UNIVERSITY OF CALIFORNIA 
 AT LOS ANGELES
 
 THE 
 
 PRODUCTION OF ELLIPTIC INTERFERENCES 
 IN RELATION TO INTERFEROMETRY 
 
 BY CARL BARUS 
 
 Hazard Professor of Physics, Brown University 
 
 WASHINGTON, D. C. 
 
 Published by the Carnegie Institution of Washington 
 1911
 
 CARNEGIE INSTITUTION OF WASHINGTON 
 PUBLICATION No. 149 
 
 PRESS OF J. B. LIPPINCOTT COMPANY 
 PHILADELPHIA, PA.
 
 Physics 
 
 Library 
 
 PREFACE 
 
 In connection with my work on the coronas as a means for the study of 
 nucleation, I came across a principle of interferometry which seemed of suf- 
 ficient importance to justify special investigation. This has been under- 
 taken in the following chapters, and what appears to be a new procedure 
 in interferometry of great promise and varied application has been devel- 
 oped. In case of the coronas there is a marked interference phenomenon 
 superposed on the diffractions. The present method is therefore to consist 
 in a simplification or systematization of this effect, by bringing two com- 
 plete component diffraction spectra, from the same source of light, to inter- 
 fere. This may be done in many ways, either directly, or with a halved 
 transmission or reflecting grating, or by using modifications of the devices 
 of Jamin, Michelson, and others for separating the components. 
 
 In the direct method, chapters II and III, a mirror immediately behind 
 the grating returns the reflected-diffracted and diffracted-reflected rays, to 
 be superimposed for interference, producing a series of phenomena which are 
 eminently useful, in addition to their great beauty. In fact the interfer- 
 ometer so constructed needs but ordinary plate glass and replica gratings. 
 It gives equidistant fringes, rigorously straight, and their distance apart 
 and inclination are thus measurable by ocular micrometry. The fringes are 
 duplex in character and an adjustment may be made whereby ten small 
 fringes occupy the same space in the field as one large fringe, so that sudden 
 expansions within the limits of the large fringe (as for instance in mag- 
 netostriction) are determinable. This has not been feasible heretofore. 
 Length and small angles (seconds of arc) are thus subject to micrometric 
 measurement. Finally the interferences are very easily produced and are 
 strong with white light, while the spectrum line may be kept in the field as 
 a stationary landmark. The limiting sensitiveness is half the wave-length 
 of light. 
 
 The theory of these phenomena has been worked out in its practical bear- 
 ings, advantageous instrumental equipment has been discussed, and a 
 number of incidental applications to test the apparatus have been made. 
 In much of this work, including that of the first chapter on a modification 
 of Rowland's spectrometer, I had the assistance of my son, Mr. Maxwell 
 Barus, before he entered into the law. 
 
 The range of measurement of an instrument like the above is neces- 
 sarily limited to about i cm. and the component rays are not separated. 
 To increase the range indefinitely and to separate the component rays, the 
 grating may replace the symmetrically oblique transparent mirror of
 
 IV PREFACE. 
 
 Michelson's adjustment, for instance. In this way transmitted-reflected- 
 diffracted and reflected-transmitted-diffracted spectra, or two correspond- 
 ing diffracted spectra returned by the opaque mirrors M and N, may be 
 brought to interfere. In both cases the experiments as detailed in chapters 
 IV and V have been strikingly successful. The interference pattern, how- 
 ever, is now of the ring type, extending throughout the whole spectrum 
 from red to violet with the fixed spectrum lines simultaneously in view. 
 These rings closely resemble confocal ellipses, and their centers have thesame 
 position in all orders of spectra; but the major axes of the ellipses are liable 
 to be vertical in the first and horizontal in all the higher orders of spectra. 
 
 Again there is an opportunity for coarse and fine adjustment, inasmuch 
 as the rings have the usual sensitive radial motion, as the virtual air-space 
 increases or decreases, while the centers simultaneously drift as a whole, 
 across the fixed lines of the spectrum, from the red to the violet end. 
 
 Drift and radial motion may be regulated in any ratio. The general 
 investigation shows that three groups, each comprising a variety of inter- 
 ferences, are possible and I have worked out the practical side of the theory 
 of the phenomenon. Transparent silvered surfaces are superfluous, as the 
 ellipses are sufficiently strong to need no accessory treatment. Consider- 
 able width of spectrum slit is also admissible. Finally, the ellipses may be 
 made of any size and the sensitiveness of their lateral motion may be regu- 
 lated to any degree by aid of a compensator. In this adjustment the drift 
 may be made even more delicate than the radial motion, thus constituting 
 a new feature in interferometry. 
 
 The interesting result follows from the work that the displacement of 
 the centers of ellipses does not correspond to the zero of path difference, but 
 to an adjustment in which /* Xd/z/ciX (where n is the index of refraction 
 and X the wave-length) is critical. 
 
 It is obvious that the transparent plate grating may be replaced by a 
 reflecting grating. Thus the grating may be replaced by a plate of glass, 
 as in Michelson's case, and the function of the two opaque mirrors may be 
 performed by two identical plane reflecting gratings, each symmetrically 
 set at the diffraction angle of the spectrum to the incident ray. In this 
 case the undeviated reflection is thrown out, whereas the spectra overlap 
 in the telescope. Finally the grating may itself be cut in half by a plane 
 parallel to the rulings, whereupon the two overlapping spectra will interfere 
 elliptically, if by a micrometer one-half is slightly moved, parallel to itself, 
 out of the original common vertical plane of the grating. 
 
 Throughout the editorial work and in the drawings I have profited by the 
 aptitude and tireless efficiency of my former student, Miss Ada I. Burton. 
 But for her self-sacrificing assistance it would have been difficult to bring 
 the present work to completion. 
 
 CARL BARUS. 
 
 BROWN UNIVERSITY, PROVIDENCE, RHODE ISLAND.
 
 CONTENTS. 
 
 CHAPTER I. On an Adjustment of the Plane Grating Similar to Row- 
 land's method for the Concave Grating. By C. BARUS AND M. BARUS. 
 
 Page 
 
 1 . Apparatus. Figs, i and 2 1-2 
 
 2 . Single focussing lens in front of grating 2-3 
 
 3. Adjustments. Figs. 3 and 4, A, B, C 3-6 
 
 4. Data for single lens in front of grating. Table i 6 
 
 5. Single focussing lens behind grating 6 
 
 6. Data for single lens behind grating. Table 2 6 
 
 7 . Collimator method 7 
 
 8. Data for collimator method. Table 3 7 
 
 9. Discussion 8-9 
 
 10. Reflecting plate grating 9 
 
 11. Rowland's concave grating. Table 4 10-1 1 
 
 1 2 . Summary 1 1 
 
 CHAPTER II. The Interference of the Reflected-Diffracted and the Dif- 
 fracted-Reflected Rays of a Plane Transparent Grating, and on an 
 Interferometer. By C. BARUS AND M. BARUS. 
 
 13. Introductory. Figs. 5, 6, 7 I 3~ I 5 
 
 14. Observations. Table 5 1617 
 
 15. Equations I 7~ I 9 
 
 1 6. Differential equations 1920 
 
 17. Normal incidence or diffraction, etc 20 
 
 1 8. Comparison of the equations of total interference with observation 20-21 
 
 19. Interferometer. Figs. 8 and 9; table 6 21-25 
 
 20. Secondary interferences 25 
 
 21. Summary of secondary interferences 26 
 
 22. Convergent and divergent rays 2627 
 
 23. Measurement of small horizontal angles 27-28 
 
 24. Summary 28 
 
 CHAPTER III. The Grating Interferometer. By C. BARUS AND M. BARUS. 
 
 25. Introductory. Figs. 10 and 1 1 29 
 
 26. Apparatus 30 
 
 27. Adjustments 3 1-32 
 
 28. Angular extent of the fringes. Tables 7 and 8 3 2 ~34 
 
 29. Test made by magneto-striction. Figs. 12 to 14 ; table 9 34-38 
 
 30. Summary 38 
 
 CHAPTER IV. The Use of the Grating in Interferometry; Experi- 
 mental Results. 
 
 3 1 . Introductory. Fig. 15 39-4 1 
 
 32. Special properties. Fig. 16 4144 
 
 33. Elementary theory. Fig. 17 44~47 
 
 34. Compensator. Table 10 47-48 
 
 CHAPTER V. Interferometry with the Aid of the Grating; Theoretical 
 Results. 
 
 PART I. INTRODUCTION. 
 
 35. Remarks on the phenomena. Figs. 18 and 19 49~5i 
 
 36. Cause of ellipses 5 1-53 
 
 37. The three principal adjustments for interference. Figs. 20, 21, 22, 23. . 53-55
 
 VI CONTENTS. 
 
 PART II. DIRECT CASES OF INTERFERENCE. DIFFRACTION ANTECEDENT. 
 
 38. Diffraction before reflection 55~56 
 
 39. Elementary theory. Figs. 24 and 25 56-58 
 
 40. Equations for the present case. Fig. 26; table 1 1 58-61 
 
 4 r . Interferometer 6 r 
 
 42. Discrepancy of the table 62 
 
 PART III. DIRECT CASES OF INTERFERENCE. REFLECTION ANTECEDENT. 
 
 43. Equations for this case. Tables 12, 13, and 14; figs. 27 and 28 62-66 
 
 44. Divergence per fringe. Tables 15 and 16; fig. 29 66-70 
 
 45. Case of dtl/dy, dX/dy, etc. Fig. 30 70-72 
 
 46. Interferometry in terms of radial motion 72 
 
 47. Interferometry by displacement 72 
 
 PART IV. INTERFERENCES IN GENERAL, AND SUMMARY. 
 
 48. The individual interferences. Figs. 31, 32, 33 73~75 
 
 49. The combined interferences. Table 17 75~?6 
 
 50. Special results 76-77
 
 CHAPTER I. 
 
 ON AN ADJUSTMENT FOR THE PLANE GRATING SIMILAR TO ROWLAND'S 
 METHOD FOR THE CONCAVE GRATING. By C. Barus and M. Barus. 
 
 1. Apparatus. The remarkable refinement which has been attained 
 (notably by Mr. Ives and others) in the construction of celluloid replicas 
 of the plane grating makes it desirable to construct a simple apparatus 
 whereby the spectrum may be shown and the measurement of wave-length 
 made, in a way that does justice to the astonishing performance of the grat- 
 ing. We have therefore thought it not superfluous to devise the following 
 inexpensive contrivance, in which the wave-length is strictly proportional 
 to the shift of the carriage at the eyepiece; which for the case of a good 
 2 -meter scale divided into centimeters admits of a measurement of wave- 
 length to a few Angstrom units and with a millimeter scale should go much 
 further. 
 
 Observations are throughout made on both sides of the incident rays and 
 from the mean result most of the usual errors should be eliminated by 
 symmetry. It is also shown that the symmetrical method may be adapted 
 to the concave grating. 
 
 In fig. i A and B are two double slides, like a lathe bed, 155 cm. long 
 and ii cm. apart, which happened to be available for optical purposes, in 
 the laboratory. They were therefore used, although single slides at right 
 angles to each other, similar to Rowland's, would have been preferable. 
 The carriages C and D, 30 cm. long, kept at a fixed distance apart by the 
 rod ab, are in practice a length of >^-inch gas pipe, swiveled at a and 6, 169.4 
 decimeters apart, and capable of sliding right and left and to and fro, 
 normally to each other. 
 
 The swiveling joint, which functioned excellently, is made very simply 
 of ^-inch gas-pipe tees and nipples, as shown in fig. 2. The lower nipple 
 N is screwed tight into the T, but all but tight into the carriage D, so that 
 the rod ab turns in the screw N, kept oiled. Similarly the nipple N" is either 
 screwed tight into the T (in one method, revolvable grating), or all but 
 tight (in another method, stationary grating), so that the table tt, which 
 carries the grating g, may be fixed while the nipple N" swivels in the T. 
 
 Any ordinary laboratory clamp K and a similar one on the upright C 
 (screwed into the carriage D} secures a small rod k for this purpose. Again 
 a hole may be drilled through the standards at K and C and provided with 
 set screws to fix a horizontal rod k or check. The rod, k, should be long 
 enough to similarly fix the standard on the slide 5, carrying the slit, and be 
 prolonged further toward the rear to carry the flame or Geissler tube appa- 
 
 1
 
 2 THE PRODUCTION OF ELLIPTIC INTERFERENCES \ 
 
 ratus. The table tt is revoluble on a brass rod fitting within the gas pipe, 
 which has been slotted across so that the conical nut M may hold it firmly. 
 The axis passes through the middle of the grating, which is fastened cen- 
 trally to the table tt with the usual tripod adjustment. 
 
 FIG. i. Plan of the apparatus. 
 
 2. Single focussing lens in front of grating. I shall describe three 
 methods in succession, beginning with the first. Here a large lens L, of 
 about 56 cm. focal distance and about 10 cm. in diameter, is placed just in 
 front of the grating, properly screened and throwing an image of the slit 
 5 upon the cross-hairs of the eyepiece E, the line of sight of which is always 
 parallel to the rod ab, the end b swiveled in the carriage C, as stated. (See 
 fig. 2.) An ordinary lens of 5 to 10 cm. focal distance, with an appropriate
 
 IN RELATION TO INTERFEROMETRY. 6 
 
 diaphragm, is adequate and in many ways preferable to stronger eyepieces. 
 The slit, S, carried on its own slide and capable of being clamped to C when 
 necessary, as stated, is additionally provided with a long rod hh lying under- 
 neath the carriage, so that the slit 5 may be put accurately in focus by the 
 observer at C. F is a carriage for the mirror or the flame or other source of 
 light whose spectrum is to be examined; or the source may be adjustable 
 on the rear of the rod by which D and 5 are locked together. 
 
 Finally, the slide AB is provided with a scale 55 and the position of the 
 carriage C read off by aid of the vernier v. A good wooden scale, graduated 
 in centimeters, happened to be available, the vernier reading to within one 
 millimeter. For more accurate work a brass scale in millimeters with an 
 appropriate vernier has since been provided. 
 
 Eyepiece E, slit 5, flame F, etc., may be raised and lowered by the split 
 tube device shown as at M and M' in fig. 2 . 
 
 FIG. 2. Elevation of standards of grating (g) and eyepiece (). 
 
 3. Adjustments. The first general test which places slit, grating and its 
 spectra, and the two positions of the eyepiece in one plane, is preferably 
 made with a narrow beam of sunlight, though lamplight suffices in the dark. 
 Thereafter let the slit be focussed with the eyepiece on the right, marking 
 the position of the slit ; next focus the slit for the eyepiece on the left ; then 
 place the slit midway between these positions and now focus by slowly
 
 4 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 rotating the grating. The slit will then be found in focus for both positions 
 and the grating which acts as a concave lens counteracting L will be sym- 
 metrical with respect to both positions. Let the grating be thus adjusted 
 when fixed normally to the slide B or parallel to A . Then for the first order 
 of the spectra the wave-length X = d sin B, where d is the grating space and B 
 the angle of diffraction. The angle of incidence * is zero. 
 
 Again let the grating adjusted for symmetry be free to rotate with the 
 rod ab. Then 6 is zero and \ = d sin i, 
 
 In both cases, however, if 2* be the distance apart of the carriage C, 
 measured on the scale ss, for the effective length of rod ab = r between axis 
 and axis, 
 
 \ = dx/r or (d/2r)zx 
 
 so that in either case X and x are proportional quantities. 
 
 FIG. 3. A, B, C, Diagrams relative to conjugate foci. 
 
 The whole spectrum is not, however, clearly in focus at one time, though 
 the focussing by aid of the rod hh is not difficult. For extreme positions a 
 pulley adjustment, operating on the ends of h, is a convenience, the cords 
 running around the slide A A. In fact if the slit is in focus when the eye- 
 piece is at the center (0 = 0, * = o), at a distance a from the grating, then for 
 the fixed grating, fig. 4, 
 
 where a' is the distance between grating and slit for the diffraction corre- 
 sponding to x. Hence the focal distance of the grating regarded as a con- 
 cave lens is/' = ar 2 /*. For the fixed grating and a given color, it frequently
 
 IN RELATION TO INTERFEROMETRY. 5 
 
 happens that the undeviated ray and the diffracted rays of the same color 
 are simultaneously in focus, though this does not follow from the equation. 
 Again for the rotating grating, fig. 3A, if a" is the distance between slit 
 
 r 2 -x 2 r 2 - x 2 
 
 and grating a" = a - - , so that its focal distance is /" = a 3 It follows 
 
 also that a'Xa" = a. For a = 80 cm. and sodium light, the adjustment 
 showed roughly/' = 650 cm., f" = 570 cm., the behavior being that of a weak 
 concave lens. The same a = 80 cm. and sodium light showed furthermore 
 a' = 91 and a" ' = 70.3. 
 
 Finally there is a correction needed for the lateral shift of rays, due to 
 the fact that the grating film is inclosed between two moderately thick 
 
 ' 
 
 " 
 
 FIG. 4. Diagram of adjustment for concave grating. R, p a , and p' 
 are measured from G, p and p " from G'. 
 
 plates of glass (total thickness t 
 shift thus amounts to 
 
 tx 
 
 .99 cm.) of the index of refraction n. This 
 
 xi i _i \b 
 
 '\vTT^3 vV-sV^/a 
 
 But since this shift is on the rear side of the lens L, its effect on the eyepiece 
 beyond will be (if /is the principal focal distance and b the conjugate focal 
 distance between lens and eyepiece, remembering that the shift must be 
 resolved parallel to the scale ss] 
 
 where the correction e is to be added to ix, and is positive for the rotating 
 grating and negative for the stationary grating.
 
 6 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 Hence in the mean values of 2* for stationary and rotating grating the 
 effect of e is eliminated. For a given lens at a fixed distance from the eye- 
 piece (b/f i) is constant. 
 
 4. Data for single lens in front of grating. In conclusion we select a 
 few results taken at random from the notes. 
 
 TABLE i. 
 
 Stationary grating. 
 
 Rotating grating. 
 
 Line. 
 
 Observed 
 
 2X' 
 
 Shift. 
 
 Corrected 
 ax 
 
 Line. 
 
 Observed 
 
 3X' 
 
 Shift. 
 
 Corrected 
 ax 
 
 c 
 
 132.60 
 118.90 
 98.23 
 
 87.87 
 
 - .26 
 -23 
 -.19 
 
 -.16 
 
 132.34 
 118.67 
 98.04 
 
 87.71 
 
 $:':: 
 
 Hydrogen 
 violet. . 
 
 132. 10 
 118.45 
 97.90 
 
 87-50 
 
 + .26 
 23 
 19 
 
 .16 
 
 132.36 
 118.68 
 98.09 
 
 87.66 
 
 5E::: 
 
 Hydrogen 
 violet.. 
 
 The real test is to be sought in the corresponding values of 2X for the 
 stationary and rotating cases, and these are very satisfactory, remembering 
 that a centimeter scale on wood with a vernier reading to millimeters only 
 was used for measurement. 
 
 5. Single focussing lens behind the grating. The lens L, which should 
 be achromatic, is placed in the standard C. The light which passes through 
 the grating is now convergent, whereas it was divergent in 2. Hence the 
 focal points at distances a', a" lie in front of the grating ; but in other respects 
 the conditions are similar but reversed. Apart from signs, 
 
 r 2 
 a' = a 5 , for the stationary grating 
 
 a" = a -j 2 for the rotating grating 
 
 / " .V 
 
 The correction for shift loses the factor (&// i) and becomes 
 tx, 
 
 As intimated, it is negative for the rotating grating and positive for the 
 stationary grating. It is eliminated in the mean values. 
 
 6. Data for single lens behind the grating. An example of the results 
 will suffice. Different parts of the spectrum require focussing. 
 
 
 TABLE 2. 
 
 
 Grating. 
 
 , Line. 2*' 
 
 Shift. ax 
 
 Stationary 
 
 D 2 118.40 
 
 + . 13 118.53 
 
 Rotating 
 
 D 2 118 65 
 
 . 13 118.52 
 
 
 
 * |
 
 IN RELATION TO INTERFEROMETRY. 
 
 The values of 2x, remembering that a centimeter scale was used, are 
 again surprisingly good. The shift is computed by the above equation. 
 It may be eliminated in the mean of the two methods. The lens L' at C 
 may be more easily and firmly fixed than at L. 
 
 7. Collimator method. The objection to the above single-lens methods 
 is the fact that the whole spectrum is not in sharp focus at once. Their 
 advantage is the simplicity of the means employed. If lenses at L' and 
 at L are used together, the former as a collimator (achromatic) and with a 
 focal distance of about 50 cm., and the latter (focal distance to be large, say 
 150 cm.) as the objective of a telescope, all the above difficulties disappear 
 and the magnification may be made even excessively large. The whole 
 spectrum is brilliantly in focus at once and the corrections for the shift of 
 lines due to the plates of the grating vanish. Both methods for stationary 
 and rotating gratings give identical results. The adjustments are easy and 
 certain, for with sunlight (or lamplight in the dark) the image of the slit 
 may be reflected back from the plate of the grating on the plane of the slit 
 itself, while at the same time the transmitted image may be equally sharply 
 adjusted on the focal plane of the eyepiece. It is therefore merely necessary 
 to place the plane of spectra horizontal. Clearly a' and a" are all infinite. 
 
 In this method the slides S and D are clamped at the focal distance apart, 
 so that flame, etc., slit, collimator lens, and grating move together. The 
 grating may or may not be revoluble with the lens L on the axis a. 
 
 8. Data for the collimator method. The following data chosen at ran- 
 dom may be discussed. The results were obtained at different times and 
 under different conditions. The grating nominally contained about 15,050 
 lines per inch. The efficient rod-length db was ^=169.4 cm. Hence if 
 i/ C= 15, osoX. 3937X338.8, the wave-length \C=2X cm. 
 
 TABLE 3. Stationery rotating grating. 
 
 Lines. 
 
