DISPLACEMENT INTERFEROMETRY BY THE AID OF THE ACHROMATIC FRINGES PART III BY CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON WASHINGTON. 1919 CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No. 249, PART III is' 77 PRINTED BY J. B. LIPPINCOTT COMPANY AT THE WASHINGTON SQUARE PRESS PHILADELPHIA, U. S. A. CONTENTS. CHAPTER I. — The Displacement Interferometry of Long Distances. PAGE. 1 . Introduction 7 2. Apparatus. Fig. i 7 3. Rigorous equations. Figs. 2,3 8 4. Ocular micrometer. Fig. 4 II 5. Collimator micrometer. Fig. 5 15 6. Half-silvered films 1 6 7. Direct observations 18 8. Indirect observations. Fig. 6 19 9. Ellipses and hyperbolas 21 10. Compensators. Figs. 7, 8 23 1 1 . Number of fringes visible 25 12. Separate adjustable auxiliary mirrors. Figs. 9, 10, n, 12 25 13. Types of achromatic fringes 27 CHAPTER II. — The Interferometry of Small Angles. Methods by Direct and Reversed Superposed Spectra. 14. Introductory 29 15. Method with prism. Figs. 13, 14, 15 29 16. Estimate 30 17. Equations 31 18. Observations. Prism-prism method. Fig. 16 34 19. Interference from rough surfaces. Figs. 17,18 36 20. Reversed rays. Figs. 19, 20, 21 37 2 1 . Second method. Figs. 22, 23 40 22. Equations 41 23. Observations. Figs 24, 25 43 24 . Reversed rays 46 25. Fringes from rough surfaces 48 26. Direct interferences without cleavage prism. Fig. 26 49 CHAPTER III. — The Elastics of Small Bodies. 27. Introductory method. Fig. 27 53 28. Apparatus. Figs. 28, 29, 30 54 29. Preliminary observations. Figs. 31, 32, 33 55 30. Rods in metallic sheath. Figs. 34, 35, 36, 37, 38, 39 59 31. The same. Thinner rods, hard rubber. Figs. 40, 41, 42, 43 59 32. The same. Brass 61 33. The same. Glass. Fig. 44 63 34. The same. Steel 64 35. Modifications of apparatus. Figs. 45, 46 64 36. Observations. Figs. 47, 48, 49 65 37. Apparent yield within the apparatus 66 38. Ocular micrometer. Collimator micrometer 68 39. Summary. Figs. 50, 51 . . 69 CHAPTER IV. — Experiments in Gravitation. I. GRAVITATIONAL ATTRACTION. 40. Introduction 73 41. Equations 73 42. Observations. Floating system. Figs. 52, 53 74 43. Expeditious fringe detection 75 44. Heavy needle in air 76 45. Light needle in air. Figs. 54, 55, 56 76 46. Summer experiments 79 3 4 CONTENTS. II. USE OF THE RECTANGULAR INTERFEROMETER IN CONNECTION WITH HORIZONTAL PENDULUM. PAGE. 47. Introductory 83 48. Apparatus. Figs. 57, 58, 59 83 49. Equations. Fig. 60 84 50. Observations. Fig. 61 86 51. Observations continued. Figs. 62, 63, 64, 65 88 CHAPTER V. — The Interferometry of Vibrating Systems. 52. Introductory 91 53. Telephonic apparatus. Figs. 66, 67 91 54. Observations. Figs. 68, 69 92 55. Binlar systems. Figs. 70. 71 94 56. Further observations. Figs. 72, 73 96 PREFACE. The present report is chiefly devoted to the investigation of methods of research in which displacement interferometry, conducted by the aid of the achromatics discussed in the preceding report, gives promise of fruitful applications. Thus, in Chapter I the method of measuring small angles hitherto suggested is given a practical test. The general theory of the sub- ject in its bearing on the two possible methods is developed at some length and a variety of interferometer devices, with mirror, ocular, and collimator micrometers, are instanced. Unfortunately, it was not till after the end of these experiments that I detected the method of reducing the fringes to the smallest number possible, practically to a single fringe; otherwise the work would have been more satisfactory throughout. As the achromatic fringes can not (in general) be found without first finding the corresponding spectrum fringes and, conversely, since for each type of spectrum fringes (direct or reversed) a corresponding group of achro- matic fringes may be associated, I have devoted Chapter II to spectrum fringes differing in their manner of production. The endeavor here has been to obtain interferences from distant slender luminous objects, without the aid of a slit. Partially at least the work has succeeded, but not as far as I hoped. The experiments are very difficult. The work in the third chapter was undertaken at the request of Prof- W. G. Cady, of Wesleyan University, in the endeavor to obtain the elastic constants of small bodies. The application of the displacement method proved at once to be astonishingly easy in a case where a degree of rough handling is inevitable; but there lurked in the elastic apparatus some dis- crepancy, both of viscosity and hysteresis, the nature of which escaped detection even after many attempts to locate its origin. Chapter IV contains applications of the rectangular interferometer using achromatic fringes to geophysical problems. A method for the determina- tion of the Newtonian constant is worked out. Again, the same interfero- meter is associated with the horizontal pendulum for the detection of small changes in the inclination of the earth's surface. Series of observations extending between January and August are recorded. Finally, in the last chapter, I have investigated corresponding methods for the interferometry of vibrating systems. The luminosity of the achro- matic fringes lends itself easily to this purpose and it was merely necessary to 5 6 PREFACE. design an appropriate vibrating telescope. To test the method, a study is made of the vibration of telephonic apparatus. For the first time I have obtained clear-cut interference vibration curves for two identical telephonic systems joined directly in series, while these forms subsided completely when the telephones were joined differentially. Such a system, from another point of view, is an electric dynamometer capable of appreciating an average alternating current well within a microampere. It could, moreover, be synchronized with an external impulse by aid of the Lissajous curves with the same accuracy as two tuning-forks. CARL BARUS. Providence, August iQi8. CHAPTER I. THE DISPLACEMENT INTERFEROMETRY OF LONG DISTANCES. 1. Introduction. — Methods for the measurement of small angles and of long distances were broached at the end of the last report. It is the purpose of the present chapter to continue the work experimentally, with a view to further development. It will therefore be desirable to collect the useful equations in this place in relation to the form of apparatus to be adopted, as well as to deduce the consequences of these equations in relation to their bearing on displacement interferometry in general. Throughout the chapter the work is done chiefly with the aid of the achromatic fringe groups, as I have called them; but these, as a rule, can be found only by means of the spectrum fringes, wherefore the latter become of coordinate importance. 