the accepted wave-lengths. A revolvable arm, with its axis at the grating and 125.5 c 111 - long, was therefore installed for the direct measurement of the diffraction of the grating. The results obtained for the wave-lengths of the lines taken showed that no mistake had been made in their selection. To endeavor to obtain further evidence, the values of B were computed for the mean data of table n, by using the standard values for HQ i in case of the C and F lines and the AAT/e given by the observations. The results so obtained are given in the last columns of table 12, for each of the four series and the C and F lines at normal temperature and pressure. The mean results are thus #oXio 14 =i.i9 from observations with sunlight directly between C and F lines. B X io 14 =2.25 from standard ;u i = .0002918 at C line. B X io 14 = 1.52 from standard no i = .0002950 at F line. The B values of table io show a march to be referred to temperature and pressure. So the present unsatisfactory differences are probably pressure- temperature effects beyond the discrimination of the method. One reason for this discrepancy which suggests itself is the possible distor- tion of the glass plates at the end of the exhaust tubes during the exhaustion. There may be a residual temperature effect due to the heating of the air by the beam which passes twice through it, above the indicated temperature of the surrounding tube of iron. But as Mo -i is already too large compared with standard values, this would make the case worse. Similarly, a larger thermal coefficient than the normal value (1/273) would further increase REVERSED AND NON-REVERSED SPECTRA. 131 the data for /i i. For the case of the F lines, the B values found by com- parison with the standard MQ i (last column of table 12) might be taken as correct within the error of method. Nothing, however, has been found to account for the correspondingly large values of .Bo for the C line. 75. Distortion of glass absent. To test the effect of possible distortion of the end plates of the tube, a shallow cell was constructed but 0.8 cm. deep, closed by plates of the same glass. The diameter of the tube was identical with that of the long refracting tubes. Tests made with the electric arc and sodium line gave the mean values Plenum. . Vacuum . 5-25; 6.6 cm. 6.4 cm. Thus the effect of exhaustion is 0.0002 1 cm. The long tube gave, on the aver- age, 0.039 cm - f r *38 cm. of length. Hence the air effect should be 0.039X0.8/138 = 0.00022 which is practically identical with the value found, ceptible distortion referable to the glass plates. Hence there is no per- TABLE 13. Dispersion of air. D and F lines. Single Tube, length 138 cm. Barom., 75.40 at 28. =74 cm. Temp. 20. St. refers to standard data. A = po i. AN St. Mean St. A to I Line. i&AN IO" e io 14 5o XX i o 6 loM loM io"B ,F io"B<,,D cm. cm. F D 39-4 *8 2 312.8 I.OT. 8 2.24 F io 6 Diff. from St. 48.61 295-0 304.0 9.0 2.IO 19.2 F 39-o 310.1 1.38 D F D 38-3 39-0 38.3 304.6 310.5 304-6 1.49 D io 6 Diff. from St. lo'Diff. FandD 58.93 292.7 2-3 299.4 -6.7 4.6 I. 7 8 1.83 2.06 '2.06 76. Further observations with sunlight. In the absence of other than inferential reasons to account for the difficulties met with, a final series of observations was made between the D and F lines and a single tube, with the results given in table 13. The mean value of B found directly, viz, 1.70 Xio~ 14 , would be admissible; but the corresponding values of ^o i as com- pared with the standard values are again too large and worse than above. The same is true of the values of BO found by comparison of AN/e with the standard values of /u i, and their coefficients come out differently for the C and F lines. In fact, the discrepancy of /to i is now about 3 per cent, whereas observations for &N/e should not be in error more than (2 X 1/400 = 0.0025) 0.5 per cent. There is thus something variable at the limit of appli- cation of the present method which has persistently escaped detection. I have thought that a distortion associated with the form of the interference pattern in passing from C to the F line may be in question, as the discrepancy 132 REVERSED AND NON-REVERSED SPECTRA. varies in different, otherwise satisfactory experiments; or the failure to com- pletely exhaust the tube may leave a small error which becomes appreciable in J9. 77. Conclusion. If allowance is made for the fact that AN" at the micrometer is measured for air, at barometric pressure p' and absolute tem- perature T, the equation for /JL O i at normal conditions would be ~~$ 1*1$ i-p'AN/ep X* where the correction factor iANp'/pe would not appreciably modify the results. It is difficult to see, therefore, why the promising results of 4, which are quite as near the standard data as the method warrants, did not bear consis- tent fruit in the sequel. The direct values of B X io 14 are usually too small, sometimes too large, and range from i to 2 . On the other hand, H Q i usually conies out too large, whereas it should be correct to a few tenths percentage. None of the causes examined, temperature, pressure, thermal coefficient, flexure of glass, etc., quite account for such a result. If BoX io 14 is computed from standard results for ju i and observed at different spectrum lines, the data are nearly correct for some lines, but too large for other lines, so that a single constant does not reduce the series. It does not seem probable, however, that equation (i) is inadequate; for the results obtained with equal care at dif- ferent times for the same MO i or B are not in accord. The discrepancies, in other words, are not persistent in value and are therefore due to some inci- dental cause which has not been detected. It has seemed to me that the change in shape of the interference pattern on passing from red to violet, which in case of ordinary glass mirrors is marked, may be responsible for some of the difficulties encountered. This pattern, which for optically flat surfaces would remain elliptical, becomes more and more irregular as the distances, e, of the mirror and grating are increased. The distorted image shrinks laterally from red to violet fully one-half, so that it is not certain that the center of figure is actually a fiducial point. The question, however, would have to be tested. CHAPTER XI. THE CHANGE OF THE REFRACTION OF AIR WITH TEMPERATURE. 78. Apparatus. In the earlier report (Carnegie Inst. Wash. Pub. 149, III, Chap. 15, p. 223) I began some experiments on the change of the refrac- tive index of air with rise of temperature. The question is interesting, inas- much as the temperature coefficient has not in most investigations been found identical with the coefficient of expansion of air, as Lorentz had obtained it and as would otherwise be anticipated; but a value, over 3 per cent larger, first put forward by Mascart, seems preferable. My earlier work was left unfinished, however, because the design of the apparatus, in which the refrac- tion tube was heated in an independent annular steam-bath, was unsatis- factory. It seemed to be impossible to reach the temperature of the steam in that way, even after half a day's waiting. In the present work, therefore, the apparatus is modified, so that the steam may play directly on the long refraction tube. In this way the temperature difficulty was quite eliminated. 93 J The tube containing the air column was made of inch brass gas-pipe, 71.7 cm. long (between windows) and 2.5 cm. in internal diameter (A, fig. 93, which shows one end of the apparatus) . The ends were closed with the usual brass caps a, in which round windows, about 2 cm. in diameter, had been cut on the lathe. The ends were closed by plates of glass g, secured between two jackets of rubber and " vulcanized" fiber. L shows the axis of the beam of light. BB is the steam chamber, steam entering at 5 and leaving by a similar tube at the other end of the apparatus. Steam is thus directly in contact with the tube. The projecting end of A is inclosed by a recess packed with wadding, 133 134 THE INTERFEROMETRY OF CC. As the walls of the brass pipe were thick and the ends relatively short, there seemed to be no objection to this arrangement. Care was taken to conduct the escape steam and hot gases away from the interferometer. The displacement interferometer was of the linear type described above, the mirrors M and N and the grating G being attached directly to the wall of the pier and without an intervening rail. Unfortunately the pier in a large city is also in incessant vibration, so that the interference patterns quiver. It is this insuperable difficulty which has prevented me from reaching results as accurate as were anticipated. A few of the data, however, will be added as an example of the efficiency of the method. TABLE 14. Refraction of air at different temperatures. Tube, 71.7 cm. long, 2.5 cm. in diameter. Barometer. Temp. P- lo'XA* Barometer. Temp. P- xo'XA* 77.04 cm. 19-7 75-3 19-3 75.12 cm. 19.9 74-3 19-3 22 19.4 20 19.1 I 19-3 VI 19.0 19.2 76.25 cm. 21.7 75-5 19.1 20 19-3 76.98 cm. 100.4 75-5 15-0 II 19.2 18.7 15.0 19.4 VII 15-0 19.4 19-3 77.22 cm. 19.1 75-7 19.8 76.55 cm. 100.2 75-3 15-0 20 19.7 15-0 IV 19.8 Ill 15-4 19.7 19.7 76.25 cm. 22.2 75-5 19.1 75.12 cm. 99-7 74-3 15.0 20 19.2 20 15-0 V 19.4 VIII 14.9 19.6 15-0 19-5 I5-I 79. Observations. The data are given in table 14, where the temperature and barometric pressure are shown in the first column, the differences in the pressure p between the plenum of air and the exhausted air in the refraction tube in the second column, while the third shows the values of AAT, the dis- placement of the micrometer corresponding to p, as found in successive inde- pendent experiments at the temperature given. For such long distances between grating and mirrors the ellipses are visually distorted, and much depends on finding a satisfactory sharp interference pattern. This was the case, except in series 3 and 5, when for incidental reasons (outside tremors) the patterns were disagreeably flickering. The observations are usually for room air, as the special drying of air in series 3 and 5 made no perceptible hfference. At 100 care must be taken to obviate convection currents of air, so far as possible. The endeavor was made to keep p as nearly as possi- ble at^the same value, apart from the barometer pressure, which does not enter into the equations. In series 4 the values of AAT are relatively large, REVERSED AND NON-REVERSED SPECTRA. 135 but quite consistent with each other. The reason for this could not be made out. But for the inevitable tremors the observations would all have been acceptable. 80. Computation. Since the ends of the air-tube are perpendicular to the beam of light (I) AW where AAT is the difference of the displacements of the micrometer in the presence and absence of air in the tube, e the effective length of the air column, and j the index of refraction of the air for the given wave-length X. The equation presupposes a knowledge of the dispersion of air dn/d\; but, as this is small, the term may be temporarily omitted. If X is constant, it corre- sponds to a constant correction of AAf throughout the experiments. Again, if we have an equation of the form of Mascart's, /Z76 referring to o C. and normal barometer, and NQ to the absence of air in the tube, -i = p -i 76 i + erf N n -N, AA/7, where a and /3 are two constants. If the tube is not quite exhausted (SB remaining), the observations for a plenum (barometric pressure, B) and exhausted air being made at the same temperature, AfrB-W __ or nearly B N B -N 9 Thus if one neglects the small correction i ftB of dB (3) AAfc-AW,-*^ the micrometer displacement AA/ in case of complete exhaustion at the barometric height, B, and the displacement &N B -. S B corresponding to partial exhaustion B 8B, are proportional to those pressures. Since dB was quite small, this equation was assumed, and p 8B is thus nearly the height of the mercury column of the partially exhausted tube. In the table this is briefly called p, and differs from the barometric height. Finally, for two partial pressures p and p' and temperatures t and t' of the AAT and AJV' being the micrometer displacement corresponding to p, t, and p', t', respectively. Hence if care be taken to make p = p', nearly, i + at (*' <) =adt AJV " AN' ~AN-&N'Nd 136 THE INTERFEROMETRY OF if, for brevity, t'-t = 8t and AN- AN' = dN; or If p is not quite equal to p', P(p'p) may still be neglected, but AN/p and AN'/p r must replace AN and AAT', orAN(i-8p/p) replace AAT where 5 = -/. On applying equation (5) to series i, 2, 3, for which p is nearly constant, $t = 'jg.$ 5Af= 0.00423 01 = 0.00380 applying it to series 4, 5, 7, similarly, *=79.8 SN= 0.00404 01 = 0.00404 The mean value is thus a = 0.00392. The reason of this difference is found in series 4, where AN" is excessive. In fact, if we compare percentage errors of a and 8N da/a AN N = so that an error of 5 per cent in dN would be an error of over 5 per cent in a. For the case where the fringes tremble this is inevitable. If the mounting were without tremor, however, dN should be guaranteed to sXio' 5 cm., corre- sponding to the evanescence of a single interference ring, so that a should be determinable to i per cent, even in case of a tube of the length 71.7 cm. given. If tt is small or *' small, equation (5) becomes, approximately, This equation may be used to find the successive values of AN in the table, if the second, for instance, is supposed to be correct. It appears that the first and fifth differ about equally (0.0001 cm.) from the second, but the error of the fourth (0.00028) is excessive. Hence if this second datum be taken as the mean of series 1,2,5, and combined with the two data for 100, AAT= 19.28 SN -0.421 52 = 78.6 t'= 100.3 a = o.oo38s This is the more probable result of table 14 and would agree with Mascart's value, 0.00382. Somewhat later, the independent series of observations 6 and 8 were carried out. The interference pattern at 99.7 was exceptionally quiet and clean, but at lower temperatures this was not better than usual. The results are 4 .i 5 5*= 79. somewhat below the preceding value. 81. Final experiments at 100. Somewhat later, at a time when the labo- ratory was relatively quiet and after the same effective improvements had been made in the mounting of the interferometer mirrors, the experiments REVERSED AND NON-REVERSED SPECTRA. 137 at 100 were repeated. The optical measurements were satisfactory, or at least just short of the counting of interference rings for measurement. The arc lamp, moreover, which is unsteady, would scarcely suffice for this purpose. The results obtained were as follows (table 15) : TABLE 15. Refraction of air at different temperatures. Bar. Temp. P icfaN Bar. Temp. P IO'A#' 75.6 cm. 21.8 74.0 19-3 76.72 cm. 100.3 74.0 I5-I 19.2 20.5 154 19-3 15.0 19.1 I5-I 76.65 cm. 21.8 74.0 19.6 15-5 19.4 15.0 19-5 15-2 19.2 154 - 19-3 15-2 15-3 15-2 78.5 If the mean values of AW and AW' be taken and a computed io 3 AW=ig.22 io 3 AW'=is.22 6W=4.oo 8t-- the result is 01 = 0.00361 As these experiments were the smoothest and were made under the most satisfactory conditions, they are probably the most trustworthy. I have not, therefore, been able to obtain evidence for a value of a (between o and 100) greater than the coefficient of expansion of gases, though it must be confessed that the method in its present surroundings is not sufficiently sensitive to furnish a definite criterion. Later results at low temperatures (series 3) like the above series 4, table 14, again gave a high result for AW, in each case consistently. It is probable that the interference pattern changes between the case of a plenum and of highly exhausted air, owing either to flexure of the glass ends or to some other cause, or possibly depending only on the form of the pattern which happens to appear. In such a case the lines of symmetry for W (plenum) and W (ex- haustion) would differ, introducing a systematic error very difficult to obviate. Thus different values of AW often follow a difference of adjustment of the mirror at the micrometer, while all cases for the same adjustment are practi- cally identical. 82. Experiments at red heat. To investigate the feasibility of such experi- ments, an inch steel tube (bicycle tube), 68 cm. long, with flanges brazed on at the ends, and an exhaustion tube near the middle, was heated in an organic combustion furnace to low red heat. The ends just projected outside the furnace and were closed by plate-glass windows with a jacket of asbestos between (applied wet and dried); or, finally, with a jacket of aluminum cement, clay, plaster, etc. These short but relatively cold ends are, of course, 138 THE INTERFEROMETRY OF an objection to the method, but no better device was found. Even so, the windows frequently cracked and had to be replaced. Such an apparatus naturally leaks, particularly at low temperatures, where the viscosity of air is relatively small, so that the experiments as a whole are merely tentative. To maintain the exhaustion as high as 70 cm., it was necessary to keep the air-pump at work. To reduce this annoyance the exhaustions were at first not carried above 60 cm. of mercury. With the interference fringes, however, no serious difficulty was experienced after the tube had taken definite shape. Distortion of fringes was inevitable, but centers of symmetry for measure- ment were always available. The first experiments were made without exhaustion, at low and high temperature (low red heat). The difference of displacement 8N between cold (2 5) and hot was (for instance) in two different experiments 25 ioW =35.0 cm. 35.2 cm. red hot ioW=a8.s 28.6 or io z 5N= 6.5 6.6 at atmospheric pressure. The 8N so obtained makes no allowance for the change of refractive index of the hot glass ends, nor for any displacement or rotation or warping of the ends during the course of the experiment, which required a lapse of an hour or two. In the next experiment, therefore, the method of exhaustion was attempted, the partial vacuum used being about 16.6 cm. when the full barometer read 76.64 cm. Thus p = 6o cm. An example of the results obtained is given in the following data. Cold Tube. Pressure 76.6 cm. ioW =34.8 cm. 34.7 cm. Pressure 16.6 22.7 22.5 P= 6o.O IO 3 AAT=I2.I 12.2 Red-hot Tube. Pressure 76.6 cm. ioW =24.6 24.5 26.8 26.0 cm. Pressure 16.6 20.1 19.4 20.0 20.0 P = 60. io 3 AA/" 4.5 5.1 6.8 6.0 In the two experiments at the end readjustment was necessary, as the red-hot tube warped during the exhaustion. In the last case the glass cracked. The first two data should therefore be taken, so that io 3 AAr=i 2 .icm. io 3 AAr' = 4 .8cm. ioW=7. 3 cm. = 25 = 6o cm. If equation (5) above is solved for t' the result is or if a= 1/273 This result is certainly small, as one would estimate the temperature (red heat) at several hundred degrees higher. Unfortunately the relatively cold REVERSED AND NON-REVERSED SPECTRA. 139 ends of the tube and the leakage at the windows both contribute to a low value of t' '. But these do not seem to be adequate reasons. It is more probable that the longitudinal radiation of the air on the one hand and the value of i /a = 2 73 assumed (if this is too small) may be the chief causes for the low value of t'. It is not, of course, possible to come to any further decision; but the experiments are distinctly unfavorable to the large value of a (small i /a) above considered. The method is not adapted for very high temperatures, since equation (7) may be written and therefore, since r'AAT = rA/V, where (r referring to absolute temperature) AJV' rapidly reaches the limit of accurate measurement. 83. Further experiments at high temperatures. A variety of experiments were now made to obtain a more nearly tight joint at the ends, by using various clays, aluminum, etc., as cements, but without success. Finally, an improvement was obtained by using plaster of paris in the way shown in figure 94. A is the end of the hot tube in the combustion furnace F. The flange / is set somewhat back, so that packing of plaster p may secure the window g to the end of the tube. The plaster is put on wet and allowed to dry thoroughly. Lying outside of the furnace, 94 it is never heated to redness. The joint is at first fairly good, though it gradually deteriorates at high temperatures, and must be replaced. In this way the following results were found: TABLE 16. p. io*&N p. lo'XAW p. lo'AAT Just below red heat 74.5 12.4 Cold tube (22) 74.0 17.6 Low red heat 73.5 8.9 12.5 17.7 8 -5 12.0 17.8 8.6 18.1 8.8 17.8 Thus, from the first and second series, t'= 154; from the first and third series, '' = 330. As in the first experiments tried, both of these data are much too low. Here they can hardly be referred to the leak, since this was smaller. The ends are exposed not more than i or 2 cm. each, or a total length of about 70 cm. of tube. Some adjustment is needed at the mirrors, to place the slit images in coinci- dence for the case of an exhaustion, as compared with a plenum of air. This adjustment is slight, but unfortunately its effect on AN' can not be estimated. 140 THE INTERFEROMETRY OF Cooling of gas as resulting from longitudinal radiation might be suggested, but, as it was not encountered in the case of the steam tube, it would not seem to be menacing here. Finally, it will be seen from equation (8) that the effect of a leak is to make AN' too small. It will be larger as the vacuum is more perfect. Hence t r should be too large for this reason. A small t' can not be due to a leak. The exhaustion effect, since the gas expands into a vacuum, can not be serious. None of these incidental difficulties seem adequate to account for the large temperature discrepancies consistently obtained. All things considered, it seems to me most probable that the temperature coefficient, as the gas enters the region of red heat more fully, continually decreases, and that this is the real explanation of the low temperature values obtained. The apparatus was now taken apart and provided with a fresh jacket. After drying, the cold apparatus again appeared in good condition. The results with the barometer at 75.55 cm. were Cold (22) p io 3 AAT Red hot p icMW 73.0 17.7 73.0 8.0 17-8 17.8 Unfortunately the glass cracked after the first experiment at red heat. The data for AW (cold) agree almost exactly with the preceding results. The high temperature would be t' =383, again enormously too low. Nevertheless, if the values of a were in question, as the temperature must have been at least 850, this would come out as low as 01=0.0015. The misgivings already enumerated apply here as before. As the experiments are very laborious they were abandoned at this point, for it did not seem that further work would materially enhance the result; nor was it thought necessary to actually measure the high temperatures. 84. Flames. In the earlier report on the refraction of flames an abnormally low result of n was obtained for the ignited gases. I have since repeated this work with additional improvements. It appears that it is quite possible to look through the peak of the blue case (symmetrically) without destroying the interference pattern as a whole, though this naturally quivers excessively. The last of the new results showed for the presence (A/ 7 ) and (A/") of the flame the micrometer readings : N', flame 0.029 0.029 0.029 0.029 0.029 N, air .0297 .0295 .0294 .0296 .0296 Hence the mean difference is 0.00056 cm., or per centimeter of breadth (2.3 cm.), 8N = 0.00024 cm. If the space occupied by flames were vacuum, the difference would have been 0.000268 per linear centimeter. Thus AW' = 0.00002 cm., which lies within the error of observation, but is otherwise quite of the order to be expected for the hot gases in question. REVERSED AND NON-REVERSED SPECTRA. 141 85. Conclusion. Though the experiments made are of a tentative charac- ter, the inference seems warranted that, so far as my work goes, the tem- perature coefficient a of air at low temperature is identical with the coeffi- cient of expansion of gases. At high temperatures the value of a seems to decrease rapidly, in proportion as the gas is more highly ionized at red heat. It has occurred to me that such ionization might load the gas in relation to the light-wave passing through it, and that the observed excess of index of refraction over the value anticipated at high temperatures might be explained in this way. But air ionized by the X-rays shows no such effect. Neither does the refraction of flames at high temperatures, so far as can be made out, show a large value of the refractive index of the ignited gases. It is difficult to see how the experiment at red heat can be improved, unless a quartz tube is made for the purpose. But even here the difficulty of obtain- ing adequately plane parallel ends and a tube of sufficient breadth is formid- able. The attempt to grind in reentrant glass cylinder-like stoppers at the end of the tube was thought of, but did not succeed. CHAPTER XII. ADIABATIC EXPANSION OBSERVED WITH THE INTERFEROMETER. 86. Introductory. In the preceding report 1 I tested a number of receivers in which air was expanded adiabatically, by passing one of the component beams of the displacement interferometer through the air contained. The vessels then used were not very satisfactory, being, as a rule, not long or capacious enough to insure trustworthy results. Moreover, the interferometer did not at that time admit of the introduction of long or bulky apparatus, whereas in the new form a length of almost 150 cm. is available. The main purpose of the research will thus be to ascertain how long and thin a tube may be made to be serviceable for expansion experiments. Furthermore, it seemed worth while to repeat the work preliminarily with a large, staunch tank since found in the laboratory. This was a heavy cylinder of cast brass, about 27.1 cm. (inside) and closed by plates of heavy glass, each 0.56 cm. thick and 20.3 cm. apart (inside), the whole containing a volume of air of about 11,713 cubic centimeters, to be increased to 12,800 cubic centimeters, because of the efflux pipe. The expansion pipe was 2 inches in diameter and closed by a 2>- inch brass stopcock, with a plug practically floating in oil to prevent the ingress of air from without. The glass plates were secured by iron bolts, a layer of resinous cement (equal parts of beeswax and resin) between glass and the flat end faces of the cylinder being introduced to prevent leakage. To expand the gas in the receiver, the 2 -inch pipe communicated with a tall, galvanized iron boiler used as a vacuum chamber, 29.4 cm. in diameter and 147 cm. high, thus containing a volume of 99,800 cubic centimeters, or 100,200 cubic centimeters with the influx pipe. It was in communication with a large air-pump and provided with a mercury gage for the measurement of the partial vacuum produced by the pump. The air flowing into the air- chamber after exhaustion was dried in the usual way and the influx controlled by a fine screw stopcock. There was a special opening for a thermometer. Vacuum and air-chamber were rigidly connected by a brass union with a rubber washer. There was no appreciable leakage so far as the atmosphere without was concerned. The 2-inch stopcock, however, was not quite tight within, so that air passed very slowly from the air to the vacuum chamber, in proportion as their pressures were different; but as the air-chamber is in service, either at atmospheric pressure (the influx cock being open) or, after exhaustion, at approximately the same pressure as the vacuum chamber, this leakage was of no appreciable consequence. Otherwise the interference pat- tern would not have been stationary. While this apparatus was not long enough to fully realize the advantages of the method of displacement interf erometry for the purposes in question, Carnegie Inst. Wash. Pub. 149, Part II, Chapter IX, 1912. REVERSED AND NON-REVERSED SPECTRA. 143 it was useful for testing the ring method in comparison with the former. The equivalent of a vanishing interference ring is here not immediately given in terms of the wave-length of light, since the rings move through the spectrum. With the exception of a few incidental experiments of my own, optic methods of the present kind have not hitherto been used. They are here par- ticularly applicable, since the number of the rings vanishing in a given region of the spectrum has merely to be counted after the sudden exhaustion and during the period of slow influx of air. Succeeding parts of the chapter will refer to other available forms of ap- paratus with similar ends in view, and the additional purpose of ascertaining how long and narrow an apparatus may be shaped, without seriously inter- fering with the adiabatic measurements; for if the apparatus is increased indefinitely in length and diameter, it is obvious that the suddenness of the exhaustion through any available pipe will be more and more impaired. The same is true if the apparatus, for a given (sufficient) length, is too narrow, though for a different reason. TABLE 17. Values of 7. Bulky air chamber, F=99,8oo cub. cm., = 11,620 cub. cm. = 1.116. C=952.6; 1+^=1.0341; 0=20.3 cm - Series. t Po P 10'AJV J Number of rings. 7' I c. 22.4 cm. 75-88 cm. 56.38 56.88 cm. I-I5 95 02 I.I5 1-39 1 44. 30 29 2Q 1.49 1.50 I 5O 1.02 Ill 29 1.50 II 22.4 75-88 47-68 1.53 i-33 1-45 i-47 1.29 i-53 1.38 1.36 46 4 6 46 4 6 1.58 1.58 1.58 1-53 III 22.8 21. 2 75-70 38.50 1-95 2 25 1.38 14. '6l 'do 1-54 I 54. 23-6 24.0 2.05 1.91 .29 .42 61 62 1.53 1.50 IV 2 22. 5 2 22. 5 23-6 21.6 76.75 30-35 2.65 2.2O 2-73 2.J.Q .28 65 .21 I.VJ 77 76 78 78 1.58 i-59 i-54 i.sj. 1 Count broken owing to flicker of arc ; obtained from rhythm. 2 Sunlight. 87. Experiments with short, bulky air-chambers. An example of the data obtained is given in table 17, where the ratio of specific heats, 7, computed directly both from displacement of ellipses and 7' from interference rings, is shown in detail. The original pressure of the air-chamber is that of the barom- eter, pQ. The pressure of the vacuum chamber is given under p. The dis- placement, A7V, from four independent observations in each case and the number of interference rings vanishing from exhaustion to plenum are the data chiefly of interest. It has not been possible, according to the table, to * Carnegie Inst. Wash. Pub. 149, Part II, 83, 9.129; 85, p. 135. 1912. 144 THE INTERFEROMETRY OF place the micrometer with an accuracy of more than 0.0002 cm. or 0.0003 cm. in successive cases, AN being the difference of two readings, each uncer- tain to 10^ cm. But the effect of this is to throw out 7 by about the same number of tenths, so that the roughness of values in the table is inevitable. On the other hand, however, 7 obtained by displacement is usually too small, whereas the value computed from the evanescence of rings is always much too large. Thus in the first series there should have been an evanescence of 3 1 rings, in the second of about 50 rings, in the third of 64 rings, in the fourth of 85 rings. The reason for this discrepancy is very hard to determine, but will be considered in the next paragraph. The mean values of 7 from displace- ment and from rings are usually more nearly correct than either, as if the errors were equal and opposite in the two cases. The error is, in some way which has not been made out, associated with the placing of the micrometer. Thus, without apparent cause, the micrometer reading with a plenum of air may differ by several 10"* cm., so that if these discrepancies are in opposite directions the value of 7 shows such large divergences as in series 4, for in- stance. In other words, the error appears to be extraneous to the method of experiment. It has been suggested that the number of vanishing rings observed is approx- imately about 10 per cent too small throughout, and that the corresponding data for 7, though excessive, are nevertheless of the same order of value. Experiments were made to determine whether the change of wave-length, X, influenced this result. This was done by allowing the center of ellipses in one case to move from the D line towards the red, in the other from the yellow into the D line. The mean wave-length would in the last case be smaller, and one may estimate the former as -X D AAT where AN is the displacement of mirror which passes the center of ellipses from the C to the D line. This was found to be io 3 AAT = 28.1 cm. Hence AAT - Even in the final case, therefore, where io 3 A]V=2.s, \ D would not be in error by more than 0.5 per cent. Using sunlight and at = 76.75 cm. = 48.65 cm. 7 = 294.5 (abs. temp.) the number of rings R were counted when the ellipses traveled into the D line and from the D line, respectively, with results of which the following are examples : From D line, tf = 47 4 6 46 Mean # = 46.3 Into D line, # = 45 47 46.5 46 Mean # = 46.1 Indifferent, R = 4 6 46 Mean = 46.0 These results agree with the second series of table 17, and there is thus no appreciable difference. REVERSED AND NON-REVERSED SPECTRA. 145 One may note that the results for 7, when rings are counted, are consistently too large, but always of the same order. In fact, if R were increased by the re- duction factor r 76/2 73^0, the values of 7 would all be nearly correct; but there is no reason for such a correction. Moreover, since the data for 7 obtained from AN (ellipses brought back to fiducial position) and from R (ellipses dis- placed) are each separately consistent with each other, the discrepancy can not be due to leakages of air, as these would affect both measurements in the same way. The only source of error which is not common to both (apart from the displacement of ellipses) is the possible distortion of the glass upon exhaus- tion; for, in case of AN, measurement is made at a plenum and at maximum exhaustion only, but at varying pressures for the case of rings. Thus if the rings needed are supposed to increase in the ratio of i.3 1.6 2.0 2.5 roughly, an approximate adjustment of the two sets of observations would also be obtained. Moreover, the effect of flexure would be an increase of the path of the beam in glass and so counteract the negative effect of decreased density. 88. Effect of strained glass. To detect the possible effect of the inward flexure of the two plates of glass, a metallic ring about 25 cm. in internal diameter was provided. To this, two glass plates of about the same thickness (0.8 cm. each) as in the above vessel were cemented free from leakage and kept in place by clamps. The distance apart of the two plates within was but 1.8 cm., so that the micrometer displacement due to exhaustion of air was reduced to a small value. Hence, if the flexure of the glass plates due to exhaustion and the reverse were optically appreciable, it should here be detected. To compute the residual air effect for the lamella of air, e 1.8 cm. thick, we may write (1) Ctfo = p/(M-i)=A/U-i) where C= 952.6, t? is the temperature of the isothermal experiment, M and ju the index of refraction of air at the pressures p and po. Furthermore, (2) ni=fJoiAN/e if AN is the micrometer displacement for the pressure difference ppo at t?o- Finally, if n is the number of rings vanishing or of fringes passing at the sodium line, then Thus ilpp =Bp, then 58.93 Ctto ^ 146 THE INTERFEROMETRY OF so that n and dp are proportional quantities. The following results were found : sp No. of rings observed. No. of rings computed. Middle of glass plate . . 4 cm. above middle.. . . 