 2*' 
 
 2X 
 
 D, 
 
 / "8-30 \ 
 
 \ 118.08 / 
 
 IlS.IQ 
 
 D t 
 
 f 118.27 I 
 I 118.05 / 
 
 118.16 
 
 Rowland's value of D z is 58.92 X lo" 6 cm. ; the mean of the two values of 
 2x just stated will give 58.87 X 10"* cm. The difference may be due either 
 to the assumed grating-space, or to the value of R inserted, neither of which 
 was reliable absolutely to much within o.i per cent. 
 
 Curiously enough, an apparent shift effect remains in the values of zx for 
 stationary and rotating grating, as if the collimation were imperfect. The 
 reason for this is not clear, though it must in any case be eliminated in the 
 mean result. Possibly the friction involved in the simultaneous motion 
 of three slides is not negligible and may leave the system under slight strain 
 equivalent to a small lateral shift of the slit.
 
 8 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 9. Discussion. The chief discrepancy is the difference of values for 2* 
 in the single-lens system (for D t , 118.7 an d 118.5 cm., respectively) as com- 
 pared with a double-lens system (for D 2 , 1 18.2 cm.) amounting to 0.2 to 0.4 
 per cent. For any given method this difference is consistently maintained. 
 It does not therefore seem to be mere chance. 
 
 We have for this reason computed all the data involved for a fixed grat- 
 ing 5 cm. in width, in the two extreme positions, fig. 3, C, the ray beinj^ 
 normally incident at the left-hand and the right-hand edges, respectively, 
 for the method of 6. The meaning of the symbols is clear from fig. 3, C, S 
 being the virtual source, g the grating, e the diffraction conjugate focus of 
 5 for normal incidence, so that b = r is the fixed length of rod carrying grat- 
 ing and eyepiece. It is almost sufficient to assume that all diffracted rays 
 b' to b" are directed toward e, in which case equation (i) would hold; but 
 this will not bring out the divergence in question. They were therefore not 
 used. Hence the following equations (2) to (5) successively apply where 
 d is the grating space. 
 
 cot 6' = (big + sin 0)/cos 6; cot 0" = (bjg - sin 0)/cos 6 ( i ) 
 
 a = 6/cos 2 0; a' = a" = V g 2 + a 2 (2) 
 
 sin i' = sin i" = g/a' ( 3 ) 
 
 - sin i' + sin (6 + 6') = Xjd; sin 6 = Jl/d; sin i" + sin (0 + 6"} = l/d (4) 
 
 cos 2 *'/a' = cos 2 (0 + 0')V; cos 2 i "la" = cos 2 (6 + 6") /b" (5) 
 
 Since 8, g, \, d, b are given, d' and 8" are found in equation (4), apart from 
 signs. If 81 and 5i" be the distance apart of the projections of the extremi- 
 ties of 6' and 6, b and b", respectively, on the line x, 
 
 di =g+(b b'} sin 0-6' sin i' d" = g + (b" 6) sin b" sin i" (6) 
 
 If fo' and &* be the distance apart of the intersections of the prolongation 
 of b' and b, b and b", respectively, with the line x 
 
 <V = sin (0 + 0')(&cos0/cos (0 + 0')-&') 
 <V = sin (0-0")(&"-&cos0/cos (6-6"}) 
 
 Given 6= 169.4 cm., 6 = 20 22', about for sodium, #=5 cm., the following 
 values are obtained : 
 
 0' = i36' 0=192. 7 cm. 0"=i34' a' =a"= 192.8 cm. 
 
 i'=z" = i3o' 6' = i66.ocm. r = 6=i69.4cm. 6" = i72.4cm. 
 
 whence 
 
 #,' = i. 92 cm. o 2 "=i.74cm. 
 
 These limits are surprisingly wide. If, however, they should be quite wiped 
 out on focussing, for any group of rays and symmetrical observations on 
 the two sides of the apparatus, this would be no source of discrepancy. 
 The effect of focussing the two parts of the grating may, in the first instance, 
 be considered as a prolongation of b' till it cuts x, together with the corre-
 
 IN RELATION TO INTERFEROMETRY. 9 
 
 spending points for the intersection of b" with x. Thus the values 85' and 
 fa" are here in question and they are 
 
 2 ' = i.97cm. o 2 ' ^,' = .05 cm. 
 
 whence 
 
 d 2 " = i.65 cm. d l "=d 2 =.og cm. 
 
 are the conjugate foci for the extreme rays of the grating, respectively, 
 beyond the conjugate focus of the middle or normal rays b, on x. Hence 
 the mean of the extreme rays lies at .07 cm. beyond (greater 0) the normal 
 ray, and the X found in the first instance is too large as compared with the 
 true value for the normal ray. 
 
 The datum .07 cm. may be taken as the excess of 2X, corresponding to the 
 excess of angle for a grating one-half as wide and observed on both sides 
 (2*), as was actually the case. Finally, since not the whole of the grating 
 is not in focus at once a correction less than .07 cm. for 2x must clearly be 
 in question. This is quite below the difference of several millimeters 
 brought out in 4 and 6. 
 
 To make this point additionally sure and avoid the assumption of the 
 last paragraph, we will compute the conjugate focus of the central ray (dif- 
 ferent angles 6) on the b' focal plane parallel to the grating and to x and on 
 the b" focal plane parallel to x. The computation is simpler if the central 
 ray is thus focussed than if the extreme rays are focussed on the x plane. 
 The distance apart will be 
 
 d s ' = g - b' cos (0 + 0'} (tan (d + 0'} - tan 6} 
 d s =g-b" cos (6 - 6"} (tan - tan (0 - 6"} ) 
 
 Inserting the results for 0/ 0/' b' b" g 
 
 o 3 ' = .06 <V' = ~~ -4 
 
 Both the b foci thus correspond to large angles. Their mean, however, 
 may be considered as vanishing on the intermediate x plane. 
 
 Thus it is clear that the effect of focussing is without influence on the 
 diffraction angle and much within the limits of observation. It is therefore 
 probable that the residual discrepancy in the three methods is referable to 
 a lateral motion of the slit itself, due to insufficient symmetry of the slides 
 A A and BB in the above adjustment. This agrees, moreover, with the 
 residual shift observed in the case of parallel rays in 8. The remedy 
 should present no difficulty. 
 
 10. Reflecting plate grating. The adjustment of the plane grating if 
 cut on specular metal is nearly identical to the above, except that the col- 
 limator is fixed as a whole in front of the grating, either to the slide carrying 
 the standard of the grating, B, or else quite in front of the cross slide A A, 
 fig. i above, so as to give clearance for the to-and-fro motion of the rail, R. 
 This admits of measurement of x on both sides of the slit, so that 2X, the 
 distance apart of the two symmetrical positions for a given spectrum line, 
 is again observed.
 
 10 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 II. Rowland's concave grating. For the case of the concave grating, 
 the accurate adjustment for symmetrical measurement on both sides of the 
 slit is not feasible, because the slit and eyepiece would have to pass through 
 each other. It is possible, however, to find conjugate foci at different dis- 
 tances from the grating in the normal position, which approximately answer 
 the purposes of measurement. Rowland's equation 
 
 (cos i/pi/R) cos t+(cos d/p'i/p') cos 6 = 
 
 where p and p' are the conjugate focal distances for angles of incidence and 
 deviation i and 6, may for = o be written 
 
 p/cos 2 * ~ R/cos i ~ |0 ~~ }f 
 where p is the normal distance of the eyepiece, so that 
 
 P* Po 
 
 If in fig. 4 (page 5) the slit S is put at p'>R from the grating G (normal 
 position), the image is at E at the end of p from G, where p <R<p'. If 
 po be used as a rail instead of R and put at an angle of incidence *', for the 
 eyepiece at E' or E", p cos * > p. But this excess need not be so large as 
 to interfere with adequately sharp focussing. 
 
 Table 4 gives an example, in which the difference of p and p ' in the nor- 
 mal position is even over i foot, an excessive amount, as the distance neces- 
 sary for clearance need not be more than a few inches. The grating has 
 14,436 lines to the inch and a radius about R = 191 cm. 
 
 TABLE 4. Conjugate foci of the concave grating. 7? = i9i cm., 14,436 lines to 
 inch, 5683 lines to cm. D=. 000,176. /> = 166 cm., p = 198 cm., p /0 = 32 cm. 
 i/p t i/R = .000,788, 0=o, sin i = \/D. 
 
 i 
 
 P 
 
 (0 cos i Diff . 
 
 (/./cos ) 
 
 Fraun- 
 hofer t 
 
 lines, j 
 
 
 cm. 
 
 cm. 
 
 
 
 
 166.0 
 
 166.0 .0 
 
 27500 
 
 B 
 
 22 59' 
 
 5 
 
 10 
 
 165.3 
 
 163 .2 
 
 165.3 -o 
 163.5 -3 
 
 27500 
 27400 
 
 c 
 
 D 
 
 21 54' 
 19 34' 
 
 ('5 
 
 159-6 
 
 160.3 - -7 27300 
 
 E 
 
 17 26' 
 
 \ao 
 
 I 54-7 
 
 156.0 1.3 27100 
 
 F 
 
 I602' 
 
 25 
 
 148.5 
 
 150.4 i. 9 26800 
 
 G 14 10' 
 
 3 
 
 140.9 
 
 143.7 -2.8 26500 
 
 
 
 35 
 
 132.2 
 
 136.0 3-8 26000 
 
 
 .... 
 
 40 
 
 122.3 
 
 127.1 4.8 25500 
 
 
 The greater part of the visible spectrum is thus contained between i = 15 
 and i = 2o. It follows that the excess of p cos ip lies between 7 and 13 
 mm. Hence the eyepiece may be placed at a mean position corresponding 
 to 10 mm. and give very good definition of the whole spectrum without 
 refocussing, as I found by actual trial. Within i cm. the focus is sharp 
 enough for most practical purposes. If the distances p and p ' are selected 
 so that eyepiece and slit just clear each other the definition is quite sharp.
 
 IN RELATION TO INTERFEROMETRY. 11 
 
 The diffraction equation is not modified and if 2x corresponds to the 
 positions -\-i and i for the same spectrum line, 
 
 It is therefore not necessary to touch the eyepiece and this is contributory 
 to accuracy. 
 
 If Rowland's equation is differentiated relatively to p and p',dp = 
 
 (P V 
 -- : ) dp ' where the factor dp '/p ' 2 is constant. Hence dp varies 
 Po cos ^/ 
 
 as (p/cos *) 2 given in the table. If, furthermore, a comparison is made 
 between dp and dp this equation reduces to 
 
 V dpjdp = (R p (i cos i) )/R cos z 
 which becomes unity either for i = o or for p =R (Rowland's case). 
 
 12. Summary. By using two slides symmetrically normal to each other 
 and observing on both sides of the point of interference, it is shown that 
 many of the errors are eliminated by the symmetrical adjustments in ques- 
 tion. The slide carrying the grating may be provided with a focussing lens 
 in front or again behind it, if the means are at hand for actuating the slit 
 which is not sharply in focus on the plane of the eyepiece carried by a 
 second slide throughout the spectrum at a given time. It is thus best to 
 use both lenses conjointly, the latter as a collimator and the former as an 
 objective of the telescope in connection with the eyepiece. It is shown that 
 a centimeter scale parallel to the eyepiece slide with a vernier reading to 
 millimeters is sufficient to measure the wave-lengths of light to few Ang- 
 strom units r while the wave-lengths are throughout strictly proportional 
 to the displacements along the scale. The errors of the three available 
 methods and their counterparts are discussed in detail. The method is 
 applicable both to the transparent and the reflecting grating. 
 
 It is furthermore shown that, in case of Rowland's concave grating, obser- 
 vation may be made symmetrically on both sides of the slit, by providing for 
 reasonable clearance of slit and eyepiece passing across each other, although 
 one conjugate focal distance is now not quite the projection of the other.
 
 CHAPTER II. 
 
 THE INTERFERENCE OF THE REFLECTED-DIFFRACTED AND THE DIF- 
 FRACTED-REFLECTED RAYS OF A PLANE TRANSPARENT GRATING, 
 AND ON AN INTERFEROMETER. By C. Barus and M. Barus. 
 
 13. Introductory. If parallel light, falling on the front face of a trans- 
 parent plane grating, is observed through a telescope after reflection from 
 a rear parallel face (see fig. 5) , the spectrum is frequently found to be inter- 
 sected by strong, vertical interference bands. Almost any type of grating 
 will suffice, including the admirable replicas now available, like those of 
 Mr. Ives. In the latter case one would be inclined to refer the phenomenon 
 to the film and give it no further consideration. On closer inspection, how- 
 
 ever, it appears that the strongest fringes certainly have a different origin 
 and depend essentially on the reflecting face behind the grating. If, for 
 instance, this face is blurred by attaching a piece of rough, wet paper, or by 
 pasting the face of a prism upon it with water, so as to remove most of the 
 reflected light, the fringes all but disappear. If a metal mirror is forced 
 against the rear glass face, whereby a half wave-length is lost at the mirror 
 but not at the glass face in contact, the fringes are impaired, making a 
 rather interesting experiment. With homogeneous light the fringes of 
 
 13
 
 14 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 the film itself appear to the naked eye, as they are usually very large by 
 comparison. 
 
 Granting that the fringes in question depend upon the reflecting surface 
 behind the grating, they must move if the distance between them is varied. 
 Consequently a phenomenon so easily produced and controlled is of much 
 greater interest in relation to micrometric measurements than at first 
 appears and we have for this reason given it detailed treatment. It has 
 the great advantage of not needing monochromatic light, of being appli- 
 cable for any wave-length whatever, and admitting of the measurement of 
 horizontal angles. In chapter 5 it will be shown that these interferences 
 may also be produced by Michelson's or by Jamin's apparatus. 
 
 FIG. 6. Showing corresponding triplets of diffracted rays for a single incident 
 ray, and each of the cases 0,'>* and 2 '<t of fig. 5. 
 
 When the phenomenon as a whole is carefully studied it is found to be 
 multiple in character. In each order of spectrum there are different groups 
 of fringes of different angular sizes and usually in very different focal planes. 
 Some of these are associated with parallel light, others with divergent or 
 convergent light, so that a telescope is essential to bring out the successive 
 groups in their entirety. At any deviation the diffracted light is neces- 
 sarily monochromatic, but the fringes need not and rarely do appear in 
 focus with the solar spectrum. If the slit of the spectroscope is purposely 
 slightly inclined to the lines of the grating, certain of the fringes may appear 
 inclined in one way and others in the opposite way, producing a cross pat- 
 tern like a pantograph. The reason for this appears in the equations. 
 
 In any case the final evidence is given when the reflecting face behind 
 the grating is movable parallel to it. The interferometer so obtained is
 
 IN RELATION TO INTERFEROMETRY. 
 
 15 
 
 subject to the equation (air space e, wave-length X, angle of incidence *', of 
 diffraction #') 5e = X/2(cos 6' cos i), and is therefore less unique as an 
 absolute instrument than Michelson's classic apparatus or the device of 
 Fabry and Perot. Its sensitiveness per fringe, 5e, depends essentially upon 
 the angle of incidence and diffraction and it admits of but i cm. (about) of 
 air-space between grating face and mirror before the fringes become too 
 fine to be available. But on the other hand it does not require monochro- 
 matic light (a Welsbach burner suffices), it does not require optical plate 
 glass, it is sufficient to use but a square cm. of grating film, and it admits of 
 very easy manipulation, for painstaking adjustments as to normality, etc., 
 are superfluous. In fact it is only necessary to put the sodium lines in the 
 spectrum reflected from the grating and from the mirror into coincidence 
 
 FIG. 7. Superposition of the cases of figs. 5 and 6 for 
 
 both horizontally and vertically with the usual three adjustment screws on 
 grating and mirror. Naturally sunlight is here desirable. Thereupon the 
 fringes will usually appear and may be sharply adjusted upon a second trial 
 at once. 
 
 When the air-space is small, coarse and fine fringes (fluted fringes) are 
 simultaneously in focus, one of which may be used as a coarse adjustment 
 on the other. Finally, the sensitiveness per fringe to be obtained is easily 
 a length of one-half wave-length in the fine fringes and one wave-length in 
 the coarse fringes, though the latter may also be increased almost to the 
 limit of the former. The fine fringes compatibly with their greater sen- 
 sitiveness vanish within about a millimeter. Their occurrence with the 
 coarse fringes makes it possible to investigate sudden displacements within 
 the latter.
 
 16 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 1. 
 
 r^ 
 
 I- 
 II 
 
 o o 
 
 11 
 
 Ii 
 
 i s 
 
 I 
 I 
 
 
 ouble 
 ut no 
 
 s-c ,v 7 r-a 
 3 S sj'J'g, 
 
 J^p C'EH wT ^ *S "3 ' 
 J 111 ll II -" SS: 
 
 OOO^ O-- ...io 
 
 00 00 ro r^oo rcoo N r>. OO 
 
 .^00 3 
 
 U-J U-) 1/1 IT) IT) IO ' 
 
 s 
 
 : 2 
 
 1 
 
 o usC J " i>4>:!E'^ ~ J^^s o : . -! 
 
 1 1.5 O ol3--o b- S ' " 
 
 O ja.QS ooa.Q> >QZf 
 
 : : : : :~TTT T rT^TT: H -T 
 
 y... &*' W' 
 
 ^QCDC^ QCC^^ - H ^{ 
 
 do s p 
 
 . O .00 . .0 
 
 VO-ss'rf 1 - O .' - 
 
 A ii n A ii ii A v ii n
 
 IN RELATION TO INTERFEROMETRY. 17 
 
 14. Observations. The following observations were made merely to 
 corroborate the equations used. The general character of the results will 
 become clear on consulting the preceding abbreviated table chosen at ran- 
 dom from many similar data. An Ives replica grating with 15,000 lines to 
 the inch (film between plates of glass .46 cm. thick) was mounted as usual 
 on a spectrometer admitting of an angular measurement within one min- 
 ute of arc. Parallel light fell on the grating, fig. 5, gg (see p. 13), under 
 different angles of incidence, i, and the spectrum lines were observed by 
 reflection (after reflection from gg and the rear face ff) at an angle of 
 diffraction 6' in air, both in the first and second order of spectra, and so 
 far as possible on both sides of the directly reflected beam. In view of 
 the front plate, the angle i corresponds to an angle of refraction r, within 
 the glass and the angle 6' similarly to an angle of diffraction 6, respect- 
 ively. Hence r>9 2 or di<r denote the sides of the ordinary ray on which 
 observation is made. As a rule these were as nearly as possible in the 
 region of the D line passing toward E. Finally 50 denotes the angle 
 between two consecutive dark fringes, observed and computed as speci- 
 fied. Similarly de will be reserved for changes of thickness e of the glass 
 and 8e f for changes of the air space in case of an auxiliary mirror M. 
 
 For i = o the number of groups of lines was a single set in each order; 
 but only the end of the spectrum could be seen. Measurements refer 
 (about) to the C line. For i = 45 several groups were too close together, or 
 too faint for measurement, and the same is true for i= 22.5. An estimate 
 of divergence is all that could be attempted on the given spectrometer. 
 The case 61 >r was usually not available, but for 2 = 22.5 two sets were 
 found in the first order, one being the normal set. The fringes in all cases 
 decrease in size from red to violet, but less rapidly than wave-length. 
 
 Whether they are convergent or divergent for a given set of fringes, as 
 for instance for the strong set, depends on the position of the grating. Thus 
 the divergent rays become convergent when the grating is rotated 180 
 about its normal. It is therefore definitely wedge-shaped. In fact when 
 the auxiliary mirror M is used, the fringes may be put anywhere, either in 
 front of or behind the principal focal plane, by suitably inclining the mirror. 
 
 1 5. Equations. If we suppose the film of the grating gg to be sandwiched 
 in between plates of glass each of thickness e, it will be seen that triplicate 
 rays pass in the direction ti, (0/>*) or of fe, (0 2 '<O'), which will necessarily 
 produce interference either partial or total. With respect to t\ t the only 
 light received comes either from DI by direct diffraction at gg, or from RDi, 
 by reflection from the lower face/, and thereafter by diffraction at gg; or 
 from DRi, by diffraction at gg and reflection at jf. Similarly the light along 
 h comes in like manner either from D 2 or DR 2 or RD^. With regard to the 
 angles of incidence and refraction or of diffraction within the glass or out- 
 side of it, we have the equations for the first and second order of spectra 
 (D being the grating space).
 
 18 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 sin *'I = M sin r\ (i) 
 
 sin 0i ' =fji sin 0i (2) 
 
 sin Oz' = n sin 2 (3) 
 
 sin r sin 02 = X/^MI or = 2\/Dn (4) 
 
 sin 0i sin r = X/.D/u, or = 2\/.D/* (5) 
 
 sin * sin 2 ' = \/D, etc. (4') 
 
 sin 0/-sin i = \/D, etc. (5') 
 
 where ju is the index of refraction of the glass, found to be equal to 1.5265 
 for sodium light, by breaking off a small corner of the glass of the grating 
 and using Kohlrausch's total reflectometer. 
 
 Furthermore, for the occurrence of interference along t\ we shall have, 
 for the case of the three rays in question, respectively combined in pairs, 
 if the wave fronts be taken in the glass plate FFgg 
 
 n\ = 2en cos r (6*) 
 
 nX= 2e/z(i sin 0i sin r)/cos t (7") 
 
 n\ = 2en( i cos (0i r))/cos 0i (8*) 
 
 where n is the order of interference and a whole number, e the thickness of 
 the lower glass plate, as may be found from a consideration of wave fronts 
 and need not be deduced here. It will be seen that, whereas equations 6 
 and 7 present cases of partial interference, equation 8, which is virtually the 
 difference of 6 and 7, should be a case of total interference, giving strong 
 fringes and possibly useful for interferometry. 
 
 Again for k the equations will be similar if corresponding wave fronts be 
 taken in the glass plate FFgg 
 
 HA = 2jJ. COS r (60 
 
 nA=2ejji(i sin 2 sin r)/cos 0, (7') 
 
 n/. = 2eii(i - cos (r - OJ )/cos 0, (80 
 
 the same as in the preceding case when 0i is replaced by 2 and 0\ r by 
 r 02. To this extent there is no essential difference between the cases. 
 Therefore may be used indiscriminately for either. 
 