2. Apparatus. — This is an interferometer of the Jamin-Mach type (fig. i) with four vertical plate mirrors, M,Mr, N,N', in parallel and at 45° to the horizontal beam of impinging white daylight L, from the country beyond. Three of these mirrors are half -silvered, viz, M,N,N', while M' is or may be opaque. The equal distances ab = cd = b constitute the base-line (6) of the appa- $ . ratus and the rectangle abed will be called / ;/ K • ^p ;/ r ^ the ray parallelogram. Its area is 2Kb, /», vicl/:/' : dl/' where ad = bc = 2R. Each of the mir- rors must be provided with thf£e adjust- ment screws for fine motion, so that the mirrors are each slightly revolvable about a vertical and a horizontal axis. In case of the mirrors N, M', these screws must be convenient to manipulate (thumb-screws), for adjustment here will frequently be necessary. The opaque mirror M' is on a Fraunhofer micrometer slide with its screw in the direction of the normal n' to M' and adapted to readings of at least icr4 cm. In the figure S, S' may be regarded as slides of a lathe-bed on which the carriages P and Q carrying the mirrors may move longitudinally. Any distance ab = cd is thus available for a base-line. The telescope is at T and receives both the light from L after two reflec- tions from the paired mirrors and the direct light from K through the half- silvers. It is desirable that the two beams from K and L be of about equal intensity. The pair of mirrors N and Nf is on a vertical axis A, so that it may be rotated as a whole. The amount of rotation may be read off either directly on a divided circle corresponding to the axis A or indirectly by the displacement of M' on the screw at nf in relation to the observed sweep of interference fringes. 7 Vi or" ol 8 DISPLACEMENT INTERFEROMETRY BY Thus there are three objects seen in the telescope: the direct landscape from K, the reflected landscape from L, and the achromatic interference fringes due to the partial beams abc and adc when the apparatus is in adjustment. To find the latter the method pursued in the preceding report suffices. It is first necessary to find the spectrum interferences when L is a beam of intense white light from a collimator and fine slit and the telescope is provided with a direct-vision prism. These are to be centered by moving the micrometer at M' and adjusting the pair of mirrors M'N. The spectro- scope is now removed (swiveled out) and the slit broadened, whereupon the intense achromatic fringes will appear covering the position of the image of the originally fine slit. If not vertical, the fringes may be made so by further slightly rotating M' and N on a horizontal axis, in the absence of compensators. The collimator at L is then removed and the fringes will be found super- posed on the foreground. If they are not bright enough from the light of the landscape, they may be given any intensity by reflecting (by way of adc and abc} a narrow horizontal strip of skylight from white paper into the telescope from L. Intense fringes will be seen transverse to the strip. 3. Rigorous equations. — In addition to the sides of the ray parallelogram b (base) and 2R (R radius of rotation) we shall have to consider the following angles or angular increments: A a the angular rotation of the paired mirrors, A0 the corresponding angular displacement of the fringes, AJV the linear dis- placement of the micrometer mirror in a direction normal to its face, and A

2A(f> cos i ~AN ~ AN /An ~ X Moreover, if 5 is the angle at the apex of the distance triangle on the base 6, (4) A s = (5) And since the distance d = b/2S = b/2 A a, from (5), bR F (6) 2 AN cos * THE AID OF THE ACHROMATIC FRINGES. 9 so that the sensitiveness is from (6), f x (7) w = — »(AAO = (a) It will next be desirable to deduce the above fundamental equations more rigorously than has thus far been done. Figure 2 is supplied for this purpose, and represents the more sensitive case where, in addition to the mirrors MM,' NN' (all but M being necessarily half-silvers), there is an auxiliary mirror, mm, capable of rotation (angle a) about a vertical axis a. The mirrors, M ---- N', in their original position, are conveniently at 45° to the rays of light, while mm is . There are but few fringes visible, however, and hence such an equation has but the specified limited application for distances close together, if fringes only are to be used. An example of the use of the latter equation follows, A0 and A

2 (sin 0o+0 cos 00)2+e2a2/v = e*Kp? where K = 4 (/x2 — sin2 i] . This is an ellipse if n at the center corresponds to a maximum value, in terms of the variables 0 and a, so long as K and /* are considered constant. But as n and therefore K vary with X, though slowly, it is true the equation is more complicated. When the center of ellipses is not in the field, but passes through the ver- tical plane corresponding to the center of the field of view, the ellipses may soon become appreciably straight lines in their visible contours, and the fringes must rotate in one direction or the other, according as the center is above or below the field. Rotation will be rapid when the vertical axis of the ellipses is relatively long. To bring the center into the field (for a proper value of N), the angle a must be zero, i. e., the two corresponding opaque mirrors which reflect the interfering beams must be rotated on a horizontal axis towards each other, or from each other, until a = o, or the horizontal plane through the field is a plane of symmetry. Furthermore, since the fringes necessarily move toward or from the center of ellipses with change of N, the motion of fringes will necessarily be oblique if the center of ellipses is obliquely outside the field of view. In the limit, if the center is in the vertical plane specified, horizontal fringes will rise or fall. Finally, if n passes through a minimum instead of a maximum, the fringes will be roughly of the hyperbolic type. At the center of ellipses in case of spectrum fringes, n is therefore a maxi- mum relatively to points of the spectrum in the same vertical or transverse line of homogeneous color. This maximum is due to obliquity, the hori- zontal one to change of X. In the case of fringes produced with white light (without dispersion), like the colors of thin plates generally or the achro- matic fringes discussed elsewhere, the center of ellipses (which are now cir- cles) is an absolute maximum, horizontally or vertically, i. e., relative to points in all directions from the center and for each color. The center of THE AID OF THE ACHROMATIC FRINGES. 23 spectrum ellipses, therefore, has no direct relation to the center of white light fringes; for the latter occur only when the rays pass the plate normally. On the other hand, when the white-light fringes are straight lines correspond- ing to very oblique incidence of interfering rays, the spectrum fringes are none the less perfect ellipses. It is finally necessary to account for the coincidence in adjustment of the center of spectrum fringes and the achromatic fringes, as the latter overlie the coincident white slit-images from which the superposed spectra are produced by the grating. This is easily seen to be referable to the fact that interferences with white light can only be visible if the light in the region of interference, when analyzed spectroscopically, contains but few dark bands. Since the number of bands in the spectrum is least near the center of ellipses, and is further reduced on making them as large as possible, the relation is obvious. In the case of strong, large achromatic fringes, a single fringe virtually occupies the whole spectrum. The light is either white or black. The displacement of the center of ellipses with the angle of incidence for a given adjustment may be computed from the original equation for centers N =e (cos r-\-2 J?