8 cm. above middle.. . . At edge [30 cm. 45 [60 45 45 45 7.2 10.0 13.2 9.8 10.0 10.5 6-5 9-7 13.0 9-7 9-7 9-7 The observed data are the means of 5 or 6 trials. As it is difficult to observe the rings without interruption in an agitated laboratory, there is no doubt that observed and computed values are coincident. The first and last rings are not easily counted, and individual data were found to agree with the computed results perfectly. Finally, if the glass strain were effective (for there is actual flexure), it would be shown in the observations made by pass- ing the beam through different parts of the plate of glass, between the center and the edge. No consistent difference was found. Hence an appreciable strain effect is also absent, and the reason for the discrepancy in the two sets of values from AA 7 and from n, in table 17, remains outstanding. In the preceding report * the equation for the value of 7 log f /ft 89. Equations. is deduced as or Here p and p are the pressures in the air-chamber (barometric) and the vacuum chamber respectively, before exhaustion, t? the original tempera- ture of the air, AAT the displacement of the micrometer corresponding to the shift of the ellipses on exhaustion. If the air-chamber is quite tight, A7V may be taken at any time. C and i +x are the optic constants C = ft/#o(/io i) =952.6 for dry air, being the optic gas constant, if MO i replaces p , the normal density of the gas. To allow for the dispersion of air an empirical equation (convenient in the present calculation), was constructed. The deduction assumes that the centers of ellipses are brought back again to the fiducial line D, of the spectrum, the micrometer displacement in question being AAT. * Carnegie Inst. Wash. Pub. 149, Part n, pp. 166-168. REVERSED AND NON-REVERSED SPECTRA. 147 In the case where rings are counted, however, the center of ellipses leaves the D line by a short distance, less than one-tenth of the interval between the C and D lines. In such a case, if n* = A*+bJ\ z for air and n g = A g +b g /\* for glass, the micrometer displacement to bring the ellipses back again from X' to X should be e & and e g being the lengths of air and glass* in the beam. Here an( ^ n are ^ ne numDer of rings * Thickness of glass plates of air-chamber, 1.3 cm.; of the plate of the grating, 0.7 cm. 148 THE INTERFEROMETRY OF vanishing when a plenum of air and the adiabatically exhausted air, respec- tively, are introduced into one of the beams. Since /*> i = n<>X/2e = po/Cdo this is equivalent to AAT=(n.-)X/2 90. Experiments with long tubes. Diameter, one inch. The difficulty encountered in the case of the preceding experiments was the small value of the displacement AAT obtained. As a consequence, every little incidental disturbance produced a large effect in 7. It is the purpose of the present experiments to remedy this defect by using long tubes by which AN may be increased over ten times. It was particularly of interest, moreover, to begin with relatively thin tubes, and inch gas-pipe suggested itself for the purpose. The value of 7 to be expected will necessarily be too small, as the air must undergo reheating before the exhaust cock can be closed. The question, how- ever, is whether consistent values of 7 will be found, even for these extreme conditions and for large variations of pressure. Obviously the window plates will not produce discrepancies, as has been directly shown in paragraph 88. The gas-pipe installed was 143.4 cm. long within. To make the junction with the vacuum chamber, a straight pipe of the same diameter and about 75 cm. long was needed between the main pipe and the 2^-inch stopcock. The connecting pipe, together with the tube itself, is probably the chief cause of the resistance to flow and the low value of 7 found, but it was not possible to shorten it. The large stopcock inevitably leaked slightly when the pressures were different in the two chambers; but immediately after exhaustion this made no appreciable difference, as the two pressures are then nearly the same. In fact, no rings vanish from the spectrum from this cause. Just before exhaus- tion, however, after closing the gas-pipe by the fine influx stopcock, appreciable leakage is shown by the spectrum. Hence the exhaustion must be made immediately after the influx cock is closed. Some low results at the outset are referable to this difficulty. The tube was, as usual, filled with dry air after exhaustion. The results are given in table 18, in the same way as in the preceding case. The experi- ments themselves were throughout satisfactory, no difficulty being encountered at the interferometer. The work, moreover, is equally trustworthy at low and at high exhaustions, a result which is rather surprising. In the latter case, as the total displacement, AAT, is over 0.0276 cm., the 7 contained should be correct within i per cent. Only one attempt was made to find AAT by the march of the interference fringes. Fully 276 were observed, and it is here necessary to count the fringes passing the D line, since the ellipses are displaced throughout the greater part of the length of the spectrum; but this introduces no inconvenience whatever. The difficulty is due to the time needed in counting so many evanescences; for dunng this interval the electric lamp is liable to flicker seriouslv, or some REVERSED AND NON-REVERSED SPECTRA. 149 commotion will occur in the laboratory or without, tending to make the count uncertain. The rings disappear temporarily during the tremor. In a quiet laboratory, however, and with sunlight replacing the arc light, this would be a method of precision. Thus, for instance, at the highest exhaustions used, over 900 fringes would have to pass the D line, a datum from which 7 could be accurately obtained. TABLE 1 8. Values of 7. Iron gas-pipe, i inch internal diameter. =952.6. e= 143.4 cm. i+x=i.034i. Series. 1 Po P IO*AN y No. of rings. y I c. 19.2 cm. 76.84 cm. 57-84 cm. 8-75 8.60 8-35 8.40 .18 .21 24 .24 276 1.28 II 19.1 76.28 48.08 12.95 i"\ I s .20 .18 1 3 IS .18 .... 13-25 I -3 10 17 10 13-25 17 III 19.2 76.28 38.98 18.05 18.10 17.95 17.80 .14 .14 .15 .17 IV 19-3 76.28 29.78 22.70 22.80 15 .14 22 6S .IS 22.80 .14 V 19-3 76.28 20.88 27.85 27.60 27.60 .12 .14 .14. If we compare the mean results for 7 with the exhaustion used (pressure p in the vacuum chamber, full barometric pressure p in the air-chamber), the results decrease slightly as the vacuum is higher. Thus If p= 58cm. Then 7= 1.23 48 cm. 1.18 39 cm. 30 cm. 1.14 21 cm. 1-13 which is what might have been expected, except that the rate of decrease is much less than would be surmised. There seems thus to be no objection to the use of high exhaustions, which in turn give a better value of 7 from the large range of A A/" obtained. The low mean value of 7 obtained has been referred to the resistance of the inch piping to the outflow of air. It is probably not due to the stop- cock, as incidental differences in the speed of opening and closing would otherwise have shown a marked effect. One may conclude that the air in the long inch gas-pipe expands adiabatically with a coefficient 7 between r.i and 1.2, in case of such exhaustions as the above. 150 THE INTERFEROMETRY OF 91. The same. Diameter of tube, two inches. The experiments were now continued by enlarging the diameter of the tube to 2 inches. Brass gas-pipe, 1.35 cm. long, to be closed with thick glass plates, was at hand. To connect the same with the vacuum chamber, a similar 2-inch pipe, 115 cm. long, as far as the 2^4-inch stopcock, was necessary. Moreover, as this was in the way of the light received from the grating, the beam was reflected by an offset con- sisting of two silver mirrors in parallel. No difficulty was found with this arrangement, and the sodium line was in view to give evidence if any acci- dental displacement should occur. Unfortunately, the ellipses obtained were somewhat irregular open forms (i.e., half ellipses), and the endeavor to secure small closed patterns did not succeed. This annoyance depending chiefly on the parts of the mirror and grating used, and on shifting accessories, is not easily controlled. The indi- vidual measurements of AN are therefore not as good as those recorded in table 1 8, where a displacement of jo" 4 cm. was assured. They suffice, how- ever, for the present purposes. The new data are given in table 19, t being the temperature of both cham- bers, po the initial normal pressure of the air-chamber (2-inch pipe), and p that of the vacuum chamber. TABLE 19. Values of 7. Brass gas-pipe, 2 inches internal diameter. C=952.6; i+x= 1.0341. =135.3 cm. (V+v)/v= 1.049. Series. t Po P lo'AJV y I C. 17.0 cm. 75.66 cm. 56.46 cm. 7-15 7-35 7-25 7-55 1-35 1.30 1-33 1.27 II 17.0 75-66 47-36 10.85 IO OS 1-33 I ^1 II. 10 11.05 1.29 1.30 III 16.1 75-86 38.36 14-93 15-15 15-10 15.38 i-3i 1.28 1.29 1.26 IV 16.2 75-86 29.36 19.53 19-57 19.27 19-75 1-25 1-25 1.28 1.23 V 16.4 75-86 20.36 23.71 23-95 23-70 23.75 1.27 1-25 1.27 1.26 The effective value of y in these experiments is, for the lower exhaustions, above 7=1-3, showing a considerable improvement over the data for the inch tube, which were not much above 7=1.1. This was to be inferred, of course ; but it was not expected that the increment of 7 due to increased diameter REVERSED AND NON-REVERSED SPECTRA. 151 would be so rapid. It would seem to be probable, therefore, that if a 4-inch tube were used the conditions for obtaining a trustworthy value of 7 would be nearly met. As the exhaustions in a successive series are gradually increased (initial partial vacua from = 56.46 cm. to = 20.36 cm. in the vacuum chamber), the observed values of y gradually but slowly decrease, the mean values being cm. to 75.9 cm.) = 56.46 7= 1.32 38.36 1.29 29.36 1.25 20.36 cm. 1.26 where the fourth value is too small, for incidental reasons. This general result is also to be expected; but it is rather remarkable that with such high exhaustions as those finally used the decrease of 7 is not more marked. The work, as a whole, progressed smoothly throughout, the only interfer- ence with precision being the incidental occurrence of open ellipses. To obtain other patterns would have required longer additional adjustment than the work at the present stage seemed to warrant. 92. The same. Diameter of tube, four inches. The first experiments made with the 4-inch tube are given in table 20. The completed apparatus showed a slight leak, which could not be detected after long searching. The tube was therefore admitted for a tentative series of experiments. The exhaust pipe here, as above, was rigid and straight, but only 2 inches in diameter, with a 2X-i ncn stopcock. To exhaust the air-chamber, the handle of the cock was suddenly jerked over an angle 180 between the two closed positions. The plug virtually floated in oil, as shown elsewhere. TABLE 20. Values of y. Brass pipe, 4 inches internal diameter. C=952. 6; i+x= 1.0341; = 126.9. (V-\-v)/V=i.ng. Small leak in apparatus. Series. t Po P 10' 'AN y C. cm. cm. cm. I 19.9 76.15 57-00 5-90 1.42 6.10 1.37 6 2\ 1.34 6 IS I -?6 II 23.0 75-79 38.39 12.40 1.36 12.73 1.32 12.60 1-30 12-75 I-3I III 23-2 75-79 2939 15.50 1.40 1571 i-37 15.77 1.37 As a whole, the results are disappointing; and they are irregular, for mean readings could not be made because of the leak. They are, nevertheless, interesting, inasmuch as with some of the above data they point out a special source of discrepancy. It will be seen that the 7 values tend to decrease in successive measurements, beginning with a high value, which is here nearly 152 THE INTERFEROMETRY OF correct. This can not be referred to the temperature of the 4-inch tube, because the initial optic density is necessarily measured. It must therefore be due to the temperature of the vacuum chamber. It follows, therefore, that the time allowed in these experiments, between observations, though sufficient for establishing the initial temperature of the air-chamber, is not sufficient for the much larger vacuum chamber. The two chambers are thus no longer at the same temperature, a condition which the equations implicitly assume. The apparatus was now taken apart and thoroughly overhauled. After reassembling the parts, the chamber was found free from leakage. As the exhaust pipe was in the way of the beam of light entering the telescope, the offset, consisting of two parallel mirrors firmly adjusted, was used without annoyance, here as above. The work throughout progressed smoothly, though the ellipses were again not as satisfactory in form as would have been desirable. TABLE 21. Values of 7. Data as in Table n, 4" brass pipe. Series. / P P lO'Atf y I c. 16.5 cm. 75-90 cm. 56.80 cm. 6-34 6.15 6.27 6. to 1-33 1-37 1-34 I ^5 II 16.6 75-90 47.60 9.25 Q 4.O 1.38 I 16 Q 7Q I ^4. 16.7 76.19 47.89 9.20 9-55 9-3i 9-45 1-39 1-33 1-37 i-35 III 16.9 76.19 38.69 12.87 12 70 1-34 i 16 12 80 I 35 12 76 I ^5 IV 17.0 76.19 29.69 16.26 35 16 25 28 V 17.1 76.19 20.69 20.00 19.60 19-73 19.69 35 .40 38 39 Table 21 contains the results. Changes in the values of AAT in a given series are most likely referable to the form of the interference pattern, indi- rectly to the flickering of the electric lamp. There seems to be no evidence to associate them with the manner in which the 2^-inch stopcock is opened and closed. This was merely jerked around 180, between the two closed positions of the plug, and, so far as can be seen, the rate of motion is adequate. The successive observations show no consistent difference, as was the case REVERSED AND NON-REVERSED SPECTRA. 153 in the preceding table. Hence this discrepancy has been eliminated. What is most interesting is that the 4-inch tube shows no consistent difference in the 7 values for high or low exhaustion. Thus the mean values under increas- ing exhaustion, p, are = 56.8 47.6 38.7 29.7 20.7 7 = 1.35 1.36 1.35 1.36 1.38 Accidentally the highest value of 7 belongs to the highest exhaustion. The chief anticipation of the work (i.e., that with a 4-inch tube the true value of 7 would appear) has not been fulfilled. The value obtained is still much below normal, successive results ranging as follows: Diameter of tube 2.5 5.0 10.0 cm. Mean 7 1.17 1.29 1.36001. Diameter of exhaust pipe 2.5 5.0 5.0 cm. The relatively small increase between the tubes 5 cm. and 10 cm. in diam- eter is disappointing. At the rate obtained from the first two experiments (see fig. 95) a 3-inch tube should have been nearly sufficient. At the rate established by the last two observations, however, a tube at least 5.5 inches U 1* 1-0 t _u- I / *^~ ^ < I < Sbioun ..,wwA s #- 95 in diameter would be needed to obtain trustworthy values of 7. These differ- ences are possibly due to the exhaust pipe, which in case of the last observation does not increase in size. Hence a 3-inch pipe with a 4-inch stopcock may be estimated as being adequate for 7 measurement, provided the exhaust pipe is straight and clear throughout. The observations were broken off at this point, with the object of searching for some means of obtaining a more sensitive and regular interference pattern. If the method is to be ultimately successful, then icr 4 cm. on the micrometer must be guaranteed. If the ellipses are not quite regular or not closed, this is not the case. A more sensitive method of defining optic density is thus in question. CHAPTER XIII. MISCELLANEOUS EXPERIMENTS. 93. Effect of ionization on the refraction of a gas. It seemed interesting to test this question carefully, although a negative result was to be expected. Accordingly one component beam was surrounded by a thick iron tube, while the other was allowed to travel freely in air, along a path energized by the X-rays. For this purpose the X-ray bulb was placed near the grating and the radiation directed toward the mirror N, the beam GM being inclosed. A thick sheet of lead, i foot square, was placed behind the bulb to additionally screen off radiation along GM. Under these circumstances the ionization along GN must have been enormous by comparison with GM. Quiet ellipses were pro- duced in the interferometer, and the effect of opening the X-ray current and closing it again, alternately, was observed. Not the slightest deformation of the ellipses or any motion of the fringes could be detected. An ionization effect is therefore wholly absent. It might have been supposed, for instance, that the ions present might load the wave of light and produce an appreciable result in the interferometer (cf. fig. 92). Since a shift of o.i of a ring would probably have been detected, AAT = 0.000005 cm. would have produced a perceptible effect. Hence, since ni is, roughly, equal to A N/e, the value of the ionization effect could not exceed The ionization effect can not, therefore, exceed o.oi per cent of ju~ i- To further test this question, the iron tube, i inch in diameter and 138 cm. long, was provided with a fine axial wire about 0.02 cm. in diameter, passing through central holes in the glass plates at the end. The ends of the wire were drawn tight by hard-rubber rods on the outside, so that the tube became a cylindrical condenser. All holes were sealed hermetically with resinous cement. The interference fringes were clearly producible. The poles of an induction coil were now connected with the inner wire and the tube, respectively, to alternately change the condenser and discharge it, with the object of strongly ionizing the air within. On partial exhaustion the whole tube became luminous, on account of the discharge, in the usual way. The best results were obtained with a plenum of air when but two storage cells actuated the coil. Under these circumstances no sparks passed from core to shell of the iron condenser tube, while the air within was intensely ionized by the silent discharge. On closing the current, from 0.5 to i per cent of the rings was swept inward at once. On opening it, the rings again emerged. This inward motion, however, was in the same sense as the effect of a decrease of 154 REVERSED AND NON-REVERSED SPECTRA. 155 density, such as would result, for instance, from rise of temperature or from partial exhaustion. Hence the effect observed, though very definite, would correspond to a temperature effect due to electrical currents traversing the air. One should expect the effect of ionization, if appreciable, to be the reverse of this. With voltages high enough to produce sparks in the tube, the inter- ference figures naturally show violent agitation or quiver. If the displacement in question is one ring and 5 denotes differences, S(AAT) = 30X10-* cm. If only temperature changes, one may write, roughly, A./V.r = constant, r referring to absolute temperature, whence 40 X 10 if results found for a similar tube, above, be taken. Thus ?>T = 2.2Xicr 7 degrees centigrade is the average temperature incre- ment, for the whole length of the tube. When but a single cell was used to energize the coil, no effect could be recog- nized. In case of two cells, moreover, when the plenum of air was replaced by a partial vacuum of i cm. or less, so that an arc was seen, no effect was observable, although the reddish light colored the field of the telescope. There are two points of view, however, from which the assumption of a tem- perature effect is not admissible. If the pipe is closed, so that the density of the air contained remains unchanged, there is no difference in the phenomenon. But there should not, for the case of constant density, be any effect, unless the nature of the gas is changed. Again, the effect is instantaneous and not increased on keeping the circuit closed. The simple explanation in terms of temperature made above must therefore be taken with reservation. At all events, the effect of ionization would be small and equivalent to a dilution of the gas of but --- X 2.2 X io~ 7 or about icr 9 273 of its density, when sparks are about to occur. 94. Mach's interferences. It is frequently necessary to use the interferom- eter in such a way that but one ray passes in a given direction ; i.e., the rays are not to retrace their path. Interferometers of this kind are treated above, but Mach's design offers advantages, which will be presently pointed out. As a rule, in using these interferometers, the center of the elliptic interference pat- tern is remote and the lines are hair-like and found with great difficulty. These annoyances are overcome when the apparatus is put together as follows: In figure 96, L is the vertical sheet of light from a collimator impinging on the strip of plate glass gg, half-silvered on one side, toward or near the ends. The pencil L is thus reflected to the opaque mirror N and transmitted to the opaque mirror M (on a micrometer), and then reflected to the other end g' of the glass strip gg' . Thereafter, both the pencils, Mg' and Ng', are available; 156 THE INTERFEROMETRY OF but it is generally more convenient to use the former (Afg') reflecting it from the plane opaque mirror m to the telescope at T. When L came from sunlight, or from an arc light, etc., the white images of the slit were very bright. After putting them in coincidence, horizontally and vertically, by aid of the three adjustment screws on the mirror M, Ives prism-grating G may be placed in front of the objective of the telescope. A very brilliant spectrum thus appears, and the fringes are easily found by moving the micrometer slide which carries M to the proper position. In my apparatus gg' was about 50 cm. long and gM=gN about 2 meters. The telescope is sufficiently near M to manipulate the micrometer, the mirror m being so placed that the beam just misses the strip gg'. ctif 96 The interference pattern, found at once and satisfactorily centered, consisted of large, broad circles. On moving the micrometer M from evanescence on one side of the center to evanescence on the other, the slide was found to have moved over about 2 mm. With a stronger telescope to magnify the fine, hair- like fringes, this distance would have been larger. It is interesting to compare this datum displacement with the datum found in the case of the phenomenon above, where a range of over 0.5 cm. (double path-difference) was observed. In the present experiment the range is smaller, because the interference pattern falls below the limit of visibility before the possibility of interference is exhausted. Mg' slides along g r when M moves. 95. A Rowland spectrometer for transmitting and reflecting gratings, plane or concave. In the above experiments I had occasion to examine a variety of gratings, and it was therefore desirable to devise a universal instrument by which this could be accomplished without delay. The method chosen is sim- ilar to that previously described,* but its details have been greatly simplified, on the one hand, and made more generally applicable, on the other. It seems permissible, therefore, to give a brief description. * Carnegie Inst. Wash. Pub. No. 149, Chapter I, 1911. REVERSED AND NON-REVERSED SPECTRA. 157 In figure 97, GG' and HE' are double slides like the carriage bed of a lathe, each about 1.5 to 2 meters long and 10 cm. wide, rigidly fastened together. They are placed at right angles to each other on a flat table, the vacant distance between G' and HH' being less than a meter. For ordinary purposes they need not be screwed down. A, B, D, K, are flat carriages, or tables, provided with screw sockets for supporting the different standards, and capable of sliding to and fro with a minimum of friction. A carries the micrometer slit 5. B and C are joined by the Rowland rail R, whose length is thus equal to the radius of the concave grating to be examined, or nearly so, so that the ends of R are on vertical axes at b and d. B also supports the table C (somewhat enlarged in the side elevation, fig. 98) , on which the table t of the grating g may be adjusted on its leveling screws. To secure a common axis, b, e, the rod at ace is twice bent at right angles. Moreover, if c is turned to one side, the supporting rod e may be screwed into the vacant socket b at the end of R. For the case of fig- ure 98, the angle of diffraction 6 is varied and \=D sin 6, where D is the grat- ing space. For the other case (c being turned aside and C screwed into and turning with 6) the angle of incidence is varied and X = D sin i. This is much simpler in form than the early method used. 97 Finally, the table C carries the essentially new addition to the apparatus (shown in front elevation in fig. 99), viz, the long slot ff, adapted to support the right-angled reflecting prism E and at the same time to allow free play to the rail R within _/f. Figure 99 then shows the progress of the rays (turned 90 to the front in a horizontal plane) from the slit or collimator, S. They are doubly reflected at E, return in a vertical plane and then impinge on the grat- ing at G. The rays thereafter pass along the rail R (fig. 97) and are examined by a strong eyepiece at d (not shown), rigidly but adjustably attached to the near end of the rail. The displacement of K along HH' is accurately measurable on a parallel scale with vernier (not shown) . If Xi and x% are the two symmetrical readings on opposite sides of the virtual slit image at 5 (fig. 97), and R the radius of the concave grating, and x = xt Xi sin , or sin 158 REVERSED AND NON-REVERSED SPECTRA. If a plane grating is used, a weak lens L is attached to the rail R and moves with it, so that its focus is in front of the ocular d (with cross-hairs). In this case 5 is a collimator. If a transmitting grating is examined, the collimator 5 (fig. 99), etc., are merely to be lowered, and the prism E is superfluous. It need not even be removed. Naturally, it is in the interest of accuracy to have all the standards like e and h as short as possible. Finally, in the equation ^ = ~5 itD=io*d, the values d and R are usually of the same order (175 cm.) for gratings with about 15,000 lines to the inch. In this case we may make the rail length R = d, whence \=X/2 Even in case of the concave grating, when ultimate precision is not aimed at, some variation of the distance SS' = 2SE, nearly, is admissible without destroying the definition. The carriage D with the prism E may be moved fore and aft on the slides GG' until the focus at d is sharp. The values of x are usually of the order of 100 to 125 cm., so that an accuracy of Angstrom units is easily obtainable without special refinement. THE INTERFEROMETRY OF REVERSED AND NON-REVERSED SPECTRA PART II By CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON WASHINGTON, 1917 CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No. 249, PART II PRINTED BY J. B. LIPPINCOTT COMPANY AT THE WASHINGTON SQUARE PRESS PHILADELPHIA, U. S. A. CONTENTS. CHAPTER I. Methods for Reversed and Non-Reversed Spectrum Interferometry. PAGE. 1 . Introductory 9 2. Apparatus. Figs. I, 2, 3 9 3. Measurements. First- and second-order spectra. Tables 1,2. Figs. 4, 5 II 4. Continued. First-order spectra 13 5. Continued. Second-order spectra. Table 3 14 6. Theory. Fig. 6 14 7. Compensator measurements. Sharp wedge 15 8. Continuation. Revolving plate. Table 4. Fig. 7 16 9. Continuation. Air column. Tables 5, 6 18 10. Continuation. Babinet compensator. Fig. 8 19 11. Micrometer displacement of the second grating, G' (fig. 3). Table 7 20 12. Prism method. Reflection. Table 8. Figs. 9, 10, II 21 13. Prismatic refraction. Figs. 12, 13 25 14. Prism methods without grating. Figs. 14, 15, 16 26 15. Displacement parallel to rays. Table 9 28 16. Breadth of efficient wave-fronts. Apparent uniformity of wave-trains. Rotation of fringes. Figs. 17, 18 30 17. Film grating. Figs. 19, 20 32 18. Non-reversed spectra. Figs. 21, 22, 23, 24 34 19. Non-reversed spectra. Restricted coincidence. Figs. 25, 26 38 43 45 48 52 55 20. The same, continued. Homogeneous light. Dissimilar gratings. 21. The same, continued. Duplicate fringes. Figs. 27, 28, 29. 22. The same. Prismatic adjustment. Figs. 30, 31.. 23. Apparent lengths of uniform wave-trains. Tables 10, II. Figs. 32, 33, 34. 24. Normal displacement of mirrors (rf=o). Figs. 35, 36, 37 ) Figs 25. Diffraction at M, N, replacing reflection. Table 12". Figs. 38, 39 57 26. Experiments with the conca 27. Polarization. Figs. 40, 41.. 26. Experiments with the concave grating 59 CHAPTER II. The Interferences of Inverted Spectra. 28. Introductory 62 29. Apparatus. Non-inverted spectra. Fig. 42 62 30. Apparatus and results for inverted spectra. Figs. 43, 44, 45 64 31 . Wave-fronts narrowed. Table 13. Figs. 46, 47, 48 65 32. Inverted spectra. Further measurements. Table 14 67 33. Rotation of fringes. Fig. 49 69 34. Range of displacement varying with orientation of reflector P' 70 35. Range of displacement varying with dispersion 72 36. Spectra both reversed and inverted. Figs. 50, 51 73 37. Experiments with the concave grating 74 38. Conclusion. General methods. Fig. 52 75 39. Displacement interferometry. Equations. Fig. 53 77 40. Continued. Reversed spectra, etc. Tables 15,16 80 CHAPTER III. Elongation of Metallic Tubes by Pressure and the Mesurement of the Bulk Modulus by Displacement Interferometry, 41 . General method and apparatus. Fig. 54 84 42. Remarks on the displacement interferometer. Figs. 55, 56 85 43. Observations. Thick steel tube 86 44. Further experiments. Tables 17, 18. Fig. 57 87 45. Brass tube. Tables 19, 20. Figs. 58, 59, 60, 61 90 46. Thin steel tube. Table 21 92 47. Conclusion. Thermodynamic application. Fig. 62 93 3 CONTENTS. CHAPTER IV. Refr activity Determined, Irrespective of Form, by Displacement Inter- ferometry. PAGE. 48. Introductory 95 49. Preliminary experiments. Figs. 63, 64 95 50. Apparatus 96 51. Equations. Table 22 97 52. Observations. Tables 23, 24. Figs. 65, 66 99 53. Dispersion constants. Tables 25, 26. Fig. 67 100 54. Further observations. Tables 27, 28. Figs. 68, 69 102 55. Conclusion. Table 29 105 CHAPTER V. Displacement Interferometry in Connection with U-Tubes. Jamin's Interferometer. 56. Introduction. Figs. 70, 71, 72 107 57. Apparatus. Michelson interferometer 107 58. Equations 109 59. Observations no 60. Jamin's interferometer. Figs. 73, 74, 75, 76 ill 61. Vertical displacement of ellipses. Figs. 77, 78 1 14 62. Displacement interferometer. Jamin type. Tables 30, 31, 32, 33. Fig. 79 .... 115 63. Broad slit interferences. Achromatic fringes 120 64. Wide slit. Homogeneous light. Sodium flame. Figs. 80, 81, 82, 83, 84, 85 121 65. Vertical displacement. Table 34. Fig. 86 126 66. Angular displacement of fringes. Table 35. Fig. 87 128 CHAPTER VI. The Displacement Interferometry of Small Angles and of Long Dis- tances. Complementary Fringes. 67. Parallel rays retracing their path. Figs. 88, 89, 90, 91, 92 130 68. Groups of achromatic fringes. Table 36. Fig. 93 133 69. Measurement of small angles without auxiliary mirror. Table 37 134 70. Complementary fringes. Figs. 94, 95 135 71. Equations 138 72. Separated rigid vertical system. Figs. 96, 97 139 73. The displacement interferometry of long distances 141 74. Theory. Table 38 142 PREFACE. In the present volume I have pursued the work on the interferences of reversed and non-reversed spectra, begun in my last report (Carnegie Inst. Wash. Pub. No. 249, 1916), in a variety of promising directions, such as the original investigation suggested. It will be remembered that the reversal (180) here contemplated takes place on a transverse line of the spectrum (i.e., a line parallel to the Fraunhofer lines), which thereby becomes a line of symmetry for the phenomena. The apparatus has been extensively modi- fied, so as to admit of measurements relating to individual fringes. The object of such quantitative work, however, is to furnish a guide for the devel- opment of the experiments and to corroborate equations, not to collate standard data. These could hardly be satisfactorily obtained, moreover, unless the work were done with optical plates and mirrors, whereas the work in this volume and the preceding has been done with ordinary window-plate and usually with film gratings. A large part of Chapter I is devoted to the treatment of prismatic methods, developed with the additional purpose of securing a greater intensity of light. A very curious intermediate case between the interferences of reversed and non-reversed spectra is the pronounced interference of spectra from the same source, but of different lengths (dispersion) between red and violet. The phenomena of crossed rays find a parallel occurrence in the present paper, in the behavior of duplicated fringes, when similar gratings or prisms disperse and subsequently recombine a beam of white light. A type of fringes is detected which depends merely on the grating space and is independent of wave-length. An interesting question as to the limits of micrometer displace- ment within which fringes of any kind are discernible (observations which were at first supposed to be due to the degree of uniformity of interfering wave-trains) is eventually shown to be a necessary result of dispersion. Finally, the direct interference of divergent rays obtained from polarizing media is exhibited. In Chapter II the interferences of inverted spectra, a subject merely touched in the preceding volume, are given greater prominence. In this case one of the two spectra from the same source is inverted (180) relatively to the other on a longitudinal axis (i.e., an axis normal to the Fraunhofer lines), which thus becomes a line of symmetry. In the development of the subject, spectra half reversed and spectra both reversed and inverted are treated successfully. In the latter case the conditions of interference are fulfilled at but a single point in the whole area of the spectrum field; and yet the phenomenon is pronounced and not very difficult to realize. The limits of micrometer displacement within which interferences may be obtained are again determined. At the end of the chapter it was thought useful to collate available equations in the treatment of phenomena of the present kind. I 6 PREFACE. The third and fourth chapters are incidental applications of the displace- ment interferometer and contain experiments on the expansion of metal tubes by internal pressure and on a promising method of measuring the refraction of glass, irrespective of form. To carry out the experiments in the last case with requisite rigor, optic plate-glass apparatus would unquestionably be essential. Nevertheless the tentative data obtained are noteworthy. I have begun in Chapter V the development of displacement interf erometry in connection with the older Jamin-Mach interferometer, an instrument which has certain peculiar advantages and is in a measure complementary to the Michelson interferometer. The work was undertaken in connection with the micromeasurement of the difference of heights of communicating columns of liquids, though the latter had to be abandoned in consequence of the excessive tremor in a laboratory surrounded by active city traffic. I shall hope, however, to carry out such work elsewhere. The chief result of Chapter V is the detection of the achromatic inter- ferences, as I have called them for convenience interferences which are ultimately colors of thin plates seen at oblique incidence; but with the new interferometer, and obtained with white light, they are peculiarly straight and vivid and resemble a narrow group of sharp Fresnellian fringes with the central member nearly in black and white. They are capable of indefinite magnification and their displacement equivalent is a fraction of a mean wave- length per fringe. Notwithstanding their strength and clearness, they are so mobile in connection with micrometric displacement that in general it would be almost hopeless to attempt to find them but for the fact that they coincide in adjustment with the centered ellipses or hyperbolae of the spectrum fringes of the displacement interferometer. The fine white slit-image which is dispersed to produce the latter carries the achromatic fringes when the slit is indefinitely broadened or removed. Once found, moreover, they are not sensitive to small differences of adjust- ment if a change of focal plane is admissible. The chapter shows a curious method for the measurement of vertical displacements, possibly available for the detection of ether drag, which, though just insufficient in connection with the spectrum fringes, would be promising in connection with the achro- matic fringes. Finally, the chapter contains some repetitions of the old experiments of Fizeau on the periodic evanescence of fringes due to the sodium lines. Curiously enough, the achromatic fringes also show periodic recurrence sometimes, which as yet remains unexplained. The peculiar adaptability of the new interferometer to the measurement of small angles, either in a horizontal or a vertical plane, is developed in the final chapter. The ratio of the angular displacement of fringes to the angle to be measured (*.