 If, however, the wave fronts be taken in the glass plate jffgg the equations 
 become 
 
 nA = 2ejj.cosr (6) 
 
 n>l = 2*/( cos 0, (7) 
 
 HA = 26/J. (COS T COS 0|) (8) 
 
 with three other corresponding forms for 0<r, as before. The latter are 
 the true equations and equation 8 is the case of total interference. 
 
 These circumstances are at first quite puzzling, but they are due to the 
 fact that in the cases 6', 7', 8', 6", 7", 8", the wave fronts taken correspond 
 to the refracted ray only, whereas the diffracted rays subsequently undergo
 
 IN RELATION TO INTERFEROMETRY. 19 
 
 sudden changes of deviation on leaving gg. We have nevertheless carried 
 both groups of equations 6' to 8' and 6 to 8 in mind, though the latter are 
 alone certain of application. 
 
 For an air-space between gg and M the equations would be 
 
 nh = 2ecosi nX = 2ecos6 f nX=^ ie (cos 6' cos i), etc. 
 
 16. Differential equations. The quantity measured on the spectrom- 
 eter is essentially angular and preferably dd'/dn, the angular distance 
 apart of the fringes, in radians. Later we shall measure 5e or the linear 
 displacement of the parallel faces per fringe. In any measurement, how- 
 ever, we meet with embarrassment, inasmuch as n, X, n, r, 9, 9', are all 
 variable. The angle i and the thickness e and the grating space D are 
 alone given. Among these, the variation of r with /* andX must be found by 
 experiment. Fortunately, in case of the interferometer all these variables 
 are eliminated and e alone changes, subject to a given i and 9'. The n used 
 need not be known. (See paragraph 19.) 
 
 For the present purpose, as the variation of p. enters only as a correction, 
 we have been satisfied with the usual results in physical tables.* If from 
 the C to the D line 
 
 and from the B to the C line, = .013, we may write 
 
 du. dX 
 
 - ^ = - OI 57 
 
 and therefore 
 
 /;\ di 
 
 (9) 
 
 We shall write a =.015, b= i+a. 
 
 The case 9>r in the present paper is not of much experimental interest, 
 and may be omitted here. For the case of r>9 we shall have successively 
 and for the total interferences RD, DR, equation 8, 
 
 dX X cos dd 
 
 -j = . j : 2T ~T~ ( I I ) 
 
 a/i sin r b sin an 
 
 dr _ tan r cos dO 
 dn sin r b sin 6 dn 
 
 dfi _ afj. cos dO 
 dn ~s5Tr-&sin0 dn 
 
 dd' = jfcos0(sinr-sin0) dd 
 dn ~ cos 0'(sin r-b sin 6) dn 
 
 * See Kohlrausch's Leitfaden, nth edition, 1910, p. 712, light crown glass being 
 taken.
 
 20 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 and finally corresponding to equations 6, 7, 8, 
 dO^ _ _cosO * (sin r -sing) 
 dn ~ 26 cos 0' 6^sin7sin0 
 
 ^ (sin r- sing) 
 
 dn 26 cos 0' 6 cos r cos + a sin r tan r cos 
 d0' cos ^(sin r sin 0) 
 
 
 dn 26 cos 0' 6( i cos (r 0)) + a sin r sin 0( i cot tan r) 
 
 the last term in the denominator being corrective. Here dB'/dn is the 
 observed angular deviation of two consecutive fringes, so that 
 dn dn _ dn 
 
 The equation corresponding to the incorrect equation 8' would have been 
 dd' = cos 
 dn 26 cos 0' 
 
 X cos 2 0(sin r sin 0) 
 6cos 2 0(i cos(r 0)) (sinr sin0)(sinr 6sin0j atan rcos 2 0sin(r 0) 
 
 17. Normal incidence, or diffraction , etc. For the case of normal inci- 
 dence i = r = o, the equations corresponding to 6, 7, and 8 take a simplified 
 form and are respectively 
 
 _ d0 ' _ >1 sin cos 
 
 ~~ dn nha "' 
 
 _d0 '_^sin0_ 
 dn 2 6e cos 0' 
 
 d0 ' cos ^ sin 
 
 - ^s ( 1 8 ) 
 
 dn zbe cos i cos 
 
 If 0' = $ = o, for normal diffraction, which is particularly useful in Row- 
 land's adjustments as well as on the spectrometer 
 
 _ A sin r 
 
 a-o 2e ^C 1 *" 008 r )~ a s i n r tan r 
 
 for the case of total interference corresponding to equations 8 and 17. If 
 j-=-0'orr=-0 
 
 W l = A i 
 
 _ 9 2^ tan cos 0' 
 
 18. Comparison of the equations of total interference with observa- 
 tion. The partial interferences corresponding to equations 6 and 7 are 
 usually too fine to be seen unless e is very small. They amount in cases of 
 equations 15 and 16 for e = .48 cm. to the following small angles 
 
 (iS) (16) 
 
 i= o d6'/dn=.o6o' dd'/dn= .062' 
 
 22.5 .048' .050' 
 
 45 -057' -058'
 
 IN RELATION TO INTERFEROMETRY. 21 
 
 usually less than four seconds of arc and are therefore lost. The origin of the 
 fine interferences actually seen in the table is thus still open to surmise. 
 With small e and the interferometer they are obvious. 
 
 The total interferences as computed in the above table agree with the 
 observations to much within o. i minute of arc and these are experimental 
 errors; particularly so as it was not possible to use both verniers of the 
 spectrometer. The interesting feature of the experiment and calculation 
 is this, that 86' has about the same value for all incidences i from o to 45 
 and even beyond. The equations do not show this at once, owing to the 
 entrance of /* and r. But apart from a and 6 equation 17 is nearly 
 
 _ 
 dn 2 efjL i-cos(r-0) W 
 
 which is independent of r to the extent in which cos (r 0) is constant. The 
 dependence of dO'/dn on wave-length is borne out. (See paragraph 19.) 
 Finally, dd' ' /dn is independent of /* except as it occurs in a and b. 
 
 If the glass plate JJgg is removed and a mirror M used, as in the inter- 
 ferometer, the fringes may be enormously enlarged by decreasing e and the 
 measurements made with any degree of accuracy; but such measurements 
 were originally impracticable and have little further interest in this place 
 since the interferometer itself is tested in the next chapter. 
 
 19. Interferometer. The final test of the above equation is given by the 
 last part of the table for different thicknesses of glass, e = .48 and e= .77 cm. 
 The results are in perfect accord. 
 
 These data suffice to state the outlook for the interferometer. In this 
 case n and e are the only variables, so that equation 8 becomes 
 
 de = >l/2//(cos - cos r) (20') 
 
 where 8e is the thickness of glass corresponding to the passage of one fringe 
 across the cross-hairs of the telescope or a definite spectrum line. 
 
 If instead of glass in the grating above, an air-space intervenes between 
 the film of the grating and the auxiliary mirror M, fig. 5, the equation 
 reduces to 
 
 dg = _ ;__ _ = __ J _ 
 2 (cos 6' - cos *) 2 ( ]/ i-( s int-J/Z?)* - cos i) 
 
 where i and 6' are the angles of incidence and diffraction in air. 
 
 These equations 20 embody a curious circumstance. Inasmuch as B and 
 6' change as * increases from o to 90 from negative to positive values at 
 about i= 13 and 1 = 20, respectively, the denominator of either equation 
 20 will pass through zero (for air at about i= 10). Hence at this value of 
 * the motion of the mirror M produces no e effect (stationary fringes), 
 while on either side of it the fringes travel in opposite directions in the 
 telescope when e changes by the same amount. In the negative case the 
 sensitiveness for air-spaces passes from 8e .000489 to de= <*>, per
 
 22 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 fringe. In the positive case from 5e= + *> to 8e=.ooo 039, per fringe or 
 to a limit of about a half wave-length in case of 15,000 lines to the inch. 
 This limiting sensitiveness may be regarded as practically reached even at 
 1 = 40 where 8e=.ooo 155 cm. per fringe and an angle of about 1 = 45 is 
 most convenient in practice. 
 
 In addition to the large fringes the fine set appears when e is small or not 
 more than a few tenths of a millimeter. The sensitiveness of these is 
 naturally much more marked. In the two cases 
 
 de = h/2 cos i (20') 
 
 de = 1/2 COStf (20") 
 
 so that nearly A/ 2 per fringe is easily attained, but the available thickness 
 of air-space within which they are visible is decreased. 
 
 At * = 20 about, and in case of an air-space 6' is nearly o. We suggested 
 above that these fine fringes may be used as a fine adjustment in connection 
 with the large fringes, on which they are superimposed. In appearance 
 these large fluted fringes are exceedingly beautiful. The fine fringes have 
 the limiting sensitiveness of the coarse fringes, i.e., the cases for * = go 
 or 0' equal to maximum value. In different focal planes both sets of fine 
 fringes may be seen separately for small e (air wedge) . 
 
 Equation 20 shows that for smaller grating spaces D, and consequently 
 also in the second order of spectra there must be greater sensitiveness, 
 caet. par. ; but as a rule we have not found these fringes as sharp and useful 
 as those in the first order. 
 
 The limiting sensitiveness per fringe, however, follows a very curious 
 rule. If in equation 20 we put t = go, 
 
 zde= 1 W/(2-l/D) =l/Vr(2-r) in the first order 
 if r=\/D, and 
 
 2de= \ / W/4(iA/D) = XJ2\ r(i r) in the second order 
 
 D is the grating space. Both equations have a minimum, 8e =A/2 , at \/D = i 
 in the first order and A/D = .5 in the second order, beyond which it would be 
 disadvantageous to decrease the grating space. These minimum conditions 
 are as good as reached even when D corresponds to 15,000 lines to the inch, 
 as above, where roughly io 6 8e = $& cm. in the first order and io 6 5e = 33 cm. 
 in the second order. 
 
 All the conditions discussed above are summarized in fig. 8 and fig. 9 
 for the first and second orders of spectrum. 
 
 To view the stationary fringes of the first order was not practicable, 
 since they occurred for i= 10, whereas the telescopes were in contact at 
 about 20. In the second order of spectra they may be approached more 
 nearly as they occur when * is roughly 20. If the distance e is made small 
 enough, so that the three cases of equations 20, 20', 20*, are visible, the 
 appearance is very peculiar. The fringes of equation 20 are very slow-
 
 IN RELATION TO INTERFEROMETRY. 
 
 23 
 
 moving. They are intersected by the small fringes of equation 20', pro- 
 ducing the fluted pattern already discussed. Over all travel the rapidly 
 moving fringes of equation 20", producing a kind of alternation or flickering 
 which it is very difficult to analyze or interpret until e is very small,when 
 all three sets are broad and easily recognized. Sunlight should be used. 
 Nothing like these alternating fringes was seen in the first order, but we 
 used a film grating only. 
 
 FIG. 8. Charts showing dependence of on r, 0' on i, cos 0' and cos i on 
 i, 8e on r and de on i, de on X/D for the first order of spectrum. 
 
 The above equation shows finally that de is not exactly proportional to 
 wave-length, though the former decreases with the latter, as shown above. 
 
 The three equations 20 indicate finally that for i>6' all fringes travel 
 in the same direction with increasing e; whereas if 6' > i, the set correspond- 
 ing to equation 20 travel in a direction opposite to that of the sets 20' and 20".
 
 24 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 This is strikingly borne out by making the experiment for 6>i with a small 
 angle , both in the first and second order. 
 
 Table 6 contains a few data obtained by carrying the mirror on a Fraun- 
 hofer micrometer, reading to .0001 cm., toward a stationary grating film. 
 
 10 20 30 40 50 60 70 
 
 FIG. o. Similar data to those of fig. 8, for the second order of spectrum. 
 In addition, dd'/dn is shown as given by equations 15, 16, 17. 
 
 Observations were made in the region of the D lines. The grating was 
 originally between plates of glass =.48 cm. thick. Finally the plate 
 between grating and mirror was removed, the whole distance now being an 
 air-space. This has no effect on 8e, but e may then be reduced to zero and 
 the fringes enlarged.
 
 IN RELATION TO INTERFEROMETRY. 
 
 25 
 
 These data merely test the equations, as no special pains were taken for 
 accurate measurement, which neither the micrometer screw nor the special 
 adjustments warranted. Usually the micrometer equivalent of 50 fringes 
 was observed on the screw. The maximum distance e between grating and 
 mirror was .48 cm. of glass and .25 cm. of air conjointly, or within i cm. 
 In case of fine fringes mere pressure on table or screw impaired the adjust- 
 ment. Moreover these fine fringes run through the shadow of the coarse 
 fringes, and their size in consecutive spaces between the latter seems to vary 
 periodically as if they alternated between the two equations 15 and 16. 
 
 TABLE 6. Interferometer measurements. Replica grating. Air-space, often in 
 addition to glass space, e = .46 cm. 
 
 ' 
 
 e' 
 
 Coarse fringes. 
 
 Fine fringes. 
 
 Air-space.* 
 
 Observed. 
 SeXio* 
 
 Computed. 
 SeXio* 
 
 cm. 
 
 Co., 391 
 
 130 
 68 
 324 
 240 
 
 Air-space. 
 
 Observed. 
 teXio* 
 
 Computed. 
 SeXio* 
 
 22 30' 
 
 45 o 
 67 30 
 25 o 
 
 3 2 
 
 2 6' 
 21 I 
 
 35 i 
 4 7 
 8 37 
 
 cm. 
 .02 to .33 
 
 .013 to .250 
 
 cm. 
 39 
 
 72 
 322 
 241 
 
 cm. 
 
 cm. 
 
 cm. 
 
 Glass removed between mirror and grating. 
 
 32 25' 
 
 i 4 o' 
 
 .013 to .054 
 
 387 
 
 392 
 
 .006 to .025 
 
 34 
 
 (32 
 13 
 
 45 o 
 
 20 49 
 
 .032 to .064 
 
 129 130 
 
 .000 to .007 
 
 33 {32 
 
 * Approximate; contact endangering adjustment. 
 
 20. Secondary interferences. We come now to consider the minor 
 interferences which are either weaker, finer, or more diffuse than the strong 
 forms discussed. In the interpretation of these we have not met with suc- 
 cess, but some reference to them is essential. We assume that after two 
 reflections the fringes can no longer be seen. 
 
 In fig. 6 if there is light reinforcement passing in any direction t\ or h\ 
 then each incident ray /, at an angle i with the normal n, will after refrac- 
 tion be represented by the six rays A RDi DRi (ft > r) , A DR Z RD 2 (02<r), 
 as shown in the figure, under a notation similar to the preceding case. If 
 these rays are brought to a focus by a telescope there must be interferences 
 between pairs of which the fringes of D, RD, and D, DR, will usually be 
 too fine to be visible, whereas RD, and DR, will be large enough to be seen. 
 
 As the lines are not quite sharp, measurement is difficult and no other 
 fringes were therefore computed, and we have not thus far been at pains to 
 discover a reason for the large fringes in table i, either in the first order 
 (* = 22.5, 60' = 4. 7') or in the second order (4 = 45, 50' = 3.3')-
 
 26 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 21. Summary of secondary interferences. The possible partial inter- 
 ferences are naturally numerous. If we superimpose the case of fig. 5 
 on fig. 6 and draw all the rays of the first reflection for 6>r only (to avoid 
 complications), i. e., for a single direction, it will be seen that nine rays are 
 included, of which a, b, c, a', b', c', a", b", c", come from a single incident ray 
 each. Thus we can assemble these interferences from a determinant like 
 
 a b c 
 a' b' c' 
 a" b" c" 
 
 and there should be 18 cases. Most of these are identical in path difference, 
 but they have not led us to any satisfactory identification. They may 
 therefore be omitted here. 
 
 Other difficulties enter . Fringes which may be invisible if observed for the 
 glass plate may be visible if arising in the collodion film, as this is very thin. 
 Possibly some of the finer lines may arise in this way, but not probably. 
 
 The tendency of certain groups of interferences to travel in opposed 
 directions with rotation of the slit has suggested to us the possible occur- 
 rence of zone-plate action, where there would be multiple foci. But we 
 have not succeeded in establishing coincidences either for the virtual or 
 real foci in such a case. Moreover this opposition in motion has already 
 been accounted for in paragraph 19. 
 
 22. Convergent and divergent rays. What finally characterizes the 
 above groups of interferences is the difference in position of their focal 
 planes. They rarely coincide with the spectrum (parallel rays) and hence 
 do not always destroy it. If present with the spectrum the latter is wholly 
 wiped out. If the strong fringes are convergent for a given adjustment of 
 grating they become divergent when the grating is rotated 180 about its 
 normal. Hence the plates of glass are sharply wedge-shaped and to these 
 differences the location of focal planes is to be referred. 
 
 In addition to this the three regular reflections are not in the same focus 
 which shows the surfaces (collodium film) to be slightly curved. The above 
 experiments succeed best when two of the reflections are yellowish, which 
 probably means that the grating face is from the observer. 
 
 Suppose the remote glass face makes an angle dr/2 with the surface of the 
 grating. Then the DR ray of the strong interferences has its angle incre- 
 mented by d6=dr, whereas the RD ray receives an increment of but 
 
 dd = -Q dr. Hence if the DR and RD rays were parallel for parallel sur- 
 faces they would be at an angle corresponding to 
 
 d6 dr cosr cos 
 
 ~dT "7os0 
 
 where dr/2 is the angle of the wedge. Thus for the partial case of single 
 incidence, fig. 6, dB>dr if 6>r, or the issuing rays would converge; and 
 d0<dr and r> 0, or the issuing rays would diverge. If DR is negative the
 
 IN RELATION TO INTERFEROMETRY. 27 
 
 opposite conditions will hold, since dr and d6 change signs together. For 
 the case of triple incidence, fig. 5, there will be similar relations with less 
 liability to convergence. The interferences are further modified by the 
 change of thickness of glass or the variable e implied. 
 
 Fig. 7 shows that rays all but parallel will cross each other in front (con- 
 vergent) or behind (divergent) the grating, depending on their mutual lat- 
 eral positions. As a ray moves from the right to the left of the normal, the 
 phenomenon may change from divergence to convergence and vice versa. 
 
 23. Measurement of small horizontal angles. These relations are very 
 well brought out by the interferometer, in which the mirror M may be in- 
 clined at pleasure. If small values of deviation only are in question, this 
 instrument becomes a means of measuring small horizontal angles 7 be- 
 tween mirror and grating as these involve less change of focus. In fact if 
 h is the vertical height of the illumination at the mirror M and the cor- 
 responding obliquity of fringes is equivalent to an excess of N fringes 
 crossing the bottom of the cross-hairs as compared with the top for a 
 wave-length X, 7 = N8e/h; or 
 
 The question next at issue is thus the value of h. It will be noticed that 
 if parallel rays fall upon the slit, they will be brought to a focus by the col- 
 limator objective first, and thereafter by the telescope objective, placed at 
 a diametral distance D beyond it. Then if 5 is the vertical length of slit 
 used, and / and /, the focal lengths of the two objectives, respectively, it 
 follows that the length h = 5 is virtually illuminated. Hence, 
 
 NX 
 7 ~~ 2 S (cos 0'- cos *) 
 
 For since the angle 7, or a ratio, is in question, N8e/h is constant and it 
 makes no difference where the mirror M may be placed, i.e., how great the 
 absolute vertical height of the illumination h may be. 
 
 In case of this method (parallel light impinging on the slit) the illumin- 
 ation at each point of the image is received from but a single point (nearly) 
 of the mirror, whereas if the light falling on the slit is convergent, the whole 
 vertical extent of the mirror illumination contributes to each point of the 
 image in the ocular. Hence in the latter case the fringes are only sharp 
 when M and the grating are rigorously parallel, and they soon become 
 blurred when this is increasingly less true. The same observation also 
 accounts for the greater difficulty in adjustment when lamp-light is used. 
 In any case, equation 2 2 furnishes N/S. N may be obtained with an ocular 
 micrometer. 
 
 In the interest of greater precision the angle 7 may be found by actually 
 measuring the inclination to the vertical /3 of the fringes in the ocular. 
 Here if the height of image 5 in the ocular corresponds to the vertical length 
 of slit 5,
 
 28 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 H-K) 
 
 while 
 
 v,, ,;' 
 
 5 dn 
 where dd'/dn is given by equation 1 7 . Hence s may be eliminated and 
 
 (24> 
 
 . ;. 
 
 If now we further eliminate A r /5 in equation 22 by equation 24, we have 
 finally 
 
 ' a/ef t (cos 0' - cos i)dO'/dn 
 
 so that 7 is given in terms of ft, the observed inclination of fringes in the 
 ocular. To measure ft the ocular must be revolvable on its axis, so that the 
 cross-hairs may be brought into coincidence with the fringes and the angles 
 may be found. To measure M, the D lines as they remain vertical may 
 often be used, if in focus, for reference in place of vertical cross-hairs. 
 Using the d ata of the above experiments, if t = 45 
 
 N = i / C =/, = Z) (nearly) =23 cm., cos 0' cos i= .2264 
 
 S = .9cm. >l = 6oXio- 6 <f0'/dn = 493 Xio~ 6 
 
 whence 7 = 146 X lo" 6 radians, or about a half-minute of arc per fringe, and 
 = 44' per fringe. Thus ft is about 88 times as large as 7. At i =22.5, 
 7= 1.5' per fringe, ft = 45' per fringe. Naturally the sensitiveness increases 
 with the angle of incidence *. When the fringes are large, one-tenth fringe 
 is easily estimated, so that a horizontal angle 7 of a few seconds between 
 mirror and grating should be measurable. An ocular micrometer, as sug- 
 gested, would carry the precision beyond this. 
 