/X2 cos r), where r is the angle of refraction for the incidence i, e the thickness of plate, and B the dispersion constant. When N and e do not vary it may be shown that (since n = A +.B/X2) , d\_ X sinr cos i (2.B/X2 — ^ cos2 r) di 2B/\2 fj. cos2 r (2+cos2 r) To obtain an estimate 2B/\2 = 0.026 may be neglected as compared with H cos2 r and the equation given the approximate form (^ = 1.5) d\ X sin * cos i — =- - = 10 X sin i cos *, nearly. di 0.026 2.5 jj. Thus, if t' = 45°, d\/di = $\ or 0.09 X per degree of i, which is about 100 times the distance of the sodium lines. If i = o° or 90°, the shift vanishes. 10. Compensators. — When the fringes are found they may be erected as stated by rotating either pair of the diagonal mirrors of the ray parallelogram towards or from each other, usually on a horizontal axis. The fringes may be enlarged by rotating the paired mirrors on either end of the ray parallelo- gram and restoring the fringes after each small step of rotation by displace- ment AAT at the micrometer. But these processes are tedious and must be very cautiously performed or the fringes are liable to be lost. The same re- sult may be accomplished by the aid of plate-glass compensators, about 0.5 to i cm. thick, placed normally in each of the two interfering beams and originally parallel and vertical. (See fig. 7, C and C'.) In addition to rotation and enlargement, these compensators serve with further advantage in equalizing the two beams in intensity. For this purpose it is merely necessary to half-silver lightly the compensator in the stronger beam of the ray parallelogram. If the fringes are more nearly vertical (between 24 DISPLACEMENT INTERFEROMETRY BY 45° and the vertical) they may be erected by rotating either compensator or both around a horizontal axis and enlarged by rotating around a vertical axis. The two compensators should be actuated together in opposite direc- tions if there is danger of the fringes leaving the field; i. e., if the adjust- ment is considerable. Similarly, if the fringes are more nearly horizontal, and particularly when horizontal fringes are wanted, the fringes may be leveled by rotating the compensators in opposite directions around a vertical axis and enlarged by rotating around a horizontal axis. Horizontal fringes, which climb up and down the broad white slit-image and may be made quite large, are often very advantageous. The illuminated field is much more extensive in the vertical direction. For slight adjustments it is convenient to have the compensators nearest at hand rotate in the same direction as the fringes. This may be done by working either on one side or the other of the center of fringes. The com- pensators may easily be ma- nipulated by hand (without ,-' T~ "\ tangent screws) and they are most efficient when nearly normal to the respective beams. To pass from vertical to horizontal fringes one would first rotate the compensators in opposite directions around a horizontal axis until the fringes are inclined about 45°, after which the further rotations would be made in opposite directions around a vertical axis. The motion of fringes indicates the proper direction of the rotation of the plates. To account for these apparently complicated effects, it is sufficient to re- call that the compensators displace the center of fringes, usually enormously distant outside of the field of view, and besides that invisible with white light ; for fringes are visible only in the narrow strip for which the spectrum fringes are very large and centered. Hence the result of rotation around a hori- zontal axis is to change fringes (fig. 8) of the type a, through c (vertical) into b, while the center moves downward, and vice versa. Again, rotation around a vertical axis changes fringes of the type a, through c' (horizontal) into d, as the center moves from left to right, and vice versa. The effects are necessarily opposite for the two beams. If the fringes are made vertical as at c, rotation of a compensator around the vertical axis can have no effect of rotation of fringes ; for the center moves in the line of symmetry ; but the effective or differential thickness of plates (e) is changed and hence the fringes are increased or decreased in size. In view of the presence of compensators, C and C', figure 7, the original ad- justment is much simplified ; for it is necessary merely that the spots of sun- light on the mirrors at a and b, figure 7, and at a' and br, one or more meters off, be at the same level and at the same distance apart, nearly. The accu- rate adjustment at c and d for coincidence horizontal and vertical is then made with the telescope at T. When the distances are approximately equal, THE AID OF THE ACHROMATIC FRINGES. 25 fine spectrum fringes will nearly always appear. These are enlarged and centered as specified. A broad slit with the spectroscope removed will then show the achromatic fringes, which may in turn be enlarged and rectified. Horizontal fringes, though often convenient, are at fault, inasmuch as they must rotate with the displacement of the micrometer AN. This appears at once from figure 8, for the center of ellipses is shifted. Vertical fringes alone are free from rotation in relation to AN. 11. Number of fringes visible, etc. — To get the most promising conditions for observing coincidence in case of range finding, the direct and reflected images should be about equally intense. Hence, if a is the coefficient of reflection, the two equal intensities are (i —a2) = 2(1 — a) a2, or a= 1/2 It is best, however, to have a less than this and to darken the direct beam if necessary by a thin half-silver plate interposed in the beam. If a = 1/2 the images are too dark and require higher illumination of the foreground than is usually present. As for the achromatic fringes themselves, they may be obtained with clear plate and opaque mirrors almost as well as with half- silvered plate, if the supernumerary images are partially screened off. With optic plate glass they would not appear. The surprising appearance of satellites — i. e., repetitions of the group with increasing faintness — is also common with clear plate. A series of experiments were made by replacing the half-silver plates with grid-like opaque mirrors. These are easily made by removing the silver along parallel lines (using a T-square) with a sharp wet stick. The slit images were then also gridlike in appearance and the achromatic fringes occurred only on the dark bars. For clearly the superposition of beams takes place on reflection from glass only. In this way the fringes on the supernumerary slit images were identified. These occurred on the bright bars. The two phenomena are therefore complementary. The reason for different numbers of visible fringes is less easily understood. In the original experiments two achromatic fringes (black or white) , with about three green-reddish fringes on either side and rapidly fading out, were alone visible. This narrow grid is very advantageous if displacement interfero- metry is in question, for the achromatic fringes are easily recognized. Subse- quently, however, large numbers of less distinctive fringes (20 or more) were usually obtained. As a small number of fringes is as frequently obtained with clear glass as with half-silvered plate, the occurrence is not attributable to the silver. (§ 13.) A variety of experiments were made with lenticular compensators, convex or concave, in each beam. The fringes, though obtained without difficulty, were usually rounded and irregular and the results without interest. 