*., the rotation either of the paired mirrors or of an auxiliary mirror in the apparatus) may be made enormously large, and the paper shows cases in which, with strong luminous fringes, the angle to be measured is mag- nified 500 times. Moreover, this is by no means a limiting performance. Again, while angles as small as a few tenths of a second or less are measured, angles PREFACE. 7 as large as several degrees come naturally within the scope of the method. Similar remarks may be made with respect to the ratio of angular displace- ment and micrometer displacement. Given, therefore, an apparatus which measures very small angles without constraint or forced approximations, the measurement of long distances is the next result in order; for it is merely necessary to place the angle to be measured at the apex of the distance triangle on the length of the ray parallelogram as a base. This may be done in a variety of ways, some of which are shown in the chapter. The sensitiveness may again be made remarkably large. The fringes here in question are preferably the very luminous achromatic fringes. They have been identified as ultimately colors of thin plates, but they look like Fresnel's fringes. In connection with this work, however, another type of fringes was detected obtainable with a fine slit, white light, and in case of centered spectrum fringes when the ocular of the telescope (or the eye) is out of focus. These are actually Fresnellian interferences, but being made up of broad concentric hyperbolic areas, brilliantly comple- mentary in color, they resemble the lemniscates of biaxial crystals without the shadows. My thanks are due Miss Lena F. Uhlig, who has assisted me efficiently in the preparation of this volume for the press. CARL BARUS. BROWN UNIVERSITY, Providence, Rhode Island, June 1917. CHAPTER I. METHODS FOR REVERSED AND NON-REVERSED SPECTRUM INTERFEROMETRY. 1. Introductory. Thus far it has been impossible to use the fringes of reversed spectra individually, because of the tremor of the apparatus. It is therefore desirable to endeavor to obviate this annoyance as far as possible, and the end would appear to be most easily obtainable if the distances cor- responding to the same path-difference are made smaller. At the same time the results for small distances will be interesting for this very reason in contrast to the long-distance methods. Furthermore, the development of different methods, with a consideration of the peculiarities of each, will constitute an essential contribution to the theory of the phenomena; for from this the degree of importance which is to be attached to the original diffraction at the slit of the collimator (i.e., the limiting angle at the slit, within which diffracted rays must lie to be subsequently capable of interference, whether reversed or inverted) will appear in its relations to the total dispersion of the system. The slit, however fine, is still a wave-front of finite breadth. 2. Apparatus. In the first experiment, the device with two identical reflecting gratings, GG', figure i, was firmly mounted on a massive spectrom- eter, the four mirrors, m, n, M, N, being specially attached. White light received from the collimator L after two dispersions was viewed at the tele- scope T. Both gratings were on a slide ss, enlarged in figure 2, set in the direction LT of the previous figure. The carriages, figure 2, was provided with universal joints (a with a vertical axis, b and e with horizontal axes normal to each other), while the swiveling of the grating G was controlled by set-screws at d, relative to the axle at e. Unfortunately, the displacement of the mirror M, figure i (on a microm- eter), passes the corresponding pencil across the face of the grating G' and thus virtually includes a fore-and-aft motion of the latter. Thus the fringes pass, with rotation, from very fine, hair-like striations, through a horizontal maxi- mum of coarseness, back to vertical lines again, when homogeneous light and a wide slit are employed. The annoyances due to tremor, however, were not overcome. Moreover, there is difficulty in obtaining Fraunhofer lines normal to the longitudinal axis of the spectrum. This method was therefore abandoned. The design shown in figure 3, with a transmitting grating at G (grating space D = 3S2Xio-* cm.) and a stronger reflecting grating at G' (D = 20oX icr* cm.) , was next tested, M being the micrometer mirror. The mean distance of M from N was about 15 cm., from MN to G' about 10 cm. and to G 40 cm. 9 10 THE INTERFEROMETRY OF Later these distances were enlarged. First-order spectra were used and the fringes obtained easily and brilliantly, particularly with mercury light, in both green and yellow. They rotated as above, admitted a displacement M of about i cm. But they were still too mobile to be used individually. The same design, figure 3, was now mounted on a round, heavy block of cast iron B, 30 cm. in diameter and 4 cm. thick, the distance G to MN being about 20 cm. A number of screw-sockets, b, b, were drilled into B on the right and left, for mounting subsidiary apparatus. G' as before was on the universal slide (fig. 2), movable in the direction LT. The tablets t, t', etc., of G, M, N, and G' were mounted tentatively on standards of gas-pipe 1.5 cm. in external diameter and 6 cm. long. Slight pressure by the finger-tips showed a passage of several fringes across the field, but the fringes were stationary in the absence of manual interferences and in spite of all laboratory tremors. A parallel arm of the same pipe was therefore firmly attached to the stem of 3D X) N and M, each arm terminating in a fine horizontal set-screw, s, s, below, adapted to push against the rim of the iron block. In this way adequately stationary conditions and an elastic fine adjustment for superposed longi- tudinal spectrum axes were both secured with advantage. Similar elastic adjustments have been recently applied. It was now possible to manipulate the micrometer at M by hand; but a glass-plate compensator C, rotated by a tangent screw over a graduated arc, was also convenient. Later other types were attached, including an air-compensator, in which path-difference was secured by exhausting the air within a closed pipe provided with glass-plate ends. These contrivances were eventually superfluous, however, as it was found that on reducing the rotation of the micrometer-screw the latter could be used at once. In case of homogeneous light and a wide slit, fringes were visible in an ordi- nary telescope for a play of over 2 cm. of the micrometer-screw, passing, how- ever, between extremes of fineness. The slit images are not of equal breadth, if first- and second-order spectra are superposed; but if the longitudinal axes are coincident, any position of the narrow image within the broader produces a wide vertical distribution of fringes, usually more or less horizontal. They are very easily found. The sodium flame is too feeble for use. The mercury REVERSED AND NON-REVERSED SPECTRA. 11 arc is unfortunately too flickering, so that the fringes jump about and are useless for measurement. Excellently sharp quiet fringes are obtained with sunlight (white), in which the cross-hatched interference pattern is nearly linear at the line of symmetry of the reversed spectra. The fringes climb very decisively up and down this line with the motion of the micrometer, reduced as suggested. The electric arc and a Nernst filament are equally available as a source of light. Finally, by suitably rotating the grating G', figure 2, on the axis e, by aid of the set of screws d, fringes whose distance apart is over one-third of the width of the telescope field may be obtained quite sharply. As this distance represents but 30 X io~ cm., there is no diffi- culty of realizing icr* cm. in case of these long fringes. 3. Measurements. First- and second-order spectra. The steadiness of the fringes, even in an agitated location, induced me to make a few measurements for orientation. Accordingly, the Fraunhofer micrometer, reading to icr 4 cm., was provided at its screw-head with a light wooden wheel w, figure 4, about 10 cm. in diameter and 3 mm. thick. A groove was cut in the circumference of the wheel, so that a silk thread t could be wrapped around it. The other end of the thread was wound around a brass screw s, about 6 mm. in diameter, turning in a nut, preferably of fiber, which was fastened to the edge of the table by a small brass clamp. In this way it was possible to control the motion of individual fringes crossing a fiducial line in the field of the telescope. This simple device worked surprisingly well, a smoothly running micrometer being presupposed. In fact, it was possible to set a fringe to a few millionths of a centimeter. Later the micrometer-head was grooved and a finer turning- screw suitably attached to the base B of the apparatus. (Cf. 70, below.) The fringes should be widened as far as convenient, by rotating the grating on the axle e, figure 2, by aid of the set-screws d. In this case they climb up or down the transverse strip as 5 in figure 4 is slowly rotated. Fringes moving horizontally are not serviceable, because they are too near together. It is not difficult to obtain the single vertical line, black or bright, on suitable rotation about e. On either side of this transitional adjustment the fringes move vertically (climb or fall) in opposite directions for the same micrometer 12 THE INTERFEROMETRY OF displacement. The arrow-shaped forms are also often satisfactory, and may be obtained by adjusting the two bright patches on the reflecting grating into coincidence, by the eye, in the absence of the telescope. The grating G' is moved fore and aft for this purpose on the slide s, figure 2, until the two bright strips become one. In making the first adjustment, I incidentally combined the first-order spectrum from N with the second-order spectrum from M, as shown in figure 5, under the impression that the wider D groups from the latter were due to slight curvatures of mirrors. The fringes were nevertheless easily found and showed no anomalies, except that observation had to be made near M or N. It appears from figure 5 that the equations for this case imply sin i + sin 0" 2 = 2\/A sin * sin 0' 2 = X/A and sin where the angles i and 6 are equal (0' 2 = 0*2). Thus sin t = 0=X/2A, A being the grating constant (A = 2ooXio~ 6 cm.). Hence sin i = 0.4420 sin 02 = 0.1423 * = 26i4' 2 = 8n' while from the first grating, A = 352X10-* cm., 0i = 938'; whence <7 =i-j-e 1 = 3S52 / 8=i 0=i6 3 6 / -- 1 = =i 3 The trial readings given in table i of the micrometer for a passage of fringes each were found without special precautions, which is equivalent TABLE i. Scale parts. io- cm. 19.7 21.0 22.2 23-4 2 4 .6 25-8 27.0 28.2 9-85 10.5 II. I 11.7 12.3 12.9 13-5 14.1 to an average of io- 6 X3o.i cm. per fringe. As the line of symmetry lay very near the two AA doublets, this is obviously an approach to half a wave- length. For accurate work AA and D\D' 2 should be superposed, in which case the fringes would lie between and actually correspond to their mean wave-length. A number of measurements like the above were now made with different types of fringes. The mean values successively taken from 3 or 4 batches of 20 fringes each were 10^ = 29.4, 30.0, 31.2, 30.6, 29.7, 30.1, 30.0, 30.0, 30.8 cm. The results were less decided when long fringes were used. The mean value of the I0 sets is thus 5^Xio 6 = 3 o.i 9 cm. per fringe. Actual or approx- mate coincidence of the D lines made no appreciable difference. REVERSED AND NON-REVERSED SPECTRA. 13 In the following results the reflection from the mirror M, figure 5, was used in the first order and from N in the second order, after leaving G' . Obser- vations were made near N, figure 5. The displacement corresponding to 80 fringes was successively 0.0024, 0.0024, 0.0024, 0.0024 cm., so that the mean value io 6 50 = 30.0 cm. agrees with the above. Similar trial observations (combined first order from N and second order from M) were made with red light near the C line in series of 6 with a mean value 5eXio 6 = 34.o cm. Again, near the b line (green), giving 5eXio 6 = 27.2 and 27.8 cm. per fringe. These should therefore be distributed in terms of wave-length, as in table 2. TABLE 2. lo'A Ratio. io6e Ratio. C D b 65.6 cm. Si-7 I. II I. 0.88 34.0 cm. 30.2 27-5 1.13 0.91 and they are as nearly as may be expected in the ratio in question, seeing that the total displacement for 60 fringes does not exceed 0.004 cm. For accurate data it would be necessary to count many hundreds of fringes, and to correct the 8e values by multiplying by sec (0 2 0i)/2. I have not done this, as the red and green fringes are not so distinctly seen as the yellow. 4. Continued. First=order spectra. The apparatus was now readjusted in such a way that first-order spectra were available from both mirrors. This puts the grating G' at a greater distance from the line M and N than before, for the angle 2 is smaller. A series of trial results were investigated in the same manner as above, the mean values from 4 successive pairs of 80 fringes each being, in three repetitions, 5eXio 6 = 2g.o, 29.4, 29.5 cm. and from 5 pairs of 100 fringes each, This makes an average of 50Xio' = 29.25 cm. somewhat smaller than half a wave-length of the D light used. Unfortu- nately, the screw at 5 (fig. 4) here worked jerkily, to which the low value is probably due. In this case sin 6 1 =\/D i , where > 1 = 352Xio- cm. or 0'i = 938'; sin 0' 2 = X/A, where A = 200X10-* cm. or 0' 2 =i79'> whence , de= oo per fringe; i.e., fore-and-aft motion would produce no result. This is necessarily TABLE 7. '= increasing; d = decreasing. No. of fringes. Total displace- ment. Mean 8e displace- ment per fringe. *3 0.0234 cm. 0.0088 cm. 3 267 3 293 4 .0302 .0080 4 317 4 3i8 4 346 4 330 d 4 .0285 .0073 4 285 4 287 4 315 4 288 the case when but a single grating is used, as in the earlier methods. In the case of two gratings, however, it is not only difficult to make a perfectly symmetrical adjustment of mirrors and grating, but it would not be of any REVERSED AND NON-REVERSED SPECTRA. 21 special advantage. Hence the fore-and-aft displacement e of the grating G' will probably be accompanied by a slow motion of the fringes, from which the angle a

, is essential (? = 18, about), as a large divergence of rays would not be accommodated on the interfer- ometer, figure 3 . The fringes were found without difficulty, in the second order, the arc lamp being used. They are also easily distorted, if the edge of the prism is not parallel to the rulings of the grat- ing. In such a case the symmetrical arrow-shaped forms become one-sided and, as it were, curved or faintly fringed beyond the limits of the strip. To get the best adjustment, the lamp should shed about the same amount of undeviated light from both faces' of the prism, on a screen temporarily placed behind it. The illuminated strips on the grating must coincide to the eye while making the fore-and-aft adjust- ment. Finally, the grating is to be slowly rotated on the axis normal to itself, until fringes of satisfactory shape and size appear. Naturally this is done through the telescope, and a readjustment of the longitudinal axes 22 THE INTERFEROMETRY OF of the spectra is necessary after each step of rotation. Fringes so obtained are as good as those obtained by any other method. The range within which the fringes are sharp is small, not exceeding 2 mm. of displacement of the micrometer mirror, M. A partial reason for this will appear from figure 9 and results from the fact that the illumination on the grating due to M moves laterally across the stationary strip due to N. Clearly if the latter were also on a micrometer it might, in turn, be displaced relatively to the direction of M and restore the fringes to full brilliancy. The range in this case may be increased till either illuminated strip gets beyond the edges of the grating. This test will presently be made. If the prism angle is

= i8i2' 0==366' 5 = 17 54'; 5/2 = 8 57' Hence _ 25 X 0.1556 0.808 if e = 0.2 5 cm., as found. Thus the rays of the same origin, or rays capable of interfering, are found in a vertical strip on the grating, not more than i mm. wide. It is interesting to note that the fringes vanish by becoming coarser and wider, corresponding to the narrowing of effective edges in contact. The attempt to produce these fringes with homogeneous (sodium) light and a wide slit again failed, although much time was spent in the endeavor. Even with a narrow slit and accentuated sodium lines (impregnated arc) the phe- nomenon may be produced between the doublets, however close together, but it fails to appear with the same adjustment when two corresponding lines coincide. I was only able to produce it in a continuous spectrum, between the two doublets and with a fine slit. It is very important to ascertain the reason. REVERSED AND NON-REVERSED SPECTRA. 23 Both mirrors, M and N, were now placed on micrometers moving nearly normal to their faces. Beginning with a coincidence of the illuminated strips on the grating, the M micrometer was moved until the fringes disappeared. The N micrometer was then moved in the same direction, until the reappearing fringes passed through an optimum and finally vanished, in turn. There- after the M micrometer was displaced again, always in the given direction, and the same cycle repeated, etc. It was possible to pass through about 8 cycles with each micrometer before the illumination reached the edge of the grating, each cycle corresponding to a displacement of about 2 mm. for a single mirror; but a total displacement of 2.5 cm. was registered, which would obviously have been increased much further if the grating had been wider. The data given in table 8 give a concrete example : TABLE 8. Position <&N. Advance otN. Remarks. Position olN. Advance otN. Remarks. 2.42 cm. Broad arrows 1.75 cm. Vertical lines 2.20 J-54 0.22 cm. M advanced 0.21 cm. M advanced 2.2O Narrow arrows 1-54 Vertical lines 1.98 .... i-37 .24 M advanced 1*7 M advanced I. 9 8 i-75 Upright lines, in- clination changed 23 M advanced As both mirrors move in the same direction, the two illuminated strips on the grating gradually separate until they are quite distinct. Meanwhile the fringes pass with rotation from the original sagittate forms to very fine hair- like striations ; whereas the part of the spectrum within which the former occur is less than the distance apart of the sodium lines (doublets), the hair- lines are visible within a strip of spectrum many times as broad as the sodium doublet. Ten such lines may be visible. In good adjustments the sagittate forms are seen to be a nest of very eccentric, identical hyperbolas, as in figure 10, A, arranged or strung on the same major axis. The vertices a are therefore thick and pronounced, but taper rapidly down into hair-lines, 6, &', on both sides. Frequently but half of the coarse vertices, a, abundantly fringed on one side, 6 or V, appear. Nevertheless this does not seem to be an exhaustive description of the phenomena; for it is not uncommon, when partial hyper- bolas appear, to find the striations (which are always faint) in the same direction on both sides, as in figure 10, B\ i.e., the striations are apt to be non-symmetrical on the two sides, as if they constituted a second diffraction phenomenon superimposed on the first phenomenon. Roof -shaped forms (fig. 10, Q, strongly dotted, are also common, often irregularly awned. Figure 1 1 may be consulted to further elucidate the subjects under con- sideration. G is the grating, PP' the principal plane of the objective of the telescope, a and 6 are two rays interfering at the focus /, and leaving the grating parallel and symmetrically placed to the axial ray of. The passage 24 THE INTERFEROMETRY OF of the coarse sagittate phenomena into the hair-like striations, as a and b move farther apart, may then be accounted for in accordance with the general theory of diffraction; i.e., if the distance apart of a and b is d and the principal focal distance of is R, - = d ~ R where z is the distance between the two fringes of wave-length X. Hence z will increase as d decreases, agreeing with the effect of fore-and-aft motion, or with the effect of simultaneous, large (2.5 cm.) displacement of both mirrors, neither of which destroys the symmetry of the interfering rays. 10 The motion of a single mirror, M or N, for instance, does destroy the sym- metry, and it was shown in 12 that the limiting range of displacement of 0.25 cm. moves either a or 6 0.096 cm. out of symmetry. The interferences thus vanish without much changing in form or size, and vanish in all focal planes. The breadth of the blades of light aa' and bb', figure n, capable of inter- fering is x on the grating and x cos = 0.096X0.808 = 0.078 cm. normally. Since the rays are parallel after leaving the collimator, this would be about half the breadth of the effective beam on the objective of this appur- tenance. Thus 2X0.0776 = 0.155001., increased by the width of the refracting edge of the prism, is the width of the strip of white light which, after separa- tion by the knife-edge of the prism, furnished the two component beams which potentially interfere on recombination. It is reasonable to suppose that the elements of these beams come from a common source and that the width in question is produced by the diffraction of the slit. This datum is more appropriately reduced to the angle at the slit a, within which lie the rays capable of interfering with each other after the interferom- eter cleavage. As the collimator used was / = 22 cm. from slit to lens, a = 2x cos 6/1 = o.i 5 5/2 2 = 0.0070 Hence the angular width of the wedge of white light, with its apex at the slit of the collimator and containing all the rays which can mutually interfere, is about 0.007 radian, or less than half a degree of arc. One would infer that a long (I) collimator (i.e., one with weak objective) is advantageous, as the blade of parallel rays issuing is proportionally wide and the range of displace- REVERSED AND NON-REVERSED SPECTRA. 25 ment at M or N larger. Similarly, divergence subsequently imparted by dispersion (prism, grating), before the rays reach the mirrors, M, N, should have the same effect. The results obtained for dispersion bear this out, but not those for a long collimator. Moreover, the width of the slit, so long as the Fraunhofer lines do not vanish, is of no consequence. It thus seems tenable (to be carefully investigated below) that the positive effect of dis- persion has a deeper significance, bearing directly on the structure of the interfering wave- trains i.e., the length of the coordinated, uniform wave- train is possibly greater as the dispersion to which the wave-train has been subjected is greater. Two parts of it will therefore fit over a correspondingly longer range of path-difference. A number of other results point in the same direction. Thus, I may again point to the impossibility of obtaining fringes with homogeneous light and a wide slit, whereas two identical sodium lines (D\ and D'\), superposed, show the interferences strongly. The lines actually become helical in shape and much broader. The range of displacement of N may be decreased from 0.25 cm. to o.io cm. by narrowing the beam emerging from the collimator with a slotted screen, while the fringes themselves are coarsened by this process. With the screen removed the fringes are not only sharper and finer, but apparently they may be seen to slowly move laterally across the fiducial sodium lines. This is in accord with the increased range of displacement of the mirror. The observation, however, is complicated by the fact that the sodium doublets are not quite in the same focal plane. The fringes must, in a reduced case, lie midway between them, in the line of symmetry of the spectra. 13. Prismatic refraction. The method indicated in figure 12 (P, prism; M,N, mirrors; G, grating; T, telescope) was next tested for small distances and the experiments begun in the third order of spectra of the grating G. The prism was a small right-angled sample, with faces only about i cm. square; but it sufficed very well. Its distance from the grating being about 13 cm. and the illuminated spots on the mirrors 18.8 cm. apart, the mirrors were nearly normal to each other. In fact, as 6 in the third order is about 62 and i' about 28, 5 = 34 and a = 90. Hence, on displacing the micrometer mirrors M or N, the illuminated strips move relatively rapidly across the face of the grating. Nevertheless, the fringes are easily found and controlled. Their range of visibility is larger than in the cases of the preceding paragraph. They remain in view for normal displacement of M of 3 to 4 mm., passing from hair-like striations, through sharp arrows, back to the hair-like forms. The range has thus been increased by the dispersion. The arrows are of the type shown in figure 13, with reentrant sides and part of the outline accentuated. In the second order of spectra from G, the phenomena were much the same, but far more brilliant. The arrows were now evenly wedge-shaped and very slender. The fringes entered as nearly vertical hair-like striations, and, after 26 THE INTERFEROMETRY OF passing the optimum, vanished as inflated arrows. The range of visibility was, as before, about 3.5 mm., so that the change of order has not had any further marked effect, such as might be anticipated. As in the preceding paragraph, if the impinging collimated beam is narrowed, the range of visi- bility decreases; in fact, the arrows themselves are reduced to slightly oblique lines. Within the limits given the fringes are well adapted for interferometry. First-order spectra are not available because of the large value of i' in the case of the right-angled prism. Taking the results of the last two paragraphs together, the increase of the range of displacement is due to the dispersion of the prism. The breadth of the pencil, diffracted at the slit, after leaving the collimator and prism, 12 13 increases. It was shown in the earlier report that inversion of spectra on a longitudinal axis does not preclude the possibility of interference. Taken as a whole, therefore, the present results have a direct bearing on Huyghens's principle. 14. Prism methods without grating. A more interesting method, in some respects, in which the grating is entirely dispensed with, is shown in figure 14. L is the beam of white light from a collimator, P a refracting prism (here with a 60 prism angle), M and N the opaque mirrors, with either or both on a micrometer, P' a silvered reflecting prism (here right-angled). The telescope is at T and should have high magnification. The rays L are refracted into a b c and a' b' c' and the two spectra observed by the telescope at T. Each of the prisms should be on 3 adjustment screws, as well as the mirrors. P must be revolvable slightly around a vertical axis and capable of fore-and- aft motion. P' is preferably a large prism placed on a tablet. The rays b and V are made collinear before P' is inserted, and both the rays c and c' must come from near its edge. REVERSED AND NON-REVERSED SPECTRA. 27 The fringes are strong and large and lie within a remarkably wide trans- verse strip. This may be 10 or 20 times as wide as the D\ D z doublets, which, in view of the small dispersion, are hardly separated. For the same reason, moreover, the range of displacement of M within which fringes are visible rarely reaches 0.5 mm. Within this the fringes grow from the fine hair-lines, usually oblique, to their maximum coarseness. Apart from the small range of displacement, these fringes are available for measurement. If both mirrors M and N are on micrometers, they may be brought forward or the reverse, alternately, and the range increased 5 or 10 times. To change the form of the fringes, the first prism, P, may be tilted slightly on an axis parallel to LT, figure 14. The fringes then pass through a maximum in the vertical direction (linear phenomenon). Fore-and-aft motion of P rotates the fringes, partially, toward the horizontal; but, as a rule, the com- 14 ponent beams b and b f pass beyond the edge of P' and the fringes vanish. Just before this (the spectra separating), the strip within which the fringes lie widens enormously. In other words, the breadth of the phenomenon depends on diffraction, not on dispersion, so that even though the prism P scarcely separates the V lines, the striated strip has about the same width as when it is produced by highly resolving gratings. It is preferable to use sunlight directly (without a long-focus condensing lens), as there is a superabundance of light. The best results are attained with a large collimator. A spectacle lens with a focal distance of i meter is excellent. The range of displacement of M is not increased, but the spectra and fringes become very sharp. If, with the large collimator, the spectra are just separated in the field of the telescope, by fore-and-aft motion of P, a magnificent display appears, resembling a thick, twisted golden cord. With further separation confocal elliptic fringes often cross the gap, as in figure 15. Here a and j8 are graphs suggesting the wave-lengths of the two spectra, g being the gap or deficient overlapping. The appearance in the telescope is shown at 7, 5 and S' being the spectra. When the fringes are erect, huge vertical furrows may lie in the gap. When the gap is closed, the linear phe- nomenon reappears. These enlarged fringes vanish, however, within 0.25 mm. of displacement at M. On the other hand, when the spectra are made 28 THE INTERFEROMETRY OF to overlap considerably (by fore-and-aft motion of P) the fringes become fine and vertical and the parallel blades of light, which interfere at the focus of the telescope, are 0.5 to i cm. apart at the objective. In further experiments, screens s, s' (fig. 14) were placed in the paths of the pencils b b', so that they were compelled to pass through vertical slits 0.5 mm. wide in the screens. In this way the interfering rays were identified. The first vertical hair-line fringes came from rays about 5 mm. behind the edge of the prism P'. Hence the pencils were here about 1.2 cm. apart when they entered the telescope. The largest and last of the fringes came from close to the edge of P'. The experiment was varied as follows : Supposing both screens 5 and s' placed as far to the rear as the visibility of fringes per- mits; let the former, s, be slowly pushed forward. The fringes then contract from the very broad set, figure 16, case i, to the strong and narrow set 2 (which is a mere line for a full wave-front), and then expand again to case 3. If, now, s is left in place and s' moved forward slowly in the same way, the identical contraction and expansion, cases i, 2, 3, are reproduced. The screen 5' may then be left in place and 5 in turn slowly moved forward with the same results, etc. (there may be 6 alternations), until finally the effective parts of the pencils b and b' are beyond the edge of the prism P'. In case 2 the two slits s and 5' are obviously symmetrical to the interfering rays, whereas in cases i and 3 the diagonally opposite edges of the slits 5 and s' limit the effi- cient pencils to a sheet. If the edge of the prism were truly a knife-edge, the last fringes would be very large, since the distance cc', figure 14, would vanish. If the fringes are vertical (obtained by tilting P around an axis parallel to LT), the case 2 is given by 2 or 3 strong vertical lines, whereas i and 3 consist of 10 or 20 lines, all of about the same width and distance apart. If the slits 5, s' are finer (i mm.), the fringes are throughout sharper. A single displacement, i, 2, 3, corresponds to about 2 mm. When the edge of P' is approached the case 2 often shows vertical strands of fringes, a strong central strand, and two or three fainter ones either side of it. The cases i and 3 are not stranded. A similar result (passage of case 2 into 3, fig. 16) may be produced by moving P forward, the case 3 appearing just before the pencils b b' leave the edge of P'. Again, when M is moved rearward, when both b and b' are near the edge of P', the cases 2 and 3 are obtained. In general, the width of the dif- fraction pattern increases without changing the size of fringes, as the width of the available wave-front decreases. A similar result will be described in connection with figure 48, Chapter II. Naturally, if the displacement is considerable, it is accompanied by some rotation of fringes. 15. Displacement parallel to rays. It now becomes of importance to test the range of displacement as modified by the angle of reflection, increasing from 5 = o. It is therefore desirable to make a few direct measurements. The angle at P, figure 14, was found to be about 49 45', so that the total angle at M is 8 = 40 1 5'. M and N are both on micrometers, with the screws normal REVERSED AND NON-REVERSED SPECTRA. 29 to their faces. P' is on a micrometer with its screw parallel to bb f , so that this prism is shifted right and left. The range of displacement was found at M, about 0.04 cm.; # = 2X0.04X0.939 = 0.076 cm. P', about y o.o'j cm.; 2^ = 0.140 cm. where x = ze cos (90 0)/2 and zy are the corresponding path-differences between the inception and evanescence of fringes. With a very fine slit, 2y was possibly smaller (see fig. 17). The question at issue is thus, in the first place, how the value of zy compares with oc ; for in the former case the angle 5 is effectively zero. In other words, when M is displaced from M to M', over a distance e, the pencil b, figure 17, changes to bi, and is soon lost at the edge of P f , whereas, when P is displaced in the direction W, over a distance y, the rays 6 and b' do not change their point of impact at the prismatic mirror P'. If PP represents the principal plane of the objective of the telescope and F its principal focus, there should be no accessory effect for the case y as compared with the case x. Results bearing on this subject are given in table 9, in which the displace- ment e, observed at M and at N, as well as the displacement y at P', are recorded when a plate of glass of thickness E is inserted normally to the rays b, b'. The corresponding air-path difference computed from E, n, B, and X should be z, nearly. This is about the value (zy) observed, remembering that to set the micrometer, fringes of a particular pattern must be selected. The rotation of fringes being but 90 or less, there are no fiducial horizontal lines. TABLE 9. Reversed spectra. Refracting prism. 6 = 49 45 = 2ecos(90-0)/2;z=E Oi- 5=4.6X10-" (assumed). Detail. E M z e X 2 y cm. cm. cm. cm. cm. cm. Mirror right, plate right. . . 0.736 1.526 f 0.3901 1+-OI95 \ 0.204 f .199 [ .206 0.381 0.400 .410 Mirror left, plate left .736 1-526 = .4096 .208 391 f .410 \ .406 Mirror right, plate right. . . 736 1.526 = .4096 .201 .205 .202 j -381 f .406 I -402 Mirror left, plate left 736 1.526 = .4096 .207 .389 .402 .207 .208 .400 Mirror right, plate right. . . 736 1-526 = .4096 .205 .207 } .387 .406 Mirror left, plate left 736 1.526 = .4096 .206 .208 ] -389 .406 Mirror right, plate right. . . 