 24. Summary. It has been shown that the interferences here in question 
 take place in accordance with the equations (6), (7), and (8) above. They 
 therefore occur in triplicate, but not necessarily in the same focus. Their 
 superposition gives rise to fluted interference patterns of which the coarse 
 fringes (8) are less sensitive to micrometer displacements than the fine 
 fringes (6) and (7), on the one hand, while they persist over a comparatively 
 greater range of path difference, on the other. The eventual sensitiveness 
 of both approaches the wave-length of light. The adjustment made is 
 essentially self-compensating, in a way that excludes the wedge-angle of the 
 plate; and the lines are rigorously straight and vertical, to the extent in 
 which they maybe regarded as parts of the periphery of ellipses, whose centers 
 are infinitely distant on the same horizontal. As a whole, therefore, these 
 interferences are special cases of the phenomenon described in chapters IV 
 and V, where they may be made to pass through the zero of air-space. 
 
 The lines are inclined for a wedge-shaped air-space in a manner admitting 
 of the measurement of very small angles.
 
 CHAPTER III. 
 
 THE GRATING INTERFEROMETER. By C. Barus and M. Barus. 
 
 25. Introductory. In view of the perfection which has been attained 
 in the construction of film gratings and of the simplicity of the instru- 
 mental equipment needed, we have been at some pains to put the type of 
 interferometer recently described* into practical form. This is shown in 
 the accompanying diagram, in which fig. 10 is a photographic view, the 
 parts of which are easily recognized by aid of the simplified plan and ele- 
 vation in fig. n, A and B. 
 
 <* o 
 
 FIG. ii. A. The same (fig. 10) in plan. B. The same in elevation. 
 
 The attachment for magneto-striction is purely incidental. The small 
 increments of length in question were thought to offer an excellent test of 
 the availability of the interferometer for micrometric length-measurements 
 when these are of the sudden type and unlike the regular expansions. 
 
 * Science, xxxi, p. 394, 1910; Phil. Mag. (6), xx, p. 45, 1910; above chapter II. 
 
 29
 
 30 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 26. Apparatus. In fig. n, A is the collimator, B the telescope for view- 
 ing the interference patterns reflected from the grating gg and mirror M 
 over the revolvable table CC. These are the essential parts. The remain- 
 der is the incidental mounting just referred to, O being the helix containing 
 an iron rod, soldered at its ends to brass or copper tubes, / and t r , the latter 
 very light and movable in the V-groove N, the former / heavier and 
 clamped in the upright P. 
 
 The whole is supported on a frame of quarter-inch gas piping and con- 
 nections uFvR, the feet being at uvR, only the latter appearing in the figure. 
 Under the grating is an attachment * with four or eight screw sockets, W, 
 respectively at right angles to each other, into which a variety of apparatus 
 may be screwed, in the absence of the removable pipe pp. When p is not 
 used it is replaced by a foot. 
 
 Telescope B and collimator A are carried on T-pieces, with nipples at 
 the same height (about 8 cm. or less) above the iron frame, and B may be 
 slightly inclined about a horizontal axis. It may also be provided with a 
 micrometer eyepiece, as shown in fig. 10. 
 
 The table, CC, carrying the grating, is revolvable on an upright H and 
 may be clamped to fixed verniers, D, also carried at d by the upright H. 
 The table is cut away on the further side so that vertical apparatus may 
 approach closer to the grating in GG. It would even be an advantage to 
 use (not much more than) a semicircular table with verniers DD alter- 
 nately in use, set at an angle. 
 
 On the table two parallel brass guides determine the motion of the slide, 
 rr, actuated by the micrometer screw E. 
 
 The grating is mounted with the film on a glass plate gg, and exposed on 
 the side away from the observer, facing the mirror M . It is adjusted in a 
 rectangular shallow brass case, GG, with the side toward M and the top 
 open. The figure shows three adjustment screws actuating the rear of the 
 glass plate, gg, and the springs ss, which push the plate to the rear, passing 
 through slits in the sides of GG. There is one spring for the top and one for 
 the bottom of the grating. The capsule GG is firmly attached to the slide 
 rr, so that the grating may be moved fore and aft by the micrometer screw. 
 
 For some purposes the mirror M, similarly adjustable by three screws, 
 may be attached to the plate CC, free from the slide rr. In the figure the 
 mirror is on the light brass tube t', which makes a prolongation of the iron 
 rod, here to be tested for magneto-striction. The silvered face of M fronts 
 the grating. 
 
 The coil, O, used is wound on a slender annular chamber or double tube, 
 through which cold water may be kept in circulation (see fig. 10), to obviate 
 changes of temperature of the rod. The uprights P, T, T, N, in figs. 10 
 and ii, moreover, are adjustable up and down as well as around the rod 
 pp, by clamps shown in fig. 10, in which other unessential details may be 
 seen, including the hose for supplying and removing water. 
 
 * In a later form a special base R' was provided below \V.
 
 IN RELATION TO INTERFEROMETRY. 31 
 
 27. Adjustments. The micrometer screw controlled at E has a double 
 purpose. In the first place it is an immediate check on all measurements 
 of length increment made. It shows, moreover, whether the displacements 
 are expansions or contractions, since it may compensate any motion of the 
 mirror. In the second place, since the collimator A and telescope B are 
 clamped at a fixed distance apart approximately, it enables the observer 
 to put the part of the spectrum needed into the center of the field. For the 
 angle aMc may here be considerably increased or decreased, and both 
 telescopes are revolvable about the vertical. They may also be raised 
 or lowered, moved right and left and rotated about the horizontal, being 
 grasped by an ordinary clamp (not shown). The head E of the screw, 
 moreover, as well as the adjustment screws of the grating, are at all times 
 within easy reach of the observer. After an initial rough adjustment (the 
 direct reflections from mirror and grating being put into coincidence), the 
 adjustment screws on the grating are manipulated until the three spectra 
 seen in the telescope B coincide both horizontally and vertically. For this 
 purpose the D and E lines of the spectrum are useful and sunlight is prefer- 
 able. In the absence of sunlight a small electric arc lamp with the rays 
 issuing nearly parallel suffices equally well, supposing the apparatus is in 
 adjustment. When telescope and collimator are fixed, motion at the 
 micrometer screw naturally does not displace the spectrum line X, on 
 which the telescope has been focussed, so that the interferences recorded 
 pass through a definite part of the spectrum. 
 
 The three sets of interferences available are subject to the equation 
 
 de" ' = XJ2 cos i (i) 
 
 de'=Xl2COsd (2) 
 
 de = A/2(cosO cosi) (3) 
 
 where X is the wave-length of light used, i and B the angles of incidence and 
 of diffraction in air, and where be, be' , de", represent respectively the incre- 
 ment of air-space between mirror and grating per fringe passing the cross- 
 hairs. It is therefore necessary to know the angles 6 and i, and for this 
 purpose the verniers D and the revolvable plate CC graduated on its edge 
 are provided. 
 
 Let the mirror or grating plate be turned until the reflected image of the 
 slit coincides with the slit itself. Here the observer must be able to look 
 at the inner face of the slit in A through the hole h in the tube. Then let 
 the angle be read off. Thereafter let the plate C be undamped and turned 
 until the slit is seen sharply on the cross wire of the telescope C and a second 
 reading made. The angle so observed is (*-f-0)/2=a. We may therefore 
 write 
 
 sin i sin 0=2 cos (t+0)/2 sin (i 6)/2=A/D 
 
 where D is the grating space. Thus
 
 32 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 from which (id)/2=b may be found. Hence i = a+b; B=ab. An 
 eccentric position of the grating is of no consequence. 
 
 In the given apparatus the inner angle of diffraction has been utilized. 
 This brings A and B closer together for a sufficiently large angle i to secure 
 the best results. 
 
 In a later construction of the apparatus on an independent foot under W, 
 uv is a long smooth rod and A and B are attached by clamps admitting of 
 motion right and left, up and down, and rotation about the horizontal 
 and vertical. 
 
 28. Angular extent of the fringes. In chapter II the equations were 
 worked out (I.e.) for the more complicated case of a medium of thickness e 
 and refractive index p. For the case of an air-space the equations become 
 much simplified. Corresponding to equations i, 2, 3, if 6<i and sin * sin 
 B=\/D 
 
 dO' = /.- i 
 
 dn 2eD cos / cos 6 
 
 dn 2eD i sin i sin 
 
 dd = * - - (6) 
 
 dn 2eD i cos (i 0) 
 
 where dB/dn is the angle subtended per fringe for the wave-length X, the 
 grating space D, the thickness of air-space e, at an angle of incidence i and 
 of diffraction 6, in air. 
 
 These three sets need not be in focus at once. Equations 4 and 6 are 
 usually easily put in focus together. Thus in the above case roughly, 
 
 = 5o24' # = 24 44' ,*/>=. 352 
 
 whence for X = 6oXio~ 6 cm. and e=i cm., 
 
 dO' _ o6 , d6" ^ df) = _ g , 
 
 dn dn dn 
 
 In case of the set of equations 4, 5, 6, therefore, there should be about 
 6 to 7 small fringes to one large fringe. This is about the order of values 
 usually observed. When e is small the change of wave-length with dd must 
 be considered. To obtain a given ratio k of small and large fringe diver- 
 gences, one may write for the cases 4 and 6, for instance, 
 
 , _ i cos (i 6) 
 
 '' * COS ~(* - 0) + COS~ (t +0)" ( 7 ' 
 
 Equation 7 is not easily treated. If, however, 8 is computed in terms of 
 i and expressed graphically, k may then also be expressed in terms of i; and 
 thus the angle of incidence i for any ratio of size of fringes, k, in question, 
 may be roughly adjusted. Table 7 shows these results.
 
 IN RELATION TO INTERFEROMETRY. 
 TABLE 7. Ratio of large and small fringes. 
 
 33 
 
 i 
 
 i/k 
 
 \ 
 
 i/k' 
 
 i 
 
 9 
 
 i/k 
 
 i/k' 
 
 
 
 -20 l' 16.4 
 
 i6-5 
 
 SO 
 
 24 27' 
 
 6.0 
 
 7.0 
 
 IO 
 
 io 17 15.6 
 
 16.6 
 
 60 
 
 3 56 
 
 3-4 
 
 4-4 
 
 20 
 
 - o 34 14.8 
 
 15-8 
 
 70 
 
 36 o 
 
 1.6 
 
 2.6 
 
 3 
 
 +831 12.3 
 
 13-3 
 
 80 
 
 39 15 
 
 .6 
 
 i-5 
 
 40 
 
 16 54 9.1 
 
 10. 2 
 
 90 
 
 40 24 
 
 .0 
 
 I .0 
 
 Thus it appears that at ^ = 37 about there should be ten small fringes 
 to one large fringe. In a general way, moreover, the ratio of small to large 
 fringes gives an estimate of the value of i. 
 
 Similarly equations 5 and 6 give 
 
 I ~ cos *- 
 
 - _ 
 
 2 - (cos (i - 6} - cos (*+0)) 
 
 from which the data also given in table i , follow. 
 
 Here there will be ten small to one large fringe when i is 40.5 roughly. 
 
 The preceding equations 4, 5, and 6 are essentially approximate, inas- 
 much as the rates are taken for the finite quantities. If we return to the 
 fundamental equations 
 
 2 COS I = tt/ (8) 
 
 2ecosO = n'/> (9) 
 
 sin sin i = XjD (io) 
 
 for 6 is greater than i, where n and n' are unequal distributive whole num- 
 bers; and if we put e/D = a, we find 
 
 . 
 tan 2 = a 
 
 4<3 
 
 + a 
 
 (II) 
 
 which are free fromX, whereas sin i and sin 6 essentially contain X. As the 
 angular width of a fringe is 
 
 M* 
 
 T~ dn 
 dn 
 
 those corresponding to equations 4 and 5 may therefore be expressed as 
 
 -dn 
 
 0, = 2 a I - 
 
 Jn "' '- 
 
 02=2(1 n(n- 
 
 */n 
 
 4(in tan i 4a 2 
 
 (12) 
 
 _ 
 2a tan i) 
 
 for a given space e in a = b/D and a given angle of incidence i in tan *. 
 These integrations are easily made, but the results are too diffuse to be
 
 34 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 worth discussing here. Equation 12, moreover, is a restatement of the fact 
 that in these interferences n\ is constant throughout the spectrum. 
 
 Equations 8, 9, and 10, however, admit of the graphic treatment of the 
 problem. For if we put A=nD/ze, they may be written according as 
 9>i or 6<i 
 
 sin0 l - -+sin* ( 14 ) 
 
 /i 
 
 <S> 
 
 t+A . 
 
 where the distributive number n in A takes the values of the successive 
 whole numbers for a dark band in the respective spectra corresponding to 
 equations 14 and 15. The coincidence of dark bands then determines the 
 position of the coarse fringes. 
 If i- 50 24' 
 
 2=.icm. D= .000,169 cm. # = 24 44' 
 then 
 
 A = .0017 M, nearly 
 
 With these data table 8 was computed. Similar results might be computed 
 for 6<i, but I have abandoned it because, as is now evident, the practical 
 demands are sufficiently met by the method of paragraph 28. 
 
 TABLE 8. Showing tf, and n, " 2 and n'. H observed about 75. 2^ = i cm., * = 5o.4, 
 > i. tie corresponds to the mean value 8. 
 
 n 
 
 ". 
 
 
 10'fo, 
 
 , 
 
 I 
 n' 
 
 '. 
 
 io*8e a 
 
 i 
 
 1650 
 
 86. 
 
 1 
 
 76 
 
 84.2 
 
 30 
 
 83.4 
 
 19 
 
 82.5 
 
 1700 
 
 82. 
 
 3 
 
 46 
 
 81.2 
 
 400 
 
 Sl.S 
 
 18 
 
 80.5 
 
 175 
 
 80. 
 
 
 
 36 
 
 79.1 
 
 500 
 
 79.6 
 
 i7 
 
 78.8 
 
 1800 
 
 78. 
 
 2 
 
 3 
 
 77.6 
 
 600 
 
 77-9 
 
 16 
 
 77.1 
 
 1840 
 
 77- 
 
 
 
 23 
 
 76.7 
 
 700 
 
 76.2 
 
 15 
 
 75-5 
 
 1870 
 
 76. 
 
 3 
 
 30 
 
 75-8 
 
 800 
 
 74-7 
 
 14 
 
 74-0 
 
 1900 
 
 75- 
 
 4 
 
 22 
 
 75-0 
 
 900 
 
 73-3 
 
 12 
 
 72.7 
 
 IQ40 
 
 74- 
 
 5 
 
 20 
 
 74-2 
 
 IOOO 
 
 72.1 
 
 
 
 1970 
 
 73- 
 
 9 
 
 2O 
 
 73-6 
 
 20OO 
 
 64 . o 
 
 
 
 2OOO 
 
 73- 
 
 3 
 
 18 
 
 73-2 
 
 
 
 
 
 2O2O 
 
 73- 
 
 
 
 20 
 
 72.8 
 
 
 
 
 . . . 
 
 2O40 
 
 72. 
 
 6 
 
 15 
 
 72.5 
 
 
 
 
 . . . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 29. Test made by magneto-striction. The very small elongation pro- 
 duced when iron is magnetized offers an excellent test of the above appa- 
 ratus. The attached water-cooled helix has already been described. The 
 rod of Swedish soft iron was quite within the helix, which surrounded it 
 closely without contact, the rod being prolonged by light copper tubes 
 soldered to its ends.
 
 IN RELATION TO INTERFEROMETRY. 
 
 35 
 
 A large number of experiments were made for trial, brief examples of 
 which are given in the following table and charts, where if the current is 
 in amperes the magnetic field 
 
 H=io8i gauss 
 
 and the elongation 5e=io$Xio~ 6 cm. per large fringe. Thus if n is the 
 number of. fringes and ^ = 28 cm. the effective length of the iron rod, the 
 absolute elongation is da= 105^X1 o" 6 cm. and the elongation per unit of 
 length, 
 
 or if small fringes N are taken, 
 
 dp =. 54^X10-" 
 
 Elongations from lo" 6 cm. to io~ 4 cm. were measured without difficulty, 
 though such measurement, without micrometer attachment, is essentially 
 an estimate. Count was made of the number of small fringes between the 
 large fringes. 
 
 TABLE 9. Magnetostriction elongations. Coil* (separated by thin sheet of flow- 
 ing water from rod) length 37 cm., 3200 turns, H = io8t gauss (*' in amperes). 
 Swedish iron rod, diam., 2r = .64 cm., length / = 30.6 cm., free length 28 cm. 
 Elongations, 10*^ = 105 cm., io 6 &?i=33 cm., io 6 <te 2 = 47 cm. Seven small fringes 
 to one large fringe. 
 
 H 
 
 Fringes. 
 
 ^Xio" 
 
 H \ Fringes. 
 
 5Xio 8 
 
 gauss 
 
 small 
 
 
 gauss small 
 
 
 13 
 
 .0 
 
 
 
 98 6.0 
 
 320 
 
 22 
 
 i. 5 
 
 79 
 
 56 i 7- 
 
 37 
 
 36 
 
 4.0 
 
 2IO 
 
 36 5-o 
 
 310 
 
 44 
 
 6.0 
 
 320 
 
 23 2.0 
 
 I 10 
 
 62 
 
 7.0 
 
 37 
 
 9 -o 
 
 
 
 62 
 
 6.5 
 
 350 
 
 - 9 -5 
 
 26 
 
 44 
 
 5-5 
 
 300 
 
 -23 3.0 
 
 160 
 
 36 
 
 5- 
 
 270 
 
 -36 j 5-5 
 
 300 
 
 13 
 
 .0 
 
 
 
 56 6.0 
 
 320 
 
 -13 
 
 5 
 
 26 
 
 -99 5-0 
 
 270 
 
 22 
 
 3-5 
 
 190 
 
 + 99 5-o 
 
 270 
 
 -62 
 
 7.0 
 
 37 
 
 69 j 6.0 
 
 320 
 
 + 62 
 
 7.0 
 
 37 
 
 56 6.0 
 
 320 
 
 '3 
 
 5 
 
 26 
 
 19 i 1.2 
 
 64 
 
 6 
 
 .0 
 
 
 
 ii . .0 
 
 
 
 
 
 
 I" ! - 5 
 
 26 
 
 * A field of 150 gauss should more than practically saturate iron. 
 
 The data partake of the usual character of magnetic phenomena. There 
 is marked hysteresis and irregularity due to residual magnetization. Table 
 9 has been inserted merely as an example of the results, which are otherwise 
 given graphically. The maximum elongation corresponded to about 7 
 small fringes. 
 
 The curves bring out the following results consistently: 
 (i) The elongation apparently begins abruptly in small fields of 5 or 10 
 gauss, see fig. 12, etc., at b.
 
 36 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 (2) In small fields, fig. 14, A, there is nevertheless a tendency of a very 
 slow continual increase; but whether this is due to inevitable percussion 
 during magnetization can not be stated. The case may be one of true mag- 
 netic elasticity. 
 
 (3) The maximum elongation is very rapidly reached in a field of 50 to 
 100 gauss. After this the elongation now due to strong fields decreases. 
 In_the case of strong fields, however, the rod as a whole is probably dis- 
 
 SO -60 -40 20 
 
 20 40 60 80 
 
 FIG. 12. Charts showing the elongation of soft iron in millionths for 
 different fields, positive and negative, in gauss. Field applied and 
 removed for each observation. 
 
 Wari 
 
 M 
 
 80 -60 40 20 40 60 80 
 
 FIG. 13. Chart showing the elongation of soft iron in millionths for 
 different fields, positive and negative, in gauss. Field applied and 
 removed for observation.
 
 IN RELATION TO INTERFEROMETRY. 
 
 37 
 
 placed or warped. The results are less and less trustworthy and not 
 suitable for exploration by the optic method in question (figs. 12, B, and 
 14, B and C). 
 
 (4) If the field is reversed the elongation is similar, but symmetrical 
 with respect to a small positive field if the immediately preceding magneti- 
 zation was positive, and symmetrical with respect to a small negative if 
 the preceding magnetization was negative. See figs. 12, A, B, 13 (hystere- 
 sis). Thus in fig. 12, A and B, where a field of 13 gauss at b after a positive 
 magnetization produces no elongation whatever, the reversal of the field 
 to 13 gauss produces a very marked elongation of nearly 3 X io~ 7 which 
 
 10 20 SO 40 50 60 
 
 FIG. 14. Charts showing the elongation of soft iron in millionths for 
 different fields, positive and negative, in gauss. Field applied and 
 removed for each observation except in fig. D. 
 
 2091
 
 38 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 may be repeated indefinitely. Hence there seems to be a definite magnetic 
 elasticity with fields of less than 5 to 10 gauss. If this is exceeded the mag- 
 netic molecules are permanently set, even in soft iron. 
 
 (5) If the field varies continuously (liquid resistances) the elongations 
 are much smaller in given fields (fig. 14, D). 
 
 The whole subject, which is here touched incidentally, is well summar- 
 ized in Winkelmann's Handbuch, vol. 5, p. 307 et seq., 1908, by Professor 
 Auerbach. The above results agree closely with the elaborate experiments 
 of Nagaoka and Honda, * who used a special type of contact lever. Recently 
 Mr. Dorsey t has made a similar investigation, using an ingenious method 
 of his own. The present method would not have been feasible but for the 
 occurrence of large and small fringes in the field of the telescope. 
 
 30. Summary. The present chapter has shown the ease with which the 
 above phenomena may be adapted to the measurement of small displace- 
 ments, has instanced their particular usefulness, inasmuch as fine and 
 coarse adjustments are both available (recommending their use in the 
 investigation of phenomena of sudden occurrence as in magneto-striction, 
 etc.), and has given suggestions of practical forms of apparatus in addition 
 to the usual spectrometric form. 
 
 * Nagaoka and Honda: Phil. Mag. (5), xxxvn, p. 131, 1894. 
 f Dorsey: Phys. Rev., xxx, p. 698, 1910.
 
 CHAPTER IV. 
 
 THE USE OF THE GRATING IN INTERFEROMETRY ; EXPERIMENTAL 
 RESULTS. 
 