12. Separate adjustable auxiliary mirrors. — To obtain strong interfer- ences the two component rays eadT and bfcT of the rectangular inter- 26 DISPLACEMENT INTERFEROMETRY BY ferometer must not only be parallel on entering the telescope T (fig. 9), but they must be locally coincident at the mirror M; i. e., the two pencils from the collimator L (M',N,Nf being half-silvers) finally entering T, must very nearly coincide. Otherwise, even when the path-difference is annulled and there is perfect coincidence of the slit image, no fringes may be obtained. This is often a great annoyance when the mirror M' is on a micrometer screw (ri) normal to the face of the mirror; for on continuously displacing M' the y xe* N Tft e — • > ,m', ^ a* /eA"' \ d 1 /"- * 7 c-" 9 ir _v lo T7l -*fi \ a- / ^ ^ ^' / :*' ^" •-U rays cMT and dM'T separate more and more fully and the fringes soon vanish, unless a fresh adjustment for local coincidence is made. It is for this reason that the fringes are often so hard to find. The achromatics are much less sensitive to this disadjustment than the spectrum fringes; but the former are so mobile and easily lost that they have to be found as a rule by the aid of the latter. To meet this difficulty there must be one mirror available which reflects the component beam normally and which may be displaced parallel to itself ; i. e., whose micrometer screw is parallel to the incident and reflected ray. This condition is most easily secured by separating the auxiliary mirror into the parts m and m', each normal to its respective ray, while m' only is on a micrometer screw n1 ' . Under these circumstances there is no difficulty in finding the fringes after the adjustment for parallel rays and local coincidence at M' has been made once for all by actuating the mirror m' in one direction or another. Moreover, it makes no difference, within limits, how the paral- lelism and local coincidence are secured by moving any of the mirrors M,M', N,N', m,m', all of which must be on three leveling screws. Finally, if the mirror m and m' rotate as a rigid system about a common axis, it is still possible to use mm' for the measurement of small angles. If the rays a and c, which may be of any length, are very long, the adjust- ment shown in figure 10 is preferable, as the observer at T is near the micrometer screw n'. Here M, N, N' are half-silvers. THE AID OF THE ACHROMATIC FRINGES. 27 For the case, however, in which the body whose small angles of rotation are to be measured is part of another apparatus which does not admit of manipu- lation, the method may be modified as in figure n. Here all the plate mirrors M,M',N,N' are half-silvers and the rays from the collimator L form the interfering pencils LemeagdT and Lbfm'fct. The mirror m and m' meet their rays normally and m' is in the micrometer screw at n' parallel to /. The mirror m", on the axis A normal to the paper, rotates, and its small angles of displacement are to be measured. Considerable light is lost in the three penetrations of half-silver films by each ray; but in case of the achromatic fringes the light is usually in excess, so that the diminution of light is an advantage. It is more difficult, however, to find the spectrum fringes, as these require a slit. The plan of figure n is carried out more simply in figure 12, where both reflections take place at the same mirrors M and M', respectively, the compo- nent rays being Lbm"bach T and Ldem"efgi T. It is necessary to incline the parallel mirrors m' and m" on a vertical axis, in order to avoid the entrance ray L' into the telescope at T. But this separates the component rays h and i locally, so that means must be employed (compensators, rotation of the other mirrors m", N, or M') to obviate this as far as necessary. In both cases of figures ii and 12 it is therefore not easy to find the fringes, and I did not persevere in the quest because of an eye affliction contracted at the time. Similarly the system M, M',m" of figure 12 might be used if a half -silver is placed at r and the telescope at T' to the right of it. In this case the mirror m" must be in two parts, with adequate air-space inserted into the shorter ray Lb. 13. Types of achromatic fringes. — The difficulty of obtaining fringes of the strictly achromatic type (i. e., two strong fringes with a black line be- tween and the remaining fringes green-reddish and faint) in the rectangular or other interferometer, has been frequently referred to in the text above. As a rule the fringes found are more or less diffuse, non-symmetric, with large numbers (10 or more) about equally strong. Such fringes are, of course, useless in displacement interferometry. When the sharp fringes needed are obtained, their definition is independent of the particular part of any of the glass plates used, and any plate may be rotated 180° in its own plane without spoiling the sharpness of the fringes. Hence such slight curvatures or wedge- shapes as the plates may possess are without influence on the phenomenon. To further test this I devised a screw-press adapted to push the vertical edges of a plate to the rear and the middle forward, so as to give the plate marked cylindricity. Quarter and eighth inch plates were operated on, in the latter case sufficiently to give the two superposed slit-images quite unequal width; but no essential or useful improvement of the fringes was observed. The type of the fringe was not altered. Again, the symmetrical fringes may be obtained from plates thickly or thinly silvered, without essen- tial difference. 28 DISPLACEMENT INTERFEROMETRY. It follows, therefore, that the relative thickness of the glass paths traversed by the interfering beams can alone be of influence in shaping the fringe pat- tern in the manner in question. This is in consonance with the general theory of achromatic fringes, the result being a superposition of the color phenomena due to the dispersive refraction of the glass and the colors re- sulting from the wave-lengths of the interfering rays. To test this the ap- paratus, figure 10, is particularly convenient, as the fringes are easily found. Moreover, both rays, a and c, from the collimator at L, eventually pass through the plate N' before reaching the telescope at T. It is thus merely the thickness of the half -silvers M and N, both at 45°, that is here in ques- tion. If this thickness is the same, the sharp symmetrical design of but two strong fringes appears. If the difference of thickness is but little over 0.5 mm., many fringes, non-symmetric in distribution, are the rule. If the differential thickness is several millimeters there may be hundreds of fringes. If these are small they may be enlarged at pleasure; but they are always faint and useless for measurement. CHAPTER II. THE INTERFEROMETRY OF SMALL ANGLES, ETC.