434 1-533 | -2313 125 .124 } -234 / -248 \ -250 | -1- oi T c\ .248 Mirror left, plate left 434 1-533 ITV*3 = .2428 \ .126 .125 } -236 (.255 248 I -246 Mirror left, plate right 434 1-533 = .2428 \ ' .127 , .126 } .37 30 THE INTERFEROMETRY OF Furthermore, though p was determined by the total reflectometer for each of the two plates used, B, the Cauchy dispersion coefficient, had to be assumed from similar results in my earlier work. Finally, the first plate ( = 0.736 cm.) was slightly wedge-shaped and some adjustment for coincidence of spectra was needed. The second plate ( = 0.434 cm.) was optically nearly plane parallel. One may therefore conclude from these details that zy=z as nearly as could be expected. The values of x, computed from 8 and e, however, certainly fall below z, being about 6 per cent and 3 per cent short of it in the two cases, respectively; or, again, x is 0.019 cm - an d 0.014 (about 5 per cent) smaller than the mean values observed for zy. This extra 5 per cent of path-difference can not be an error of observation or of adjustment, but must be interpreted as the path- difference added when the pencil shifts towards the edge of the prism (x) instead of being stationary as in y. In case of inverted spectra, moreover (next chapter), x is usually in excess of 0, and the shift is the other way. The deficiency in x, though not equally marked, is present in observations both on the right and left sides of the prism P 1 '. 16. Breadth of efficient wave=fronts. Apparent uniformity of wave=trains. Rotation of fringes. It follows from figure 17 that if M is displaced to M' t over a distance e, the pencil b is displaced parallel to itself over 5 = 2e sin 5/2 where 5 = 90 6. The pencil c is then displaced parallel to itself over a distance * = stan '/2=s since 6 = 49 45', 5/2 = 20 7', and therefore 5 = 20X0.344 = 0.70, nearly. If the rotation of fringes is but 90, either 5 (or 5/2) is also the breadth 'of the strips, or patches of like origin, which, when sliding over each other more 17 18 or less, produce the fringes. This may be treated from a graphic point of view as follows, a theory not being aimed at : In figure 1 8, let a and b be two patches of light of like color and origin at the objective pp, figure 17, producing interferences at the focus F. Hence the fringes will be arranged in the direction/, figure 18, at right angles to the REVERSED AND NON-REVERSED SPECTRA. 31 line joining a and b. Since a and b here correspond to c and c' in figure 1 7, let a be continually displaced to the right, as indicated by the arrows, figure 18. In proportion as the positions ab, a'b', a"b", are taken, the fringes must pass by rotation from /, into /', into /", etc. i.e., over about 90. In the present experiment, c, figure 17, can never pass across c' ', for they are essentially separated by the edge of the right-angled prism P'. Hence the rotation can not exceed 90, for the vertical through a can not cross the vertical through b. This is not the case when a grating replaces P', as in figure 12; nor is it the case when, as in Chapter II, inverted spectra are treated, and the patches a and b slide along the edge of the prism. In such cases figure 18 may be continued symmetrically toward the right (mirror images), and the limit of rotation is therefore 180. All these suggestions are borne out by experiment. Moreover, if the first prism P, figure 14, is tilted slightly on an axis parallel to LT, a (fig. 1 8) will be lowered and b raised. If a and b are on the same level, the fringes are always vertical and pass through a vertical maximum, when ab is a minimum. On the other hand, if a and b are not in the same level, as in the figure, fore-and-aft motion brings the rays c and c' (fig. 17) to or from the edge of the prism P'. Hence the case ab passes into a"b", or the reverse; in other words, the fringes pass through a horizontal maximum when ab is a minimum, etc. This is also shown by experiment. Moreover, if a, figure 18, is the angle (in the observer's vertical plane) of ab to the horizontal, the horizontal distance between c and c' will be ab cos a, which is zero when a = 90, and both c and c' are at the edge. Suppose the full breadth of the strips are at the edge, so that the fringes present the strongest, coarsest, but narrowest field of case 2, figure 16. Then if either c or c' retreats until the fringes vanish, the width of the appreciably effi- cient strip cc' will be ab cos a = t=s = o.'je, nearly. This is probably the best method of estimating the width in question. Usually, however, away from the edge, the succession i, 2, 3, figure 16, is obtained. In such a case the breadth of efficient strip is t/2 =0.350. The experiment made by moving screens with slits, forward or rearward, successively, by which the appearance and evanescence of fringes may be repeated through several cycles, is next to be explained. Here it is merely necessary to remember that the spectra c and c' are reversed, or that the colors of like origin and wave-length are successively farther apart. When the screens are alternately moved, therefore, the same phenomenon is in turn produced in slightly different colors. But as ab continually increases, whereas the efficient breadth of the strips does not, the fringes soon pass beyond appreciable smallness. When, as in the earlier methods, but a single grating is used with two successive diffractions through it, the patches a and b are obviously in the same level when the longitudinal axes of spectra coincide. Hence the fringes are essentially vertical. In the experiment with screens, 5, s', figure 14, it is obvious that path- difference remains constant. The distance from the same wave-front in the 32 THE INTERFEROMETRY OF pencils b and b', figure 17, to the principal plane pp, is always the same; but pencils different in lateral position are successively selected. On the other hand, when the prism P' is moved in the direction y, parallel to bb', path- difference only is introduced, while the pencils selected remain the same. Supposing the ordinary conditions of visibility (magnification, etc.) to remain unaltered throughout, the wave-fronts are, as it were, explored in depth as to their uniformity i.e., the distance is apparently recorded throughout which a wave-train consists of identical wave-elements. Effectively, however, the rapidity with which fringes decrease in size beyond visibility is directly in question. Finally, when the opaque mirror M (or N) is moved from M to M', both effects occur together. Path-difference x = 2e cos 5/2 is introduced and the pencil is displaced from 6 to b'. The ^-effect is thus probably the same as if P' were displaced to the left and 5 were brought forward. Hence it is of great interest to determine the extent in which the values of y and x are different. It appears as if the dis- tance within which the wave-trains are uniform is definitely limited and that it increases with the breadth of effective wave-front just instanced, while both increase with the amount of dispersion to which the incident white pencil has been subjected. Diffraction at the slit of the collimator may be regarded as the first dispersion. This seems to me to be a very important observation, and a systematic investigation of the lengths of uniform wave- trains, so understood, in their dependence on dispersion, is desirable, even if the geometry of the system should prove to be adequate to explain the phenomena. 17. Film grating. The method of two gratings was now again resorted to, except that the first at G, figure 3, was a film grating. This attempt failed in my earlier work, when but a single film grating was used for the two dif- fractions, because of insufficient light.* In the present case, where two gratings (G 1 being reflecting) are employed, the method succeeded at once. The first grating constant was D = io^X 167 cm. ; observations were therefore necessarily made in the second order of G', so that the spectra are not as intense as with prisms. But the fringes are perfect and may be made as large as desirable with but two in the breadth of the spectrum, for instance. They come in and go out of range with inflation of form, and they are free from the awns seen in the preceding paragraph (with prism), probably because the light is less intense. The phenomena in general are the same in character; but the range of displacement of either mirror is enhanced, conformably with the increased dispersion at G. This range was found to be about 6 mm. under the best conditions (arrows). If both M and N are successively displaced in the same direction, the total displacement available between the hair-like fringes at the extremes is about 1.5 cm. for each mirror. * I have since obtained the fringes with a single film grating. REVERSED AND NON-REVERSED SPECTRA. 33 At these extremes the two patches of light on the grating G' may have been separated by several millimeters. The nature of the transformation from arrows to the oblique striations would be well reproduced if equidistant vertical wedges were moved from right to left, or the reverse, behind a vertical slit. The distance between G and G' was about 10.2 cm., and between the spots of light on mirror M and gratings G and G', respectively, 22.2 and 12.6 cm. This corresponds to 01 = 20.5 and #2 = 36. 2, so that 6 = 15.7 and 9. To obviate this difficulty the method was modified for prisms as in figure 20, where G is the concave grating, T a strong lens near its focus, and m and n auxiliary mirrors. If this method is used for highly dispersive gratings (G replacing P), the rays must be crossed as shown in figure 34. The fringes were found in this case when G was a film grating, but the work had to be abandoned, as the spectra were dull. The data given in table n again show marked increase of displacement with the dispersion dB/dK, though it is not proportional here. The method with two gratings lacks the brilliancy of the prism method. TABLE n. Range of displacement for different dispersive powers. Method of figure 20, full displacement. Details. m de d\ e Remarks. 60 prism and concave grating Ruled and concave grating cm. 0-357 .300 514 cm. 0.315 .282 C2O 760 760 2 880 49 45' 49 45 9 39 Bright. Too dark. Bright arrows. Film and concave grating 509 2,880 6,400 9 39 20 40 Dark. Too dark. 24. Normal displacement of mirrors (5 = o). This desideratum was secured in the original methods, in which a single grating was used for both diffractions. Rays in such a case have to cross the grating somewhat obliquely to the hori- zontal. The method, furthermore, is restricted to the linear vertical fringes, which are not useful if practical measurement is aimed at. In the methods with a right-angled reflecting prism (fig. 14), this result is easily secured by displacing the prism. In all other methods (5>o), the dis- placement of mirror is accompanied by sliding of the pencil along it. The effect, as has been shown, is not negligible. It therefore seemed desirable to devise other methods in which 5 = o, and figure 35 is a device of this kind. Here G and G' are two identical gratings, the first, G, receiving the light L from a collimator. The component pencils a, a' pass through the half-silvered plate H, and thence (b, b') to the opaque mirrors M and N, one or both on a micrometer. The pencils 6 and b', impinging normally, retrace their paths and are thereafter reflected at the plate H into c and c' . These strike the second grating G' at the proper angle of diffraction and thereafter enter the telescope T together. The path of rays is symmetrical throughout. White light is screened off at d. Reversed spectra are seen at T. Unfortunately the only identical gratings at my disposal were film gratings of high dispersion 56 THE INTERFEROMETRY OF (D = 162 X io-*) . As a result of the two diffractions and the half -silver reflec- tion, the spectra were too dull to make it worth while to look for fringes, at length. None was discernible after some searching, to my regret, for the method itself has many points of interest and with gratings of low dispersion it succeeds. Later I stripped a celluloid film from the ruled grating (D = 352 X icr 6 ) and mounted it by simple stretching. Using the original grating at G and the film at G', the fringes were found with some patience. The spectra were fairly bright (arc lamp) and the fringes reasonably strong; but they admitted of a displacement of only i .8 mm., in spite of the vanishing angle 6 = 0. Possibly this small displacement is due to the imperfect film grating. In further experiments I half -silvered plates of glass to different density. In this work I obtained adequately bright spectra and practically perfect fringes, but the range of displacement (3 = o) could not be increased above 1.8 mm. Within this interval the fringes seem to change form but little, thinning only being evident. Then they become dull and vanish. The method was temporarily modified, as shown in figure 36, where G is a ruled grating (D = 3 52X10-* cm.), G' a film grating (D = 176X10-* cm.), and m an opaque mirror. This naturally introduces an angle 8= 6' 0, which is negative when 6'> 6, or the grating G shows greater dispersion. The mirror is limited in breadth, so that the rays a, a' have free access to M and N. The fringes were found after some trouble, for the transmitting grating G' also acts as a reflecting grating if the rays b, b' fall upon it, and it is not always easy to separate these two cases, each of which will give fringes on proper adjustment. The spectra are very bright and the range was about 2 mm. The same method was now carried out with prisms, as shown in figure 37, where L is the incident pencil from a collimator, P and P' right-angled prisms, I the half-silvered plate, N and M opaque mirrors on micrometers, T the telescope. The spectra seen at T have each been four times refracted and twice reflected, at M and H. They are very bright, so that a very fine slit (here specially desirable) is available. The sodium lines are not separated. On bringing the spectra to overlap at their corresponding edges, the fringes were found. They are peculiar, inasmuch as they show the phenomenon of figure 32, but with the faint fringes curved and more prominent than hereto- fore. In other words, the faint phenomenon shifts across the field from side to side, but is enormously accentuated at the transverse strip of the linear phenomenon. Narrowing the incident pencil broadens and blackens the fringes. They may be obtained in the gap between two spectra just separated. The range of displacement within which fringes are seen was, however, very small, not exceeding 0.2 mm. This is a characteristic of these fringes and in keeping with the low dispersion. It is interesting to note that the systems figures 35, 36, 37 constitute an element of a direct-vision spectroscope. It has been shown that it can be made quite powerful. The same method, figure 37, was now used with two 60 prisms of highly REVERSED AND NON-REVERSED SPECTRA. 57 refracting glass. In contrast with the ease in which the adjustments were made with the symmetrical 45 prisms, the corresponding work with 60 prisms proved to be exceedingly difficult. True, the prisms were large, including long glass-paths. The sodium lines, now clearly separated, were markedly curved, so that on placing them near together they either assumed an O-shape or an X-shape. But the spectra were brilliant and nothing appeared to mili- tate against a successful result. But it was not until after days of searching that the fringes were found, and then only with two prisms of different, highly refracting glass. They were not quite uniform, but it seemed impossible to improve them. The ranges of displacement were found to be 0.088 to 0.093 cm. with the electric arc, 0.103 cm. with (condensed) sunlight. This is again in accord with the large increase of dispersion, the range of displacement being about five times greater than was the case with 45 prism of less refracting glass. - 35 25. Diffraction at M, N, replacing reflection. The present method of observing interferences in the zero, first, second, third, and even fourth order, successively, without essential change of the parts of the apparatus, is noteworthy. I happened to possess a plane reflecting grating (D X lo" 6 = 200) , cut into two equal parts by a section parallel to the rulings, and it was there- fore easy to devise the method. In figure 38, the incident light L from the colli- mator is separated into two component beams a and a' by the 60 prism P. This is essential here, as an abundance of light is needed (sunlight should be focused by a large lens of long focus (5 feet) on the slit). The rays a, a' are then either reflected or diffracted in any order by the plane reflecting gratings G, G' into the collinear rays b, b'. They are then reflected by the silvered right-angled prism P' and observed in a telescope at T. G and G' and if possible also P' should be on micrometers, so that 58 THE INTERFEROMETRY OF corresponding displacements e, e', normal to G and G' and y in the direction bb', may be registered. The adjustments, if symmetry were demanded, would be cumbersome; for, in addition to precise modification of the position and orientation of the prisms P, P', the grating requires fine adjustment and a means of securing parallelism of the rulings. But an approximate adjustment does very well and no pains were taken in the first experiments to secure symmetry. The spectra were intensely brilliant in the low-order work; but even in the fourth order the light was adequate. One may note that here the gratings do not reverse the dispersion of the prism P, though this is relatively small. Table 12 is an example of results : TABLE 12. Ranges of displacement varying with dispersion. Paired gratings and 60 prism. 0=46. 8=44. x=2ecosS/2. Order Observed eXio 3 *Xio 3 i d$/d\ Remarks. cm. cm. o 38 70 22 760 1 80 200 180 35i 100 12.8 31-2 } 3.490 200 35 1 2 420 420 777 1 -.<> ^ 400 380 I 721 L 3-5 J 40-5 } 6,440 3 320 300 520 520 | 573 | 962 -6.4 50.4 } 9,930 /Fringes lost at \ edge? 4 580 580 520 1 1070 1-17-6 61.6 } 14.800 ("Fringes faint. [Fringes faint. 450 | The fringes in the zero order were good and strong, not inferior to any of the others, but unfortunately too short-lived. In the fourth order the fringes are weak (although the enormous sodium doublets stand out clearly), doubt- less from excess of extraneous light. Here also it is difficult to prevent the beam from vanishing at the edge of the prism P'. Hence the anomalously small displacement, a discrepancy which is already quite manifest in the third order. The present experiments furnish a striking example of the uniform breadth of the strip of spectrum carrying the fringes, quite apart from the dispersion of the spectra. In the prism spectrum, where the sodium doublets are indi- cated by a hair-line just visible, to the fourth-order spectra, where they stand apart like ropes, the linear phenomenon has the same width. The computation of the dispersive power in these cases is peculiar. It will be seen from figure 38 that the angle 8 = 44 between the incident ray a and REVERSED AND NON-REVERSED SPECTRA. 59 the diffracted ray b is constant and is 8=6-\-i in the first and second and 8 = 6 i in the third and fourth orders. Hence in succession sin (5 i) sin i = X/Z) sin sin (5 i) sin i = z\/D sin(3-H)-r-sint = from which equations the angle i may be computed. I did this with sufficient accuracy graphically, and the values of * and 6 so found are given in table 12. Since dQdi, apart from sign, it follows that the dispersing power is -de/d\=n/D (cost+cos (")) where n is the order of the spectrum and i changes sign in the third and fourth orders. With the values of i given, the data for d 0/dK in the table were finally computed. The dispersive power of the prism was computed as above and is to be added to all the succeeding dispersive powers. Figure 39 shows the relation between the dispersive pow- ers and the path-difference x = ze cos 5/2 computed from the observed range of displacement e of the grat- ing G. The largest values of x are taken, as they are the most proba- ble. The effect of dispersion here breaks down in the third and fourth orders, as already stated, probably from incidental causes. For the rfftjfci 39 10 spectra themselves were still ade- quately bright, but the fringes were faint for some reason and I failed to make them stronger. The rate x/(dB/dK) is here about 120X10-* initially. This is larger than above, owing to the differences of apparatus used, etc. 26. Experiments with the concave grating. As there was an excellent Rowland grating in the laboratory with a 6-foot radius and a grating space D=i77Xicr* cm., it seemed worth while to obtain the interferences with it. I had hoped to doubly diffract the rays at the same grating, but there is not light enough to make this method fruitful. Accordingly the device, figure 40, promised the best results, where L is a convergent pencil of sunlight, S the slit. The pencil L is carried above the grating G to the opaque mirror m, whence it is reflected to the grating and diffracted into the component beams a and a'. These are in turn reflected by the opaque mirrors M and N (on micrometers) into the pencils b and b', nearly collinear. The latter are then reflected by the silvered right-angled prism P to the common focus F, to be observed with the lens T. The spectra are very bright and highly dispersed and are easily made to overlap on their contiguous edges (parallel to a Fraun- hofer line D, for instance) sufficiently to show the linear phenomenon of reversed spectra. 60 THE INTERFEROMETRY OF The fringes are easily found and splendid, though they are not superior in this respect to the fringes obtained by other methods. P is also on a microm- eter with the screw in the mean direction b b'. The range of displacement of P was about y = 0.5 5 cm., so that the available path-difference is considerably above a centimeter. Within this the fringes pass from fine hair-lines through a maximum and back again, apparently without rotation. To enlarge the fringes the grating G may be tilted in its own plane or a similar adjustment made at P. The longitudinal axes of the spectra (wire across the slit) are not in focus with the D line; but though hazy they suffice for adjustment. Naturally the distance FbaG is the radius of the grating R (6 feet) and the distance Sm-\-mG is R cos 6. The distances are approximately laid off and the observer at T may then push the mirror M fore and aft by the aid of a lever till the Fraun- hofer lines are sharp. Unfortunately the spectra as a whole are shifted by the micrometers, to- gether when P moves and separately when M or N moves. 27. Polarization. The two rays obtained from calc spar, if corrected for polarization, should be available for interferences of the present kind. Nat- urally an achromatized calc-spar prism (C, fig. 41) is most convenient for the 40 41 purpose. White light from a collimator, L, is doubly refracted by this prism C, and the extraordinary ray a, just missing the grating G (above or on the sides), impinges on the opaque mirror M and is thence reflected to the grating G. The ordinary ray a' is reflected from the opaque mirror N and thence also reaches the grating. These two pencils, b, b', are directly reflected by the grating (>Xio 6 = 2O o cm.) into R m and R n and diffracted into >' m and >'.. By suitably adjusting the grating between C and M and inclining it as shown, two coincident spectra may be made to pass along the common direction c to the telescope at T. These spectra are quite intense. One is somewhat REVERSED AND NON-REVERSED SPECTRA. 61 more dispersed than the other, owing to the difference of angles of incidence i, i' and to the residual or differential dispersion at the calc-spar prism, for only the a pencil has been adequately achromatized. In my apparatus the distances CM and GM were about 130 and too cm. and the distance NM roughly 30 cm. N is on a micrometer, as it is nearer to the observer at T, though it would be preferable to put M on a micrometer. To obtain interferences of the present kind, the plane of polarization of either a or a' must be rotated 90. This may be done by two quarter-wave- length micas M, in a', the first of which (set at 45 as usual) produces circu- larly polarized light and the second then erects the vibration into parallelism with the vibration of the pencil a. Other methods will presently be resorted to. With this adjustment fringes were found at T after some searching. They were large, but occurred in a transverse strip of spectrum about A #2/2 in width. .True, the spectra are without reversion; but, as stated before, one was about one-fifth longer than the other. This is probably the reason for the narrowness of the strip, for the fringes should otherwise fill the spectrum. The fringes as first found admitted a displacement of N of about 0.3 cm. only. They were hard to control, needed sharp longitudinal adjustment, and when lost were difficult to find again. They rotate from and to vertical hair- lines, through a horizontal maximum. They are always found in the line of symmetry, which for unequal spectra moves more rapidly than the Fraun- hofer lines if they are separated. The nearer quarter-undulation plate at W may be considerably rotated without quite destroying the fringes. If the double quarter-wave-length plate is suitably put in the a beam, results of the same kind are naturally obtained. If a single quarter- wave- length plate is placed in each beam, at 45, both will be circularly polarized, the position of the micrometer being intermediate. I tested this case at some length, but found no interferences, as was to be expected. Circularly polar- ized rays of the same sense do not interfere. In place of the achromatized calc-spar prism, a double-image Wollaston prism or a double-image Fresnel prism (rotary polarization) could be used. Unfortunately I had neither of these, but the latter in connection with an analyzing nicol would have been worth testing. CHAPTER II. THE INTERFERENCES OF INVERTED SPECTRA. 28. Introductory. If two identical spectra are superposed in such a way that one is rotated 180, on a transverse axis (parallel to the Fraunhofer lines), with respect to the other, it will be convenient to refer to the phenomena resulting and described in Chapter I as the interferences of reversed spectra. On the other hand, if one of the superposed spectra has been rotated 180 with respect to the other on a longitudinal axis (parallel to the length of the spectrum), we may refer to the interferences as those of inverted spectra. The absence of either of these adjustments would then be accentuated as non-reversed or non-inverted. In the case of inverted spectra, therefore, we are dealing with phenomena of virtually homogeneous light, exhibited throughout the length of the spec- trum. Such experiments were made cursorily in my first paper on the sub- ject,* but the phenomenon is very peculiar, apparently anomalous, and further treatment is therefore desirable. 29. Apparatus. Non=inverted spectra. The apparatus formerly used for long optical paths is difficult to manipulate. It has therefore been simplified in the present method and used for short distances, so as to be wholly in the observer's control. The parts of the apparatus are conveniently assembled as in a preceding experiment, except that the slit which furnishes the collimated light L, figure 42, is now horizontal i.e., at right angles to the edge of the sharp prism P (about 20 at apex). The rays a and a' (horizontal blades), after leaving the opaque mirrors M and N, are then reflected from the sides of the right-angled prism P' into c and c' . Thus far the light is white; but c and c' are now diffracted by the grating G with its rulings horizontal (par- allel to slit) and with the interposition of an auxiliary prism p (edge hori- zontal), the two spectra due to c and c' are observed by a telescope at T. The experiment succeeds best with sunlight. The triangle of rays, a, a', b r , b, is first made isosceles and horizontal by adjusting P,P', being set midway and symmetrically between M and N. The two spectra in the telescope (vertical ribands) are now made to overlap at their inner edges and the two sodium doublets placed accurately in contact, by aid of the adjustment screws on the mirrors M and N. The latter are then to be moved micromet- rically (Fraunhofer slide) until the fringes appear. The experiment is not an easy one. It is obvious from figure 42 that the two superimposed spectra are non- * Am. Journal, XL, pp. 486 to 498, 1915, 4; Carnegie Inst. Wash. Pub. 249, Chap. I. 62 REVERSED AND NON-REVERSED SPECTRA. 63 inverted or direct. Hence with a wide slit, or insufficiently homogeneous light in the direction of a given Fraunhofer line, no fringes will appear. Such fringes as may in any case be found will not, therefore, much outlast the Fraunhofer lines, so far as width of slit is concerned. Furthermore, even if the slit is narrow and the spectra coordinated, there will be no fringes obtain- able if the superimposed solar spectra are quite unbroken i.e., without in- cidental furrows in the direction of their length (normal to the Fraunhofer lines). For it is to be noticed that the slit is horizontal, and therefore there is no observable diffraction i.e., virtually no slit in the horizontal direction. If, however, the spectrum field is interrupted longitudinally by a thin (o.i mm.) wire drawn across the slit (or, much better, by very fine specks of dust lying incidentally within the slit), then these fine opaque objects will effectively replace the slit, or act analogously to a slit in the horizontal di- rection. Hence fringes will appear when the path-difference is sufficiently small, associated with the geometric shadow of the opaque objects, through- out the length of the spectrum. 42 We have here, therefore, a peculiar case of the diffraction of a rod, from two separately controlled half -wave-fronts. The fringes at one extreme of adjustment of mirror M begin with fine horizontal lines, which on moving M incline and enlarge until they gain the maximum of size in the vertical direc- tion. After this, on further motion of M in the same direction, they incline further, diminish in size, and finally become horizontal and hair-like again. They move along the horizontal axis with M, subject to the equation de __ X dn 2cos d/2 where 5/2 is the angle of incidence at M, X the wave-length, and de/dn the normal displacement of M per fringe. Within the region of overlapping spectra each longitudinal black line is covered from end to end with fringes, the strip being about of AA width or more if the line is thinner. A broad black line (thicker wire across slit) 64 THE INTERFEROMETRY OF shows the fringes at its edges only. While the spectrum may thus be covered with independent strands of fringes, they all move or change phase together. The range of displacement of the micrometer-screw is about 2 mm. When the spectra are about to separate i.e., when overlapping ceases the fringes are apt to be particularly wide, say even 4AA, and they may be seen in the gap between spectra. In general, the degree of coarseness of fringes de- pends on the adjustment for parallelism of spectra. 30. Apparatus and results for inverted spectra. The preceding apparatus may be modified as shown in figure 43 . Here L is a collimated beam of white sunlight from the vertical slit, G a transmitting grating. The component pencils a and a' of spectrum light impinge on the opaque mirrors M and N, and are then reflected in b and b' to reach the silvered sides of the right- angled prism P'. This is now placed with its sides at 45 to the horizontal and its edge in the direction of the incident beam L prolonged. Hence the two pencils b and b' are reflected vertically upward (fig. 44) and appear as two identical and parallel spectra in the field of a telescope, vertically above P'. Observation is conveniently made downward to obviate additional reflection. If the triangle a, a', b', b is horizontal and isosceles there is no difficulty. On slightly moving the adjustment screen on M, N, P' (which must be revolvable on a vertical axis, and P', or the beam L, movable up or down), to bring the two spectra into coincidence along a given longitudinal axis, and a transverse axis like the D lines, the two spectra are symmetri- cal i.e., mirror images one of another with respect to the longitudinal axis. If, now, the mirror M is moved on a micrometer, the fringes of inverted spec- tra appear when the path-differences are nearly the same. 43 a b c d e, Obtained at short distances (of the order of a foot) on the cast-iron block (Chap. I, fig. 3), these fringes are quiet and better circumstanced for obser- vation; but their characteristics are the same as found for them before (Chap. I, 4, Carnegie Inst. Wash. Pub. 249). They lie within a narrow strip at the line of symmetry of the two superposed spectra, running from end to end and in breadth about three times the distance apart of the sodium lines (sAA). When the mirror M is moved micrometrically in one direction, the fringes begin to appear as fine hair-lines (a, fig. 45) parallel to the D lines. These fine fringes gradually coarsen and rotate until they reach their max- REVERSED AND NON-REVERSED SPECTRA. 65 imum size c, when they are perpendicular to the D lines. A single black or bright line may here extend from end to end of the spectrum. Thereafter they grow finer (rotating again) in the same way until they vanish, e, as they began. The total angle of rotation is thus 180. If the two beams b and b', figure 43, are moved nearer the edge of the prism, the fringes become larger, but usually not much. At least I did not, at first, succeed in separating the fringes much beyond the DiDz distance. The width of the longitudinal interference strip remains unchanged. If the light is removed at the line of symmetry (wire across slit), the fringes are sharply outlined across the black line. Being so small, they are naturally always sharp and vivid. The range of displacement of M within which fringes are visible was about 0.2 cm. for the given grating (D = 352 X iQ" 6 ), corresponding to the complete rotation 180, instanced above. If the slit is widened, the fringes slightly outlast the Fraunhofer lines. They also lie in the principal focal plane. The remarkable feature of these fringes is the definite breadth of strip, from red to violet, within which they lie. With full wave-fronts the stria- tions look as though they were cut between two parallel lines, 3AA apart, In some adjustments, however, suggestions of concealed fringes, very faint prolongations of the strongly marked striations, are unmistakable. 