 31. Introductory. In chapters II and III a method* was described of 
 bringing reflected-diffracted and diffracted-reflected rays to interference, 
 producing a series of phenomena which, in addition to their great beauty, 
 promise to be useful. In fact, the interferometer so constructed needs but 
 ordinary plate glass and replica gratings. It gives fringes rigorously 
 straight, and their distance apart and inclination are thus measurable by 
 ocular micrometry. Lengths and small angles are thus subject to micro- 
 metric measurement. Finally, the interferences are very easily produced 
 and strong with white light, while the spectrum line used may be kept in 
 the field. 
 
 3 
 
 FIG. 15. Diagram of grating interference adjustments adapted to 
 Michelson's apparatus. L, source of white light (parallel rays), gg' 
 grating, M and N opaque mirrors. 
 
 The same method is available as an adjunct to either Jamin's or Michel- 
 son's interferometers, except that here the transmitted-reflected-diffracted 
 and reflected- transmitted-diffracted rays are brought to interfere. To 
 take the example of the Michelson type, stripped of unnecessary details, 
 let gGg' in fig. 15 be the grating or ruled surface, n its normal, L the source 
 of white light, M and N the mirrors, and E the eye. In the usual way the 
 rays from L interfere at E. 
 
 *Science, xxxii.p.Qa, 1910; cf. Phil. Mag., July 1910, and chapters II and III above.
 
 40 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 Now replace L by a slit and collimator, by a telescope focussed for 
 parallel rays. The eye at E now sees a sharp line of light. At D and D', 
 however, there must be two diffraction spectra coinciding in all their parts 
 and hence interfering rhythmically, if all adjustments are sufficiently per- 
 fected. The other two diffractions within MGg' and EGg are often lost at 
 an incidence of 45. 
 
 The attempt to produce these interferences at D, D' with replica gratings 
 is liable to result in failure; for while the transmitted system NGD shows 
 brilliant spectra, the reflected system MGE is dull and hazy. Both spectra 
 are clearly in evidence and may be brought to overlap. The film, however, 
 does not reflect in a degree adequate to the transmission. Attempts made 
 to realize the conditions of interference eventually, however, resulted in 
 complete success (chapter V). 
 
 What is strikingly feasible at once, with ordinary plate glass and a non- 
 silvered grating, * is the production of interferences between pairs of dif- 
 fracted spectra, D' and D, for instance, if returned by equidistant mirrors 
 M and A T to a telescope in the line D. Both of these spectra are very bril- 
 liant and not very unequally so and the coincidence of spectrum lines both 
 horizontally and vertically brings out the phenomenon. This, for each of 
 the cases specified, is of the ring type and not of the line type heretofore 
 discussed; but it occupies the whole field of the spectrum from red to 
 violet. One obtains brilliant large confocal ellipses with horizontal and 
 vertical symmetry, and the spectrum lines, simultaneously in focus, may 
 serve either as major or minor axes. The interferometer motion is twofold 
 in character, consisting of radial motion, combined with a drift of the figure 
 as a whole, in a horizontal direction. Naturally a fine slit is of advantage, 
 but the experiment succeeds with quite a wide slit, especially in the red, 
 much after spectrum lines vanish. Similarly the regular reflection from 
 M and A' (mirrors) will produce these phenomena along GD. 
 
 Such ellipses (I shall so call them, though they are probably ovals) as 
 have their centers in the field, are clearly due to reflection from the same 
 surface, as shown in fig. 1 5 ; curved lines or ellipses with remote centers are 
 due to simultaneous reflections of the component rays from the opposite 
 faces of the grating, since the angle of the wedge of glass can not be excluded. 
 All owe their vertical and horizontal symmetry to the vertical slit and 
 horizontal spectrum. The ellipses are identically present in the successive 
 orders of spectra at once. 
 
 These elliptical fringes thus embody with the preceding linear set the 
 common property of being duplex in character; only here the motion of dark 
 rings to or from the centers of the ellipse, as a fine adjustment, is associated 
 with a displacement of the fringes bodily through the spectrum (coarse 
 adjustment). 
 
 * My thanks are due to Prof. J. S. Ames, of Johns Hopkins University, who was 
 good enough to lend me this grating.
 
 IN RELATION TO INTERFEROMETRY. 41 
 
 This displacement may be from red to violet or from violet to red, as the 
 virtual air-space increases in thickness, depending upon the adjustment, 
 as will presently appear. In other words, for a given small interferometer 
 motion of mirror, there is in general less displacement of fringes bodily 
 than radial motion of fringes to or from the center. Slight deflection of the 
 grating, gg , is chiefly accompanied by radial motion of the fringes.* The 
 effect thus, produced is to give sharpness to the interference pattern, which 
 soon vanishes for approximate adjustments of spectra. 
 
 Similarly the micrometer screw produces a rapid passage of the pattern 
 through its maximum size, a play of the screw of o. i cm. being sufficient to 
 pass from fringes of one extreme of small size, through the maximum size 
 to the final extreme. One may note that in chapters II and III the admis- 
 sible play for the coarse fringes was about i cm., ten times greater. On the 
 other hand there are many regions of interference; in fact, different groups 
 of interferences may sometimes be in the field at once, or more usually 
 corresponding to slightly different angles between the mirrors and grating. 
 
 All admissible angles, provided that symmetry is maintained (the virtual 
 plane of the grating bisecting the angle of the mirrors) , bring out the phe- 
 nomenon. This points to the fact that the earlier phenomenon referred 
 to (I.e.), in which the center or normal ray was virtually at an infinite dis- 
 tance, is now produced near an accessible center, even if this is not actually 
 in the field. However, in the earlier case, when the angle of incidence is 
 zero, curved lines may also be obtainable. Such cases will be given in the 
 next chapter. 
 
 Experiments with a silvered grating showed no advantage, whether the 
 transparent film covered the grating or the plane face of the glass. In 
 fact, in case of good adjustment the phenomenon is so strong as to need no 
 accessory treatment; this is in fact one of its advantages. A great diffi- 
 culty in adjustment is the occurrence of stationary fringes due to the rear 
 face of the grating. These may even wipe out the spectrum lines; but 
 usually they lie at a finite focus and are not seen with a sharp spectrum. 
 Fortunately many positions are available for obtaining the ellipses, so that 
 a satisfactory one is easily found. Unless all overlapping spectra show 
 sharp lines the adjustment is very tedious. f The amount of grating space 
 used is less than i sq. cm., though of course a larger size is convenient for 
 experiment. 
 
 32. Special properties. The motion of the ellipses bodily across the 
 spectrum, corresponding to the increased or decreased virtual air-space 
 (micrometer screw), shows that whereas the vertical dimension or axes 
 (direction of fixed color) do not appreciably change,! the horizontal axes 
 
 * This changes the effective thickness of the grating. 
 
 t These difficulties are largely removed by the method of coincident white slit 
 images given in chapter V. 
 
 J Probably referable to increased refraction and decreased diffraction from red 
 to violet.
 
 42 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 grow rapidly smaller; i.e., the ellipse is more eccentric from red to violet. 
 With the grating used, however (about 2800 lines per centimeter), the 
 major axis remained vertical, i.e., the circular form was not reached 
 throughout the first order even in extreme red. 
 
 It follows from this that in the second order the major axis of the ellipses 
 for 2800 lines to the centimeter will probably be horizontal, and this was 
 found to be strikingly true. The figure, moreover, is necessarily coarser and 
 the lights and shadows in greater contrast. In the third order the ellipses 
 are drawn out horizontally to a correspondingly greater degree. Thus it is 
 clear that with a more dispersive grating the circular or even the hori- 
 zontally elongated form must occur in the first order. Indeed with 6000 
 lines to the centimeter, ellipses occur in the red having horizontal major 
 axis while the figure in the green is circular. Spectrum lines are very dis- 
 tinct, particularly in the second order. In all orders of spectra the centers 
 of the ellipses are simultaneously on the same spectrum line and their 
 vertical dimensions are about the same. Hence the spectrum lines may be 
 used instead of cross-hairs, as they are fixed landmarks among the moving 
 ellipses. 
 
 With replica gratings the center of the ellipses is usually remote, i.e., 
 reflection does not easily take place at the free grating surface. Apart 
 from this the curved lines are good and strong. Samples must be tried out, 
 in which the lines are as clear as possible in all the overlapping spectra. Six 
 gratings of this kind were examined with about the same results. The 
 centers of the ellipses could only rarely be brought into the field. Centers 
 may, however, be obtained for gratings cemented under pressure between 
 unequally thick plates of glass. 
 
 It has been stated that considerable width of slit is admissible. Moreover, 
 the focal plane of the collimator (convergent or divergent light) or the focus 
 of the telescope makes little difference, though the condition of parallel 
 rays is naturally preferable. An eye focussed for infinity sees the fringes 
 very well without the telescope and they may be also caught on a screen; 
 but coming through a slit they are liable to be dark and the advantage of 
 the telescope is obvious. The second order is particularly accessible for 
 naked-eye observation, and the light and black appearance is accentuated, 
 but they are liable to be irregular. Inclining the collimator moves the 
 spectrum across the fringes vertically, while inclination of the telescope 
 moves both equally. In this way the centers may often be found. With 
 strong tipping, however, the figure becomes distorted or open above and 
 below, as would be expected. If the light is intensified for projection or 
 naked -eye work, there is also distortion. 
 
 It is not necessary (and apparently of little advantage) to have the 
 reflections from the mirrors M and N occur at normal incidence. In fact 
 the patch of white light on the grating surface and the return patch of 
 spectrum may be over an inch apart. Inasmuch as the spectra are rigo- 
 rously coincident, it follows that (apart from, the refraction of the thick
 
 IN RELATION TO INTERFEROMETRY. 43 
 
 plate glass) the grating must be symmetrical with respect to the mirrors; 
 i.e., intersect the angle between them. Very little difference of size or shape 
 is observable between extremes of such adjustment. Thus I rotated the 
 grating about 10 (larger angles being unavailable) without appreciable 
 effect, after one of the mirrors had been correspondingly adjusted. 
 
 On examination of the effective air-distances of the mirrors M and N 
 from the grating, it is found that three positions are favorable to inter- 
 ference; in the first (case A), M is farther away than N from the corre- 
 sponding faces of the glass plate; in the second (case B) they are about 
 equidistant; finally, in the third (case C), N is further away, symmetrically 
 with the first adjustment. For each case the admissible play of mirror 
 (micrometer screw) does not much exceed i cm. for ordinary magnification. 
 The second or equidistant adjustment brings out the lines or ellipses with 
 distant centers, and there are usually three types easily found (after one 
 has appeared), by slightly inclining either mirror around a horizontal 
 axis by a tangent screw. Thus (see fig. 16) fine lines, say at 135 to the 
 
 FIG. 1 6. Interference patterns in case of self-compensation, B. 
 
 horizontal, coarse nearly circular lines, concave downward or upward as 
 the case may be, and finally broad lines say 45 to the axis, may appear 
 in succession. If the grating is rotated 180 around its normal, convexity 
 upward (fig. 16, A) changesto concavity upward (fig. 16, B), showing the 
 wedge angle of the glass plate to be effective. The micrometer-screw will 
 pass any of these forms through a succession of inclinations corresponding 
 to the eccentric intersection of a group of concentric circles by a straight 
 line. This is also shown in fig. 1 6 at A and B, as observed for a fixed air- 
 space and at C, on moving the micrometer in the A and B cases. These 
 figures cover the coincident spectrum fields only. A solitary part of the field 
 shows no interferences. For adjustment it is therefore necessary to cut off 
 the spectra sharply above and below, by limiting the length of slit. Of 
 the three or four spectra present the coincidence may then be secured with 
 least difficulty. With each such coincidence there goes a definite position 
 of the micrometer screw (interferometer), and this is the initial difficulty 
 4 -
 
 44 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 of adjustment; i.e., to coordinate the various spectrum coincidences with 
 the three screw readings. 
 
 The non-equidistant case of mirrors and grating correspond to the ring 
 types with the centers in the field. Hence both reflections take place from 
 the same face of the grating. When the distance GM is shorter, G'N longer, 
 I have noticed two or even three ellipses successively in the field, obtained 
 by slowly inclining the mirror N about the vertical (tangent screw), with 
 their centers in turn in the yellow-green, the blue, and the green of the 
 spectrum. They do not occur together. On turning the micrometer screw 
 clockwise they move from red to violet, and vice versa. 
 
 When the distance GM is longer and G'N shorter, rings also occur; but 
 only a single set was found. The spectrum lines not being clear, finding 
 them was a matter of discrimination. When the micrometer screw was 
 turned clockwise, however, they marched from violet to red, i.e., in opposite 
 direction to the preceding. The drift of ellipses corresponds with the 
 direction in which the vjolet fringes move horizontally. 
 
 It is at first an astonishing result that when the grating is reversed 
 (front face put rearward, leaving the air space unchanged) the ring type 
 is left in the field ; though the solitary ring type changes to the multiple 
 type. This is true for the case specified as A or case C. It is also true for 
 the eccentric type, case B, where line types, single or multiple, reappear at 
 slightly different mirror angles. Briefly, rotation has no effect on the march 
 of ellipses through the spectrum; but in case B it changes convex lines 
 downward to concave lines, and vice versa. Nor has reversal of grating any 
 effect on the march. The position of the grating with respect to the mirrors 
 (three places being available) alone determines this result, certainly in the 
 extreme cases A and C, and probably in the intermediate case B. 
 
 The use of a compensator in either of the component rays is always 
 accompanied by the two effects in question; i.e., there is both relatively 
 large radial motion of the fringes and relatively small displacement, within 
 the field of the telescope. One may therefore be evaluated in terms of the 
 other, the two having the stated relation of coarse and fine adjustments. 
 In case B the fringes will rotate, expand, or contract. 
 
 33. Elementary theory. The endeavor must now be made to explain 
 these results more in detail. For this purpose fig. 1 7 may be consulted. 
 
 In the grating used at an incidence of about 45 the angle of diffraction 
 in the first order was about 32 39' for the A sodium line, there being 
 about 2847 lines to the centimeter, corresponding to a grating space of 
 D = . 0003 5 1 2 cm . Thus \/D = 0.1677. The index of refraction was assumed 
 to be 1.53 for the same line. 
 
 The diagrams are drawn for an angle of incidence of 45 and for normal 
 reflection from the mirrors M and N. In case of the grating given, the 
 three positions of the grating at which interferences occur were about 0.6 
 cm. apart and are marked 124, 100, and 76 scale parts, on the micrometer
 
 IN RELATION TO INTERFEROMETRY. 
 
 45 
 
 screw normal to the grating. In other words it was convenient to move the 
 grating at the center of the spectrometer, rather than the mirrors M and N 
 adjustably attached near the edge of the plate graduated in degrees. 
 
 In group A at 124 the short air-path is to M, the long path to N; in group 
 C at 76 the reverse is true. In group B at 100, the system is self-compen- 
 sating and the air-paths about equal. 
 
 In all cases the angle of diffraction, 6, is less than the angle of incidence, i. 
 There are of course corresponding cases, 6>i, which have not been drawn, 
 because they merely duplicate the cases given, at a different angle. It is 
 assumed that after more than one direct reflection or one diffraction the 
 interferences are no longer observable. 
 
 FIG. 17. Chart of the three groups of interferences. A, B, C. Mirrors M and N 
 return the spectrum. Grating face in position shown at g. 
 
 The diagrams 124 and 76 show that the two reflections of the component 
 rays at the grating take place at the same surface; hence the occurrence of 
 centered figures or rings. On the contrary the reflections in diagram 100 
 take place at the two different faces of the grating respectively; hence 
 the angle of the grating is included and liable to produce eccentric ring 
 systems. The center may be so far off that the dark lines are nearly straight, 
 but they change their inclination as the vertical projection of the center 
 moves horizontally through the field. 
 
 Some of these cases may coalesce in practice or they may destroy each 
 other more or less. I have taken a single incident ray from which may 
 come two parallel emergent rays, which are brought to interfere by the 
 telescope. It would have been just as convenient to have taken the two
 
 46 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 corresponding incident rays which interfere in a single emergent ray. From 
 the position of the mirrors it is clear that the regularly refracted rays are 
 not returned. Only rays first diffracted at the grating (where they may also 
 be reflected) are returned by the mirrors. 
 
 As a whole we may distinguish two typical cases, those in which both 
 component rays are diffracted as in No. i or refracted as in No. 2 ; and those 
 in which one component ray is refracted and the other diffracted. If * and 
 8' are the angles of incidence and diffraction in air and r and the corre- 
 sponding angles of refraction and diffraction in glass, the glass path differ- 
 ences, Jx, in the important cases are as follows : 
 
 No. i. Jx = 2ne/cos 6=2.2 cm. No. 7. J# = Zero. 
 
 3. =2/u/cos 6=2.2 cm. 8. =Zero. 
 
 4. =Zero. 9. = 2/ztf/cos 6 = 2.2 cm. 
 6. =Zero. 10. = zjue/cos 6 = 2.2 cm. 
 
 Here M is the index of refraction and e the thickness of the glass plate of 
 the grating, and excess of path for the M ray is reckoned positive. These 
 paths must be compensated by corresponding decrements and increments 
 respectively of the air paths GM and GN. Ordinarily these path differences 
 in glass being fixed for given angles tf would fall away; but they vary essen- 
 tially with color and hence the degree of compensation is never the same 
 for all colors. 
 
 Furthermore, although the wave-fronts of the two rays are the same on 
 emergence, this does not imply coincidence of phase even in such cases as 
 Nos. i and 2, for instance; the absolute lengths of paths in glass are quite 
 different, although their differences are the same. Consequently the cases 
 i and 2 would again interfere if superimposed, one case being first diffracted 
 and the other first refracted. 
 
 Thus it is not surprising that so many cases were identified. It is also 
 apparent that the air compensations are very different and hence identifi- 
 cation is facilitated. Finally, since a vertical slit and collimator are used, 
 the section of a beam of light passing through the grating by a horizontal 
 plane consists of parallel rays; the section by a vertical plane, however, is 
 convergent. 
 
 It is interesting to find the numerical data for the above equations, 
 assuming that 
 
 = 45 0' = 32 39' *=.68cm. M= i. 5 3 (estimated) \/D=.i6jj 
 
 for the sodium line of the spectrum. The results are given with the equa- 
 tions. Their value is about 2.2 cm., which is equivalent to a displacement 
 of mirror actually found. 
 
 It follows, moreover, that the center of the ring system, order n = o, 
 must move from red to violet or the reverse, inasmuch as the compensation 
 takes place successively, at each color, in the same way. Although the 
 equations hold only for the center, and the symmetrically oblique rays
 
 IN RELATION TO INTERFEROMETRY. 
 
 47 
 
 belonging to the rings have not been given consideration, an approximate 
 computation of the motion of the ring centers may nevertheless be 
 attempted. 
 
 From the equations qualitative interpretations of the above and the fol- 
 lowing data are obtainable; but quantitatively they are too crude, because 
 they ignore the essential feature of oblique reflection from M and N. 
 
 Omitting these equations, to be fully discussed in the next chapter, an 
 example of the displacement and radial motion in a given experiment may 
 be adduced; the former was fully 16 times less sensitive per fringe than the 
 latter. The displacement is thus a coarse adjustment in comparison with 
 the usual radial motion of the fringes and this is the distinct advantage of 
 the present method for many purposes. It is like a scale division into 
 smaller and larger parts, where the enumeration of small parts alone would 
 be confusing or impossible. The ratio of the micrometer value of displace- 
 ment and radial motion per fringe may be given any value since dz/d\ oc e 
 and the lateral displacement may actually be more sensitive than the 
 radial motion, when the grating plate is thin. 
 
 The sodium lines here make admirable cross-hairs and the ocular itself 
 need have none. The conditions are the same in the second order and the 
 coarser rings and spectrum lines are often easier to count. 
 
 34. Compensator. It is not necessary, however, to use thin glass, for 
 if a compensator is provided, i.e., if the grating is on the common plane 
 between two thicknesses of identical plane-parallel glass plates, one of 
 which carries the grating, the ideal plane in question is provided. The 
 ellipses in this case would be infinite in size and their displacements infi- 
 nitely large. By partial compensation (compensator thinner) ellipses of 
 any convenient size and rate of displacement may therefore be provided 
 at pleasure. The following table gives some rough experimental data 
 where z is the advance of the micrometer screw normal to the grating, i.e., 
 the displacement of the grating to move the center of fringes from the D to 
 
 TABLE 10. Displacement of ellipses from the D to the b line, by moving the grat- 
 ing Az cm. normal to itself. t = 45, nearly. Grating: e = .6S cm., D => . 000351 
 cm., ^=1.53 (estimated). Compensation shown by negative sign. Position z 
 taken while the sodium line was the major axis. 
 
 . 
 
 Com- 
 pensator 
 thickness* 
 e'Xio 2 
 
 Position 
 of grat- 
 ing z 
 
 Microme- NQ of 
 
 placement * 
 A0Xio< Diob 
 
 Displace- 
 ment per 
 fringe 
 io*Xdz/dn 
 
 Ellipses. 
 
 Side of 
 compen- 
 sator. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 
 
 -48 
 
 5 
 
 30 12 
 
 2-5 
 
 Very large. . . . 
 
 N 
 
 -44 
 
 .08 
 
 35 
 
 
 Very large .... 
 
 N 
 
 -29 
 
 . 12 
 
 45 
 
 
 Large 
 
 N 
 
 10 
 
 .20 
 
 65 
 
 
 Mean 
 
 N 
 
 + 48 
 
 .38 
 
 US 44 
 
 2.6 
 
 Very small . . . 
 
 M 
 
 -65 
 
 .00 
 
 10 4.5 
 
 2.3 
 
 Eno'rmous. . . . 
 
 N 
 
 o 
 
 
 70 28 
 
 2.5 
 
 Mean 
 
 None 
 
 * Compensator parallel to mirror, M or N, respectively.
 
 48 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 the b lines of the spectrum, e' the thickness of the compensator placed 
 parallel to the mirror M or N as stated. 
 