— METHODS BY DIRECT AND REVERSED SUPERPOSED SPECTRA. 14. Introductory. — It occurred to me that a number of the methods treated in my papers on direct and reversed spectrum interferometry might be used directly for the measurement of small angles and possibly of the distance of the source of light. Such a procedure would have an apparent advantage, at least theoretically, of not calling for the preliminary superposition of two images of distant objects, as the superposition is inherent in the method itself. But there are large constants involved, which make the result very problematical unless these constants can be removed by a compensator. In- deed, it is also very questionable whether such interference can at all occur. A further difficulty which hampers the method is the decrease of size of objects as their distance increases. Nevertheless a progressive investigation with the object of ascertaining to what degree the experiment is feasible is worth while, and as it will be convenient to develop the methods without reference to the ulterior conditions which limit the interferences, this method has been pursued. 15. Method with prism.— Figure 13 is a sketch of one of the methods in which 5 is the distant source of light, from which rays d and d' strike the mirrors m and n, and are thence reflected to the silvered sides of the right-angled prism P. After leaving it the rays enter the spectroscope at T in parallel. If the proper angles are se- lected the prism P may be replaced by one of any angle or by a reflecting grating. Suppose now the system mPn is securely at- tached to a rigid metallic beam or rail capable of rotating around a vertical axis at its center (P). This is indicated in figure 14, where the direction of rays and the normals of mirrors have been drawn and where the angle of rota- tion a has placed mPn into the position m'Pn' . The result is that a part y of the ray d is cut off on the left side and a part x added to the ray d' on the right side, so that the path-difference, which may be assumed to have been zero originally, is now appreciably incremented, but not sym- metrically for both sides. It may be shown, however, that the rays n'P2Tz and m'P\T\ still enter the telescope in parallel and that therefore the conditions of interference have 29 30 DISPLACEMENT INTERFEROMETRY BY not been disturbed. This is the interesting feature of the method, for the angle a between the two positions of the rigid beam will also be the angle between all corresponding normals of the mirrors, as indicated in the diagram. If we take the case on the left, the angle between incident and reflected ray at P will therefore be ^2+40: for the original mirror at P and TT/ 2 -\-4a- 2 a = 7r/2 + 2Q! for the new position at PI. But the angle between the rays re- flected at m and m' respectively as 2 a. Hence if T\P\ is prolonged backwards it must intersect the line mn at the original angle \ and thus P\Ti is paral- lel to PT, Similar reasoning applies on the other side for PiTi and will still hold if the direction of the ray Sn prolonged is reversed. Finally, ir/2 may be any reasonable angle. It will contribute to a more adaptable design of the apparatus for general interferometry if the ray Sn' may also be reversed by reflection (fig. 15, mirror n") in parallel to itself, allowing a small lateral offset, similar on both sides for clearance of the mirrors. Reflection between fixed parallel mirrors on the left in d and between mirrors set at a reentrant right angle on the right, say at n" ', would accomplish this at corresponding distances for the transverse rays. Again, half -silvers may be used at m and n for reflection, which method is probably best. These details will here be disregarded. If small angles are to be measured the direct method is enormously more sensitive. 16. Estimate. — The full expression for the path-difference corresponding to the rotation of rail a will be complicated and of no interest here. It is not sufficient to regard the intercepts y and x as solely contributing to the path- difference, which would therefore be x-\-y for the direct case and x — y for the case when the ray d' is reversed somewhere at n" (fig. 15) and returned parallel to itself. It may be shown that for small angles a, if /3 is the angle between incident and reflected rays originally at m and n and b the distance mP = Pn, d the distance Sm = Sn, are sufficiently approximate equations up to the squares of small quantities to meet the interference for the direct and the reversed cases respectively. Hence, if for instance a=i° =0.0175; b=i meter = io2cm.; d= i kilom. = io5 cm; X = 6X io~6 cm., the number of fringes corresponding to each of the terms may be computed as (Direct) w = 6Xio4— io3+6o (Reversed) w = io3 — 60 In the first case over 61,000 fringes pass per degree of rotation, a= i°. This makes about 2. 9 Xio~7 or about 0.06 second of arc per fringe. But the method is insensitive as regards distances d, unless the first two terms can be compensated. In the second or reversed-ray case, the method would be relatively much more sensitive as regards d if the first term 2baz could be compensated. The difficulty lies in the occurrence of «2 in the term, whereas most compensators would act as the first power of a. THE AID OF THE ACHROMATIC FRINGES. 31 Furthermore, if the angle a is small and S is displaced over an angle a or a distance r = da to the right, the original triangle may be regarded as restored. Hence the same number of fringes roughly should pass back again. In the second case, supposing 2bo? can be removed by compensation, r = da and a = \d/2b2, nearly, or or the object should be located to 30 cm. at a kilometer for each fringe pass- ing. In this case d need not be known, since n\ and n fringes are observed to pass for the angle of rotation a in the compen- sated apparatus. The direct rays without compensation would of course give indefinitely better results if d is known; for the angle per fringe has been found as a = 2.9Xio~7 when r = da = 0.03 cm. per fringe if d is i kilometer. Unfortunately, however, the method of figure 14 can not be rigorously carried out experimentally. For in any practical apparatus the mirrors M and N would have to rotate at a fixed distance from each other, apart from the micrometers; i. e., the two mirrors rotate on a rigid radius or rail and are therefore both rotated and displaced. It is this displacement which is relatively of much importance and by it all terms involving the first order of distance d are wiped out, so that terms of the second order in b/d only remain. 