31. Wave=fronts narrowed. The longitudinal strip within which the interferences lie is very sharply limited in breadth, as has been stated. It may, however, be broadened by screening off the white beam, as it leaves the collimator, from below. When the whole length of slit is utilized, the strip may not be more than AA in width. As the vertical blade or beam of white light is cut off, more and more from below, the strip increases to a width of 4AA, and when the two inverted spectra in the field of the telescope begin to separate at the line of symmetry, the strip may be over ictDiD^ in width. It is obvious that in such a case the light comes from near the horizontal top edge of the prism. The wave-fronts are slit-like. The field within which diffraction is perceptible in the telescope increases in breadth as the colored wave-front, incident at the objective of the telescope and parallel to the spectrum, decreases in breadth in the same direction. In figure 46, a and b are the two component beams from the two faces of the prism, respectively. The focus is at /. The arrows show the effect of narrowing. The oblique rays (omitted) are similarly affected and in step with the axial rays shown. But the fringes are not changed in size. They may, however, be definitely changed in inclination. Size results from the anterior relations of the spectra (distance between paired pencils), and not from the width of wave-front. These inverted fringes admit of much magnification. With a strong tele- scope (magnification about 15) they are quite sharp only in a part of the magnified spectrum and grow vague beyond, showing that the component spectra are not quite identical after the two reflections. When not quite in adjustment, the strip is liable to exhibit separate oblique strands, lying within 66 THE INTERFEROMETRY OF the same strip or region (fig. 47)- They may become sharp by changing the focal plane of the eye-piece. When parallel to the length of the spectrum and seen in a strong telescope, about 7 lines alternately black and white may be counted. The whole lie within a strip of not much above AA width. The range of displacement of M for a rotation of 180 of fringes is about 0.25 cm. Change of size of fringes from red to violet is hardly appreciable. The same results may be obtained by placing a screen 55, figure 46, with two parallel slits, under the vertical telescope, to admit and limit the two rays c and c' in figure 44. It was found that slits 3 or even 6 mm. apart may still show fringes, though they are obviously smaller as the distance apart is greater. The most interesting results were obtained by bringing the rays a little beyond the edge of the prism, so that the spectra in the telescope 55', figure 48, are separated at some distance. A long collimator (i meter) is advantageous. In this way the character of these fringes was definitely established. They are of the elliptic type, as suggested in the figure, char- acteristic of the displacement interferometer, and the cases, figure 45, a, b, c, d, e, are thus merely the intercepts of ellipses with the distant centers, between parallels. Very coarse central fringes were obtained in the dark gap, and the displacement at mirror M, between the extreme hair-line types, was now only about 0.08 cm., all fringes filing by the coincident D lines while the micrometer shifted. Hence this method is available for displacement interferometry, the hori- zontal type c, figure 45, normal to the D lines, being used for setting the mi- crometer at M. If a plate of glass of thickness E and index of refraction p. is inserted normally into one beam, the corresponding air-path is when B is Cauchy's constant and the wave-length X. I assumed^ = 4.6 X IQ- U and B/\* = 0.0265. On the other hand, the same air-path for a normal dis- placement, e, of the mirror as given by the micrometer-screw is X = 26 COS (90 - 0) /2 = 26 COS 5/2 where is the angle of diffraction of the grating G, figure 43, and b b' is nor- mal to the direction of incident light. Hence (M I+2J3/X 2 ) = 26 cos (90 6)/ 2 A rough experiment was made with a plate = 2.2 cm. thick, ^=1-53. REVERSED AND NON-REVERSED SPECTRA. 67 The corresponding displacement found was = 0.92 cm. The computed dis- placement should be e = o.8o cm. The low value was supposed to be due to the need of readjustment (wedge-shape) and insufficient normality of plate. Further data (table 13) were therefore investigated for thinner plates, but they do not clear away the difficulties. TABLE 13. Inverted spectra. 0=9 39'. x=2e cos (90 0)/2. ^ = 1-526; 5 = 4.6X10-"; 25/X 2 = 0.0265 ; z = E (M-i)+2EB/X*. E Ob- served. e Ob- served. X e Computed from* = 2 z E Computed from e (observed) and x = z 0.736 0.298 0.282 0.295 0.455 0.431 0.451 0.268 0.268 0.268 0.3901 +0.0195 = 0.4096 0.817 0.774 0.809 o ^14 0.480 0.268 0.861 0.311 0-475 0.268 0-853 In table 13, e is the observed normal displacement of the mirror M, x the corresponding computed path-difference, z the path-difference computed for the glass of thickness E and the index of refraction p. From z the displace- ment e or from x the glass thickness E may be computed. These are given in the table. In the second and third parts of the table, the edge of the prism P' was inclined at an angle to the line of symmetry, in opposite directions. The effect of this is manifest, but it does not explain the very large values of x for the symmetrical adjustment (rays from grating to mirror making am isosceles triangle) of the next observations. These were made with care. P' r figure 43, lay at the middle of the base of the isosceles triangle of rays from G. The long collimator was used, giving two splendid spectra, and rays were raised until magnificent large, cord-like fringes appeared. The longitudinal fringe, a single line normal to the sodium lines extending throughout the spectrum, made it possible to set the micrometer to about 0.0003 cm., or 5 wave-lengths. The motion of fringes with the rotation of plate being very striking, the plate was placed in the normal position to the rays by noting the retrogression of fringes at this point. The total range of displacement between extreme hair-like striations was 0.25 to 0.30 cm. One circumstance was noticed for the first time that on limiting the blade-like beam from below (as above described) the fringes not only enlarge, but rotate i.e., path-difference is modified. It is thus difficult to ascertain the discrepancies in * in table 13, as these values should be nearly 0.41 cm. throughout, or in e, which should be about 0.27 cm. when the prism is symmetrically placed. 32. Inverted spectra. Further measurements. The curious results shown in table 13 induced me to endeavor to detect the nature of the discrepancy by displacing the prism in the direction of the beams incident upon it. To 68 THE INTERFEROMETRY OF do this effectively it was necessary to do the work at long distances (meters), in order that adequate space might be available between opaque mirrors and prism. Accordingly M, N, P', figure 43, were all placed on micrometers, with the screws normal to the faces of the mirrors and the right edge of the prism, respectively. The fringes were found without difficulty and they were large and perfect near the edge of contact of the spectra. Though the sunlight was waning, a few measurements of ranges of displacement were made. They were on the average (e at M and N, y at P) : e M = o.og$ cm. e AT = 0.097 cai - y~ 0.062 cm. and since #=20 cos 5/2 should correspond to 2y, XM = 0.145 cm. ## = 0.148 cm. 27 = 0. 124 cm. Here, as above, x>2y, or the sliding along the edge of P which accompanies e is distinctly effective, being nearly 16 per cent of zy. Next day, with a bright sun, so that much finer fringes could still be de- tected, the range could be increased to 0^=0.2 cm. or #M = -3 ccn - when the spectra were all but separated on their near edges and fringes very large. For the case of much overlapping of spectra, 0^=0.15 cm., #^=0.22 cm., were obtained. Finally, when the spectra were all but separated on the far edges (implying reflection at some distance from the edge of the prism P), the fringes were glittering, but too small to be distinctly seen. TABLE 14. Inverted spectra. Long distances. Plate. E = 0.434 cm - ; A* = 1-533 ; B =4.6X10-"; z = o.23i3+.oi 15 = 0.2428. Series, etc. Observed e Observed e Observed 2-y X X I. Ruled grating. io 6 Z) 352 cm. o 184. cm. o 186 cm. O 25^ cm. cm. o 184 o 185 O 2S^ Mean o 184 o 185 o 281 o 281 II. Do., another adjustment. . 0.185 0.185 0.186 0.184 0.184 0.246 0.246 o 24.8 0.186 O 24.8 Mean o 185 o 283 o 281 III. Do., another adjustment.. o 24.4. IV. Film grating. 10*0 = 167. V. Prism 60 0.244 0.234 0.240 O 24.4, In table 14 the results for e M , e N , and y are given, when these displacements are produced by a glass plate =0.434 cm. thick. If the coefficient of dis- persion B is assumed, the displacement computed from X, n, E would be 2=0.243 cm. Although the values of zy are larger than this, the difference REVERSED AND NON-REVERSED SPECTRA. 69 must be ascribed to the assumed value of B and to the difficulty of placing the plate normal to the rays. The method of reversion of fringes, used else- where, is not sufficiently sensitive here. The data for x are again in excess of y by about 13 per cent. Finally, a series of consecutive measurements were made of the ranges of displacement, zy, for different dispersive powers at G, figure 43, all under (otherwise) like circumstances. The mean results were for 60 prism d6/d\= 760. 2y = o.o6ocm. Ruled grating 2,880. .117 Film grating 6,400. .208 Three or more measurements, completed in each case, were in good agree- ment. The attempt was also made to use a 45 prism here, but the spectra were too small and the fringes could not be found. In each of these cases the value of 2y for the plate =0.434 cm., as shown in table 14, series III, IV, V, are virtually the same. The rapid increase of range of displacement with the dispersion of the system is thus again encountered. 33. Rotation of fringes. A word must now be given relative to the rotation of fringes, which is here throughout 180, whereas in the similar case above (Chapter I, 25, 26) the rotation was but 90. It will be seen on consulting figure 43 that if M moves micrometrically, normal to itself, the pencil b will slide fore and aft, along the edge of the reflecting prism P'. Thus 6 may be either in front of or behind b' or coplanar with it in a vertical plane. It will not generally be collinear. This is an essential part of the explanation. <8 4 In figure 49, let a and b be the two patches of light of like color and origin, which produce interferences. The fringes will therefore be arranged in the direction /, normal to the line ab. Now suppose a is moved toward the right or b toward the left, or both, parallel to the edge of the prism, as the arrows in the figure suggest. Then the fringes will successively take the trends of which cases i, 2, 3, 4 are typical examples. In other words, they will be markedly accelerated and retarded in passing through the cases 2 and 3 re- spectively. This is precisely what takes place and suggests why the case between 2 and 3 may be used as a fiducial mark in interferometry. If a and 6 also move vertically, in figure 49 there will be no essential difference, unless the latter motion is large. In such a case the rotation may become 1,2,2,1. The displacement of 6 parallel to itself, for a normal displacement e of the mirror M, will be, as above, figure 17, if 5=0' 6 = 90 9 s = 2e sin '/2=s = 2e sin d/2. If 0=939', t=s = 2eXo.8i = 1.626. Thus if = 0.25 cm., 2 = 0.4 cm., and since / is twice the width of the patches or strips sliding over each other, the width of this strip would be 0.2 cm. It is the sliding of the pencil b along the edge of P' which introduces addi- tional path-differences whenever P 1 is not symmetrical and its edge not par- allel to the plane a a'. It is probable also that the same restrictions as to the breadth and depth of efficient wave-fronts will apply here as before; but this should be specially investigated. The nearly circular outline of the locus of fringes, when spectra are both reversed and inverted in 36, even though homogeneous light is in question in the last case, clearly points in this direction. 34. Range of displacement varying with orientation of reflector P'. The displacement e of the mirrors M and N slides the corresponding pencil b or b r , figure 43, along the edge of the reflecting prism P', and a reason for the rotation of fringes is thus easily at hand. It does not at once appear why the right or the left displacement (y) of P' should also produce a rotation of fringes ; for here the pencils b and b' remain collinear and there is no sliding. It must thus be remembered, however, that the fringes are ultimately ellip- tic; for the axis parallel to the Fraunhofer lines is conditioned by the obliquity of rays in this plane only, whereas the axis in the direction of the length of spectrum depends on dispersion. The motion y of P' displaces these ellipses bodily through the spectrum. Hence the fringes first appear at any given Fraunhofer line in the form of hair-like striations parallel to it. These enlarge and rotate to a maximum normal to the Fraunhofer line. In fact, a single interference line may now run from end to end of the spectrum. There- after the fringes vanish in symmetrically the same manner and are last seen as fine striations parallel to the Fraunhofer line. It is possible, therefore, that the sliding of pencils which accompanies the e displacement accounts for the difference of values of x = ze cos d/2 and 2y. Before discussing this question further it seemed necessary to study the effect of different orientations of P' relative to the b b' rays. One may note, preliminarily, that a rotation of M and AT on a horizontal axis parallel to their faces, or of P' on a horizontal axis parallel to its edge, also rotates the fringes; but it seems probable that these motions are virtually equivalent to a displacement, y, of the edge of the prism. The fringes may be seen in all focal planes, at least the long line parallel to the length of spectrum. The others may often be restored by rotating the grating. The marked occurrence of fringes in the narrow longitudinal gap between two spectra (overlapping just removed by the rotation of M and N on a horizontal axis) can possibly be explained in this way. These fringes in the dark space are very sharp and luminous and seen in the principal focal plane with the Fraunhofer lines. But it will usually be found that on draw- ing the ocular out the separated spectra will overlap at their edges again, REVERSED AND NON-REVERSED SPECTRA. 71 whereas on pushing the ocular in from the principal position the separation is increased. Hence, to account for the disturbance in the ether gap, as it were, it seems most reasonable to assume that the rays cross and interference occurs after the rays have passed the principal focal plane (i.e., nearer the eye of the observer), and that the interferences occurring here are projected into the principal focal plane. Nevertheless, the fringes are so strong and sharp that the two clearly focused spectra seem to react on each other across the gap at their edges. I have pointed out similar phenomena in the pre- ceding report (Carnegie Inst. Wash. Pub. 249). The case is just as if a tele- scope or lens focused on a single or a Young's double slit (with the images sharply delineated) should show fringes. In adjusting the interferometer, figure 43, for these experiments, the fol- lowing systematic plan was pursued. By a rough adjustment with sunlight and measurement, all parts of the apparatus are first placed symmetrically to each other (as in figure). The direct beam should just graze the edge of the prism P' and the naked eye, viewing the edge from above, should see the two bright rays of the same color (reflected from M and N) contiguously near the edge. In the spectra of the same length on the two sides of the prism P', the same colors are opposite each other. On looking down on the edge of the prism with a telescope (fine slit), two sharp and clear spectra should be seen, which can be made to overlap at their edges in any amount by rotating M and N on a horizontal axis. Finally, the D lines are brought into coincidence by rotating the grating G on a vertical axis. The largest fringes are obtained by slightly raising and lowering the incident beam L to the grazing position in question. By displacing P on the micrometer, right and left, the fringes are soon found. The first experiments were made with the object of testing the effect of a slant, to the right or left, of the edge of the prism on the range of displace- ment y. With a 60 prism at P, the search was found to be too difficult and therefore soon was given up. The few data obtained were : Edge of P' sym- metrical range, y=o.o6 1, 0.050, 0.062 cm.; edge of P' toward left range, ^=0.63 cm.; showing no certain difference. P was therefore replaced by a ruled grating ( = 352X10-* cm.) to obtain greater dispersion. The ranges of displacement, y, now found were: edge of P' to left, y=o.i2o, 0.130 cm.; edge of P' symmetrical, ^=0.128, 0.127 cm.; edge of P' to right y = o.no, 0.113 cm. ; readjusted, 0.130, 0.132 cm. These differences are apparently incidental, as much depends on the light. In a darker field fringes vanish sooner. One may assume that slight inclinations of the edge of the reflecting prism P' are without consequence. The next experiment was to determine the effect of a lack of collinearity of the rays b b'. This shows itself to the naked eye looking down upon the edge of the prism from above, since by rotating M and N in contrary direc- tions around a vertical axis the bright spots of light move along the edge of the prism from front to rear, or the reverse. If regarded by a telescope, the axis of the instrument will be correspondingly inclined toward the front 72 THE INTERFEROMETRY OF or to the rear. The data obtained for the range of displacement were now: Images and telescope toward rear, ^ = 0.124, 0.140 cm.; ^=0.150, 0.142 cm.; images and telescope toward front, y= 0.146 cm., ? = 0.154 cm. These differences are again incidental. The following data were subse- quently found: Telescope inclined rearward, y=o.ioo, o.no cm.; 0.143, 0.140 cm.; telescope vertical, ^=0.150, 0.150 cm.; 0.162, 0.151 cm.; tele- scope inclined forward, ^=0.150, 0.133 cm.; 0.145, 0.162 cm. In the first experiment the illumination was insufficient, so that the finer fringes escaped detection. Hence, it is here also probable that slight depar- ture from collinearity in the rays b b', normal to the edge of the prism P' t is without consequence. Discrepancies are introduced by changes in the in- tensity of light conditions which are often hard to control. As the range of displacement is not a quantity which can be accurately ascertained, the effect of the insertion of a glass-plate compensator, 0.434 cm. thick, was determined, with a similar end in view, for different angles of the rays b b' (nearly normal) to the edge of the prism P'. The results were : Telescope inclined toward front, y = 0.122, 0.123 cm.; telescope vertical, y = 0.122, 0.122, 0.122 cm.; telescope inclined toward the rear, ;y = 0.120, 0.12 1 cm. These are the differences of the displacement corresponding to the linear central fringes normal to the sodium lines, obtained in the presence and absence of the plate. The path-difference computed above was z 0.2428 cm. This is as near zy as the observations warrant. It follows, therefore, even if the observations are in their nature not very precise, that if the rays b and b' meet at such small angles as any reasonable adjustment may introduce, the effect may be disregarded. Furthermore, that the difference between x = 2e cos 5/2 (where e is obtained by moving the mirrors M and N parallel to themselves) and zy (obtained by moving P' at right angles to its edge) is to be ascribed to the sliding along or across the edge. The rotation of fringes which necessarily occurs in displacement inter- ferometry by the shifting of the ellipses is augmented or decreased in the former case (x) by the equivalent of the sliding in question. The reason for this has clearly been suggested in connection with figure 18, Chapter I, and figure 49, Chapter II. 35. Range of displacement varying with dispersion. The interesting method in Chapter I, 25, where the opaque mirrors are replaced by two identical gratings (halves of the same grating) with the object of obtaining successive orders of dispersion, may be used in connection with figure 43 of the present chapter. It is therefore the object to find the range of displace- ment y of the prism P' when the fringes pass from the initial transverse hair-lines to the final transverse hair-lines (fig. 45), through the longitudinal maximum of size. The same difficulty inheres in this method as in the above, viz, it is not possible to state precisely when the hair-lines have vanished; but the successive orders of range of displacement are so different that inter- pretable results are obtained. The experiments in the large interferometer REVERSED AND NON-REVERSED SPECTRA. 73 proved very trying, however, because the ruled faces of the available grat- ings at M and N were but 0.5 inch square. There is thus, without refined and special instrumental equipment, considerable difficulty in adjusting the rays to this small surface. This was particularly true in the higher orders of spectra. To obtain sufficient light the resolving grating G was replaced by a 60 prism. The dispersive powers are thus the same as in 25, Chapter I. The work proceeded smoothly in the orders of o (reflection from grating face) and i. In the third, the fringes were hard to find and hard to retain, for rea- sons which I do not understand. There was abundance of light, except in the fourth order, which was abandoned for that reason. The best results Order o, y = 0.066 cm. d6/d\ = 760 if - 2 3<> 3,500 % 2, .450 6,400 3 1 -650 9,9OO Though much time was spent on this work, the results (excepting the first) are doubtless still too low. Since the path-difference zy corresponds to x = 2e cos 8/2, cceteris paribus, and since the data in x, table 12, are essen- tially half the total range (rotation but 90, while it is 180 here), y corre- sponds to x. Thus the results in table 12 are larger throughout, but the present data make a smoother series even through the third order. 36. Spectra both reversed and inverted. This is an interesting combina- tion of the two methods of investigation and not very difficult to produce. Retaining the adjustment for inverted spectra as in 30, figure 43, the light impinging on the grating G is dispersed, preferably by a direct-vision grating (with auxiliary prism). The rulings of both gratings (the prism grating g inserted as shown being between the collimator at some distance and the grating of the interferometer G, figure 43) must be parallel. If the grating constants are different (D = 167X10-* cm. film and .0 = 352X1 cr 6 ruled grating were employed), the spectra in the telescope are naturally of different lengths; for the dispersion of the prism grating g is increased on one side and decreased on the other side by the second grating G. Moreover, this decrease from the larger dispersion of the first grating g is beyond zero (achromatism) into negative values. Hence, the corresponding duplicate spectrum in the telescope is a small and a large spectrum reversed, while the inversion re- mains intact. In the experiment made, the larger D\D'z distance was some- what more than twice the smaller AA- It is now merely necessary to place any longitudinal axis (line of symmetry) of the spectra in contact, or it is but necessary that the spectra are longitud- inally parallel and overlap. The phenomenon a, figure 50, then appears at the intersection of the lines of longitudinal and of transverse symmetry. It is thus proportionately nearer the smaller A A and farther from the larger D\D' Z doublets, but always between them. If the AA He within the D'iD'z lines, the fringes lie within the AA P a i r - 74 THE INTERFEROMETRY OF The phenomenon, which should be observed with a powerful telescope, usually consists of three small elongated dots, lying within an elliptic locus, the locus usually having a transverse axis (parallel to the Fraunhofer lines) about two or three times as long as the longitudinal axis (parallel to lengths of spectra). As a rule, the width was DiD 2 and the length larger than D\D' Z , but this ratio may be changed, as above, by screening off the wave-front. The fringes are not more than one-half of DiD 2 apart and are frequently horizontal (longitudinal), however the micrometer at M is shifted. The interesting result is here again met, incidentally, that spectra, though of different lengths, are nevertheless quite capable of producing strong inter- ferences. a 50 51 In further experiments with the long collimator and very bright spectra, a variety of other forms were obtained, shown at b, c, d, figure 50. In the patterns a and 6, the elliptic outline, sometimes circular, is always evident from the enhanced brightness of the bright fringes of the spot. The arrow-shaped form, c, inclosed a bright egg, whereas d was usually sharply semicircular. As any adjustment of overlapping spectra suffices, the D lines may be quite out of the field, or the spectra may be slightly separated with the interfer- ence spot in the gap. The experiment was also made of crossing the spectra at some other angle than o or 180. To do this the rulings of the prism grating were placed at right angles to those of the interferometer grating, as in Newton's method of crossed prisms. Seen in the telescope (adjusted for inverted spectra, as above, 30) the two spectra now made an elbow with each other, figure 51, while the AA lines are still parallel and can be put in coincidence. At first no interferences could be detected in any adjustment. Later, however, on using the large collimator, strong interferences were obtained in the line of symmetry of the elbow and normal to the D lines, as shown. They have the same characteristics as the preceding and persist during a displacement of M of about 0.3 cm. 37. Experiments with the concave grating. If in the device figure 40, Chapter I, the prism P is rotated 90 on an axis parallel to b b', so that the rays move upward, the phenomenon of inverted spectra may be realized. The fringes are observed with a lens from above or reflected forward. They REVERSED AND NON-REVERSED SPECTRA. 75 were found without much difficulty and showed a range of displacement (y) of the prism P, right or left, in various adjustments, of at least ^=0.71, 0.61, 0.67 cm. larger, therefore, than with the forward prism, as was inferred. Of course, much depends upon how far the extremely fine fringes at the begin- ning and end are pursued. This method is not very convenient for the present purposes. In the first place, the distance GMPF is given, an unnecessary restriction on the adjust- ment, unless the lens T is replaced by a short-distance telescope, which has other disadvantages. In the second place, the mirrors M and N can not be used for displacement, as they move the Fraunhofer lines of the correspond- ing spectrum. In fact, M or N affords a method for the adjustment of these lines to coincidence. Finally, the amount of overlapping which is usually secured is always very partial, and if the edge of the prism P is not quite sharp and well silvered, the edges are ragged. Even in the right-and- left displacement (y) of P, the spectra are carried bodily with it in front of T. Although this is no serious objection, it is an unnecessary complication. In spite of the brilliant spectra and large ranges, I did not spend much time in developing the method. 38. Conclusion. General methods. The origin of the phenomenon of reversed spectra seems to be the slit of the collimator, the diffraction of which furnishes a patch of light, effectively i or 2 mm. in breadth, out of which the component rays are to be separated. Spectroscopically the slit furnishes the degree of homogeneous light within which the phenomenon may be developed. In the case of inverted spectra, the slit is not primarily necessary, for here the interferences occur in the direction of (or the fringes lie normally to) the Fraunhofer lines and therefore virtually in homogeneous light. The fringes are due to the continuous changes of the obliquity of rays in each separate color and thus belong to the phenomenon of a wide slit with homogeneous light. The fringes of reversed spectra owe their occurrence to the continuous change of the obliquity of rays produced by dispersion and require a spectro- scopic slit. In the case of combined inversion and reversion, the locus of fringes is not far from circular, though the major axis of the ellipse corresponds to the inversion. Obliquity and dispersion are thus about equally effective. The patch of light rays of identical origin may now be separated into two component rays by a variety of methods. The white pencil may simply be cleaved by the edge of a sharp silvered prism, or the pencil may be refracted into two beams at a blunt-edged prism, or the separation may be produced by the diffraction of a grating, by polarization, etc. Two entirely distinct pencils are thus obtained, subject to independent control, by which the phenomenon of diffraction may be generalized. In other words, in the classical experiments in diffraction, the diffracting system is rigid. Take, for instance, the following experiment, which has a close bearing on the phenomenon of this paper. In figure 52, L is a distant slit or a fine Nernst filament, P the principal plane of the objective of a telescope 76 THE INTERFEROMETRY OF with the principal focus at F, to be observed with the eye-piece. The screen 5 5' with a double slit parallel to the linear source L is placed in front of the objective. Brilliant interferences are then seen at F, which are coarser as the distance e between the slits 5 s' is smaller, and if the distance PF is r and the distance between fringes is x, the usual equation \/e=x/r is applicable. Owing to the direct application of this experiment to the above investigations, its very fundamental importance in the theory of resolution of optical instruments, etc., it seemed worth while to give it experimental treatment here. The screen s s' is conveniently made by cutting parallel lines with a sharp triangular cutting stylus and a steel T-square, on a black- ened gelatine dry plate. A number of such doublets 0.05 to 0.5 cm. apart may be ruled at a distance of about 0.5 inch apart. When the parts are as- sembled, the fringes may be seen in all focal planes F, and the fringes are in fact much more brilliant with the ocular out of focus. The enormously large diffraction of each single slit is simultaneously visible, and if two doublets are close enough together, their systems may be seen superposed. If c is small (less than o.i cm.) the fringes are in fact visible to the naked eye without a telescope. With two identical doublets close together, the fringes may be seen to be alternately in step and out, as the ocular of the telescope moves outward, until finally the diffractions of the "rod" between the doublets is strikingly manifest. This has also been generalized in the present work. In all these classical cases there is a continuous succession of pairs of corre- sponding points (one of the pair in each slit of the doublet) between which interferences occur. The line of any two such points is rigidly normal to the direction of the slits. In the above experiments with spectra, however, the two points may not only have any relation to each other, but either point may be moved at pleasure. This gives rise to the bewildering variety of beautiful phenomena, some of them useful, which I have tried to describe in the present and preceding reports. With the beams of like origin separated, it is next necessary to bring them together again. This requires at least one independently controllable reflec- tion for each beam. In the interesting group of phenomena obtained with crossed rays, two reflections may be desirable, though with a change of appa- ratus a single reflection here also suffices. Thereafter the beams may be com- pounded in a manner inverse to the one by which they were produced, for instance, by the reflection of a silvered obtuse prism, by refraction toward the edge of a prism, by a grating, by polarization, etc. To observe the recom- REVERSED AND NON-REVERSED SPECTRA. 77 bined ray, the telescope is always the more convenient instrument, since its use is not restricted to definite distances from the system. It has been shown that as a whole the above phenomena correspond very closely to the behavior of the ellipses encountered in displacement interferom- etry. Thus the cases of non-reversed spectra, of inverted spectra, etc., if we dis- regard certain exceptional accompaniments for the moment, can be at once so classified. The shift of ellipses behind a narrow slit in an opaque screen and ob- served in front of it, for instance, would exhibit all the rotational occurrences. 39. Displacement interferometry. Equations. It is thus desirable to ad- duce the equations of displacement interferometry in a somewhat different way, but in the main as taken from my earlier reports. In figure 53, G is a thick plate of glass on which the blade- shaped' pencil of white light L from a collimator impinges at an angle i. M and N are the opaque mirrors of a Michelson device. G' is a plate grating by which the white beam R (feeble spectrum) is resolved, P the principal plane of the objective of the telescope, ro the image seen through the ocular. The plate disperses the white light L into the spectrum rv, as shown in the figure. The direct reflection at R' is not used. For convenience in discus- sion the mirror N and its component ray may be rotated 180 around the trace of the grating G as an axis, into the position N', where IN becomes x+ >' t intercepted between normals. If perpendiculars be let fall from 7 to AT or N' and /' to M, their difference of length is (1) N=x+x" = ecosi+x" N is an important coordinate used throughout, below, since it is indepen- dent of color (X and M). In (i) * is the angle of incidence, R of refraction of the plate of thickness e and index of refraction M- If we draw the wave-front w, the path-length of the red ray through glass to M, for instance, is en/cos R+p. The path of the ray through air (only) to mirror M is x+x'+p+x"=e cos i+e sin i tan R+p+x" If sin i=/i sin 1? be introduced, the path-difference thus becomes, after re- duction, (2) n\ = 2N-2encosR or ( 3 ) n\ = 2 e (ju-cos (i-R)) /cos R 78 THE INTERFEROMETRY OF the first equation being more practical. N, therefore, is the differences in distances of the extremities I, I' of the normal n at the point of impact I from the mirrors N and M respectively; x" is the air-distance apart of the two mirrors (after rotation). To find the change of wave-length per fringe, dK/dn may be deduced. rfX = _ X 2 _ 2 dn N-e(ii cos R+\(d[i/dX) /cosfl) This equation has a maximum when and this is the coordinate N e for centers of ellipses on any given spectrum line X. N=N e in equation (2) gives da zB In general (J.=A+B/\ 2 ; = is adequate for experimental work. Thus Ct\ A if 5 = 4.6X10"", e=i cm., R = o, then c = 88o, or 880 sodium wave-lengths are expended in the path-difference at the elliptic center. If n' fringes pass at a given color X for a displacement AN of the mirror M (7) X = 2AAT/' If n' fringes lie between two given colors X and X' for the same position N of the micrometer mirror M or if 5 is the differential symbol ucosR i w' = 205- -- 2^5 - X X The question next of interest is the change of the angle of altitude of the ray per fringes transversely to the spectrum. Light is homogeneous, N there- fore constant. Let a be the angle of altitude of an oblique ray in the hori- zontal plane L, figure 53, impinging at I. If i' and R r are the corresponding angles of incidence and refraction, then i f , i, a make up the sides of a spherical triangle, right-angled at the angle opposite i'. Hence (9) cos i' = cos i cos a, and (i cos R r = \/ff i +cos 2 * cos* a With this introduction of R r for R, equation (2) is again applicable, since nothing has been changed in N, e, ft, X, or i. Hence (10) n\ = zN 20vV i + cos 2 * cos 2 a REVERSED AND NON-REVERSED SPECTRA. 79 To determine the change of altitude per fringe, da _XVM*~ i -j- cos 1 i cos 1 a dn e cos 2 i sin 2a and this is a maximum when a=o; i.e., in the plane of figure 53. In this case equation (2) is reproduced from (10), so that a double maximum occurs for a = o and The other practical datum is the shift of a given fringe per centimeter of displacement of the mirrors. Here n and e are constant while N, X, ^, R vary, so that (12) ' - 2 dN ze df.i 2e du This is a maximum when n = n c = --- =45/X 3 , or when N=N e . In cos R d\ other words, there is a minimum displacement relative to wave-length shift of fringe at the centers of ellipses. Equation (10) is thus inclusive. If i = o, which is nearly the case, experi- mentally, in my work and no restriction on the apparatus, and since a is always very small, equations (10), (n), (12), etc., may be simplified. Hence approximately (i = o) (13) (14) = - \V-a 2 dn tea ( <&_ X 2 2 dn (6^ dN where n is the order of the fringe at X for N, e, /f/, a. Again, n e = -2e.d[t/d\. Equation (13) admits of an interpretation in terms of the approximately elliptic locus found for constant n. The equation may be written, if fi is treated as a mean constant, Here e (a/fi) and X may be regarded as the coordinates of a curve described on the faceof theplateof glass toward the observer, so that the equation is an ellipse referred to an eccentric axis of ordinates. The axes of this ellipse are tqp/N horizontally and e vertically. Of course, the telescope converges all this to a 80 THE INTERFEROMETRY OF single white image of the slit ; but the grating G reproduces the spectrum and enlarges it, in which case, however, not absolute position but the direction of rays is the determining factor. 