 Rotation of the compensator offers the usual easy method of adjustment. 
 In the second order the first rings, on compensation, usually more than 
 fill the field. This is of course eventually the case in the first order also. 
 With very perfect compensation the ellipses are quite eccentric and the 
 lines under the limiting conditions nearly vertical and straight. Hence their 
 motion, partaking of the twofold character specified, is complicated but 
 usually opposite in direction on the two sides of the center for the same 
 micrometer displacement. The whole phenomenon may vanish within a 
 half millimeter of play of the grating, passing from fine lines through enor- 
 mous ellipses back into reversed fine lines, all nearly vertical. 
 
 The displacement of the grating by the micrometer screw is of the same 
 order per fringe, no matter whether the ellipses are large or small, and in 
 the last table it was about .00025 cm. per fringe. The displacement at the 
 mirror would exceed this. The radial motion per fringe is of the order of 
 wave-length. Naturally the position z of the grating changes parallel to 
 itself linearly with the thickness e' of the compensator, supposing other con- 
 ditions the same, so that p, z, and dz/dn all vary linearly with e' the thick- 
 ness of the compensator. 
 
 The full equations for the amounts of displacement, etc., require an 
 evaluation of dQ/dn, which in turn must take into consideration that 
 reflection from the mirrors can not in general be normal except for the one 
 color instanced above. This investigation will have to be reserved for the 
 next chapter.
 
 CHAPTER V. 
 
 INTERFEROMETRY WITH THE AID OF A GRATING ; THEORETICAL 
 RESULTS. By Carl Barus. 
 
 PART I. INTRODUCTION. 
 
 35. Remarks on the phenomena. In the earlier papers* I described 
 certain of the interferences obtained when the oblique plate gg, fig. 18, of 
 Michelson's adjustment, is replaced by a plane diffraction grating on 
 ordinary plate glass. Some explanation of these is necessary here. In the 
 figure L is the source of white light from a collimator. Such light is there- 
 fore parallel relative to a horizontal plane, but convergent relatively to a 
 vertical plane. M and N are the usual silver mirrors. A telescope adjusted 
 for parallel rays in the line GE must therefore show sharp white images of 
 the slit. As the grating is usually slightly wedge-shaped, there will be 
 (normally) four such images, two returned by M after reflection from the 
 front (white) and rear face (yellowish) of the plate gg, and two due to N. 
 There will also be two other, not quite achromatic, slit images from N or M, 
 respectively, due to double diffraction before and after reflection. These 
 will be treated below. In the direction GD there will thus be a correspond- 
 ing number of diffraction spectra, more or less coincident in all their parts, 
 and therefore adapted to interfere in pairs throughout their extent. If the 
 two white and the two yellow images of the slit be put in coincidence and 
 the mirrors M and N are adjusted for the respective reduced or virtual path 
 difference zero, the interferences obtained are usually eccentric; i.e., the 
 centers of the interference ellipses are not in the field of view. The effective 
 reflection in each of these cases takes place from the front and rear face of 
 the grating at the same time. Hence the interference pattern includes the 
 prism angle of the grating plate and is not centered. The air-paths of the 
 component rays are here practically equal. In addition to the ellipses, 
 this position also shows revolving linear interferences and (as a rule) a 
 double set is in the field at once, consisting of equidistant, symmetrically 
 oblique crossed lines, passing through horizontality in opposite directions 
 together, when either mirror M or N is slightly displaced. 
 
 If either pair of the white and yellowish images of the slit be placed in 
 coincidence when looking along EG, the interference pattern along DG is 
 ring-shaped, usually quasi-elliptic and centered. The light returned by 
 M and N is in this case reflected from the same face of the grating, either 
 from the face carrying the grating or the other (unruled) face. The corre- 
 sponding air-paths of the rays are in this case quite unequal, because the 
 
 * Am. Journ. of Science, xxx, 1910, pp. 161-171; Science, July 15, 1910, p. 92; 
 and Phil. Mag., July, 1910, pp. 45-59.
 
 50 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 short air-path is compensated by the path of the rays within the glass plate. 
 Hence these adjustments are very different, GM being the long path in one 
 instance, GN in the other instance. For the same motion of the micrometer 
 screw, the fringes are displaced in opposite directions. In one adjustment 
 there may be a single family of ellipses; in the other there may be two or 
 even three families nearly in the field at once. 
 
 If the grating were cut on optical plate glass, the adjustment for equal 
 air-path would probably be best. But with the grating cut as usual on ordi- 
 nary plate, or in case of replica gratings on collodion or celluloid films, the 
 adjustment for unequal paths is preferable. Here again one of the positions 
 is much to be preferred to the other, owing to the occurrence of multiple 
 slit images from one of the mirrors, as above specified. In fig. 19, for 
 instance, where the grating face is to the rear, there are but two images, 
 i and 2 from M, if the plate is slightly wedge-shaped; but from N, in 
 
 FIG. 1 8. Diagram showing adjustments 
 for interference. 
 
 FIG. 19. Diagram showing double- 
 diffraction. 
 
 addition to these two normal cases (not necessarily coinciding with i and 2), 
 there are two other images 3' and 4' (3 and 4 are spectrum rays), resulting 
 from double diffraction, with a deviation 6 and angle of incidence 7, 
 respectively 6 < I and 6 > I, in succession ; or the reverse. As the compensa- 
 tion for color can not here be perfect, the two slit images obtained are very 
 narrow, practically linear spectra, but they are strong enough to produce 
 interferences like the normal images of the slits, with which they nearly 
 agree in position. Other very faint slit images also occur, but they may 
 be disregarded. The doubly-diffracted slit images are often useful in the 
 adjustments for interference. 
 
 Among the normal slit images there are two, respectively white and yel- 
 lowish, which are remote from secondary images. If these be placed in coin- 
 cidence both horizontally and vertically along EG, fig. 18, the observation 
 along DG through the telescope will show a magnificent display of black
 
 IN RELATION TO INTERFEROMETRY. 51 
 
 apparently confocal ellipses, with their axes respectively horizontal and 
 vertical, extending through the whole width of the spectrum, from red to 
 violet, with the Fraunhofer lines simultaneously in focus. The vertical 
 axes are not primarily dependent on diffraction and are therefore of about 
 the same angular length throughout ; the horizontal axes, however, increase 
 with the magnitude of the diffraction, and hence these axes increase from 
 violet to red, from the first to the second and higher orders of spectra, and 
 in general as the grating space is smaller. It is not unusual to obtain 
 circles in some parts of the spectrum, since ellipses which in one extreme 
 case have long axes vertically, in the other extreme case have long axes 
 horizontally. The interference figure occurs simultaneously in all orders 
 of spectra, and it is interesting to note that, even in the chromatic slit images 
 shown in fig. 19, needle-shaped vertical ellipses are quite apparent. 
 
 It is surprising that all these interferences may be obtained with replica 
 or film gratings, though not of course so sharply as with ruled gratings,the 
 ideal being an optical plate. With thin films two sets of interferences are 
 liable to be in the field at once and I have yet to study these features from 
 the practical point of view. If the film is mounted between two identical 
 plates of glass, rigorously linear, vertical and movable interference fringes, 
 as described* by my son and myself, may be obtained. 
 
 36. Cause of ellipses. The slit at L, fig. 18, furnishes a divergent pencil 
 of light due (at least) to its diffraction, the rays becoming parallel in a 
 horizontal section after passing the strong lens of the collimator. But the 
 vertical section of the issuing pencil is essentially convergent. Hence if 
 such a pencil passes the grating the oblique rays relatively to the vertical 
 plane pass through a greater thickness of glass than the horizontal rays. 
 The interference pattern, if it occurs, is thus subject to a cause for contrac- 
 tion in the former case that is absent in the latter. Hence also the vertical 
 axes of the ellipses are about the same in all orders of spectra. They tend 
 to conform in their vertical symmetry to the regular type of circular ring- 
 shaped figure as studied by Michelson and his associates and more recently 
 by Feussner.f but in view of the slit the symmetry is cylindric. 
 
 On the other hand the obliquity in the horizontal direction, which is 
 essential to successive interferences of rays, is furnished by the diffraction 
 of the grating itself, as the deviation here increases from violet to red. In 
 other words the interference which is latent or condensed in the normal 
 white linear image of the slit is drawn out horizontally and displayed in the 
 successive orders of spectra to right and left of it. The vertical and hori- 
 zontal symmetry of ellipses thus follows totally different laws, the former 
 of which have been thoroughly studied The present paper will therefore 
 be devoted to phenomena in the horizontal direction only. 
 
 * Phil. Mag., I.e. 
 
 t See Professor Feussner's excellent summary in Winkelmann's Handbuch der 
 Physik, vol. 6, 1906, p. 958 et seq.
 
 52 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 At the center of ellipses the reduced path difference is zero; but it can not 
 increase quite at the same rate toward red and violet. Neither does the 
 refractive index of the glass admit of this symmetry. Hence the so-called 
 ellipses are necessarily complicated ovals, but their resemblance to confocal 
 ellipses is nevertheless so close that the term is admissible. This will appear 
 in the data. 
 
 If either the mirror or the grating is displaced parallel to itself by the 
 micrometer screw the interference figure drifts as a whole to the right or to 
 the left, while the rings partake of the customary motion toward or from a 
 center. The horizontal motion in such a case is of the nature of a coarse 
 adjustment as compared with the radial motion, a state of things which is 
 often advantageous in other words, the large divisions of the scale are not 
 lost ; moreover the displacements may be used independently. 
 
 The two motions are coordinated, inasmuch as violet travels toward the 
 center faster in a horizontal direction, i.e., at a greater angular rate, than 
 red. Hence the ellipses drift horizontally but not vertically. Naturally 
 in the two positions specified above for ellipses, the fringes travel in opposite 
 directions for the same motion of the micrometer screw. As the thickness 
 of the grating is less the ellipses will tend to open into vertical curved lines, 
 while their displacement is correspondingly increased. With the grating 
 on a plate of glass about e=.6& cm. thick and having a grating space of 
 about D = .000351 cm., at an angle of incidence of about 45, the displace- 
 ment of the center of ellipses from the D to the E line of the spectrum 
 corresponded to a displacement of the grating parallel to itself of about 
 .006 cm. It makes no difference whether the grating side or the plane side 
 of the plate is toward the light or which side of the grating is made the top. 
 If the grating in question is stationary and the mirror N alone moves 
 parallel to itself along the micrometer screw, a displacement of N = .01 cm. 
 roughly moves the center of ellipses from D to E, as before. This displace- 
 ment varies primarily with the thickness of the grating and its refraction. 
 It does not depend on the grating constant. Thus the following data were 
 obtained with film gratings on different thicknesses e of glass and different 
 grating spaces D for the displacements N of the mirror at N, to move the 
 ellipses from the D to the E line, as specified. 
 
 Glass grating, ruled e = .68 cm. \/D = .168 N = .010 cm. 
 
 Film on glass plate e = .57 .352 .008 
 
 Film on glass plate e = .24 .352 .003 
 
 Film between glass plates. ] ,_' .352 .003 
 
 ( e .24 / 
 
 Reduced linearly to e=.6S cm., the latter data would be AT = .oio and 
 W = .ooo, which are of the same order and as close as the diffuse interfer- 
 ence patterns of film gratings permit. The large difference in dispersion, 
 together with some differences in the glass, has produced no discernible 
 effect.
 
 IN RELATION TO INTERFEROMETRY. 
 
 53 
 
 An interesting case is the film grating between two equally thick plates of 
 glass. With this, in addition to the elliptical interferences above described, 
 a pattern of vertical interferences identical with those discussed in a pre- 
 ceding paper* was obtained. These are linear, persistently vertical fringes 
 extending throughout the spectrum and within the field of view, nearly 
 equidistant and of all colors. Their distances apart, however, may now be 
 passed through infinity when the virtual air-space passes through zero ; and 
 for micrometer displacements of mirror in a given direction, the motion of 
 fringes is in opposite directions on different sides of the null position of the 
 mirror. I have not been able, however, to make them as strong and sharp 
 as they were obtained in the paper specified. 
 
 FIG. 20. Diagram showing displace- 
 ment of mirror N. 
 
 FIG. 21. Diagram showing displacement of 
 grating G. 
 
 37. The three principal adjustments for interference. To compute 
 the extreme adjustments of the grating when the mirror N is moved, 
 fig. 20 may be consulted. Let y e be the air-path on the glass side, whereas 
 y & is the air-path on the other, e the thickness of the grating and n its index 
 of refraction for a given color. Then for the simplest case of interferences, 
 in the first position N of the mirror, if / is the normal angle of incidence and 
 ft the normal angle of refraction for a given color, 
 
 for equal paths. Similarly in the second position of the mirror 
 
 y*=yt+en/cos R 
 Hence if the displacement at the mirror N be N, 
 
 N =y*-y K = 2 en/ cos R+y K '- y 
 The figure shows 
 
 y* y K ' = 2e tan R sin / 
 whence 
 
 * Phil. Mag., I.e.
 
 54 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 The measurement of this quantity showed about 1.88 cm. The computed 
 value would be 2 X .68 X i .53 X .7 1 = i .84 cm. The difference is due to the 
 wedge-shaped glass which requires a readjustment of the grating for the 
 two positions. 
 
 The corresponding extreme adjustments, when the grating gg instead of 
 a mirror N is moved over a distance z, are given in fig. 2 1 . The mirrors 
 M and N are stationary. It is first necessary to compute y, the angle 
 between the displacement z and the side 5 in the figure, or 
 
 tan y = tan I 2e tan R/z 
 so that the semi-air-path difference of rays to the mirror M becomes 
 
 s = y*y K ' = z(cos 7 sin 7 tan 7) 
 
 We may now write, with the same notation as before, for the extreme posi- 
 tion of the grating, G and G', respectively, 
 
 R 
 
 2/cos / = 2?/i/cos R z(cos / sin 7 tan 7) 
 whence 
 
 2(1 /cos 7-f cos 7 sin 7 tan 7) = ze^/cos R 
 
 If e = o, it follows that z = o, or 7 = 7, since the parenthesis by the above 
 equation is not zero. There is only one position. The angle 7 may now be 
 eliminated by inserting tan y, whence 
 
 The 
 
 _ .. .-- 
 cos / cos R 
 
 The observed value of z for the two positions was about 1.3 cm. 
 computed value for 7 = 45 is the same. 
 
 On the other hand, when reflection takes place from the same face of the 
 grating, while the latter is displaced z cm. parallel to itself, the relations 
 of y and z for normal incidence at an angle 7 are obviously 
 
 y cos 7 = z 
 
 FIG. 22. Diagram showing displace- FIG. 23. Diagram showing displacement of 
 ment of grating for deviated rays. mirror N for different colors.
 
 IN RELATION TO INTERFEROMETRY. 55 
 
 For an oblique incidence t, where i I=a, a small angle, the equation 
 is more complicated. Fig. 22, for e = o, g and g' being the planes of the 
 grating, shows that in this case 
 
 sin a . ,. . 7 
 y = z(i+2 ---- -. sin /)/cos / 
 cos * 
 
 or, approximately for small a 
 
 y = z(i+2 sin a tan 7)/cos 7 
 
 This equation is also true for a grating of thickness e, whose faces are plane 
 parallel, for the direction of the air-rays in this case remains unchanged. 
 Finally, a distinction is necessary between the path difference 2y and the 
 motion of the opaque mirror 2N, which is equivalent to it, since the light 
 is not monochromatic. This motion 2 A 7 is oblique to the grating and if 
 the rays differ in color further consideration is needed. In fig. 23 for the 
 simplest type of interferences, in which the glass-path difference is en/cos R 
 and for normal incidence at an angle I, let the upper ray y u and the grating 
 be fixed. The mirror moving over the distance JN changes the zero of path 
 difference from any color of index of refraction H D to another of index ME, 
 while ;y ND passes to ;y NE . Then 
 
 y*l = ;VND + ,D COS # D = 
 
 Hence 
 
 J.V = -esin 7 (tan # t) -tan 
 
 COS 
 
 or 
 
 JN =e(fJL D COS RD fJL E COS .R E ) 
 
 This difference belongs to all rays of the same color difference, or for two 
 interpenetrating pencils. 
 
 If reflection takes place from the lower face the rays are somewhat dif- 
 ferent, but the result is the same. If the plate of the grating is a wedge of 
 small angle^, the normal rays will leave it on one side at an angle 7+5 
 where 5 is the deviation 
 
 The mirror N will also be inclined at an angle 7+5, to return these rays 
 normally. We may disregard d8 = <pdn, if p is small. 
 
 PART II. DIRECT CASES OF INTERFERENCE. DIFFRACTION ANTECEDENT. 
 38. Diffraction before reflection. If in fig. 18 the diffracted beams 
 (spectra) be returned by the mirrors M' and N' to be reflected at the grating, 
 interference must also be producible along GD. Again there will be three 
 primary cases: if reflection takes place from both faces of the grating at 
 once, the air-paths must be nearly equal, the grating itself acting as a 
 compensator. The interference pattern is ring-shaped, but as usual very 
 eccentric. If the reflection of both component beams takes place from the
 
 56 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 same face of the grating, the interference pattern is elliptical and centered 
 and the air-paths are unequal. The case is then similar to the preceding 
 i, 2; but there is no direct or normal image of the slit, as M and N are 
 absent. In its place there may be chromatic images of the slit (linear 
 spectra) at C due to the double diffraction of each component beam in a 
 positive and negative direction successively. But these chromatic images 
 are nevertheless sharp enough to complete the adjustments for interference 
 by placing the slit images in coincidence. It is not usually necessary to put 
 the spectrum lines into coincidence separately (both horizontally and ver- 
 tically), as was originally done, both spectra being observed. Again along 
 GE, fig. 1 8, approximately, there must be two successive positive diffrac- 
 tions of each component beam, which would correspond closely to the 
 second order of diffraction. The advantage of this adjustment lies in the 
 fact that there are but two slit images effectively returned by M' and N' t 
 and hence these interferences were at first believed to be stronger and more 
 isolated. As a consequence I used this method in most of my early experi- 
 ments, before finding the equally good adjustment described in the preced- 
 ing section. 
 
 FIG. 24. Diagrams showing interferences of diffracted rays, case H normal to 
 mirrors M and N; case oblique to mirrors. Double incidence. 
 
 39. Elementary theory. To find the path differences figs. 24 and 25 
 may be consulted. In both the grating face is shown at gg, the glass 
 plate being c cm. in thickness below it, and n is the normal to the grating. 
 M and N are the two opaque mirrors, each at an angle 9 to the face of the 
 grating. Light is incident on the right at an angle * nearly 45. In both 
 figures the rays y m and > (air-paths) diffracted at an angle S in air, are 
 reflected normally from the mirrors M and N respectively and issue toward 
 p for interference. The rays y n (primed in figure) pass through glass. 
 Both figures also contain two component rays diffracted at an angle 6 in 
 air, where 6 8 = a, and reflected obliquely at the mirrors M and N, thus 
 inclosing an angle 20. in air and 2/3, in glass, and issuing toward q. These
 
 IN RELATION TO INTERFEROMETRY. 
 
 57 
 
 component rays are drawn in full and dotted, respectively. In fig. 24 there 
 are two incident rays for a single emergent ray, in fig. 25 a single incident 
 ray for two emergent rays interfering in the telescope. The treatment of 
 the two cases is different in detail; but as the results must be the same, 
 they corroborate each other. 
 
 A 
 
 FIG. 25. Diagrams showing interferences of diffracted rays, case 9 normal to 
 mirrors M and N; case 6 oblique to mirrors. Single incidence. 
 
 The notation used is as follows: Let e be the normal thickness of the 
 grating, e' the effective thickness of the compensator when used. Let dis- 
 tance measured normal to the mirror be termed y,y n and y m being the com- 
 ponent air-paths passing on the glass and on the air side of the grating gg, 
 so that y m >y n . Let y=y m y* be the air-path difference. Similarly let 
 distances z be measured normal to the grating, so that z may refer to dis- 
 placements of the grating. 
 
 Let i be the angle of incidence, r the angle of refraction, and /u r the index 
 of refraction for the given color, whence sin i = H T sin r. Similarly if 8 is the 
 angle of diffraction in air of the y rays and S\ the corresponding angle in 
 glass, 
 
 sin # = ( esin S l (i) 
 
 and if 6 is the angle of diffraction of any oblique ray in air and 8\ the cor- 
 responding angle in glass 
 
 sin = sin (0+a) = /* sin 0, (2) 
 
 I have here supposed that d>6, so that a = 09 is the deviation of the 
 oblique ray from the normal ray in air. Finally, the second reflection of 
 the oblique ray necessarily introduces the angle of refraction within the 
 glass such that 
 
 sin (0- a)- /* sin ft (3) 
 
 If D is the grating space we have, moreover, 
 
 sin i sin = X jD and sin i sin = J e /Z> (4)
 
 58 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 For a displacement, z, of the grating, or an equivalent displacement, y, 
 of either mirror, the path difference produced in case of the oblique ray 
 will be 
 
 2^/cos a = 22 (cos (i 0)+sin (i 9)) /cos a cos i (5) 
 
 For normal rays cos a = i . If as in paragraph i the reflection is ante- 
 cedent 
 
 i = 2}/cos a = 22/cos a cos i 
 
 40. Equations for the present case. From a solution of the triangles, 
 preferably in the case for single incidence, as in fig. 25, the air- path of the 
 upper oblique component ray at a deviation a is 
 
 2y m cos 0/cos (9 a) 
 The air-path of the lower component ray is 
 
 2 cos (y n e sin# (tan 6\ tan 6>i))/cos (Q a) 
 
 The optic path of this (lower) ray in glass as far as the final wave front in 
 glass at /3i is 
 
 jUf*(l/COSd t + I/COS /3J 
 
 The optic path of the upper ray as far as the same wave front in glass is 
 
 /is sin ft 1 2 - - . (y,,, y n e sin 9 (tan 0, tan 0,) ) 
 
 e(tan0 t -tanft)[ (6) 
 
 Hence the path difference between the lower and the upper ray as far as the 
 final wave front is, on collecting similar terms, 
 
 cos sin a sin (0-a) 
 lS^ + ^cos (*--) 
 
 + e sin (0 - a) (tan t - tan t ) - , 2 . (tan O l - tan 0, ) 
 
 cos (ff a) 
 
 (sin cos 6 +sin a sin 9 sin (0 a)) (7) 
 
 Before reducing further, one may remark that if incident and emergent 
 rays were coincident, the quantity given under (6) would be zero. Hence 
 if this particular y n y m = y\, it follows that 
 
 2 y, = 2 e sin #(tan 0, - tan 0,) - ~ ^ (tan t - tan ft) 
 
 Hence the particular path difference for this case would be 
 
 /cos 0, + i /cos ft) -e cos 0(tan 0,-tan ft) /cos a 
 
 which becomes indeterminate when a = o. 
 