17. Equations. — To derive these equations certain intercepts of the rays, figure 14, in addition to x and y, b and b' may be defined. PI PZ is the trace of the vertical plane of symmetry of the right-angled prism, if rotated at an angle a to the right. In this case the reflected ray n'P^q on the right cor- responds to the reflected ray m'P\ on the left, both terminating in the common wave-front P\qs before entering the telescope. Let n'P2 = c' = b sin /3/cos a sin (f} — a)=zl/sin a cos a m'Pi = c =b sin /3/cos a sin (/3+a) =z/sin a cos a P2t = z' = b sin a sin /3/sin (/3 — a) tq = z =6 sin a sin /3/sin (/3+a) and nn' = x = b sin a/sin (/3 — a) = z'/sin /3 mm' = y =b sin a/sin (/3+a) =2/sin /3 since the original angles at the ends of the base are /3 and the rotation a. The angles between incident and reflected rays are respectively /3 — 2a at n', /3+2a at m', 90° — 20. atP2, and go° + 2a at P\. Most of the angles are indicated in the figure. The new radii m'P = b' = b sin /3/sin (0 — a); n'P = b" = b sin /3/sin 03+ a). The rays, however, do not reach the planes of symmetry, but are reflected by the faces of the right-angled prism, and this may be sketched in, in the rotated position (angle a) at Pipp'. The path of the reflected rays from n' 32 DISPLACEMENT INTERFEROMETRY BY is now n'rs and from m', m'P\ before they meet in the common wave-front P\qs. Hence the intercepts rs = v = (2+2') (cos a — sin a)/(sin a-f cos a) rP2 = w = (0-fV)/cos a (sin «+cos a) will enter in treating the path-differences. On the left the rays have not been distributed. If we take the direct case first the original path-difference SnP and SmP may be regarded zero or n and m in the same phase. On rotation, therefore (angle a), the path-difference is increased on the right by x-\-cf — w-\-v and increased on the left by —y+c, so that the total path-difference is equiva- lent to the equation riK = c' — c—(w — v)-\-x+y If the above equivalents are inserted, this equation may be reduced to sin /3 cos /3/cos a — sin2 /3 sin a + sin /3 cos a n\ = 2b sin a- sm (p-\-a) sin (p—a) in which the three terms in the numerator correspond to the respective intercepts c' — c, w — v, x-\-y. Since a and /3 are small angles, we may write sin a = a, cos a=i— a2/2. and cos 0 = b/d. Therefore the equation would, for practical purposes, become n\ = 2ba — 2baz-\-2bza/d the three terms corresponding to the xy, wv, and cc' effect. In the case of reversed ray (fig. 2) we may consider the points m' and n' in the same phase. Hence the original path-difference (« = o) is x — y. The path-difference after rotation c' — w-\-v — c. The total change of path- difference due to rotation is thus given by n\ = c' — c— (w — v) —x-\-y This differs from the preceding by the deduction of 2*. The rays again ter- minate in the common wave-front P\qs to enter the telescope. Hence after reduction sin /3 cos /3/cos a — sin2 8 sin a — sin a cos /3 n\ = 20 sin cc - sin (j3-f a) sin (/3 — a) the terms showing the cc', wv, and xy effects. The approximate form of this equation is thus practically The wv effect predominates, the cc' effect is intermediate, and the xy effect very small if d is large, as already instanced. The preceding equations may also be obtained geometrically by letting fall the normal from n (fig. 14) to the prism-mirror and prolonging the ray at s backward. In the isosceles triangle so formed the angle at the base is 45° — a. Hence in the above notation the path-difference takes the form x+2 (c'—w)cos2 (45° — a) — (2' — 2) — (c— y) THE AID OF THE ACHROMATIC FRINGES. 33 On inserting the values of the quantities as given above and reducing, an equation identical with the above appears, which for small a. is n\ = 2ba (cosec /3 — a+cot 0) If the prism has its nose at P (nearly), or in the axis of rotation, a small correction is to be added to the preceding expression. The path-difference on the right is increased by , i + cos (90° — 20) t 2 sin a cos a (cos a+sin a) cos a — sin a and increased on the left by z 'I (cos a — sin a) cos a Hence the correction is on reduction 2b sin2 a sin 0/sin 5 = 2&a2, nearly. This merely wipes out the small middle term, — a, of the above equation, leaving n\ = (2ba/sin 0) (i +cos 0) When the prism is reduced to reflectors in its plane of symmetry, as treated at the beginning of this paragraph, the equation loses the terms w — v and reduces to n\ = xJt-y-\-z-\-z'-\-c' — c, or to n\ = 2ba (i/sin /3-f-i) In the practical apparatus the mirrors m and n rotate on a fixed radius b, whereas b in the diagram elongates on the right and contracts on the left respectively to b" = b sin 0f sin 8 b' = b sin j3/sin a Hence the mirrors in the apparatus are displaced normally on the right and left by e—(b" — b) cos j8/2, inward, and e' = (b — b') cos 0/2, outward. The path-difference thus introduced is the sum of the decrease on the right and increase on the left and its value is 2e cos i, when i is the angle of inci- dence in question. Thus the correction is (after reduction) (b" -b} (cos a+cos (0-a)) + (6-b') (cos a+cos (0-f-a)) The expression may be further reduced to 260: cos 0 sin 0 . -(cos 0+cos a) sin a sin 5 when a is a small angle, or to 260: (i/sin 0 — sin /3-f-cot 0) If this quantity is deducted from the above equation for path-difference and direct rays there remains simply n\ = zba sin 0. The latter, therefore, is the equation to be used in interpreting the observation. So that generally when i = 0/2 for the micrometer at n 2 cos i A./V/Aa: = 26 sin 0 In the case of reversed rays the conditions on the left remain the same as before. But on the right the mirror n is set at an angle 0/2 to the rail 3 34 DISPLACEMENT INTERFEROMETRY BY and at right angles to its former position. Hence the normal displacement is e = (b" — b) sin 0/2 . The angle of incidence is i = 90° - (13/2 - a) . Thus the path-difference here to be deducted is 2 e cos i or 2 (6" -b) sin 0/2 . sin (0 — «) = (&" — 6) (cos a— cos(0— a)) and the total deduction from both sides is therefore (&" — &) (cos a — cos 5) + 2 (6 — 6') (cos a+cos a} This expression when reduced gives for small a zba (cot 0 — a cot 0/sin 0) or more simply 2ba cot 0 It is practically as large as the total path-difference for reversed rays found above. If, therefore, the two effects are opposite in sign, the path-difference introduced by rotation would be zero, apart from the change of glass paths and second-order effects which are relatively small. In fact, the experiments show that the rotational effect, Aa, in case of reversed rays, is relatively negligible as compared with the effect in case of rays not reversed. In other words, if from the equation for direct rays ri\ = 2ba (i/sin 0+cot 0) we deduct . . , . , / / • 0 \ a\ 2X-\-2e cos i-\-2e cos ^ = 2ba (i/sin 0-)-cot 0) the right-hand member vanishes to the second order of small quantities. 18. Observations. Prism=prism method. — In this case (fig. 19 below) a sharp-angled prism at S, with its knife-edge vertical, cleaves the beam of white light issuing from a collimator, reflecting the beams d and d' as described in my earlier papers. The system should be leveled so that all corresponding rays lie in a horizontal plane. By making the strips of light on both mirrors m and n (figs. 13, 14) coincide horizontally and vertically (using an auxiliary lens, if necessary) and then placing the prism P so that the rays mP and nP all but escape at its edge, the adjustment may be com- pleted by aid of the telescope at T. The two slit-images, which should be equally bright, are made to coincide horizontally and vertically by the ad- justment screws on m and n. If now the direct-vision spectroscope (prism grating) is swiveled in front of the objective of T, fringes will usually appear when the path-difference is annulled. For this purpose the prism P is placed on a Fraunhofer micrometer with the screw in the direction mn. The spec- trum fringes are as a rule easily found and are quite strong, but they can not be centered in the field of view, for the occurrence of ellipses presupposes the rigorous superposition of the two strips of light on the edge of the prism P, which is not possible. The fringes, if too oblique, may be erected by a plate compensator with a horizontal axis, or the prism P may be rotated on a horizontal axis. Vertical spectrum fringes are not usually wanted in these experiments, for they are to serve only as an essential aid to finding the achromatic fringes. THE AID OF THE ACHROMATIC FRINGES. 