40. Continued. Reversed spectra, etc. The equation underlying the greater number of experiments in the work with reversed spectra is of the form (17) n\ = 2e cos 5/2 where 5 is the double angle of incidence at either opaque mirror and e is the effective distance apart of the two mirrors i.e., the distance between the faces when one is rotated 180 about the axis of symmetry into parallelism with the other. In the present case, therefore, e corresponds to N in 39. The angle 5=02 0i, where 2 and Q\ are the angles of refraction of the col- lecting and the dispersing grating, respectively, so that sin 8 = \/D if D is the grating space. If the silvered right-angled reflecting prism is used for alining the separated pencils, 5 = 90 61. From the above equation the change of X per fringe is d\ _ _ X 2 _ dn ~ ~~ e( 2 cos d/2+\(dd/d\) sin 8/2) Since Xd5/dX=X/> cos = tan 6 d\ X* dn e(2 cos 5/2+sin 8/2 (tan 2 -tan 00) This is a maximum if e = o, as the quantity in the parenthesis can not vanish for very acute angles, such as and 5 must be. If 5 = o, or D l =D 2> d\/dn = -\ 2 /2e. Similarly, since e is now the micrometer variable or coordinate, and n constant, , , d\ 2\ (20) = - de e(2+tan 8/2 (tan 2 -tan 00) from which the similar conclusions may be drawn with regard to the motion of a given fringe. There is a maximum for e = o. The equation, it will be seen, is quite cumbersome, so that further treatment is inexpedient. Never- theless equation (20), if and 5 are expressed in terms of X, should admit of integration, at least approximately. The equation n\ = 2e cos (90 0)/2 for reflecting prisms needs special treatment, since 90 is not derived from 0. The coefficients after reduction become dn and d\ 2 X(>+X) 2 (>+X) >/2 In both cases there is a maximum for e = o. REVERSED AND NON-REVERSED SPECTRA. 81 Equation (22) may be integrated, and if C is an experimental constant, There remains the equation for crossed rays or achromatic conditions, i8o-20 n\ = ze cos = 2e\/D or (24) e = nD/ 2 in which there should be no motion of fringes throughout the spectrum, but for secondary reasons. It is now possible to consider the above results on the increase of the range of displacement within which interferences are visible, with the dispersion of the grating. In this case the equations in d\/de, viz, Nos. 20, 22, as well as 23 and 24, may be consulted. Since sin = X/D, all of them involve the dis- persion i/D, where D is the grating space. In the case of No. 24 the range of displacement should be indefinite, since the locus of fringes is stationary in the spectrum. It is found to be exceptionally large, but limited by special diffraction. Equation (20) is cumbersome, but otherwise similar to equation (22), which may be treated first. The displacement of any given fringe in wave-length increases with/= i/D, the number of lines per centimeter. If a fringe travels between any two wave-lengths X and X' and if D is large relative to X, equation (23) shows that approximately The range of displacement should therefore be roughly proportional to the square root of the dispersion, and one is not at once at liberty to conclude that the uniformity of wave-trains is enhanced by dispersion. In fact, if D is not large compared with X, as in the higher orders of dis- persion, the full equation must be taken. Unfortunately the data of Chapter I, 25, table 12, which are the most complete, do not easily admit of compu- tation in full. I have compared them both with e = e \/\/(D+\), in which the ratios of e observed and computed run up with D, regularly, from i to 7 ; and with 2^/dX = C(2D+X)/(D+X) 8/2 , in which the regular change of ratios for the same D (as D decreases) is again from i to 7. In other words, the observed values of e varying with D decrease enormously faster than coeffi- cients of this type, as they should. In view of the equation it follows that de e Q I ~ V dD~ 2 and the comparison of the e observed in table 12 with this coefficient is there- fore crucial. 82 THE INTERFEROMETRY OF To determine D from table 12 we have D=D'/n=i/(cos 6 n 'd6 n /d\), where is the order of the spectrum and d8 n /d\ its dispersive power, like is given in the table. In this way the data of table 15 were obtained. TABLE 15. Ratio of the range of displacement e observed in table 13 and de/dD, computed. Ratio Order. DXitf I tfe observed. VV(0+A) eoWVV(+X) 3 1,320 38 I5i 2-5 i 335 200 980 2.1 2 3 3 420 520 1 ,800 2,400 2-3 2.2 4 142 58o 2,690 2.2 In view of the character of the results for e, the ratio e/Vx/GD-hA) 3 , where e is the observed range of displacement, may be considered constant. The enormous variation of the range e with the dispersive power, as observed, must therefore be regarded as in keeping with the theory of the phenomenon, although the computation is not direct. The latter would require an integra- tion of equation (22) which may be written f" -U d\ 2.D+X \A(Z?+A) but the simpler comparison given was regarded adequate. Data bearing on equation (20) are given in table n and may after reduc- tion be written as in table 16 (62 = 19 30', D 2 = 177 X io" 6 ). TABLE 16. Range e observed. 0i Z?Xio d tan 5/2 X (tan 61 tan 0i) 0-33 52 2 3 6' 9 39' 1,320 352 16 54' 9 5i' 0.0459 .0156 The value of the term in the last column is thus small in comparison with 2 and may be neglected, as a first approximation. Hence roughly and the range of displacement should be nearly independent of the dispersion. As it is not, some corresponding principle must here be active, and this has already been found in the shift of one illuminated strip on the collecting grating relative to the other, when either of the opaque mirrors is displaced. For the same reason the effect produced by making d = o, as in 24, is not marked, so far as equation (20) is concerned. If 6 = o rigorously, the original experiment with but a single grating did in fact show large ranges of displace- ment, in view of the absence of sliding. We thus return to the special diffraction already mentioned in 38. When two spectra from the same source coincide, horizontally and vertically, throughout their extent, they will interfere at every point. The interference REVERSED AND NON-REVERSED SPECTRA. 83 will be visible within a certain range of path-difference. If one spectrum shrinks longitudinally on the other, the strip carrying fringes rapidly dimin- ishes in breadth; but interference is still marked near the transverse line of coincident wave-lengths. If one spectrum is reversed with reference to the other on a transverse axis, the interferences are reduced to a single nearly linear strip coincident with the line of symmetry. The width of this strip is independent of the dispersion of the system. It depends on the breadth of colored region which contributes rays to the strip in question. Hence, if, beginning with both ends of the spectrum, rays are cut off except those very near the line of symmetry, the linear phenomenon rapidly increases in size until all light is extinguished. This is what one would expect from the theory of the diffraction of wave-fronts broad or slender, with the generaliza- tion as to the rotation of fringes to which I have already referred. The association of the two diffractions is well illustrated by the experiments with inverted spectra. Here the edge of the reflecting prism is horizontal and normal to the interfering beams. When this edge is moved normal to itself, path-difference only is introduced. To compensate a plate 0.434 cm. thick the motion should be about 0.243 cm. The displacement found was 2^=0.247 cm., the difference being referable to insufficiently accurate dis- persive constants. When either of the opaque mirrors moves, the correspond- ing beam slides along the edge of the prism and the displacement 20=0.370 cm. of mirror was found, corresponding to the path-difference x = ze cos 5/2 = 0.282 cm. About 14 per cent more path-difference is thus needed with sliding (x) than without (zy). If now the reflecting prism is turned 90, so that the edge is vertical, the corresponding beams slide normally to or from the edge of the prism when the opaque mirrors are moved. The corresponding data were then found to be 2^=0.250 cm., x = 0.23 5 cm. Here about 6 per cent less path-difference is needed with sliding (x) than without (zy). The smaller effect in the latter case is to be expected, since the two corresponding rays slide toward each other in the same plane and can not pass through each other. In the former case they pass in marked degree across or through each other and must there- fore essentially contribute to the rotation of fringes. But the sign of the effect is precisely the opposite to what one would expect. Investigations such as these and the corresponding question of the width of strip carrying inter- ference fringes in case of crossed rays call for apparatus with better optical plate, or for more rigorous instrumental adjustment than I have been able to utilize in the present papers. It is best, therefore, to waive them for the present, however interesting the theoretical results with which they are associated.* ^ * I have since treated the outstanding difficulties of the text rigorously and the results will be given in a future report. CHAPTER III. ELONGATION OF METALLIC TUBES BY PRESSURE AND THE MEASUREMENT OF THE BULK MODULUS BY DISPLACEMENT INTERFEROMETRY. 41. General method and apparatus. About 25 years ago* I obtained satis- factory results in the measurement of pressures of the order of 1,000 atmos- pheres by the expansion of steel tubes of suitable thickness. The tube in this case was inclosed in a snugly fitting glass tube filled with water and the volume expansion measured by an attached capillary tube, the system being submerged in water to obviate thermal discrepancies. The whole subject has since been transformed by the famous experiments of Prof. P. W. Bridgman, and I merely touch it here with the purpose of testing the optic apparatus involved and with a view to the experiment explained in the final paragraph of this paper. In the present experiments I shall attempt to measure the increase of length of a steel tube due to internal pressure, by the displacement interferometer. The experiments will lead to an independent method for the measurement of the bulk modulus (Tait) and to a procedure for studying the thermodynamics of the adiabatic expansion of liquids. The interferometer used was of the linear type (fig. 54). Here L is a weak lens, about 2 meters in focal distance and 12 cm. in diameter, concentrating a beam of sunlight on the slit 5. c is the objective of the collimator, being a spectacle lens of about i meter focal distance. It is particularly advanta- geous to have rays of slight obliquity here if a brilliant and wide spectrum is to be seen in the telescope at T. H is the half-silvered plate of the inter- ferometer, N and M (on a micrometer) are the opaque mirrors, each about 2 meters from H. The rays reaching the telescope T would therefore show two white slit images from N and from M, which are to be placed in coin- cidence both horizontally and vertically by the adjustment-screws on M and * Phil. Mag. (5), xxx, p. 338, 1890; Bull. U. S. Geol. Sur., No. 96, 1892; cf. Am. Acad. Arts and Sciences, xxv, p. 93, 1890; for effect of pressure on electrical conductivity of liquids, see Am. Journal, XL, p. 219, 1890, and on the mercury pressure-gage, Am. Journal, p. 96, XLV, 1893. 84 INTERFEROMETRY OF SPECTRA. 85 N. To bring out the interferences, a direct-vision prism grating g is placed in front of the objective of T, whereupon, when the path-difference HM, HN is annulled, magnificent ellipses may be seen in the bright spectrum in the field of the telescope. The steel or other tube whose elongation under pressure is to be measured is shown diagrammatically at 1 1'. The end t', moreover, is closed by a tinned- steel plug-screw, while t communicates, by aid of a thick-walled ^-inch tube p of small bore, with the screw-compressor P described in my earlier papers (/. c.}. It is here that the thick hydrocarbon oil is forced into the tube 1 1' and the pressure measured by a Bourdon pressure-gage G, reading in steps of 10 atmospheres to 1,000 atmospheres. The parts of the interferometer are attached directly or indirectly to a brick pier in the laboratory. M is separately so attached; so is also the end / of the steel tube by the bracket at B, this being fixed rigidly. The other end, t', which must be free to expand, is to be supported on knife-edges or rollers of a vertical pendulum hanger y, the supports of which are in turn rigidly fixed to the pier. This will presently be further described. It was found that long vertical wires, supporting intermediate parts of the tube, were desirable and quite as good as more complicated arrangements. With the tube 1 1' thus fixed except as to linear expansion toward the right, the mirror N is clamped by a horizontal lateral arm at the end t', and the half-silvered plate H by a similar arm on the other side, at the end t. Thus the length HN varies with the pressure and the increment is compen- sated at M by bringing the center of ellipses back to the D lines in the field of the telescope. Accessories like water-jackets, etc., are left out of the fig- ure for clearness and will not be used in these experiments. To obviate friction, the end t' of the tube 1 1' was suspended -from a rec- tangular yoke or pendulum consisting of two vertical rods y and y', figure 55, and horizontal smooth round brass cross-rods r and r'. The latter roll on the round horizontal rods a and b suitably anchored at the same level in the pier. The rod r' supported the free end i' of the pressure tube. To counteract vibrations the rod y carries a bell-shaped damper, d, below, submerged in oil in the cup c. The middle of the tube i' is similarly damped at its center by a bell-shaped damper in oil (not shown), against lateral and vertical vibrations. 42. Remarks on the displacement interferometer. As a rule the ellipses are not seen at their best in the principal focus. The ocular must be drawn out somewhat to focus them sharply; but usually the sodium lines are still visible for guidance. No doubt this is due to the fact that ordinary glass plate was employed at M , N, and H, figure 54, or that H was not optically plane parallel. Moreover, as M is displaced, the focus of fringes changes; but as the centers of ellipses are used in measurement this is no particular disadvantage. A few trials were made with lens compensators, but the available combi- nations reduced the ellipses to horizontal sharp spindles or lines, without 86 THE INTERFEROMETRY OF passing into the hyperbolic patterns. However, a pair of plates of about the same thickness (0.7 cm.), but respectively of crown and flint glass, when in- serted in the M and N beams, gave good hyperbolas.* It was interesting to notice that in the principal focus the fringe pattern was nearly circular or roundish elliptic, but the ellipses were not strong. On drawing the ocular out, these ellipses underwent continuous deforma- tion, passing through horizontal bands finally into sharp and strong hyperbolic d forms. The fringes obtained with the ^ ** differently dispersing plates must therefore be referred to space loci in which, on passing along the axis, an elliptic cone becomes linear at the apex and symmetrically hyperbolic beyond. Thus in figure 56 the sections of the cone at a, b, c show fringes of the form d, e, f. 43. Observations. Thick steel tube. The seamless steel tube in my pos- session had the following dimensions: Length between mirrors, 160.8 cm.; diameter inside, 0.515 cm.; diameter outside, 1.278 cm. Hence the inside cross-section is 0.208 square cm. and the cross-section of steel 1.075 sq. cm. Such a tube is therefore rather adapted for measuring very high pressures, whereas the following work will not go beyond 1,000 atmospheres; but it was admitted for trial. If the case is treated as simple traction with Young's modulus taken as 2. 14X10 u , the elongation should be about i4Xio' 6 cm. per atmosphere, equivalent to a little less than half an interference ring. But this is obviously too large, as the tube expands in all directions by internal pressure. The case has been treated by Tait and will presently be referred to. The experiments made proceeded with unexpected smoothness from the outset, barring the tremors of the laboratory, which made it difficult to set the ellipses. Comparing the steel gage with a standard Bourdon gage read- ing in steps of 10 atmospheres to 1,000 atmospheres, the first six series of experiments showed an elongation AL of tube per atmosphere of io 6 AL = 7.s, 8.0, 7.0, 8.0, 7.0, 7.5 cm. In the last series the individual results were: Pressure 100 200 300 400 500 500 400 300 200 100 atmosph. Micrometer at 93 87 79 71 65 65 73 81 89 96 cm./io lo'AL = 7.4 X icr* cm. io 8 A = 7-7 X icr 6 cm. The data for decreasing pressures do not return into the preceding values for increasing pressures for incidental reasons which need not be discussed, but the mean rates are not very different. The discrepancies are probably in the setting of the Bourdon gage. They are not due to temperature here, * Later experiments showed that the flint plate was slightly curved. REVERSED AND NON-REVERSED SPECTRA. 87 even though the tube was not jacketed with water. The data, as was to be expected, show an expansion less than was computed for the case of traction, being only about half as large, or equivalent to about a quarter of an inter- ference ring per atmosphere. An arc lamp was used as a source of light and its flickering was very annoying. Tait* in the Challenger reports justifies the expression _ dx ai 2 - ao 2 36 where d%/dx is the elongation due to the internal pressure TI, in case of a tube of sectional diameters oo and ai and bulk modulus k. This merely re- places Young's modulus by the linear equivalent of the bulk modulus. Hence for the -tube of length x = 160.8 cm. and 00=0.515 cm., 01=1.278 cm., if for tool steel k = i.84X io 6 and pressure is given in atmospheres (io 6 dynes/cm. 2 ) the elongation per atmosphere is 160.8X0.265 * cm. 1.368X5- 52Xio 6 This is smaller than^the value found, probably because the tube is made of mild steel. If k be computed for the tube, since the elongation per atmosphere 160.8X0.265 which is about the order of value given by Everett^ for wrought iron. Voigt gives r.46Xio 6 for steel (see Landolt and Boernstein's Tables, 1905, p. 45). 44. Further experiments. The washer of the screw of the compressor, in the default of marine glue, was a perforated disk of pitch. This proved to be quite inadmissible for further work. The behavior of pitch was very peculiar and in itself interesting. The perforated disk was found to adhere without slip to the screw at the inner edge r, figure 57, and to the wall of the stuffing-box at its outer edge R. On turning the screw a smaller coaxial disk r' of pitch broke out of the larger disk, and the turning proceeded on this surface, r', without leakage. It was found nearly impos- sible to force the screw with the adhering pitch into the socket without serious injury to the screw. The sharp edges of the threads were in fact planed off flat and it was only by leaving fore-and-aft room for play in the stuffing-box that the compressor could be used at all. The part of the screw turning in pitch was ruined. _ ^ * Tait: Challenger Reports, n, 1882, App. A, p. 26. t See Everett tables, 1879, p. S3, containing original experiments of the author. THE INTERFEROMETRY OF The pitch washer was therefore removed and replaced by one of tallow slightly hardened with a little resin or wax, the two being melted together. This functioned perfectly within 1,000 atmospheres and was easily inserted in parts which could then be molded into a disk within the stuffing-box on forcing the gland into it. The measurements given in table 17 were made in steps of 100 atmos- pheres. AL denotes the elongation per atmosphere. TABLE 17. Series. Pressure range. io"AL (pressure increasing) . lo'AL (pressure decreasing). I 9 10 lootosooatm. 100 600 100 700 100 800 8.3 cm. 9-2 7-9 7-9 6.5 cm. 6.1 6.2 7-1 An example of the individual results may be given in case of series 9 : Pressure 100 200 300 400 500 600 700 700 600 500 400 300 200 looatm. Micrometer reading i. 01 9.4 8.6 7.9 7.0 6.2 5.5 5.4 6.3 7.0 7.7 8.5 9.1 g.Scm./io 3 The elongation here is always greater when pressures increase, although time is allowed for dissipation of temperature, than when they decrease. Four reasons may be assigned for this result: (i) the temperature increase on increasing pressure and vice versa; (2) permanent set imparted to the tube; (3) elastic warping of the tube owing to the end-thrust of internal pressures and consequent disadjustment of the interferometer; (4) vis- cosity of steel. Probably all of these discrepancies are present. That there was set I infer from the gradual displacement of the reading of the interfer- ometer for 100 atmospheres at the beginning and end of a series, though this may be due to new adjustments. The incidental displacements are particu- larly shown in the values of AL when pressure increases and are specially marked in series 7 and 8. They are also apparent in the change of form of the ellipses. As sunlight was used in the above work the annoyances of a flickering arc do not occur. The ellipses were not centered. To obtain some notion of the relation of these discrepancies we may pro- ceed as follows : The difference between the elongation per atmosphere dur- ing the phases of increasing and decreasing pressures in the four series given is, respectively, 1.8, 3.1, 1.7, 0.8 cm./io 6 , or i.SXio" 6 cm. per at- mosphere of compression. For a tube-length of 160.8 cm. and a coefficient of expansion 12 X io" 6 this is equivalent to a rise of temperature 9.3 X io" 4 , or, roughly, io" 3 C. per atmosphere of compression. Supposing the compressibility of the oil to be cfo=iooXio~ 6 per atmos- phere per cubic centimeter and the mean pressure = 500 atmospheres, the work done is pdv or 5ooXio 6 XiooXio" 6 = 5Xio 4 ergs per atmosphere per cubic centimeter at 500 atmospheres REVERSED AND NON-REVERSED SPECTRA. 89 Since the preceding datum corresponds to both increasing and decreasing pressures, the work done must be reckoned per 2 atmospheres or it will be IO B ergs. Taking the specific heat of oil as 0.5 and the mechanical equivalent as 42 X io 6 , the rise of temperature of the oil should be io 5 5 Xio- 3 C., nearly Hence the residual temperature discrepancy found, o.ooi C., would be but one-fifth of the full temperature discrepancy to be anticipated i.e., four- fifths of the heat would have dissipated by conduction, etc., during the wait- ing between successive compressions. The residual temperature is thus adequate to account for the full discrep- ancy, and if a tube of this kind is to be used as a pressure-gage, the tube should be made of "invar" or other metal without thermal expansion. True, a water-jacket surrounding the tube would improve the apparatus, but the thermal increments in question are so small that the device would not be trustworthy. If, however, a temperature discrepancy is shown by the optic gage it should also be shown by the more sensitive Bourdon gage, which is not the case. Thus a residual effect of temperature is improbable. The optical difficulties are slight and could be overcome by suspending the yoke, which in figure 55 rolls loosely on the cylinder a, b, t r , from steel pivots bearing on jeweled cups. Elastic and particularly slow viscous yieldings to persistent pressure are thus the probable reason for the errors. This also accounts for the displacement of the fiducial reading. Two further experiments were now made in which the ellipses were centered before each observation (table 18). In the second set (series 12) the tube was attached to the yoke and the latter to the hangers by soft adhesive wax, applied in the molten state. This proved quite adequate. TABLE 18. Series. Range. io s AZ< (pressure increasing). io*AL (pressure decreasing). II 12 iooto6ooatm. 100 600 7.2 7-7 7.2 7-7 The zeros were regained and the results were marked improvement on the preceding series. The same mean elongation is found for increasing and de- creasing pressure; but the values are not identical in the two series. The following data give the details of series 1 1 : Pressure 100 Micrometer reading 50 43 300 34 400 28 500 20 600 15 500 21 400 28 3 00 36 200 43 100 atm. 50 cm./io 4 A few units in the cm./io 4 place are thus uncertain, in figure 58. The graph is shown 90 THE INTERFEROMETRY OF 45. Brass tube. A somewhat thinner seamless tube of soft brass was next tested within 600 atmospheres. The dimensions were: length, 161 cm.; diameter within, 0.485 cm. ; diameter outside, 0.960 cm. Hence 161X0.235 (0.9216 0.2352)3^ 18.36 AL The ellipses were centered throughout and the yoke was given additional stability by soft adhesive wax, as above. The tube showed extraordinary variability, but during the trials under increasing and decreasing pressure and in the lapse of time the viscous changes somewhat subsided, as will be seen from table 19. TABLE 19. Series. Range of pressures. ioAL (pressure increasing). io 6 AL (pressure decreasing) . i 2 3 4 lootosooatm. 100 500 100 500 100 600 13.6 cm. 13-5 13-7 15.6 cm. 16.5 17.0 16.7 Mean. 13.6 16.7 Disregarding the first series, which was preliminary, the remaining data are consistent. Hence k from the mean io 6 AL=is.2 is relative to atmos- pheres ,18.36 15.2 I.2lXlO Voigt's value for brass is but 0.61 X io 6 , i.e., but half this. Throughout these experiments the reading for 100 atmospheres wandered continually, creeping over 7 X io' 3 cm. during the time interval of the experiments. The following individual data show this for the fourth series: Pressure 100 200 300 400 500 600 600 500 400 300 200 100 atm. Micrometer reading 320 301 290 274 263 250 252 270 290 305 321 336 cm./io* One would naturally refer this to the viscosity of the brass tube, but, curiously enough, the march is a contraction. Apparently the tube continually con- tracts in the lapse of time under internal pressure. The contraction occurs, however, for the case of a tube which was not rigorously straight. Optically, apart from the tremor of the laboratory, one would have no fault to find with the measurements, allowing a micrometer accuracy of a few io- 4 cm. Interference rings could easily have been utilized, but this would have required two observers. Two more series of experiments were made with the brass tube (table 20), in one (5) of which it was supported only at the ends with its original curva- REVERSED AND NON-REVERSED SPECTRA. 91 ture convex upward; in the sixth series the tube was additionally supported in the middle on a large pendulum-like wire or hanger. TABLE 20. Series. Range of pressure. io 6 AL (pressure increasing) . ioAL (pressure decreasing). 5 (free) 6 (supported)... iooto6ooatm. 100 500 12.3 cm. 13-6 15.2 cm. 15-2 The individual results are shown in figures 59 and 60. It will be noticed that an error was introduced when pressures passed from the increasing to the decreasing phase at the highest pressure, and this was particularly the case when a slight leak developed. If the highest pressure is omitted, io 6 AL = 14.7, both for increasing and decreasing pressures in the case of the sus- pended tube. Hence k = i.25oX io 6 , not differing essentially from the above. The unsupported tube is highly subject to viscosity. WO SCO The large effect resulting from viscosity is also shown in the initial and final readings (100 atmospheres) in figures 59 and 60. It is in both cases again an apparent contraction and is present even in the supported tube. The viscous effect in the lapse of time is directly given in the following inde- pendent measurements of apparent contraction. The tube was kept charged with an internal pressure of 100 atmospheres. Time 9 h 3O m io h 5 I2 h I2 Length, L 0.0173 0.0183 It is difficult to understand, however, how anything of the nature of vis- cous longitudinal contraction can occur under internal pressure. One might 92 THE INTERFEROMETRY OF suppose that the cylindrical tube as a whole is gradually progressing toward an ultimate spherical form, but this seems far-fetched. It is more reasonable to suppose that the viscosity is simply flexural. The tube is curved slightly convex upward and therefore the end mirrors N and H, figure 54, rotate con- tinually towards each other on a horizontal axis, under the end-thrusts of the internal pressure. The component beams and HNH and HMH are thereby each modified in length. Though it is difficult to specify why the former should be shortened relative to the latter, such a result is easily conceived. 46. Thin steel tube. The steel tube in 43 and 44 was adapted for high pressures only, showing but 0.2 interference ring per atmosphere. In con- trast with this a thin steel tube was now inserted, adapted for lower pressures. This was more sensitive than the Bourdon gage, the other tube being on the whole less so. The dimensions were: length, L=i6i cm.; diameter inside, 00 = 0.799 cm.; diameter outside, 01 = 0.951 cm. The outside diameter was calipered. A short length was then cut off and slit open and the wall thick- ness similarly found (0.076 cm.). The tube was not quite straight. Snugly fitting brass plugs were carefully soldered into the ends, and these were then tapped to receive the tubes conveying pressure. The observations shown in table 21 were recorded, the steps of pressure being 50 atmospheres in the first two and 100 atmospheres in the last two series. TABLE 21. Series. Pressure range. io 6 AL (pressure increasing) . io 6 A (pressure decreasing). i 2 3 4 5oto20oatm. 50 300 100 400 ioo 400 89 cm. 95-3 97-3 95-2 100.1 cm. 95-5 100.0 100.7 At 400 atmospheres the tube developed a slight leak at the ends. At 750 atmospheres one of the soldered end-plugs was blown out. It is remarkable that the plugs held so well. An example of the individual data may be given for the second series. In figure 6 1 these data are shown, positively. Pressure 50 ioo 150 200 250 300 300 250 200 150 ioo 50 atm. Micrometer reading 260 210 164 115 67 23 24 67 120 163 212 266 cm./io* Very little viscosity is, therefore, in evidence, but there is some displace- ment or irregularity, probably in the reading of the Bourdon gage. Pressure increments and decrements slightly rotated the mirrors in opposite directions around both a vertical and a horizontal axis. These were compen- sated by adjustment at the mirror M before each reading. As the mirrors inclined towards each other for pressure increments the tube must have been slightly convex upward, and therefore successively straightened as pressures REVERSED AND NON-REVERSED SPECTRA. 93 increased. The mean elongation per atmosphere is io 6 AL = 9; cm. and the bulk modulus may be computed as oo 2 L 0.6384X161 ~ a 2 i-a 2 3 AL (0.9044-0.6384) X3XQ7 X In spite of the large difference of dimensions, this datum is of the same order of value as the above (k= i.sgXio 6 ) for the thick tube, particularly as the present AL, from the occurrence of flexure, is probably slightly large. A tube of this kind with well-sealed ends (brazed probably), quite straight, and supported at different points of its length by wire pendula, should make a good pressure-gage within at least i ,000 atmospheres. An individual read- ing about id- 4 cm. per atmosphere or over 3 interference rings would be uni- formly available throughout. 47. Conclusion. Thermodynamic application. The data given show that an independent method of measuring the bulk modulus of metals is quite within the province of the displacement interferometer. The annoyances encountered, resulting from the viscosity of the metal or from warping, may be considered eliminated in the mean of the pressure-increasing and pressure- decreasing phase of the experiment. The tubes should be supported at vari- ous points along their length. Even the temperature discrepancy, if sufficient time is allowed between the successive steps of pressure, seems not to be of serious effect on the mean data. For the measurement of pressure, how- ever, the device is promising. In such a case the tube section should be chosen to correspond with the pressures to be meas- ured. Within 1,000 atmospheres a steel tube about i cm. in diameter, with walls about 0.75 mm. thick, gave fair results, showing the evanescence of about 3 inter- ference rings per atmosphere. Such a tube must be rigorously straight, well supported, and if possible of non-expanding (temperature) steel. It is interesting to consider the case of the adiabatic expansion of liquids in relation to such a gage. The available thermodynamic equation is where A is the temperature increment corresponding to the adiabatic com- pression Ap at the temperature 0, in case of a liquid whose coefficient of ex- pansion is a, density p, and specific heat C f ., /is the mechanical equivalent of heat. In an apparatus like figure 62, in which P is the screw compressor (with tinned or waxed screw 5) filled with the liquid in question, G the Bour- don gage, pressures may be suddenly applied without leakage by turning the 94 INTERFEROMETRY OF SPECTRA. screw 5. These are also measurable at the interferometer gage g, H being the half-silvered and TV an opaque mirror, as in figure 54. The subsidence of pressure due to cooling adiabatic compression is, however, also measur- able at g in terms of the delaying pressure. For we should have (v denoting volume, k the bulk modulus) Av Av 8p = A0; =~ v v k Sp or, apart from signs, A0= , where 5p is the subsidence of pressure due to the cooling A0, after adiabatic compression. Hence the original equation takes the form _ka z e Ap p ~7p~ *P an equation for measuring the specific heat of constant pressure of the liquid 2p-\-Ap at temperature and pressure . The observations thus consist in 2 measuring Ap (in displacement) and 8p in interference rings, both at the gage g. One may estimate the value of dp per atmosphere of Ap, for alcohol, by way of illustration. Here in c.g.s. units, fc=i.2iXio 10 ; a=i.iXio' 3 ; 0=20; = 0.79; 6^=0.58 Hence i.2iXio 10 X(i.iXio- 3 ) 2 X2oXio dynes *" 42XxoeXo.79Xo.s8 ^^''^nT* or 6^ = 0.015 atm. per atm. of Ap Hence, if 8p = 2oo atmospheres, the above gage would show about 10 rings for dp. Similarly a pressure dropping adiabatically from 1,000 atmospheres would show about 50 rings, residually, after closing. The present research was planned to be pursued at much greater length; but owing to the annoyances of a quivering pier, which are particularly marked during the term weeks, and to the injury of the screw before the tal- low washer was inserted, it was thought wise to discontinue it at present. If the micrometer is to be set to io- 4 cm., the ellipses must be reasonably quiet, and in case of long-distance interferometry such a condition can be realized only during the summer months. CHAPTER IV. REFRACTIYTTY DETERMINED IRRESPECTIVE OF FORM BY DISPLACEMENT INTERFEROMETRY. 