 To return to path differences, equation (7), the coefficients of y m y* = y 
 (where y is positive) and of e may now be brought together, as far as
 
 IN RELATION TO INTERFEROMETRY. 
 
 59 
 
 possible. If the path difference corresponds to n wave-lengths X, we may 
 write after final reduction 
 
 nXe= 2y cos a+ 20/xe cos a(i/cos 6 1 cos (O l X ) /cos t ) 
 
 + efie( i + cos (0i - ft) )/cos ft (8) 
 
 which is the full equation in question for a dark fringe. It is unfortunately 
 very cumbersome and for this reason fails to answer many questions per- 
 spicuously. It will be used below in another form. 
 
 The equation refers primarily to the horizontal axes of the ellipses only, 
 as e increases vertically above and below this line. The equation may be 
 abbreviated 
 
 nXo = zy cos a + eZ + eZ 6 
 
 where Z and Z & are the coefficients of e in the equation. In case of a par- 
 allel compensator of thickness e' we may therefore write, y being the path 
 difference in air, 
 
 nX 6 = 2y cos'a + (e e') (Z e Z e ) (9) 
 
 If e = e', i.e., for an infinitely thin plate e = o, 
 
 (10) 
 
 (n) 
 
 (12) 
 
 Again for normal rays y m and y n , a = o and 0i = 0i = ft. Hence 
 
 and for a parallel compensator of thickness e' 
 
 If /x e = i =n e , equation (8) reduces to 
 
 nX = 2 cos a(;y -f e/cos #) , 
 clearly identical with the case e = o for a different y. 
 
 TABLE n. -Table for path difference , a(y m ya)cosa+eZQ + eZg = - 
 Light crown glass. E rays normal to mirrors. Grating space, D, = 
 
 aycoa a+eZ. 
 .000 3512 cm. 
 
 7 = 45. 0-33 Si'- 
 
 
 
 
 
 
 Spectrum lines = 
 
 B 
 
 D 
 
 E 
 
 F 
 
 G 
 
 AX 10' = 
 
 68.7 
 
 58.93 
 
 52.7 
 
 48.61 
 
 43 .08 cm. 
 
 9 = 
 
 i .5118 
 30 46' 
 
 32 38' 
 
 i .5186 
 33 Si' 
 
 1 .5214 
 34 40' 
 
 1.5267 
 
 35 46' 
 
 0, = 
 
 19 47' 
 
 20 51' 
 
 21 31' 
 
 21 57' 
 
 22 30' 
 
 a = 
 
 - 3 5' 
 
 - i 13' 
 
 o o' 
 
 o 49' 
 
 1 55' 
 
 o\~&i = 
 
 - i44' 
 
 - o 40' 
 
 o o' 
 
 o 26' 
 
 o 59' 
 
 PI 
 
 23 25' 
 
 22 17' 
 
 21 30' 
 
 21 I 3 ' 
 
 20 49' 
 
 0i & = 
 
 - 3 38' 
 
 - I 26' 
 
 o o' 
 
 o 44' 
 
 i 41' 
 
 2 cos a = 
 
 1.9970 
 
 1.9996 
 
 2 .OOOO 
 
 1.9998 
 
 i . 9996 cm. 
 
 Ze = 
 
 -0383 
 
 .0176 
 
 .OOOO 
 
 - .0095 - 
 
 .0219 cm. 
 
 Z& = 
 
 3.2100 
 
 3 2404 
 
 3 2648 
 
 3 -2805 
 
 3 .3038 cm. 
 
 Path difference * n/ = \ 
 
 -1.9970) 
 
 i .99963 
 
 >-2. 0000) 
 
 -i.ggg&y- 
 
 1.9996)- ) cm 
 
 ( 
 
 + 3 24830 
 
 + 3 .2580* 
 
 + 3 26486 
 
 + 3.2710*? + 
 
 3 . 2819? J 
 
 Path difference) 
 "=3 . 2648 cm. > 2Ay = .0116 
 
 .0062 .0000+ .0059 + .0177 cm. 
 
 The same for ) 
 <?=.68cm. / 
 
 = 0079 .0042 
 = .0107 - .0056 
 
 .0040 + 
 .0049+ 
 
 .oi2ocm. 
 .oi56cm 
 
 * The purpose of these data is merely to elucidate the equations. AAT refers to 
 the displacement of either opaque mirror, M or N.
 
 60 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 In view of the complicated nature of equation (8) , I have computed path 
 differences for a typical case of light crown glass,* as shown in table 1 1, for 
 the Fraunhofer lines B, D, E, F, G, supposing the E ray to be normal (a = o). 
 The table is particularly drawn up to indicate the relative value of the 
 functions Z e and Z e . It will be seen that Z = Z e -fZ fl in terms of X, is a 
 curve of regular decrease having no tendency to assume maximum or 
 minimum values within the range of X, whereas y cos a passes through a flat 
 maximum for a = o. This is shown in the curves of fig. 26. 
 
 3*60 
 
 50 
 
 52 
 
 48 
 
 
 
 - -X X 10 
 
 46 48 50 5 545658606264 
 
 76 if 
 
 FIG. 26. Chart showing path difference in centimeters (to be annulled 
 at E line), in terms of wave-length. 
 
 The path difference, which is the difference of the ordinates of these curves, 
 thus passes through zero for a definite value of e and y, and this would 
 at first sight seem to correspond to the center of ellipses. That it does not 
 so correspond will be particularly brought out in the next section, but here 
 this tentative surmise may be temporarily admitted to simplify the descrip- 
 tions. It is obvious that for a given e the value of y (air-path difference) 
 which makes the total path difference zero, varies with the wave-length, 
 hence on increasing y continuously the ellipses must pass through the 
 spectrum. It is also obvious that, if the grating is reversed, the path 
 difference will change sign, caet. par., and the ellipses will move in a con- 
 trary direction, for the same displacements y of the micrometer screw, 
 at the mirror M or N, respectively. 
 
 If e= i cm. and ^ = 3.2648 cm., the path difference will be zero for the E 
 ray and (barring further investigation as just stated) one is inclined to place 
 the center of ellipses there. The table shows the residual path differences 
 in the red and blue parts of the spectrum. Fig. 27 has been constructed for 
 
 * Taken from Kohlrausch's Practical Physics, nth edition, 1910, p. 712.
 
 IN RELATION TO INTERFEROMETRY. 61 
 
 the same condition. It follows, therefore, that in proportion as e is smaller 
 y will be smaller for extinction at the E line. Hence their differences will 
 be smaller and the number of wave-lengths corresponding to the path 
 difference of the Fraimhofer lines will be smaller. In other words, the major 
 axes of the ellipses will be larger. Thus the ellipses will be larger as e is 
 smaller, the limit of enormous ellipses being reached for e = o or e = e' of the 
 compensator. 
 
 This result may be obtained to better advantage by reconstructing equa- 
 tion (8), and making the path difference equal to zero for the normal ray. 
 This determines 
 
 sB (13) 
 
 in terms of e, and the path difference now becomes 
 
 nX 9 e{ /e cos a cos (0^ &i) +/ie(i + cos (6 l &))} /cos t (14) 
 
 This is zero for a = o, Q\ =/3i = Si at the E line. The other values are given 
 in the table in centimeters and in wave-lengths. Path difference diminishes 
 as e; the horizontal axes of the ellipses increase with e. If observations 
 are made very near the E line (normal ray) the following approximate form 
 may be used : 
 
 but this rapidly becomes insufficient unless 0i S\ and 0i /8i are very small. 
 
 4 1 . Interferometer. If in equation (8), y alone is variable with the order 
 of fringe n while a, 6, S, Q\, Q\, /Si, e, X, /*, are all constant (i.e., if the number 
 of fringes n cross a given fixed spectrum line like the D line, when the mirror 
 is displaced over a distance y) it appears that 
 
 dn ~ 2 cos a 
 
 where a refers to the deviation of X fl from X e normal to the mirror. If 
 a = o, dy/dn o, the limiting sensitiveness of the apparatus, which appears 
 for the case of normal rays. 
 
 If the grating is displaced along z 
 
 dz _ Xe cos i 
 
 dn 2 cos a cos (i 8)+ sin (i 9} 
 
 where cos a = i for normal rays. In both cases the displacement per fringe 
 dy/dn and dz/dn vary with the wave-length. Hence if the ellipses are 
 nearly symmetrical on both sides of the center, i.e., if the red and violet 
 sides of the periphery are nearly the same, or the fringes nearly equidistant, 
 the smaller wave-length will move faster than the larger for a given dis- 
 placement of mirror. In other words, the violet side will pass over a whole 
 fringe before the red and consequently the ellipses, as a whole, must drift in
 
 62 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 the direction in which violet moves. It is at first quite puzzling to observe 
 the motion of ellipses as a whole in a direction opposed to the motion of 
 the more reddish fringes when these are alone in the field. 
 
 42. Discrepancy of the table. The data of table i are computed sup- 
 posing that the path difference zero corresponds to the center of ellipses. 
 This assumption has been admitted for discussion only and the inferences 
 drawn are qualitatively correct. Quantitatively, however, the displace- 
 ment of about JAf=.oc>3 cm. should move the center of ellipses from the 
 D to the E line of the spectrum, whereas observations show that a displace- 
 ment of either opaque mirror of about JA r =.oi cm. is necessary for this 
 purpose; i.e., over three times as much displacement as has been computed. 
 The equation can not be incorrect ; hence the assumption that the center of 
 ellipses corresponds to the path difference zero is not vouched for and must 
 be particularly examined. This may be done to greater advantage in con- 
 nection with the next section, where the conditions are throughout simpler, 
 but the data are of the same order of value. 
 
 FIG. 27. Diagram showing interferences of reflected rays, 
 subsequently diffracted. Case /, normal to mirrors; 
 case i = 7 + a oblique to mirrors. Rays R refracted, D 
 diffracted. 
 
 PART III. DIRECT CASE; REFLECTION ANTECEDENT. 
 
 43. Equations for this case. If reflection at the opaque mirrors takes 
 place before diffraction at the grating, the form of the equations and their 
 mode of derivation are similar to the case of paragraph 40, but the variables 
 contained are essentially different. In this case the deviation from the
 
 IN RELATION TO INTERFEROMETRY. 63 
 
 normal ray is due not to diffraction but to the angle of incidence, and the 
 equations are derived for homogeneous light of wave-length X and index of 
 refraction /* 
 
 In fig. 27 let y^y' and y m = y be the air-paths of the component rays, 
 the former first passing through the glass plate of thickness e. Let the 
 angle of incidence of the ray be /, so that y m and y n are returned normally 
 from the mirrors M and N, respectively, these being also at an angle / to 
 the plane of the grating. Let i be the angle of incidence of an oblique ray, 
 whose deviation from the normal is * 7 = a. Let R, r, and ft be angles of 
 refraction such that 
 
 sin = sin (J+a) =/x sin r sin/ = /zsin.R sin (/ a)=/*sin ft (18) 
 
 The face of the grating is here supposed to be away from the incident ray, 
 as shown at gg in the figure. 
 
 TABLE 12. Path difference, 2(ym yn)cosa + eZ 1 eZ i + eZ 3 . Light crown glass. 
 a = 3 throughout. Grating space .000351 cm. 7 = 45. e = icm. 
 
 Spectrum 
 
 lines 
 
 B 
 
 D 
 
 E 
 
 F 
 
 G 
 
 / ) 
 
 < 10' = 
 
 68.7 
 
 58-93 
 
 52.70 
 
 48.61 
 
 43 . 08 cm. 
 
 
 i: 
 
 i .5118 
 27 53' 
 
 2 7 49' 
 
 1.5186 
 27 45' 
 
 i .5214 
 27 42' 
 
 1.5267 
 27 35' 
 
 
 r = 
 
 29 26 
 
 29 22' 
 
 29 18' 
 
 29 14' 
 
 29 8' 
 
 
 A = 
 
 26 16' 
 
 26 12' 
 
 26 8' 
 
 26 5' 
 
 26 o' 
 
 
 a = 
 
 3 o' 
 
 T. O' 
 
 3 o' 
 
 } o' 
 
 3 o' 
 
 r 
 
 -/? = 
 
 i33' 
 
 i33' 
 
 i33' 
 
 i33' 
 
 i33' 
 
 r 
 
 ~~A = 
 
 3 10' 
 
 3 10' 
 
 3 10' 
 
 3 9' 
 
 3 8' 
 
 
 z = 
 
 3 .4160 
 
 3.4218 
 
 3-4272 
 
 3-43i8 
 
 3 .4403 cm. 
 
 
 Z 2 = 
 
 656 
 
 
 767 
 
 808 
 
 896 cm. 
 
 
 Z 2 = 
 
 689 
 
 647 
 
 799 
 
 841 
 
 928 cm. 
 
 Path difference, 
 a = 3, e = i cm., 
 y = i cm. 
 
 H 
 
 -1.9992 
 + 3-4193 
 
 -1.9992 
 
 + 3-4252 
 
 -1.9992 
 + 3-4304 
 
 -1.9992 
 + 3-4352 
 
 -1.9992 
 + 3-4436 
 
 If path differ- "I 
 
 
 
 
 
 
 
 ence is annulled ! 
 at line, = 3, [ 
 
 2Ay - 
 
 .0111 
 
 - .0052 
 
 .0 
 
 + .0048 
 
 + .0132 
 
 e = i cm. 
 
 
 
 
 
 
 
 J= C o S ?i'cm i } 2 * y = '" - 0111 ~ - 0052 ' + - 48 + - 0132 
 
 The data are merely intended to elucidate the equations. 
 
 The values Z are nearly equal. So also the cases for = 3 and a = o. 
 
 Fig. 28 has been added to accentuate the symmetrical conditions when 
 a compensator parallel to the grating and of the same thickness is employed. 
 
 Then it follows as in paragraph 40, mutatis mutandis, that the path dif- 
 ference is (if y = y m - y ) 
 
 f 2 cos a 2 cos a cos (r R) i + cos (r 8.) \ 
 n\= -zycosa + fie < - D - -+- 
 
 t cos R cosr cos r J (19) 
 
 = 2y cos a + 0(Z, Z 2 + Z 3 )
 
 64 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 where n is the order of interference (whole number). This equation is 
 intrinsically simpler than equation (8) since /i r = /u, as already stated, and 
 since a is constant for all colors, or all values of /x and A, in question. R and 
 r replace S\ and B\. 
 
 In most respects the discussion of equation (19) is similar to equation (8) 
 and may be omitted in favor of the special interpretations presently to be 
 given. If M= i, equation (19) like equation (8) reduces to the case corre- 
 sponding to e = o, with a different y normal to the grating. 
 
 All the colors are superposed in the direct image of the slit, R n and R in 
 fig. 27, seen in the telescope, and the slit is therefore white. This shows 
 
 FIG. 28. Diagram showing symmetrical interference for compen- 
 sated grating, gg. Rays oblique to mirror. 
 
 also that prismatic deviation due to the plate of the grating (wedge) is inap- 
 preciable. The colors appear, however, when the light of the slit is analyzed 
 by the grating in the successive diffraction spectra, D n or D, respectively. 
 In equation (19) n is a function of X and hence of the deviation 6 produced 
 by the grating, since sin (7 a) sin 6=\/D. 
 
 The values of equation (19) for successive Fraunhofer lines and for 
 = 3, have been computed in table 1 2 for the same glass treated in table 1 1 , 
 the data being similar and in fact of about the same order of value. The 
 feature of this table is the occurrence of nearly constant values o.=iI,
 
 IN RELATION TO INTERFEROMETRY. 
 
 65 
 
 r R and r ft, throughout the visible spectrum. Hence if the following 
 abbreviations be used 
 
 A = cos a = .9986 B = cos (r R) = .9996 C = i +cos (r /?,) = i .9984 
 
 A, #, C are practically functions of a only and do not vary with color or X. 
 Furthermore, if the path difference is annulled, at the E line for instance, 
 for the normal ray, since r = ft = R, a = o, so that R = R E is the normal angle 
 of refraction, 
 
 o = y 4- 2ne/cos R E 
 
 and equation (19) reduces to 
 
 n\ = ieA (///cos R fj. cos R E ) + e(C 2AB) l u/cos r (20) 
 
 where zeA and e(C 2AB) are nearly independent of X and the last quan- 
 tity is relatively small. The two terms of this equation for e= i cm. show 
 about the following variation: 
 
 TABLE 13. 
 
 B 
 
 D 
 
 E F 
 
 G 
 
 .0111 
 
 .000009 
 
 .0052 
 . 000004 
 
 .0 +.0048 
 .0 + .000005 
 
 .0132 
 
 .000013 
 
 Hence for deviations even larger than a = 3, the path difference does not 
 differ practically from the path difference for the normal ray. Thus it 
 follows that the equation 
 
 nX = 2y -f 2///cos R (21) 
 
 is a sufficient approximation for such purposes as are here in view. 
 
 Finally, for e = .68 cm. (the actual thickness of the plate of the grating) , 
 y and N, the semi-path difference and the displacement of the opaque 
 
 mirror, will be 
 
 TABLE 14. 
 
 
 B 
 
 D 
 
 E F G 
 
 y = 
 Ay = 
 
 Wo = 
 AJVo = 
 
 1.1630 
 - .0038 
 + .0014 
 - .0052 
 
 i . 1650 
 .0018 
 + .0007 
 - .0025 
 
 I. 1668 I. 1686 LI730 
 
 .0 + .0016 + .0045 
 .0 - .0005 - .0018 
 .0 + .0021 + .0063 
 
 where 5A/o is the color correction of dy and JN = dy dN determines 
 the corresponding displacements of the opaque mirror; or more briefly 
 
 N =y e/j. tan R sin R = zen cos R 
 The data actually found were 
 
 C DBF 
 
 JN = .0148 .0093 .o +.0057 
 
 (22)
 
 66 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 These results are again from 2 to 4 times larger than the computed values. 
 True the glass on which the grating was cut is not identical with the light 
 crown glass of the tables; but nevertheless a discrepancy so large and irreg- 
 ular is out of the question. It is necessary to conclude therefore that here, 
 as in paragraph 42, the assumption of a total path difference zero for the 
 center of ellipses is not true. In other words the equality of air-path differ- 
 ence and glass-path does not correspond to the centers in question. It is 
 now in place to examine this result in detail. 
 
 44. Divergence per fringe. The approximate sufficiency of the equation 
 n\=2fM/cos R 2y = zfjL cos R 2N (23) 
 
 makes it easy to obtain certain important derivatives among which dd/dn, 
 where 6 is the angle of diffraction corresponding to the angle of incidence 
 / and M the order of interference, is prominent. 
 
 If e and y are constant and if n, \, M and R are variable, the differential 
 coefficients may be reduced successively by the following fundamental 
 equation, D being the grating space, M and r corresponding to wave-length 
 X and angle of diffraction 6. 
 
 dR=-tanRd/n (24) 
 
 d\=-D cos B dd (25) 
 
 (26) 
 
 where, as a first approximation, a = .015, is an experimental correction, 
 interpolated for the given glass. Incorporating these equations it is found 
 that 
 
 d0 = __ __ cosK 
 
 dn 2Dcos0 en-ycosR+aefi(i-ta.n 2 R) * 7 ' 
 
 
 If the path difference n\ is annulled 
 
 dd Q = ___ cos/? 
 
 dn 2 D cos ae//(i-tan 2 /?) 
 
 which is the deviation per fringe, supposedly referred to the center of 
 ellipses. 
 
 These equations indicate the nature of the dependence of the horizontal 
 axes of ellipses on i/D (hence also on the order of the spectrum), i/e, i/n, 
 and i/a, where the meaning of a, here apparently an important variable, 
 is given in equation (26). If instead of the path difference y the displace- 
 ment N of either opaque mirror is primarily considered (necessarily the 
 case in practice), the factor (i tan 2 R) vanishes. 
 
 Table 15 contains a survey of data for equations (27) and (28). The 
 results for dB /dn would be plausible, as to order of values. The data for 
 d$/dn, however, are again necessarily in error, as already instanced above, 
 paragraph 42. They do not show the maximum at e and the X effect is 
 overwhelmingly large.
 
 IN RELATION TO INTERFEROMETRY. 
 
 67 
 
 TABLE ic Values of dO/dn and displacements N. 
 
 refer to centers of ellipses. 
 
 = .68 cm. N c and y c , etc. 
 
 Spectrum line 
 
 B 
 
 D 
 
 E 
 
 F 
 
 G 
 
 Equation 28. .. 
 
 de o /dn = 
 
 i' 27* 
 
 i' 5" 
 
 53". 
 
 46" 
 
 en* 
 
 $ 
 
 27... 
 
 dd/dn = 
 
 3' 4 
 
 1 5 2 
 
 i 20 
 
 59 
 
 6 
 
 
 dO/dn = 
 
 6' 53* 
 
 -10' 38" 
 
 oo 
 
 + 8<47" 
 
 + 3' Si" 
 
 
 ' j 
 dfi 
 
 dX 
 
 281 
 
 445 
 
 622 
 
 793 
 
 1140 
 
 
 d9/dn = 
 
 i' 54" 
 
 i' So" 
 
 QO 
 
 + 2' 21* 
 
 + 40* 
 
 
 
 - i' 44" 
 
 -i' 44" 
 
 oo 
 
 + 2' n* 
 
 + 36* 
 
 . . 
 