35 16 It is a curious fact that although the ellipses can not be produced nor the slit much widened, apparently achromatic fringes do occur in white light for a micrometer placement at P such as should produce centered ellipses. Moreover, as the white slit-image is linear, the achromatic fringes are pref- erably made to run transverse to it. They are then exceedingly brilliant, extending much beyond the slit-image, and they travel up and down it with the motion of either micrometer at P or at n, (A./V) , or with the rotation of the rail (Aa) . As there are but four or six fringes with but one or two strong and brilliant, they make an exceedingly sensitive index for measurement. The occurrence of achromatic fringes may also be detected in the solar spec- trum, as all the Fraunhofer lines (homogeneous light) become helical and broad from the cross-hatching due to the fringes. Here with homogen- eous light the fringes are indefinite in number and follow each other contin- uously, whereas with white light but one or two intense black-white fringes appear. Though the achromatic fringes are by far the most brilliant part of the phenomenon, they rarely occur without streamers. The general appearance is roughly suggested in figure 1 6, where ss is the white slit-image in the telescope and a the achromatic fringes moving up and down 55 when A./V or A« change. In the lateral glare of the field, however, fan- shaped or radiating coarse fringes bb are seen, intersected with very fine hairlike fringes cc. Probably there is also an intermediate group. These streamers are very useful to register the approach of the achromatic fringes, which move so rapidly that they are easily lost. A few measurements or rather estimates were made to coordinate the values of A./V of the micrometer displacement at n and the corresponding rotation Aa of the rail necessary to annul this displacement. To do this the achro- matic fringes were placed on the cross-hair, or better, on the image of the cross-hair at the slit of the telescope, and both readings were taken. They were then displaced by rotating the rail and restored by moving the microm- eter. To measure the rotation an index was placed at the end of the rail (radius 27 cm.) moving over a millimeter scale observed with a lens. The constants of the triangle, figure 13, were 6 = 20 cm. d = 62 cm. /3 = 7i.3° * = 35-6° Corresponding readings were found as follows in two separate adjustments: 10* AN I04Aa 10* AN io4Aa 0 O 0.0 o 204 9 26.9 7 493 19 76.0 30 Mean A7V/Aa = 26.5 cm. /radian 79.8 132.5 33 56 178.3 74 1/9-3 74 204.0 85 Mean AN/Aa = 2^ cm. /radian 36 DISPLACEMENT INTERFEROMETRY BY Since AAr/Aa = - ^ = 23.3 the observed data are above the computed 2 cos i values, but not more so than the difficulties of these measurements on an improvised apparatus imply. A much more refined method for finding Ac* is, of course, essential. 19. Interference from rough surfaces. — The question now at issue is whether the interferences can be retained when the collimator is removed and the light comes directly from a ground-glass surface or a Nernst filament. The spectrum fringes go at once when the slit is widened ; not so the achromatic sets. Having produced them clearly with sunlight, I found that a ground- glass screen or a scratched mica film could be placed at c or b or a, figure 17, whereas 5 is the slit and L the coliimating lens. The fringes should be transverse, as in figure 16, as vertical fringes are too easily confounded with the white slit-image. The slit was now broadened or quite removed; but the fringes, though less prominent from excess of non-interfering light, remained in place distinctly and without other change. On removing the lens, however, the fringes invariably vanished . I now replaced the sunlight by the light of a Nernst filament, under the impression that ground glass might to a small degree still behave like plate glass. The same experiments were made, the filaments at e (fig. 17) replacing the ground glass. In this case, however, I first removed the lens L and it was then seen that the two washed slit-images were not superposed, as is otherwise obvious; but it accounts for the failures of the experiments with sunlight. Superposing the two vague images both out of focus, a position was soon found in which the achromatic fringes appeared brilliantly. The slit could now be widened or removed at pleasure, yet the fringes persisted strongly, but with loss of brilliancy. It is thus possible to obtain these achromatic fringes directly from the Nernst filament and without a collimator; but they are so mobile, with change of Aa and AJV, that to find them it is necessary first to produce the spectrum fringes with collimator and spectro-telescope ; then to find the achromatic fringes on removing the spectroscope; next to remove the lens of the collimator and adjust for superposed images; and finally to remove the slit. These non-collimated achromatic fringes are best seen in a par- ticular focal plane of the telescope and they change their focal plane with displacement (Aa, A/V). They practically cover the whole width of the washed slit-image. They usually measure about 0.5° in width, but the streamers may extend laterally five times further, depending on the adjust- ment. When pronounced, the slit -images may even be separated as in figure 1 8, while each alone retains the achromatic fringes. This puzzling phenom- enon, which I had previously obtained, is probably due to the intersection and interference of rays in a region in advance of the plane of vision. Finally, as THE AID OF THE ACHROMATIC FRINGES. 