48. Introductory. Some time ago* I made a number of experiments on the use of curvilinear compensators in connection with the displacement interferometer. It is obvious that the curvature in such a case must be very small, so that single lenses for the purpose are not easily obtained. The use of a doublet of two lenses of the same glass, but respectively convex and con- cave, meets the case fairly well, the necessary refracting power being received by spacing the doublet. Lenses of about i diopter each gave the best re- sults, bringing out fringes of quasi-elliptic and hyperbolic symmetry in great variety. Later it appeared as if plates of different varieties of glass, as for instance crown and flint, if placed in the two component beams MH, NH, figure 54, would produce the same phenomena. The flint plate used, however, proved to be inadequately plane, so that the result is in doubt. More recently I have endeavored to secure similar results by submerging the lens (convex or concave) in a liquid of about the same index of refraction. This method would seem to be interesting in other respects, for it is probable that the index of the solid may be determined in this way irrespective of form.t If, for instance, the liquid and the solid have the same index, one would be tempted to infer that the latter may be removed or inserted with- out displacing the center of ellipses at the particular wave-length under con- sideration. The index of the liquid in place is then determinable by the interferometer to a few units in the fourth place. If experiments of the present kind are to be accurate, it is obvious that the walls and cavity of the trough in which the lenses are to be submerged must be optically plane parallel; otherwise some compensating adjustment must be made at the opaque mirrors of the interferometer, and for this no adequate allowance is at hand. It did not, however, seem worth while to provide expensive apparatus before the method had been worked out in de- tail. Accordingly the present experiments were conducted with troughs of ordinary plate-glass put together by myself, and little attention will be given to absolute values of index of refraction, as such. 49. Preliminary experiments. The first experiments were made on a large linear interferometer (see fig. 54) with distances of nearly 2 meters between * Carnegie Inst. Wash. Pub. No. 249, 1916, chapter ix;c/. Amer. Journ. Science, XL, PP t 2 MrT 3 R 8 W 9 Cheshire (Phil. Mag., xxxn, 1916, pp. 409-420) has recently used Tc-pler's method for the same purpose with marked success. 96 THE INTERFEROMETRY OF the mirrors. The rays in such a case are all very nearly parallel. Sunlight, arc light, and the Nernst filament were each available for illumination. If the latter is used, the adjustment must be made by aid of the two white slit images, which are to coincide horizontally and vertically. Otherwise the sodium lines are available. With a very long collimator (2 meters) and a wide single-lens objective (10 cm. or more), the Nernst filament may be used directly in place of the slit. If the beam passing the objective is not wider than i cm. (opaque slotted screen), very perfect ellipses may be obtained. On inserting the trough with a thickness of 1.3 cm. of CS 2 solution normally into the MH beam, the original very large ellipses were reduced in size and rounded as usual to smaller circles. Submerging a convex lens (i diopter) into the liquid until the beam passed symmetrically through it changed these circles to very long horizontal spindles. A concave lens similarly pro- duced horizontally very eccentric hyperbolae. With water in the trough, only the convex lens showed observable fringes, these being very long, practically linear horizontal spindles. All these fringes lie considerably in front of the principal focal plane of the telescope (fig. 54, T), and the abnormal forms are necessarily relatively faint. They change in shape and intensity with the focal plane observed. 64 On mixing CSz with kerosene (about equal parts), types of fringes shown in figure 63, but with many more lines, were obtained. This is a combina- tion of both spindles and hyperbolae. Probably three layers of liquid are chiefly in question, viz, kerosene, kerosene + CSa, CS 2 , and the three stages of form and the sinuous lines correspond to them. Fringes were sharp only if viewed in front of the principal focal plane of the telescope. By submerging convex or concave lenses, the hyperbolic parts or the spindles of the fringes could often be removed. Similar results were obtained with mixture of CS 2 and sweet oil, though this solution is more homogeneous. 50. Apparatus. To obviate the tremor of apparatus which is inevitable in the case of the long-distance interferometer, the parts were now screwed down at short distances in the cast-iron block B, figure 64. Here the ranges MH, HN of half-silvered plate H, and opaque mirrors M, N, did not exceed REVERSED AND NON-REVERSED SPECTRA. 97 14 cm., but this gives ample room for the manipulation of the trough t placed normally in the beam MH. White light enters by way of the collimator SC at any convenient angle 6 (as this does not enter into the equations), and = 6o was used. The opaque mirrors M (and preferably also AT) are on micrometers with screws normal to their faces, and each must be provided with adjusting-screws relatively to horizontal and vertical axes. An elastic fine adjustment is desirable. The block contains a number of screw-sockets, 6, for attaching subsidiary apparatus. The trough t should preferably be at- tached to an independent supporting arm, not connected with B, and be revolvable about two axes normal to each other. In such a case the position normal to the beam of light may be found from the reverse of motion of the interference rings, while the trough is slowly rotated in a given sense. The telescope T (relatively much enlarged in the diagram) is not attached to the block. It is to be used both as a simple telescope for the adjustment of the white slit images to horizontal and vertical coincidence, and as a direct- vision spectroscope. The most convenient attachment for this purpose is the direct-vision prism grating G (film grating) just in front of the objective of T. Two perforated thin disks of brass are useful for this purpose, one disk being firmly attached (like the cap) to the objective, the other to the flat face of the grating with the prism outward. A swivel bolt a, between the disks, thus allows the observer to throw out the grating and use the telescope. A stop arrests the motion of the grating when it is rotated about a, back again, for viewing the spectrum. This plan worked very well, and the el- lipses obtained were magnificent. It was almost possible to control the microm- eter M manually, and all hurtful quiver is absent. The fiducial line to which centers of ellipses, etc., are to be returned is always the sodium doublet pres- ent in sunlight and the arc and artificially supplied by an interposed burner in case of the Nernst lamp. The telescopic lens need not be more than 2 cm. wide, and cross-hairs are not needed. For measuring dispersion the Fraun- hofer lines B, C, D, E, b, F were used. 51. Equations. The useful equations for present purposes are given in a preceding report,* and the following cases only need be repeated here. If e is the thickness of glass plate of index of refraction n for the wave-length X, and if the equation M=^+/X 2 , where A and B are constants, be taken as sufficient, (i) n-i=AN/e-2B/\* where AAT is the displacement of the micrometer at the opaque mirror M or N due to the insertion of the plate normally to the component beam in ques- tion. To determine //, B must be known at least approximately. It may be measured in the same adjustment, however, if two Fraunhofer lines are used fidutially. Let 8N be the displacement of micrometer to pass the center^of * Carnegie Inst. Wash. Pub. No. 229, 1915. 4<>, 4', 42- THE INTERFEROMETRY OF ellipses from wave-lengths X to X'. Then if e' is the thickness of the half- silvered plate H, and R the angle of refraction within it, (2) dN = If Circles. 104 THE INTERFEROMETRY OF same trough but with different solutions, dilute and nearly concentrated. To place the trough normal to the beam, the reflection of the latter from the face of the trough was made to coincide with the spot on the half-silver. This is inadequate for fine work, but the interferometer method of retro- gressing fringes is not applicable unless the trough is separately mounted. From the data obtained the constant B was computed from pairs of Fraun- hofer lines, B and E, C and b, D and F. Apart from the effect of the thick- ness e, it should be correct in case of the solutions to about a few tenths per cent. In case of the concentrated solutions, however, the ellipses already begin to move sluggishly and are much smaller. 70 From ju I+2.S/X 2 and B, the corresponding indices of refraction in table 27 are easily computed ; but these data are of little value here, because the thick- ness e of the ordinary plate-glass used is not constant to the degree necessary. It is, however, worth while to exhibit the B for the different mercury-iodide solu- tions in terms of either p or, what is equally serviceable and more convenient, in terms of /z i+K; for this quantity is directly given by the displacement measurement bN/e. If we regard the mean B for the B to E lines, and C to b lines as applying to the D line, the coordinated values are: (2) (4) (8) (9) do) H i+K 0.5384 0.5869 0.8732 0.8736 0.6194 BXio 11 15.63 18.37 40.99 40.98 20.94 For solutions of small concentration the ratio of B and n i+K changes but slowly (29 to 34) as shown in figure 69, so that B may be predicted from the latter, always remembering that our equation with two constants (n=A +B/X 2 ) is inadequate at the outset. REVERSED AND NON-REVERSED SPECTRA. 105 To take the case of the submerged glass-plate series 5, if e' be the thick- ness of plate and e" of solution, so that *'+*"=*, the thickness of trough, Thus, 0.5204 of the data of series (3) added to 0.4796 of the data of series (4) should reproduce series (5). The results are given in table 28. TABLE 28. B C D E b F 3 1 Computed 5 Found . Difference. . . . 0.5561 .5465 0.5621 5525 0-5791 5697 0.6052 .5962 0.6104 .6016 0.6338 .6247 .0096 .0096 .0094 .0090 .0088 .0091 These differences are nearly constant and due to the orientation of the glass plate with which its effective thickness (assumed 0.293 cm -} will vary. 55. Conclusion. It appears, therefore, that the expectation of recognizing the equality of refraction of a submerged solid and a solution, at any given wave-length, from the fixity of the fringes in the presence and absence of the solid has not been fulfilled, at least for the mercury-potassic iodide solu- tion. The reason is found in the enormous difference of the dispersions of the solution and ordinary glass. When the ellipses are not displaced, n~n' = 2(B'B)/\ 2 , and this difference may even approach a unit in the first decimal of p.. The troughs in which such experiments are to be made must be optically plane parallel, as otherwise an inadmissible error due to thickness of plates is introduced. With such a trough, however, the ease and accuracy with which the dispersion constants may be found, at least for the solution, are noteworthy. When the solution is more refracting than the glass, it is curious that the ellipses are not seriously distorted or vague, even when the symmetrically submerged solid is lenticular. Hence the equation just stated is available for a wide variation of form. Furthermore, if AAf is the displacement at the micrometer corresponding to the presence and absence of glass of the thickness e, But as /*', B' for the solution are known, M and B for the glass may both be computed from observation at a number of wave-lengths, X, provided M = A+B/\ Z for glass, which is sufficiently nearly so to the fourth place of decimals. Hence if &N/e+n'+2B'/\ 2 =x is known, 106 INTERFEROMETRY OF SPECTRA. and /i=#+2.B/X 2 . Data of this kind are given in table 29, the absolute values depending wholly on the correctness of the thickness E of the solution and e for the submerged lens. The ellipses were excellent, sharp, and strong. /t i+2.B'/X 2 is found directly and AN is negative as the solution is stronger. TABLE 29. Refraction of lenses. D line. Thickness of solution, =0.563 cm.;K' = n' i-f-2-B'A 2 ; 2.B/X 2 = 0.0260 for glass. nl=K'AN/e. Lens. e K' AWXio 3 M Remarks. I diopter 2 diopters. . . . 5 diopters 10 diopters cm. 0.138 .200 .248 447 0.6140 .6140 .6140 6343 cm. 7.8 10.7 18.7 *20.5 I-53I5 1-5345 1.5126 1-5625 Ellipses strong. Ellipses strong. Ellipses vague. Ellipses vague. * Probably not adequately centered. But such a method is not as convenient as was anticipated at the outset, while it may fall short of the needed accuracy if rigorously carried out. The only escape out of the difficulty would be the use of a liquid of about the same dispersion constants as the glass. In this case the corrections would be minimized and the experimental work simpler. CHAPTER V. DISPLACEMENT INTERFEROMETRY IN CONNECTION WITH U-TUBES. JAMIN'S INTERFEROMETER. 56. Introduction. A variety of constants in physics may be found from the relative heights of two communicating columns of liquid. This is, for instance, the case in the classical experiment of Dulong and Petit on the thermal expansion of liquids. Again, if one of the tubes is subject to a special force acting in the direction of its axis, this force in its bearing on the liquid may be evaluated from the resulting difference of heads of the columns. Thus one tube may be surrounded by a magnetizing helix and the effect of the axial magnetic field on the liquid in question (i.e., the susceptibility) found from the displacement of its surface by the presence and absence of the field, etc. It seemed to me worth while, therefore, to test whether it would be possible to measure small displacements of this kind by passing the two component beams of a displacement interferometer axially through the two columns respectively, and to measure the differential effects in question in terms of the resulting displacements of fringes. cfl 70 57. Apparatus. Michelson interferometer. The interferometer first used was of the same form as that described above ( 2, fig. 3), B, figure 70, being a heavy iron block, i foot in diameter and 1.5 inches thick, on which the mirrors M, N (the latter and preferably both on micrometers) are securely mounted with the usual direct rough and elastic fine adjustment for hori- zontal and vertical axes. A beam of parallel white rays L arrives from a collimator (not shown) and impinges on the half-silver plate //, to be reflected and transmitted at a convenient angle 6 (about 60), thus furnishing the two component beams which are to traverse the limbs of the U-tube. 107 108 THE INTERFEROMETRY OF The vertical columns of this tube are shown at C and C' (with accessory mirrors removed), and they are joined to the capillary tube p near the bottom of C and C'. Details will be given in connection with figures 71 and 72. The ray HM strikes a mirror symmetrically at 45 to the vertical below C, is thence reflected upward along the axis a, striking another mirror above, also symmetrically at 45 and parallel to the former, whence it is reflected to the opaque mirror M. The latter reflects the ray normally back, so that it retraces its path as far as H, by which plate it is now transmitted to be observed by the telescope at T. Similarly the transmitted component ray HN is guided by suitable reflectors at 45, so as to take the path Ha'Na'HT, thus passing axially (a') through the tube C'. It is necessary that the U-tube CpC' be mounted independently of the block B on suitable bracket or arm attached to the pier. Otherwise any manipulation at N will disturb the surfaces of water in C and C'. Ordinary clamps admit of raising or lowering or rotating CC' satisfactorily, always providing that it shall not touch B. The telescope at T is also mounted apart from B on the table below. The direct-vision prism grating g is placed im- mediately in front of the objective and swiveled as described in figure 64, Chapter IV, so that either the white slit images or their spectra may be seen in the field of view, according as g is rotated aside or is in place. In figure 71 a front sectional elevation of one of the shanks of the U-tube is given with all appurtenances, and a similar sectional elevation at right angles to the former is added in figure 72 for the top of the tube. In figure 7 1 the mirrors m' and m are on horizontal axes and the component ray coming from behind the diagram strikes m' below, is reflected axially upward through CC, impinging on the mirror m (also on a horizontal axis) , whence it is reflected horizontally toward the front of the diagram. The ray a and mirror m are given more clearly in figure 72. The lateral capillary tube appears at p and the tube C is closed below with a plate of glass e, cemented in place. To mount the mirrors m, m', snugly fitting rings r and r' encircle the tube C near its top and bottom and can be fixed by the set-screws 5 and s'. In virtue of these rings, the mirrors m, m' may be rotated at pleasure around the vertical axis a of CC. The horizontal axis of the mirrors m, m' rotates at pleasure in the vertical arms A, A' of square brass tube. A, A' m turn may be slightly swiveled about the horizontal axis b, b', in a rigid lateral projection of the rings r, r' . Thus m, m' are capable of rotation around three axes normal to each other and adequately clamped in any position. The component ray HN may be adjusted to the center of the lower mirror m' by placing the collimator L and then guided axially by m', m, N as described, each being adjustable. The component ray HM may be similarly adjusted to the center of the lower mirror m' (at 45) by slightly rotating the half- silver plate H (on horizontal and vertical axes) and then guided axially by m r , m, M. As a whole the adjustment is difficult, though it need not be much refined. Clear white slit images in the telescope T are an adequate criterion. In the absence of a liquid in CC, figure 71, the fringes are easily found after REVERSED AND NON-REVERSED SPECTRA. 109 careful preliminary measurement, and they are strong and satisfactory. When this adjustment is given, the presence of liquid in CC, if the two columns are of nearly equal length, does not much modify the adjustment. In fact, the fringes were found much more easily than I anticipated, and in quiet surroundings they are strong and fine. It is necessary, however, that the tube CC should be of sufficient width to avoid all curvature due to capillarity, at least in the axis. Tubes 2 cm. in diameter and 10 cm. long of thin brass were first tried, but proved to be too narrow. No sharp slit images could be obtained with reasonable care as to setting the mirrors. Thereafter tubes 4 cm. in diameter were used, but even these are somewhat too narrow. Slit images, however, were sharp and parallel and could be easily brought to coincide. With the wide tubes, however, the mobility of the liquid in CC increases enormously, so that only under exceptionally quiet conditions could the fringes be seen, and never quite without quiver. The wind beating on the house, for instance, threw them nearly out of view, so that only a suggestion of their presence remained. In spite of the very promising beginnings, there- fore, it became a serious question whether, with the apparatus as here devised, the purposes of the research could be reached in this laboratory. Finally, the flickering of the arc lamp may be a grave inconvenience; for if the columns C, C' as usual are virtually prisms, the coincidence of spectra will for this and other reasons be destroyed by the displacement of the arc. 58. Equations. Some estimate of the increments to be anticipated may be given here, and expressed in terms of the Dulong-Petit experiment. If a is the mean coefficient of expansion of water at the temperature in question and A// the increment of the head H corresponding to the temperature difference A* between the columns, (i) &H=aHM Again, if A/V corresponding to AH is the displacement of centers of ellipses at the wave-length X, and n the index of refraction of water, so that M = A +B/\*, nearly, AAT Hence A* may be computed as AAT Since the value of AAT is within iQ- 4 cm. and H= 10 cm. in the above appa- ratus, we may further write at mean temperatures (25) a = 2 . 5 Xio^ M =i. 3 33 = io- ll X 3 .i 2 /X*= 0.018 at the> line. Thus Ai-i-f- 2 B/X 2 =o.35i and A*= 10^/0.351 X2.sXio- 4 Xio = o.ii4. other words, in case of tubes 10 cm. long, the effect of a difference of tempera- 110 THE INTERFEROMETRY OF ture of about o.i degree between the tubes should be easily observable by mere displacement, whereas a difference of less than 0.03 would be equivalent to the passage of one interference ring. Again, from equation (2), if AN= icr 4 cm., then AH = io~*/o.35i = lo^Xa.S cm., or about gX icr 6 cm. per vanishing interference ring are the displacements to be anticipated. These are equivalent to pressures of about 0.3 and o.i dyne per square centimeter. 59. Observations. A large number of observations were made with the apparatus described, but as under present surroundings the fringes always quivered violently, no quantitative results of value were obtained. Naturally such experiments imperatively demand a laboratory remote from traffic, since the undulation of mobile liquid surfaces is introduced in addition to the tremors of solid appurtenances. An attempt was made to register the pressure near an electrically charged point, but no results could be obtained. Again, though the attraction of an electrically charged surface for the free surface of water in either tube was recognized, on using adequately high potentials to measure the forces the surface became troubled and the fringes vanished. In this case, if p is the pressure, V the difference of potential in volts, and d the distance apart of surfaces in centimeters, /=4.4Xio~ 7 X(V/c0 2 dynes/cm. 2 if 7=0.3 can just be determined by the displacement method, and if V = 80 volts (roughly) is the smallest potential difference discernible for quiet fringes. Finally, Dulong and Petit 's experiment gave very definite results even for small ranges of temperature, subject to the conditions stated. By surrounding the top of the tube C, figure 71, with a close-fitting helix, the upper face of which reached just below r, while the surface of liquid within, w, lay at its center, an attempt was made to detect the susceptibility k of water, etc. If the field of the coil be written roughly H=o.4irin/l, where i is the current in amperes, n the number of turns, and / the length of the helix, we may write Since p = i, the increment of head, h, becomes Hence if roughly k = io-*, g=io 8 , i=i am., //=3S as in the helix used, A/* = 44 2 X io- 9 = i. Thus if i=io amp&res, A/t = 2Xio~ 4 cm., nearly, and easily determinable by the displacement of ellipses, or from interference rings. The experiment was tried, but the quiver of rings was such as to admit of no decision. In case of the magnetic solutions, k is of course much larger; but under the circumstances it did not seem worth while to attempt further work. This will be done with other apparatus in the course of this paper. REVERSED AND NON-REVERSED SPECTRA. Ill 60. Jamin's interferometer. The ease with which the Michelson inter- ferometer may be adjusted and its remarkable adaptability have led to its general preference over the older form of Jamin. Nevertheless, the latter furnishes two parallel rays which for such purposes as the present are desir- able. Hence if the four faces of the interferometer be separated in the manner suggested by Mach (shown in fig. 73), a very available form of interferom- eter is obtained. Here M and N are half-silvered plates, M' and N' the opaque mirrors. The white light L impinging from a collimator thus fur- nishes the two component beams ac and bd, which are observed with the tele- scope at T, after passing the direct-vision prism grating g. If either mirror M' or N' is displaced a distance e, moving parallel to itself, the path-differ- ence 2e cos 6 is introduced with the corresponding shift of ellipses. The U-tubes C, C', with their helices H, H', and connecting pipe p are now con- veniently installed as shown. But the trouble with the arrangement is the difficulty of adjusting the Jour surfaces. Not only are the centers of ellipses liable to be remote from the center of the field, but it is often hard, without special equipment, to even find the fringes. If, however, the device which I suggested in the preceding report is adopted i.e., if (fig. 74) the half-silvered plates M, N are at the ends of a single strip of plate-glass, so that rays terminating in M, M' N N' after adjust- ment necessarily make a rhombus-like figure symmetrical to MN the fringes are found at once; for they appear when the white slit images in T coincide horizontally and vertically and the rays bd and cd intersect in the common point d. Hence the mirrors M', N' should be on carriages D, F, adapted to move on parallel slides 5, 5'. M, N may also be put on a carriage E, though this is not necessary. S, 5' need not be parallel to ac or bd. If the mirrors M' and N f are wide, considerable latitude of adjustment is thus obtained. If MN is half-silvered on the same side (*.*., toward N') a compensator is needed in ac or cd if path-difference is to be annulled (symmetry). If, how- ever, M is half-silvered on the N' side and N on the M' side, no compensator is required. In the latter case, however, if ordinary plate-glass is taken, M and N are not quite parallel and the ellipses will be eccentric. This, however, is not necessarily a disadvantage, unless the strip MN is excessively wedge- shaped. 112 THE INTERFEROMETRY OF The ellipses obtained are usually long vertically i.e., quite eccentric so that the fringes soon become straight and the rotation is extremely rapid whenever the center of ellipses is out of the field. It is therefore possible to adjust relative to horizontal fringes (parallel to the shadow of wire across slit), as these incline very obviously for a displacement of less than io~* cm. and rapidly become vertical. For this reason it makes little difference in practice whether the half -silvers are on the same or on opposite sides, or whether obser- vation be made at T (cd prolonged) or at T' (bd prolonged). Moreover, the plate MN may be conveniently constructed, as in figure 75, of two mirrors m, n, attached to the clear strip of plate-glass g by aid of strong steel 76 clips at c, c'. With the half-silvers 5, 5', on the same side, the wedge-angle of the glass is excluded. For shorter diagonals, the plan of figure 76, with the silver surfaces s, s' held together by clips at c, is preferable. If the mirror M', figure 74, is displaced a distance e, where a glass-plate compensator of thickness E and refraction constants n and B is introduced normally either into ab or bd, the equation is easily seen to be, at wave- length X, where B is the angle of reflection at M. Using the plate = 0.434 cm. treated above, the first member is 0.2428 cm. Values of e of 0.2420, 0.2409, 0.2427 were roughly obtained. Hence the mean value of 6 should be about 60, as it actually was. The occurrence of this angle and the shift of the beam bd along the mirrors M' and N are the main objections to the method of figure 74, for the rhom- bus is not necessarily perfect. If the ends of the plate MN are silvered on the same side, the compensator must have double the plate-thickness to annul path-difference. Finally, the half-silvering does not, for large 6, suffi- ciently exclude the reflection of bd from the naked face of the plate, so that the fringes are never quite black. These difficulties may be met by making MN, figure 74, the short diagonal of the rhombus and using the strip, figure 76. In such a case 6 at M' is small, and in view of the nearly normal reflec- tion at M and N relatively little reflection comes from naked glass, sliding is largely avoided, and no compensator is necessary. In this case the fringes for no path-difference are actually strong black horizontal lines on a colored ground and far enough apart that o.i fringe could easily be estimated. A test experiment with the above plate showed = 0.1244 cm., corresponding to the small angle 6, a little over 12. When the U-tube CC', figures 71 and 74, is introduced, the strip MN will have to be at a considerable angle (about 45) to the horizontal, so as to raise the N end about 15 cm. above the M end, corresponding to the height of m above m' in figure 71. The new condition, however, in no way changes REVERSED AND NON-REVERSED SPECTRA. 113 the general procedure. In case of figure 74, the mirror N f must be high and M' low. This is usually less convenient than the case when both mirrors are high (C placed at G) or where both mirrors are low (C r placed at G'). In the former case, again, the rays have considerably diverged in a vertical plane and the fringes are less marked. If C' is at G' the whole of each com- ponent beam may be caught and passed through the respective shanks of the U-tube. The fringes are strong, easily found, and large, so that the cen- ter of ellipses is not far outside of the field of the telescope. It is obvious that to facilitate adjustment the mirrors m and m', figure 71, must be nearly parallel. They are made so by the aid of a broad beam of sunlight and then clamped firmly in position at about 45 to the respective axis of the U-tube. Finally, if the connecting-tube p is nearly horizontal when in place, the fringes are usually found at about the same position of the micrometer (at M') after the liquid is introduced into the U-tube. Here it is advantageous and usually permissible to make a part of the connection p of flexible rubber tubing. But unless the free surface w, figure 71, is very nearly parallel to the plate e, the center of ellipses is liable to be far outside of the field of the telescope and the fringes correspondingly small. The difficulty of adjust- ment for large fringes is now considerable, because of the two mobile liquid surfaces in the U-tubes. For this reason I did not attempt to make measure- ments, although the fringes themselves were surprisingly steady and strong and would have been quite available, apart from laboratory tremors. Slight changes in density, due to solution or temperature changes in one shank of the U-tube, were well recorded, after stirring, with curious effects of surface- tension and viscosity. The fringes being very clear, a number of other promiscuous experiments were tried. Thus, a tube with plate-glass ends and filled with water was made the core of a powerful magnetic helix. The tube, 26 cm. long, was placed in one of the component beams and compensated by a column of glass in the other. Good fringes were easily found; but not the slightest displacement could be detected by alternations of presence and absence of the magnetic field. The water was now replaced by a solution of nickel sulphate. Fringes were again easily found and strong in the green, but the effect of the magnetic field was quite as inappreciable as before. Magnetic fields were thus totally ineffective. In a set of experiments of a different kind the attempt was made to observe the gradual deposition of silver on plate-glass. Bottger's solutions were poured into a plane-parallel clean glass trough normal to one of the component beams (of, fig. 74) and compensated by a plate of glass in the other (W). Large fringes were produced by setting the micrometer and observed during the formation of the two silver films on the opposite faces of the trough, until they became quite opaque. It was astonishing to find that fringes were still faintly visible long after a highly reflecting mirror had been deposited. But no displacement larger than a fraction of a fringe could be detected, showing the extraordinarv thinness of the silver film even when practically opaque. 114 THE INTERFEROMETRY OF Similarly, the silver deposit on plate-glass was removed in parallel strips, so that the film had the appearance of a grid. The results when this plate was placed normally in one of the component beams were the same. 61. Vertical displacement of ellipses. If the fringes are too small when horizontally centered by the micrometer, the center of ellipses may be brought into the middle of the field of the telescope by sliding one component beam vertically over the other without appreciably changing the direction of the rays. In other words, one illuminated spot at d, figure 74, is to move vertically relative to the other by a small amount. This may be done by placing a thick plate-glass compensator, such as is shown in figure 77, in each of the component beams abd and acd and suitably rotating one plate relative to the other, each on a horizontal axis. Very little rotation is required. In the same way elliptical fringes may be changed to nearly linear horizontal fringes when desirable. If the fringes are to be sharp the slit must be very fine. When sunlight is used with a slit not too fine, each of the coincident sodium lines (AA) frequently shows a sharply defined helical or rope-like structure, the dark parts in step with the fringes of the spectrum. It looks like an optical illusion of slanting lines or a shadow interference of two grids (fringes and sodium lines respectively); but later experiments showed it to be an inde- pendent phenomenon. (Cf. 63 et seq., 68, 70.) a V Hi ( 77 78 The first result is particularly interesting, inasmuch as it is thus possible to displace the centers of ellipses not only horizontally as usual relative to the fixed sodium lines in the spectrum, but also vertically relatively to the fixed horizontal shadow in the spectrum due to the fine wire across the slit. The following experiment was made to coordinate the vertical displacement of the component rays and centers of elliptic fringes: A glass plate d = 0.705 cm. thick was placed nearly normally in the beam ac, figure 74, and provided with a horizontal axis and graduated arc. The amount (*) of rotation of the plate, corresponding to the vertical displacement of one central fringe in the telescope (i.e., passage of fringe a into b, into c, in the duplicate spectrum 5, fig- 78), was then found to be, if * is the angle of incidence, i h No fringes 3.5 0.0149 cm - One fringe 5.0 .0214 Two fringes 6.5* .0281 where h is the corresponding vertical displacement of the rays ac, figure 74, and computed from (p index, r angle of refraction) h = d(sin i cos i tan r) REVERSED AND NON-REVERSED SPECTRA. 115 Thus the vertical displacement of rays corresponding to the vertical semi- axes of the central ellipse or one fringe is between 0.0065 and 0.0067 cm. *.*., on the average below 7 X io' 3 cm. Hence h=NXo.oo'j for N such central fringes. It was difficult to get a closer result, owing to quiver. The interesting question is now suggested, in how far such an arrangement would fall short of being able to exhibit the drag of the ether in a rapidly rotating body, should such drag occur. In figure 73, let aft be a cylinder of glass with plane-parallel ends, capable of rotating on the axle 75. If / is the length of the cylinder, /x its index of refraction, and r the distance of either component ray (ac, bd) from the axis yd, n the number of turns per second, and V the velocity of light, we may write, using the above excessive estimate, N being the number of fringes displaced, = 27rwr/M/F since ac rises while bd falls. If n=2oo, r=iocm., /=ioo cm., 3.5 x 10-3x3 x io' It would thus be necessary to estimate about one-sixtieth of a fringe, which is just beyond the limit of certainty, even if nr can be increased and / multi- plied by reflection. The device suggested is nevertheless of interest and deserves further consideration. It will appear much more promising in connection with the achromatic fringes described below. 