 35--- 
 
 '.'.'.'.'.'.'.'.'... ^y c = 
 
 I.I779 
 . 0142 
 
 i .1811 
 .0110 
 
 i . 1921 
 
 .0 
 
 i . 1981 
 -f .0060 
 
 i .2090 
 + .0169 
 
 
 N = 
 
 .9235 
 
 .9274 
 
 939 1 
 
 9456 
 
 9578 
 
 
 AN C 
 
 - .0156 
 
 .0117 
 
 .0 
 
 -f .0065 
 
 + .0187 
 
 34*.- 
 
 (observed) A N c = 
 dd/dn = 
 
 - -0153 
 
 .0098 
 
 - i' 54" 
 
 .0 
 
 oc 
 
 + .0057 
 
 2' 2l" 
 
 '48* 
 
 Since equation (27) is clearly inapplicable, giving neither maxima nor 
 counting the fringes, it follows that in this equation y>en/co& R; i.e., the 
 centers of ellipses are not in correspondence with the path difference zero. 
 In other words the air-path difference is larger than the glass-path difference 
 
 FIG. 29. Chart showing dispersion per fringe in terms of wave-length. 
 
 "Interpolated between D and G by fi = a +bi +cA 3 where b= - .00273, c = -0000197.
 
 68 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 in such a way that de/dn is equal to c for centers (here at the E line), but 
 falls off rapidly toward both sides of the spectrum. 
 
 There is another feature of importance which must now be accentuated. 
 In case of different colors, and stationary mirrors and grating, y is not 
 constant from color to color, whereas 
 
 2N = 2y2e/ji sin/? tanR 
 
 is constant for all colors, as has been shown above. Thus the equation to 
 be differentiated for constancy of adjustment but variable color loses the 
 variable y and becomes 
 
 n\ = 2n cos R zN (29) 
 
 N here is the difference of perpendicular distances to the mirrors, M and 
 N, from the ends of the normal in the glass plate, at the point of incidence 
 of the white ray. Performing the operations 
 
 dl X 2 cos R 
 
 dn~ 2(ycosR-efi(i +a)) 
 d0 tf cosR 
 
 
 dn 2DcosO e{i(i +a) y cos R 
 Hence maximum dd/dn = <*> at the centers of ellipses which occur for 
 
 y= c ' os ^ (3 2 ) 
 
 The value of dd/dn for the different Fraunhofer lines above, if a = .015 is 
 still considered constant and e= .68, is given in table 15. 
 
 These results show that the distance apart of fringes on the two sides of 
 the center of ellipses is not very different, though they are somewhat closer 
 together in the blue than in the red end of the spectrum, as observed. 
 There is thus an approximate symmetry of ovals, and dd/dn falls off very 
 fast on both sides of the infinite value at the center. This is shown in fig. 29. 
 
 The observed angle between the Fraunhofer lines D and E for the given 
 grating was 4,380". The number of fringes between D and E would thus 
 be even less than 4,38o"/638" = 6.7 only, which is itself about 4 times too 
 small. The cause of this is then finally to be ascribed to the assumed con- 
 stancy ofa = (dn/n)/(d\/\), and a discrepancy is still to be remedied. 
 We may note that a does not now enter as directly as appeared in 
 equation (27). 
 
 By replacing a by its equivalent, equation (31) takes the form 
 
 dd X 2 i 
 
 dn = 2 Dcos6 I*I7~~^yl"" (33) 
 
 and a definite series of values may be obtained by computing dn/d\; but 
 as all experimental reference here is, practically, not to path differences
 
 IN RELATION TO INTERFEROMETRY. 69 
 
 but to displacements of the movable opaque mirror N, the form of the 
 equation applicable is 
 
 dd _ _ P i 
 
 dn~2Dcosd ' i ^- ~ (34) 
 
 (A du\ 
 ucosR .5 JT) A 7 
 cos R dX I 
 
 e 
 
 To make the final reduction, I have supposed that for the present pur- 
 poses a quadratic interpolation of n, between the B and the G lines of the 
 spectrum, would suffice. Taking the E line as fiducial, I have therefore 
 assumed an equation for short ranges, corresponding to Cauchy's in simpli- 
 fied form 
 
 in preference to the more complicated dispersion equations. From the 
 above data for light crown glass we may then put, roughly, 
 
 6 =.456X10-* dfjL/dl=- 2 b/l 5 
 
 Thus I found the remaining data of table 15. The results for dQ/dn agree 
 as well with observations as may be expected. The ovals resemble ellipses, 
 but are somewhat coarser on the red side, as is the case. 
 
 TABLE 16. Values of dX/dy, dd/dy, etc. e = .68 cm. 
 
 
 
 
 
 B 
 
 D 
 
 E 
 
 F 
 
 G 
 
 
 Equation 
 
 36 
 38 
 
 d^/dy 
 dXJdN 
 
 ._ 
 
 .00372 
 269 
 
 .00318 
 229 
 
 .00283 
 205 
 
 .00260 
 189 
 
 .00229 
 167 
 
 cm. 
 cm. 
 
 
 39 
 
 dX/dN 
 
 = 
 
 - .017 
 
 - .031 
 
 QO 
 
 + .030 
 
 + .015 
 
 cm. 
 
 
 42* 
 
 dX/dN 
 
 ^r 
 
 
 - -0055 
 
 OO 
 
 + .0081 
 
 + o3 i 
 
 cm. 
 
 
 42 
 
 dX/dN 
 
 ma 
 
 - .0044 
 
 .0050 
 
 00 
 
 + .0075 
 
 + .0023 
 
 cm. 
 
 
 42 
 
 dd/dN 
 
 = 
 
 -14-6 
 
 - 17.0 
 
 QO 
 
 + 26.0 
 
 + 8.1 radians 
 
 The centers of ellipses are thus defined by the semi-air-path equation 
 
 ? cos/ ' (35) 
 
 or the corresponding equation in terms of N c . The trend data for JN C 
 agree fairly well with observation, except at the D line, which difference is 
 very probably referable to the properties of the glass, since the grating was 
 not cut on light crown. 
 
 The number of fringes between the D and E lines now comes out plaus- 
 ibly, being less than 4, 380" '/ 104" = 42. It is difficult to count these fringes 
 without special methods of experiment; but the number computed is a 
 reasonable order of values, about 25 to 30 lines being observed. 
 
 Some estimate may finally be attempted as to the mean displacement 
 of mirror 5N per fringe, between the D and E lines. As their deviation 
 is 6= i 13' and the displacement from D to E, JN e , 
 
 dd JN C )? i JA/* C 
 
 
 _ __ _ 
 
 0/(dd/dn) ~dn 6 2 D cos~0 JW C 
 
 * Constants interpolated between D and G by n = a+bl + cP when 6= -.00273. 
 c = . 0000197.
 
 70 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 if the value of dB/dn for the D line be taken. Thus 
 
 5N>\*/(2D8 cos 0) = X 2 /> sin 26, nearly 
 
 Hence 8N is independent of the thickness, e, of the plate of the grating, as 
 I showed* by using a variety of different thicknesses of compensator. 
 Since X = .000059 cm., D = . 0003 51 cm. 
 
 dN >. 0003 1 cm. 
 
 The values found were between .00033 and .00039, naturally difficult to 
 measure, but of the order required. 
 
 FIG. 
 
 f * I " w ' ^ '~ ' - >r I 
 
 30. Chart showing dispersion (dl/dN), per cm. of displacement of 
 mirror in terms of wave-length. 
 
 * American Journal of Science, xxx, 1910, p. 170.
 
 IN RELATION TO INTERFEROMETRY. 71 
 
 45. Case ofd\/dy, and dO/dy, etc. If in equation (23) e and ware constant 
 while n, R, y, and X vary, the micrometer equivalent of the displacement of 
 fringes may be found. Here 
 
 which coefficients are given by equations (24), (25), and (26). In this way 
 
 ^= ^ cos ^ (**) 
 
 dy efjL(i+a(i-tan 2 R))-ycosR 
 
 and 
 
 de/dy = - (d\/dy)/D cos 6 
 
 If y = efi/cos R, where y is variable, the motion is supposedly referred to the 
 centers of ellipses. Thus 
 
 dd n X cos R 
 
 "_ (77) 
 
 dy D cos 6 aefi(i tan 2 R} 
 
 Values are given in table 4 and hold for the glass of tables 11-15. These 
 results are, as usual, many times too large and they contain no suggestion 
 of opposed motion on the two sides of the center. 
 
 If we consider the displacement of the mirror N instead of the path differ- 
 ence and for a given color write 
 
 N=y e sin I tan R 
 then 
 
 dN efi(i +a) y cos R 
 
 dy ~e t u(i + a(i tan 2 jR)) j> cos /? 
 and hence in 
 
 dX X cos R 
 
 ri\f = (3) 
 
 the factor (i tan 2 R) is removed from the equation. But the data are 
 not much improved. 
 
 The equation n\ = 2e/j. cos R N gives, on differentiation and reduction, 
 
 dX _ X 
 
 'dN~y-ett(i+a)/aa*_R 
 
 To be consistent with the preceding paragraphs, it is therefore necessary 
 to put y /*(i+a)/cos R = o, for centers of ellipses, so that (d\/dN) = (X> 
 If a is constant this supposition leads to results which are throughout out 
 of the question. The values of d\/dN may, however, be found by inserting 
 the data for dn/d\ given in the preceding paragraph. 
 Centers thus correspond to 
 
 eX d,n 
 
 cs (40) 
 
 or 
 
 dX = * 
 
 d N \ - efi cos~R + e\ (dftjdK) /cos R
 
 72 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 so that if d\/dN= , the maximum at the centers of ellipses, the simulta- 
 neous effect at X' will be (as the mirror has not moved) 
 
 
 D , DA / x dft /' dn'\ 
 
 e(ttcosR ft cos R') e[ n ., ... -...-) 
 
 Vcos R dX cos /^ x d/T / 
 
 
 If the center of ellipses is at the E line the values of table 4 hold. The 
 motion on the blue side of the E line is thus larger than the simultaneous 
 motion on the yellow side, conformably with observation. 
 All the results are given graphically in fig. 30. 
 
 46. Interferometry in terms of radial motion. Either by direct obser- 
 vation or combining the equations (34) and (41) for dX/dn and d\/dN the 
 usual equation for radial motion again results 
 
 dn 2 
 
 where N is the displacement of mirror per fringe. This equation is best 
 tested on an ordinary spectrometer, by aid of a thin compensator of micro- 
 scope glass revolvable about its axis and placed parallel to the mirror M. 
 The change of virtual thickness e' for a given small angle of incidence i may 
 then be written. 
 
 , 
 
 nearly. If / = o, dP = (dl) z . Therefore 2de'= dP/^. In a rough trial for 
 e'=. 0226 cm., dl = . 053 radians, ^=1.53, one fringe reappeared. Hence 
 
 /i^' = ^X 1.53 X27 X io~ 6 = 21 X io- 6 cm. 
 which is of the order of half the wave-length used. 
 
 47. Interferometry by displacement. In a similar experiment the dis- 
 placement of ellipses due to the insertion of the above glass, e' = .0226 cm., 
 was from the D line to about the G line. If AN is the displacement of the 
 mirror N, to bring the center of ellipses back to the same line, D or E, we 
 may write n = i -\-JN/e'. I found at the E line /* = i .53 , at the D line n = 
 1.53 ; special precautions would have to be taken to further determine these 
 indices. 
 
 Thus there are two methods for measuring /u, either in terms of the radial 
 motion of the fringes or, second, in terms of the displacement of the fringes 
 as a whole. Moreover, paragraph 44 may be looked upon reciprocally, 
 as a method for measuring dn/d\, directly.
 
 IN RELATION TO INTERFEROMETRY. 
 
 73 
 
 PART IV. INTERFERENCES IN GENERAL, AND SUMMARY. 
 
 48. The individual interferences. In figs. 31, 32, 33, gg is the face of the 
 grating, M and N the opaque mirrors, and I the incident ray. 
 
 As the result of reflection from the top face, the available air-path being 
 >> m and y m ' ', there must be two images of the slit seen in the telescope directly, 
 viz, a and c, fig. 31. Of these c will be more intense than a, which is tinged 
 by the long path in the glass. These two rays together, on diffraction, will 
 produce stationary interferences whose path corresponds to the equation 
 
 n\ = 2e/j. cos R 
 The optical paths of the two rays are 
 
 reflected-refracted, I', 2y m -\-en(2 cos R sec R) 
 
 refracted-reflected, II', 2y m '+T > e(j. sec R = 2y m +en(3 sec R~4 sin R tan R) 
 
 If the plate of the grating were perfectly plane parallel, the slit images 
 a and c would obviously coincide. 
 
 FIG. 31. Diagram of interferences reflected from same face 
 of grating plate. Non-compensated. 
 
 The directly transmitted rays, however, after reflection from N give rise to 
 four images of the slit in case of a slightly wedge-shaped plate, the one at 
 
 a, fig. 3 1, being white, that at c yellowish, the distances apart being the same 
 as in the preceding case. Besides this there are two images of the slit at 
 
 b, figs. 32 and 33, which result from double diffraction at the lower face, the
 
 74 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 case shown corresponding to 8<i first with 9>i thereafter, while the other 
 image corresponds to 6 > i and 6 <i in the two successive diffractions. These 
 slits show strong chromatic aberration but are nevertheless linear and 
 useful. The optical paths for the three lower rays are 
 
 transmitted, I. 2y n +en/cos R 
 transmitted-refracted, II. 2y n +en/cos R+zep cos R 
 transmitted-diffracted, III. 2y n +en/cos R-\-ien cos t 
 
 These three equations determine the three stationary interferences 
 nX = 20/i cos R X = 2en cos Q\ X = ten (cos B^ cos R) 
 
 Jlf 
 
 FIG. 32. Diagram of interferences reflected from both faces 
 of grating gg. Excess of glass path on N side. Self- 
 compensated. 
 
 discussed in the preceding paper*. If the grating is between two identical 
 plates of glass, these stationary fringes may become movable ; for, inasmuch 
 as there is complete symmetry between the rays to be reflected from the 
 mirrors M and N, the stationary fringes become identical for both plates 
 of glass, from either of which they may be reflected, before or after dif- 
 fraction. Hence the motion of either opaque mirror changes the phase and 
 moves the fringes, which are now linear, vertical, and nearly equidistant. 
 
 *C. and M. Bams.
 
 IN RELATION TO INTERFEROMETRY. 
 
 75 
 
 They may also be regarded as confocal ellipses of infinite size, the visible 
 parts of their peripheries lying close together in the field of view. They 
 pass through infinity in this field, when the virtual path difference is zero. 
 
 FIG. 33. Diagram of interferences reflected from both 
 faces of grating plate. Excess of glass path on M 
 side. Self-compensated . 
 
 49. The combined interferences. The five equations given by the 
 first group I' and II' when combined with the second group, I, II, III, lead 
 to six other interferences, all of them of the movable type and useful for 
 interferometry. If for simplicity the path difference is zero, they may be 
 written, if y = y n y m and y'= y n y m ' and y corresponds to the glass-path 
 difference, N as usual referring to the displacement of the opaque mirror: 
 
 IV, Reflection at bottom, equation I-II', y ' = en/c,Qs R, N = en cos R 
 V, Reflection at bottom, equation II-I', y = en/cos R,N = encosR 
 
 VI, Reflection at top, equation III-I', ^ O = ^M(COS 0i+sin R tan R), 
 
 N = en cos 0i 
 
 VII, Reflection at top and bottom, eq. II-II', y ' =e\i sin R tan R, N = o 
 VIII, Reflection at top and bottom, eq. III-II', y ' = en(sec R-cos 00, 
 
 N = en(cos Rcos 0i) 
 IX, Reflection at bottom and top, eq. I-I', y = en sin R tan R,N = o 
 
 Hence the fringes of the interference VII and IX are identical through- 
 out the spectrum, when the mirror N moves. The remaining fringes are 
 elliptic and eccentric, because reflected from two faces of the thin wedge.
 
 76 
 
 THE PRODUCTION OF ELLIPTIC INTERFERENCES 
 
 Nos. VII and IX are parallel lines which pass symmetrically from negative 
 to positive obliquity, or the reverse, respectively, through horizontality, 
 in opposed directions. 
 
 The results of these equations have been computed for light crown glass 
 and the Fraunhofer lines B, D, E, F, G above and in the following table 17. 
 
 TABLE 17. Diffracted-reflected and reflected-diffracted rays. e = .68 cm. D = 
 .000351 cm. 7 = 45, nearly. Light crown glass, y^y^-y^. Change of faces: y- N. 
 
 
 Line of Spectrum B D E 
 
 F 
 
 G 
 
 
 ioA= 68.7 58.93 52.70 
 
 48.61 
 
 43 -08 
 
 
 = 1.5118 1.5153 1.5186 
 R= 27*53' 27*49' 27*45' 
 
 1-5214 
 27 42' 
 
 1-5267 
 27 6 35' 
 
 
 0, = 19 47' 20 51' 21 Jl' 
 
 21 57' 
 
 22 30' 
 
 
 (correction) 2A = .5088 -574 -559 
 
 5050 
 
 -5023 
 
 I 
 
 . Reflection from a single face. y Q =eft cos(R 1 )/cos 
 
 R: 
 
 
 
 nl = o;- 2y = 2.3030 2.3128 2.3199 
 
 2.3252 
 
 2-3334 
 
 
 E*=o; 2AJ = .0169 - .0071 .0 
 
 + -o53 
 
 + -0135 
 
 
 correction 2 <5A r = 4 .0028 + .0014 .0 
 
 .0009 
 
 .0036 
 
 
 2AA r = .0197 - .0085 .0 
 
 4- .0062 
 
 4 .00171 
 
 II 
 
 . Reflection from one face. y =efi/cos R: 
 
 
 
 
 + 2y = 2.3261 2.3301 2.3337 
 
 2-3369 
 
 2.3426 
 
 
 4- 2A.y = .0076 - .0036 .0 
 
 4- . 003 2 
 
 4- .0089 
 
 
 + 2AA = .OIO4 - .0050 .0 
 
 4- .0043 
 
 4- .0126 
 
 III 
 
 . Reflection from both faces. y = en (sec R cos 0j) : 
 
 
 
 
 + 2^0 = .39 J 3 -4043 .4124 
 
 4177 
 
 -4242 
 
 
 + 2AV = .0211 .Oo8l .0 
 
 + -0053 
 
 4 .0118 
 
 
 4-2AA r = - .0239 - .0095 .0 
 
 4- .0062 
 
 + .0154 
 
 IV 
 
 . Reflection from both faces. y t = ep sin s tan R: 
 
 
 
 
 + 2y = .3682 .3870 .3985 
 
 .4061 
 
 4151 
 
 
 4- 2Ay = .0303 - .0115 .0 
 + 2AAo= .0332 .0130 .0 
 
 4- .0075 
 4- .0084 
 
 4 .0166 
 4 .0202 
 
 V 
 
 . Same as II, but reflection from top face : 
 
 
 
 
 - 2y = 2.3261 2.3301 2.3337 
 
 2-3369 
 
 2.3426 
 
 
 2Ay = .0076 .0036 .0 
 
 4- . 003 2 
 
 4 .0089 
 
 
 2AA 7 = .OIO4 - .OOSO .0 
 
 4 .0043 
 
 4 .0126 
 
 50. Special results. The table shows the following positions for the suc- 
 cessive interferences, i.e., for zyN when the reflection changes from the 
 lower to the upper face. N = 2eu tan R sin R. 
 
 Reflection from bot- 
 tom, y m <y n 
 
 Reflection from top, 
 
 [Cases I-IP, y n = 1.1650, Equation IV, Position, 1.1650 
 
 II-IP, .2537, VII, .2537 
 
 Ill-IP, .2021, VIII, 2021 
 
 I- P, - .2537, IX, .3537 
 
 II- P - 1 . 1650, V - 6576 
 
 ?-' I III-P, -1.2166, VI, - ^092 
 
 Hence the total play of the mirror N should be 1.1650 ( .7092) =1.8742 
 cm., with which the observed datum 1.88 agrees as nearly as the experiment 
 warrants. Again the fringes of equations VII and IX are simultaneously 
 in the field, which is true, these being a series of lines which change their 
 obliquity in opposite directions symmetrically. The two ellipses of equa- 
 tions V and VI, however, were found nearer together than the com-
 
 IN RELATION TO INTERFEROMETRY. 77 
 
 puted value. Thus their distance apart in the D region corresponded to 
 J;y = .oo75 cm.; in the E region it corresponded to J^ = .oo5o cm., whereas 
 the above data require .0516 cm. Here, however, there is some confusion. 
 For instance, the spectra may coincide, whereas the doubly diffracted 
 white and colored images of the slit do not. Moreover, there are three 
 ellipses in this region of y, since the double diffraction for 6>i also gives an 
 ellipse, not included, because not belonging to the above enumeration. 
 The motion of ellipses from the D to the E line has different values. The 
 set toward the violet corresponds to JN = .oogo cm.; the set toward the 
 red to JN= .0070 cm. There is thus a different speed of fringes relative to 
 N, and closer contact in the violet than in the red. The latter are also more 
 nearly circular than the former. Equation IV gives the strong solitary 
 interferences useful in practice and treated in detail in the earlier parts of 
 this paper. The treatment of the other cases, being less important, may 
 be omitted here. 
 
 My thanks are due to Prof. Joseph S. Ames, of Johns Hopkins University, 
 for his kindness in lending me the glass-diffraction grating by which the 
 above equations were tested. I hope at some other opportunity to work 
 with a grating whose refraction is known throughout the spectrum and 
 also to endeavor to obtain the phenomenon as clearly from film gratings 
 (replicas) as has been possible for the linear series in the preceding paper 
 (I.e.). Thus far the above phenomena as obtained from film gratings were 
 not strong and sharp enough for measurements of precision but in a series 
 of experiments which has since been completed I will show how this desid- 
 eratum may be realized.
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 Los Angeles 
 This book is DUE on the last date stamped below. 
 
 Form Ly-50m-ll,'50U554)444
 
 000702610 7 
 
 UNIVERSITY of CALIFORNIA 
 
 A wnwi .