62. Displacement interferometer. Jamin type. These considerations induced me to devote further study to the Jamin type of interferometer (fig. 73). The mirrors M, N' were put on one pair of long slides (1.5 meters long) parallel to ac and the mirrors M', N on similar slides parallel to the former. In this way any distance ac or bd was available. The beams were about 1 6 cm. apart, corresponding to a normal distance between the end mirrors (NN', MM') of about 12 cm. But these distances could also be increased from nearly zero (M and M' nearly contiguous) to about 20 cm. in view of the width of mirrors used. The angles at a, b, c, d were each about 45, so that a rectangle of rays is in question. (See figure 88 or 93 below.) The adjustment proved eventually to be greatly facilitated by using a horizontal beam of sunlight with weak condenser-lens and collimator. A thin wire is to be drawn across the dit. M and M' are first set for parallelism in the absence of N and N', by adjusting the images of the slit at the same level (horizontal) on a distant wall. The images or shadows of the wire specified on the wall are to be equally far apart, with the beams ac and bd at the mirror. The mirrors N and N' are next put in place with the distances acd and abd about equal. The two images seen in the telescope at T (g removed) are then made to coincide both horizontally and vertically by adjusting N and N , 116 THE INTERFEROMETRY OF and these are then slid by a small amount on their slides (direction ac) until the rays are coincident at d to the eye (light strips on the mirror coincide). If, now, the grating g is inserted, very fine oblique fringes will usually be seen. These may be enlarged to a maximum by moving the micrometer controlling the displacement M' normal to itself. Somewhat coarser horizontal lines are thus obtained. Finally, the distant centers of the ellipses are brought into the center of the telescope by aid of the thick glass compensator, like figure 77 (the equiva- lent air-path of the other ray being correspondingly lengthened) by rotating the glass plate on a horizontal axis.* It is desirable to have an excess of glass- path in one beam, as otherwise the ellipses are so large as to be unwieldy. The ellipses so obtained with common plate-glass and a film grating at g were magnificent. A rough test of the displacement interferometer was made by using the above plate-glass of thickness =0.434 cm., where z=E(jji i) -f-2.B/X 2 = o.2428 cm. In two experiments agreeing to within icr 4 cm., 20 = 0.3448 cm. were the displacements obtained. Assuming that = 45, 26 cos 8 =0.2438 cm. This agrees with z as nearly as may be expected, unless 6 is specifically measured. Experiments were now made (as above) with thick plate-glass compen- sators inserted in one component ray (bd) only, to determine the rotation of compensator (i) necessary to raise the center of ellipses in steps of half the diameter of the first ring (see a, b, c, fig. 78). The initial angle i is already large and shows the rotation of compensator from the vertical needed to bring the ellipses into the field. Two sets of experiments were made with plates respectively d = 0.965 cm. and d = 0.705 cm. in thickness, with the results given in table 30. TABLE 30. Experiment I. d j h Afc 0.965 cm. 9.0 IO.I 11.4 0.0529 cm. .0596 .0676 0.0067 cm. .0080 Experiment II. d i h Afc 0.705 cm. 2.6 3-3 3-9 o.oioocm. .0139 .0165 0.0039 .0026 The equation of the preceding section is used for h. The first case shows about the same order of sensitiveness (M per half-ring). In the second case, for the thinner plate, the sensitiveness has been more than doubled. This * The same result may be obtained in the absence of the compensator by rotating N and N' on a horizontal axis, successively by small amounts, into parallelism with M and M'. REVERSED AND NON-REVERSED SPECTRA. 117 is in a measure not unexpected, because the amount of displacement, caet. far. (i.e. t the mobility and size of ellipses), increases in marked degree as the plate compensator is thinner. But apart from this AA is in some way, yet to be stated, associated with the obliquity of rays in a vertical plane. In case of the rotating compensator, vertical and lateral displacement of centers of ellipses go together. It is therefore next in order to determine the ratio of vertical and lateral displacement. The equation deduced in 8, which follows easily from figure 77, may be put in the form (for a single passage of light through the plate) n\ = 2e(sw?i/2 /* sin 2 r/a) or into the approximate form for small angles n\=e(n i)f/2fji From the former equation di \ dn 0(sin i cos i tan r) or approximately, again, di dn X/z Hence, if At corresponds to x fringes, di = x-~r =x ~, - . dn ei(ji i) roughly Again, for the corresponding normal displacement AN of the micrometer at the opaque mirror, #X = 2AJVcos 6 Hence __ ei(n i)A* _ aA/V cos 6 X/x ~^~~ The data given in table 3 1 were found from successive positions of the plate = 0.705 cm., while the center of ellipses moved as in figure 78, at the D line. TABLE 31. Center at i At X loW io*AN X Bottom.. .. 13-6 19 cm. ... = 45 Middle.. .. 12.5 i.i 18 28 9 22 x = i .27 /A* Top -s I.O 15 35 7 17 x = 24,oooA^V Bottom.. .. 7-5 in ioA = 59;A = i-53 Middle.. .. 6.7 0.8 7 "5 4 10 Top . . 5-9 0.8 6 119 3 8 In view of the small values of AAT and A* and the estimated n and B, the two sets of values of x are no more divergent than would be expected. The 118 THE INTERFEROMETRY OF values of * are very different in different adjustments, because the ellipses may also be raised and lowered by rotating one of the opaque mirrors, as M, around a horizontal axis, though in this case with rapid loss of sharpness. Here the rotation of both mirrors of a pair, M' and M for instance, by the same amount, is required as an equivalent to the rotation of the compensator, as has been stated. The aim is to render both parallel pairs themselves parallel. If x\ is eliminated from the two equations on the preceding page, i n=- The above data are inadequate for evaluating n, but they nevertheless indicate a value entirely too low. It seems, therefore, as if some essential term has been left out of sight. This is also to be inferred from the values of x, which differ systematically on the two sides of the table 31. TABLE 32. Refraction of a glass plate, e = 0.434 cm. ^ = 1.53. Rotation around vertical axis. Jamin type displacement interferometer. = 45. i AATXio 3 At i AWXio 3 M cm. cm. + 12.5 2.60 156 - 6.5 .60 10.5 1.65 147 - 8.5 1. 20 "56 8-5 1.25 151 -10.5 i-75 52 6-5 .70 -12.5 2.60 56 4-5 45 -14-5 3-35 51 2-5 05 -16.5 4.60 57 0.0 .00 -18.5 5-6o 55 - 4-5 30 80 yr 100' w In view of the discordant results obtained here and elsewhere with this type of rotating compensator, the provisional parts of the apparatus were improved by mounting a more accurate graduated circle with vertical axis and tangent screw. A good plane-parallel plate of thickness e = 0.434 cm. and refractive index ^=1.53 was then adjusted normal to the ray passing through it, by noting the position of reversal of motion of the ellipses both when the plate was rotated around a vertical and around a horizontal axis. The data of table 32 and figure 79 were found while the plate was rotated on its vertical axis both in a clockwise and counter-clockwise direction from the normal position. In the former case (left side of curve) the normal position showed the same constants before and after. In the latter this was not quite the case, the micrometer being not sufficiently refined for such purposes. The results for JLC were computed from the full equation AATcos0=*sin 2 -Ui They are as good as the small values of AN and the impossibility of obtaining the zero of i with sufficient sharpness admit, and they show that the latter cause adequately explains all the irregularities encountered. REVERSED AND NON-REVERSED SPECTRA. 119 The equations are liable to be cumbersome in the cases of greatest interest. I therefore proceeded experimentally to obtain a limit of Ah or Ax per fringe, using thinner glass compensators. The results are given in table 33. The ellipses were now so large and distorted that it was difficult to define the center of irregular rings. The transverse displacement is therefore largely referred to the top, middle and bottom of the spectrum band, which took up about one-third of the height or diameter of the field of the telescope. The angular width of the latter being about 3, the corresponding angular height of the spectrum is thus about 1.0. In view of the large rings, moreover, the displacement AAT at the micrometer of the mirror M' is difficult to obtain and the data given are estimates. Experiments like the present must be made with optic plate-glass, so that sharp rings nearly circular may be obtained, if the data are to be quite satisfactory. In table 33, A/t thus corresponds to a transverse displacement of one component ray parallel to itself, equivalent to a displacement of the centers of ellipses of about 0.5 at the sodium line. TABLE 33. Vertical (transverse) displacement of ellipses. Glass-plate compensators n = 1.53. Horizontal axis. Angle of telescopic fields ; angular height of spectrum about 1.0 or 0.175 radian. Vertical diameter of first fringe in excess of height of spectrum. Fringe centers at e f h A&Xio 4 A#Xio 4 Top of spectrum Middle of spectrum . . Bottom of spectrum. . Bottom of spectrum. . Middle of spectrum . . Top of spectrum cm. 0.300 .300 0.4 '9 2.6 2-3 i.i o.o cm. 0.0007 34 47 .0042 20 oo cm. -26 +13 +22 00 20 cm. B .. O-434 40.8 0.1274 + 15 o M 40.2 .1259 00 7 T B .4. -74 K .1217 .0213 -42 + 12 22 O M 7.6 .0200 00 T f 7.1 .0187 13 2 B 4^4 6.5 .0170 +29 M 5-4 .0141 oo 2 T 4.1 .0112 29 7 B 474. f 3.3 .0087 + 19 o M 2.6 .0068 oo 4 T 1.5 .0039 29 4 B .O2O 20.0 .00253 + 14 M 9.4 .00115 00 T -3-6 -.00044 -16 The mean result of all data is here about AA = 0.002 cm., and this is not influenced in a discernible way, either by the thickness of plate e or by the rotation angle of the compensator i. The smallest value M = 0.0012 cm. appears incidentally and not when the system of four mirrors is most nearly in parallel adjustment. The transverse displacement of ellipses changes sign with the sign of the rotation of i in all cases and is independent of the normal position. The longitudinal displacement reverses at the normal position. 120 THE INTERFEROMETRY OF 63. Broad slit interferences. Achromatic fringes. Some allusion has been made above to a type of interferences totally different in size from the regular fringes and seen in the broadened slit. These were finally isolated and show exceedingly interesting properties. They appear to best advantage, in the absence of the spectroscope, in the broad white field of a very wide slit. The latter may be removed. They have the appearance when vertical of regular Young or Fresnellian fringes, very sharp and fine, achromatically black and white at the middle of the grid, colored and fainter outward. They are vertical when the enormously larger spectrum fringes discussed above are centered. Like these, they partake of displacement here through the broad white slit image, and this displacement is extremely sensitive in relation to the displacement of the opaque mirror M' (fig. 73) to which it is due. Thus a displacement of AAT= 10"* cm. of the latter corresponds to a march of fringes through about 0.017 of the telescopic field of 3; i.e., to 0.05. This comprises two fringes or A/V = 5Xio" 6 cm. per fringe. Now, these fringes are so sharp and luminous that it should be possible on proper magnification to measure a few hundredths of this with an ocular micrometer. It is from this point of view that I regard the new fringes important. They supply the fine fiducial mark in displacement interferometry for which I have long been seeking. They appear in a white field, thus requiring no spectrum resolution nor monochromatic light. Moreover, the source of light need not be intense. To have a distinctive name for these fringes which will be much used in the work following, I shall refer to them under the term " achromatic fringes." If not too large, the central fringes are straight and almost quite black and white. The displacement of fringes with AN" at the mirror (when wX = 2AJVcos 0) is so rapid that if they are lost it is difficult to find them, unless the centered large spectrum fringes in the spectroscope are first reestablished. The latter are easily found. A removal of the prism grating g, figure 73, and a widening of the slit show the achromatic fringes. The datum for sensitiveness may be found directly as follows: The displacement at the mirrors corresponds to about two residual fringes. Thus a single fringe (distance apart of the intensely black lines at the center which can be distinguished and used as fiducial lines for this very reason) corresponds to a displacement of mirror of AA/^=5oXio" 6 , as above. The white pattern, as a rule, appears but once and is not usually present rhythmically, as is the phenomenon in the next section for homogeneous light. As a clue to the nature of the residual fringes, one may note in the first place that they may be recovered in the principal focal plane, if the two white slit images which may have separated be put in coincidence. Without such coincidence they are seen sharply in other focal planes. Later, on more careful adjustment as to parallelism by the auxiliary normal method described below, periodic reappearance of the achromatic fringes was in fact obtained. The central set was exceedingly strong and sharp as usual. To the right and left of it similar patterns or groups rapidly decreasing in strength were discovered. Not more than two patterns on each side of the REVERSED AND NON-REVERSED SPECTRA. 121 central set could be seen. The white field between the patterns was several times the width of the fringed field. Each group as a whole resembles the fringes of the biprism, as usual, and they differ appreciably only in intensity and in their focal planes. It is difficult to account for this periodic reappear- ance; but it must be due to reflections at the half -silver surfaces. The reflec- tions from the uncovered surfaces are indeed just visible, and naturally, since they are duplicates, they also carry the fringes. But they are easily differ- entiated by the relative faintness of field and have nothing to do with the recurrences in question. Ordinary daylight is quite adequate to show the residual fringes in the complete absence of the collimator. They are superimposed on the field of view (landscape, etc.) and hence will subserve other purposes than are here given. When the adjustment for parallelism is not sharp, the fringes may often be found strong in continuously varying focal planes. 64. Wide slit. Homogeneous light. Sodium flame. A further clue to the nature of the residual fringes will be obtained when white light is replaced by homogeneous light. A strong, large sodium flame near the mirror M, figure 73 , suffices. The fringes now appear of the same size in yellow light, naturally spread over a much larger area of field. But on moving the mirror M' (A/V increasing continually) forward very gradually, the homogeneous fringes alternately vanish and reap- pear, each time, however, enlarged in size (nearly doubled but still straight) until at an intermediate position of symmetry enormous round ovals cover the yellow field. The fringes then diminish symmetrically in the same way. The following data for the microm- eter position corresponding to [the clearest demarca- tions of fringes are illustrative. At least six periods (n) are easily detected on each side of the ovals ( = o) . Thus (originally small fringes, vertical, increasing in size to huge ovals) n =6 5 4 3 2 i o.etc. AJVXio 3 = o 49 90 127 171 214 243 cm., etc. These intervals, since it is impossible to establish the maximum states of presence or absence of fringes quite sharply, are practically equidistant, as figure 80 indicates. Thus the mean period of reappearance is A7V= 0.042 cm.; or a path-difference of 2 AATcos 0=0.059 cm.; or a shift of ray paralle itself (2AJVsin 6 = 0.059 cm.) of the same amount. The reason for this rhythm can only be the two wave-lengths oi and A lines of the sodium flame, originally detected in the colors of thin pi by Fizeau. Hence a relatively enormous shift of micrometer of nearly 0.5 mm. is equivalent to the wave-length interval AX=6Xio- cm., or AX/AJV = 122 THE INTERFEROMETRY OF icr 8 /42Xicr 3 =1.4X10-*. Treating the case in terms of the interferences of thin plates and two wave-lengths, X and \+d\, AX Aw wA= constant, or while wX = constant, or = = - for each period X n n X 2 wX= = 2 AN cosO or AW=X 2 /2AXcos0 since (0 = 45) approximately, X=6oXio" 6 cm., AX = 6Xio~ 8 cm., cos = 0.71, A/V= 0.041 cm. agreeing as nearly as may be expected with the experimental datum. The apparatus thus serves incidentally for investigating such properties of spec- trum lines as Michelson in particular has detected. Finally, with the sodium arc the data of figure 81 were found, where black dots denote fringes, open cir- cles a clear yellow field. The mean trend is about AAT= 0.043 cm- per period or wave-length gained. White fringes coincided with n = 2. With white light the interference grid does not usually reappear rhythmi- cally, nor does it correspond to the zero period of figure 80 i.e., to the ovals for sodium light. It was exactly of the size of the fourth period, in yellow light; it always coincides with the central ellipses of the spectroscope as stated, but does not require sharp horizontal and virtual coincidence of the super- posed images. The reappearance of the achromatic fringes obviously depends on conditions different from the case of sodium light. In a flash of the arc, showing many sharp spectrum lines in all colors, each of the lines gives evidence of the phenomenon i.e., if the residual fringes are oblique, each such line is strongly helical in appearance. If a single compensator (i.e., a glass plate in one interfering beam) is used and the path-difference annulled, the fringes are visible again, but very rapidly grow smaller with the increase of glass-path. If compensators of nearly like thickness and glass are used in both beams, they nearly neutralize each other if at the proper angle one to the other. But there is almost always an out- standing micrometeric difference in thickness (if ordinary glass plate is used) of great importance in modifying the residual phenomenon. The following experiments, for instance, were made with a compensator 0.944 cm. thick in the rear and 0.958 cm. thick in the front beam, both of the same glass. The half -silver mirrors were 0.7 cm. thick. The wide-slit experiments are supposed to start after the spectrum ellipses first to be found (fine slit) have been cen- tered. The fringes are always clear and very sharp when the white slit images accurately coincide. With the compensators A, B at the proper angle and rotating in the same direction (fig. 82), fringes of very slight enlargement were seen with the lines nearly vertical. If the compensators A, B were rotated at a proper pace in the opposite direction to each other, A, B, figure 83, the fringes /' and /" grew rapidly smaller and turned toward the horizontal. Since the fringes are large circles and the beams here rise and fall, respectively, while a greater REVERSED AND NON-REVERSED SPECTRA. 123 glass thickness is introduced, this result is to be expected. It shows the im- portance of the differential glass-path. The preceding thick half-silver mirrors (0.7 cm.) were now replaced by thinner half-silvers, the glass plates being each about 0.3 cm. thick. The fringes after being found had not appre- ciably changed. Another pair of half- silvers of the same thickness was then installed with like results. But now, on adding the compensators (0.944 and ,/ ' & -Z -^ x X 0.958 cm.) as above, a marked enlarge- 82 ment of fringes resulted. Small differ- 7* X ential thicknesses must here have been '-T J accidentally compensated. Opposite rotations, as in figure 83, rapidly pro- duced .very fine, nearly horizontal fringes /'/". Rotation as in figure 82 left the vertical fringes nearly intact, but on passing from the position AB to A'B' very marked enlargement occurred, as follows: Mean angle of glass plates. 45 o +45 Mean angle subtended by one fringe in the telescope. 0.0015 rad. 0.008 rad. 0.0005 rad. The compensator plates were now exchanged and the fringes found after centering. They proved to be very much smaller, the angle subtended in the same telescope being only about 0.0002 radian. The preceding accidental compensator has therefore been destroyed by exchange. On passing through the normal position of one plate in figure 82, the fringes usually incline toward one side or the other. Thus there can be little doubt that the fringes in question are due to slight difference of glass-path or ex- tremely sharp glass-wedge excess in one or the other component beam. In fact, I found eventually that fringes could be enlarged by rotating the proper compensator around a vertical axis. Large fringes (up to 0.002 radian in the given telescope) are usually colored and curved and not so available as smaller fringes highly magnified. It is in this way (double rotation) that it was pos- sible to make the white fringes coincide in order with the ovals of the fringes for homogeneous light (n=o), the orders met with above being the n-2 and n = 4. The fringes resemble those of Fresnel's biprism; but as they are seen with a wide slit or in the absence of a slit only, as they coincide with the cen- tered spectrum ellipses of a fine slit and as they are a definite order (second, for instance) of the fringes seen with a flame of homogeneous light, they are necessarily referable to the colors of thin plates. Hence the equation for these fringes may be assumed to be (as may be seen from figure 84, where A and B are the compensators) n \ = (e-e') (p cos (r - a) - cos *) when 9 and *' are the thicknesses of the two half-silver plates, /* their index of refraction, i the angle of incidence, r the angle of refraction of an incident ray, and where a is the outstanding angle between the faces of the differential 124 THE INTERFEROMETRY OF glass wedge, e e' thick at the ray in question. The possibility of throwing these fringes into any order of size, their small extent, sharpness, and great abundance of light constitute their value for measurement. Thus it is furthermore obvious that the achromatic fringes must also be obtainable in Michelson's interferometer, in any order of size and in a field of white light. Tests were made with this object, beginning with the fine slit and the centered ellipses of the spectrum interferences. Removing the spec- troscope and enlarging the slit indefinitely, the residual fringes appeared. They were never so strong and clear, however, in any order, as was the case with the Jamin interferometer, although many trials with different compen- sators were made. They would not be useful for measurement. Removing white light and replacing it by sodium light, the white fringes were found to be of exceedingly high order, more than 0.7 cm. of micrometer-screw being needed before the circles of the yellow field were approached. The latter were very vague and usually not seen in the principal focal plane of the tele- scope. Since the rays retrace their path in the Michelson interferometer, the raising and lowering of the spectrum ellipses is not possible; but the residual fringes may be put in any order by changing the differential glass-path of the rays. They appear but once, not rhythmically like the sodium fringes. The equation for this phenomenon would thus be (since the rays retrace their paths) X = 20ju cos (r a) where e is the thickness of the single half -silver and a its effective wedge angle, positive or negative. / ci e: ,sfi } 84 To obtain the circular fringes in Michelson's interferometer with a wide slit and homogeneous light, the rays must rigorously retrace their path i.e., all reflection must take place at the same spot on the half -silver plate. When this is the case the fringes, even though obtained with common plate- glass and a sodium flame, are beautifully circular and sharp. This is due to the fact that so small a part of the plate is used. They are stationary and exhibit the Fizean periods due to the doublet DiD 2 , admirably. With white light the fringes are faint and useless. When the rays do not accurately retrace their paths i.e., when there are two spots of light on the half -silver one or more centimeters apart the fringes are soon linear and very small, as above. With regard to the last equation, if N is the difference of normal distances to the two opaque mirrors (M, N) of the Michelson interferometer, from the REVERSED AND NON-REVERSED SPECTRA. 125 two respective extremities of the normal to the half-silver, at the point where e incident ray impinges on it, the equation may be more completely written if is temporarily disregarded. N is independent of color X, but otherwise represents the difference of air-paths of the two interfering rays. From this equation _ _ dn 2*0 cos R- (X/cos R) ( figure 88, if the fringes escape. In fact, the achromatic fringes are so sharp and luminous that in a quivering system the fringes are distinctly seen at the two elongations, doubled by the quiver. To adequately support the mirrors m and m' reasonably free from vibra- tion, a rectangular wooden yoke F F, figure 97, is best, so long as the instal- lation is not permanent. This yoke is made of inch boarding about 18 inches high, 15 inches broad, and 3 inches deep. It is bolted below at d to the iron carriage 5, and thus capable of rotation around a vertical axis and of slid- ing normal to its face on the iron slides B B' (about 1.5 meters long) of the base. 5 and B B' are arranged like a lath-bed. The mirror m', which is horizontal, is adjustably attached to the top of the frame F F; three adjustment screws (two seen at a, a') and the strong spring 5 (pulling upward) control the position of m' with the usual plane dot- slot mechanism and axes of rotation parallel to and normal to the plane of the frame. The board carrying the mirror m may be roughly set at 45 to the vertical by rotation on the strong bolts bb' and clamped in position. The mirror m is attached to it by three adjustment screws and a rearward-acting spring, as shown in the case of m 1 '. The axes of rotation are parallel to the two edges of m, and all fine adjustment, together with the final rotation of fringes and changes of their size, are made here. The columns of the U-tubes C and C' must be supported from an inde- pendent bracket, suitably braced from the wall or from a separate pier, if liquid surfaces are in question. A part of this free support, the table, also of wood, appears at D, and holes are cut in it sufficiently large for the passage of the two beams of light a, of, which retrace their paths between m and m', passing through C and C'. The table D is provided with three leveling screws (two shown at c and c') on which a plate of thick glass G may be mounted and made accurately horizontal by aid of a spirit-level. The two columns C and C', joined by a flexible rubber pipe and provided at the bottom with glass windows, are placed on G, so that the beams of light pass through their axes. The magnetizing helix of each is shown at e and e'. It is obvious, therefore, that the two columns of liquid in C and C' are of the same height and their ends parallel, other things being equal. Hence, if the apparatus is adjusted for the achromatic fringes in the absence of the U-tube C C', it will be nearly in adjustment when the columns are introduced; and this proved to be the case. The rigid system is easily adjusted for strong REVERSED AND NON-REVERSED SPECTRA. 141 vertical achromatic fringes. The fringes are found with a few turns of the micrometer after the liquid w has been poured in. The fringes as seen through the liquid columns are still strong and clear, with scarcely any deterioration; but unfortunately in this locality they are in incessant and vigorous vibration. It is indeed astonishing that a phenome- non so sensitive can survive such relatively rough treatment. With this apparatus the endeavor was again made to determine the sus- ceptibility of water. Good fringes were first produced, but for all levels of the water-surfaces the effect of the presence or absence of a magnetic field was quite nil, so far as any appreciable displacement of fringes was concerned. They remained in place while the current in the helix was alternately closed and opened. The reason for this completely negative result I am unable to explain. In a repetition of such experiments tubes much wider than 4 cm. should be used. At this diameter the liquid surfaces still show appreciable curvature, which is an annoyance. The yoke, figures 96 and 97, is finally a considerable convenience in the meas- urement of vertical angles near the zenith. The rotation of the mirror m r around its two axes is here available. Horizontal achromatic fringes in all these cases may often be produced and used to advantage. In such a case the two adjustment screws of the mirror m' change their function, and the fringes travel up and down the wide slit image with changes of AN". As this image is more extended vertically than horizontally, the fringes are much longer in sight. Fringes are largest when the planes of symmetry of pencils of rays accurately retrace their paths. But as large fringes are apt to be irregular, a small angle of reflection at the mirror m' is preferable. 73. The displacement interferometry of long distances. In the preceding paragraphs I suggested two methods for the measurement of small angles. The first (fig. 88) used an auxiliary mirror, and, apart from corrections, the angle A a over which the auxiliary mirror m turns is (i) Aa=AA/"cosj/2.R where AN is the displacement of one of the plane mirrors parallel to itself necessary to restore the achromatic fringes to their former position in the field of the telescope, i the angle of incidence (conveniently 45) 2 # the normal distance apart (ab or cd) of the (parallel) interfering pencils in the fore-and-aft direction of the incident beam. In the second method (fig. 93) the auxiliary mirror is dispensed with and the rotation of a rigid system of paired mirrors is used. The sensitiveness is half the preceding. Suppose that the paired mirrors near the telescope (figs. 88, 93) confront but a part of the area of the objective and that the telescope can therefore look over the mirrors directly into the region * beyond, as shown at K . The * A series of small mirrors or reflecting prisms may be employed to the same purpose; or the mirrors may both be half-silvered and transparent. 142 THE INTERFEROMETRY OF telescope now contains two images, the first due to rays (K) entering it directly, the second due to rays (L) reflected into it by the mirrors of the interferometer. Suppose the object seen lies at infinity like a star, that its two images are made to coincide by adjusting the angle ex, and that the achromatic fringes have been brought into the field by adjusting the micrometer displacement N. Now let the angle a be changed by A a until the two images of an object M at a measurable distance d coincide. Displace the micrometer mirror by A N until the achromatic fringes are restored to their former position. Let b be the effective distance apart (ac or bd) of the paired mirrors in the direction right and left to the observer or transverse to the impinging rays (L), and finally 5 the angle at the apex of the triangle of sight on the base b i.e., the small angle between the present rays KL. Then (2) d = b cotg s = b cotg 2 Aa = b/2 Aa (nearly) by the laws of reflection. Hence from equation (i) (3) d=bR/ANcosi Here 2bR is the area of the ray parallelogram of the interferometer (abdc, fig. 88). Using the constants of my apparatus, let 2 = 45, R= 10 cm., 6 = 200 cm., AA/ r =io~ 4 cm., the latter being the smallest division on the micrometer. Hence d = 20oX lo/io-^Xo.yi =2.8 X io 7 cm. = 280 kilometers or about 170 miles, is the limit of measurement of the apparatus. Again, from equation (3) the sensitiveness 5(AN)/8d, since (4) 5d=(d*cosi/bR)5(AN) is inversely proportional to the square of the long distance d and the area of the ray parallelogram 2bR. Thus with the above constants, if d is 2 kilometers, Thus an object at about a mile should be located to about 30 feet. Per fringe of mean wave-length X, moreover, since 8d='hd?/2bR, the placement should be about 6 meters at 2 kilometers. I have stated the case, of course, merely for the interferometer, not for subsidiary optical appurtenances, nor for measure- ment by angular fringe displacement. 74. Theory. To account for these phenomena theoretically the equations of displacement interferometry are available; for the center of ellipses of these and the central member of the achromatic fringes correspond to the same position, AA/", of the micrometer mirror. In fact, the fine white slit image which produces the spectrum when observed through the spectroscope is the central achromatic fringe when the spectroscope is removed. We have, there- REVERSED AND NON-REVERSED SPECTRA. 143 fore, for the centers of ellipses in the spectrum fringes the equation heretofore deduced, 2AAf cos * t -i)costf =-2H cosRd\j where AAT is the displacement of the micrometer to restore the center of fringes to their original position in wave-length X, when the angle of incidence at the mirror is i, after a glass plate of thickness e and index of refraction /* is introduced in one component beam, at an angle of incidence corresponding to the angle of refraction R in the plate. This is equivalent to an equation in terms of the coordinates N, if n=A+B/\ z is assumed, 2N cos i=ep cos R+2eB/\* cos R But for the colors of thin plates we may write n\ = 2en cos R where n is the order of the fringe. If the rays, as in the present experiment, do not retrace their paths, the factor 2 is omitted, whence 2N cos i = n\+2eB/\ 2 cos R Let N and n vary together, i, R, X, e, B, n remaining constant. Then 2 cosi'AA/"=X'Aw Now let A^> be the angular breadth of a fringe in the telescope, so that In other words, the displacement A0 of the group of fringes is due to the displacement of the individual fringes as usual; but as only those achromatic fringes which coincide in micrometer value A/V with the centers of the elliptic spectrum fringes are visible, the displacement of the group is actually seen and thus determinable. It would not be so in case of sudden displacement and homogeneous light. Finally, the relative sensitiveness of the measurement of the angle Aa, directly in terms of A/V and indirectly in terms of the displacement of achro- matic fringes, must be presented. The given rough data in table 38 are as close as they could be obtained from the small displacements entering. The quantities to be compared are: AAf, the displacement of the micrometer; A a, the corresponding rotation of the auxiliary mirror in the first method ; Ay>, the angle subtended by a single fringe in the telescope, and A0, the corre- sponding angle of the displacement of the fringes in the telescope. A0 was constant throughout and measured about 3 mm. in the ocular of a telescope 33 cm. long. TABLE 38. Values of AN, A9, AV>, Aa. All angles in radians, AN in cm. A0Xio A*Xio AJVXio. AaXlo' A*/A* A0/A AvAtfXio- A^AXio 9000 9000 500 1500 730 250 g 12.3 36.0 170 500 0.365 375 0.026 .027 144 THE INTERFEROMETRY OF Here Aa is computed from equation (i), where i = 4S and 2^=10 cm., about. The fringes of width 100" and 300" were both brilliant and capable of high magnification. Thus it appears that the fringes travel faster in proportion to their width, or if A