37 the transverse arrow in figure 1 9 (indicating the right and left side of the slit- images) show, after the reflections at p m n P, although the slit-images are not reversed, the superposed rays are reversed; for these constitute the right- hand and left-hand radiation from the filament, separated by the prism p. If the slit is widened or removed, there is only one vertical line of rays (co- inciding with the position of the slit before removal) which can interfere. The remaining light does not interfere and its admixture robs the phenomenon proper of its brilliancy. A few experiments made on the nature of the achromatic phenomenon here obtained showed that the fringes are probably Fresnellian interferences. To test this the objective of the collimator was removed and strong fringes were obtained by passing the two washed images of the slit over each other laterally, by moving the corresponding adjustment screw on the mirror n. It was found that the fringes passed from horizontal maxima in size, gradually to vertical hair-lines, as the images slid horizontally from contact of their nearer edges to the contact of the further edges. The coarse fringes were even strongly present in the narrow gap between slit-images before contact. The telescope was now focussed on the slit, so that sharp linear images appeared. The fringes vanished; but it appeared that the coarse fringes corresponded to coincident sharp slit-images when observed out of focus, and the fine fringes to sharp slit-images far apart. The whole phenomenon thus depends on the distance apart of two lines of light and the interferences are observ- able before or behind their plane. 20. Reversed rays. — The apparatus was now adjusted for the reversal of the rays d' by adjusting a mirror at some place n" (fig. 15) on a fixed microm- eter and in such a way that the rays on reflection retraced their path. The mirror at n being a half-silvered plate, in turn reflected the rays toward the prism P. This modification of apparatus in- troduces very considerable path-difference, 2 nn", on the right, which must therefore be compensated on the left. It is diffi- cult to accommodate the micrometer and leveling devices at n and n" without an allowance of 5 to 10 cm. of path-excess. In my first experiments, which were merely tentative, the com- pensation on the left was secured by inserting a glass column about 15 cm. long. With this and the right-and-left microm- eter displacement of the prism, or the to-and-fro motion of the mirror n", path-difference was easily annulled and the ellipses found in the spectrum. They are centered as usual by rotating the glass compensator on a horizontal and a vertical axis, till with the occurrence of parallel rays at T the illuminated strips on the prism coincide, locally, to the eye. In view of the long glass path and therefore of considerable dispersive effect, 38 DISPLACEMENT INTERFEROMETRY BY the ellipses are small and the spectrum is filled with innumerable lines. More- over, in view of the prism separation (at p, fig. 19) the ellipses are throughout half -ellipses, all terminating in the vertical axis. For the two areas or strips of light (ab, fig. 20) seen on the face of the grating and entering the spectro- telescope are each single, being one-half of the full area of light rays capable of interference obtained at the collimator. This results in the half -ellipses e. If the prism is replaced by a half-silver plate as in the next paragraph, the strips ab1 and a'b are both double, the full areas being superposed; thus the areas ab and a'b' give rise to the full ellipses ee' . Hence, also, the vertical axis in e, being at the edge of the prism P, is not quite clear. Hori- zontal lines do not occur. These half-ellipses move with displacement of the micrometer at n', or at P, or on rotation of the rail mPn, as a body. It is difficult, however, to use them for measurement, as their vertical terminus is not sharp enough. If AA/" is the micrometer displacement corresponding to the rotation Act, we may write aAAT/Aa = o I did not succeed, however, in obtaining trustworthy results with the half- ellipses. The achromatic phenomenon can not occur when glass columns are used for compensation from the great number of lines in the spectrum. To obtain large ellipses the dispersion effect B/\2 must practically vanish. Hence an air-path compensator is to replace the glass column. This is conveniently made (as shown in fig. 21) of two pairs of parallel opaque mirrors ab and cd. The pair ab are clamped between short lengths of square brass tubing and cd similarly and at right angles (nearly) to the pair ab. Both are mounted normally to a horizontal brass table t, provided with three leveling-screws, capable of being raised and lowered and of rotating around a vertical axis. The path-excess introduced is thus equivalent to ab and cd and the ray dm is collinear with ra. This compensator not only introduces path-differ- ence, but since the mirrors are capable of rotating as a whole both around a vertical and a horizontal axis (leveling-screws), the beam dm may be moved right and left or up and down without ceasing to be parallel to ra. If, therefore, the ray entering T (fig. 19), were first made parallel, the ray d may be adjusted by the compensator until the strips of light on P practi- cally coincide at its edge. With the use of this air compensator or offset, the fringes were found with- out much difficulty and enlarged as specified. In view of the reflection at P, only half fields are returned; full ellipses or horizontal lines are not ob- tainable, as explained. But on removing the spectroscope and cautiously advancing the micrometer at N, the achromatic fringes eventually appear. In the present experiments these fringes did not take the usual and desirable form, consisting of but few fringes with the middle member in black and white. Probably because of the many reflections at mirrors (fig. 21), none of which was perfect, the fringes were now colored and present in large number THE AID OF THE ACHROMATIC FRINGES. 39 without much distinction between fringes. On being made transverse to the white image of the fine slit, they cross-hatched it from top to bottom. Nevertheless, their rapidity of motion is such that they serve quite well for measurement, the datum being more accurate than the measurement of Aa. The comparison was carried out in the same manner as before, the pres- ence of the achromatics being successively destroyed by rotation (Aa) and restored by the normal displacement (A7V) of the mirror at n. In this way the following data among many others were obtained. It is necessary to dis- place the mirrors very carefully; for if the fringes are lost they are extremely difficult to find without beginning with the spectrum fringes all over again. AaXlo3 AiVXio8 AaXio8 AA7Xio« O. o.o cm. XH-4 16.0 cm. 2.6 2.1 18.2 19.0 6.7 7-o 21.8 21.4 8.2 9-4 25-5 25-9 "•5 12.3 The range of Aa is much increased by removing the objective lens of the col- limator, and this is done after the observation marked x in the table. The fringes are perhaps even more distinct when present in the absence of the lens. The constants of the apparatus were: 6 = 21 cm.; 18 = 70.7° ; ^ = 64 cm. From this the rate AAf/Aa= 1.05 was found graphically. In the other series the rates were above 0.9. Approximate estimates of the same value were ob- tained with the spectrum ellipses and the glass column. This result again differs from the computed value, AN = o. The reason may lie in the fact that the plane of symmetry of the prism P (fig. 14) did not pass through the axis of rotation, or was not originally midway between the mirrors m and n. To test this inference (which will again be treated in the next section) the fol- lowing experiments were made: The prism P was as carefully as possible centered by the eye, so that its plane of symmetry passed through the axis of rotation. In this case the relative measurements icfi AN = 0.0 10'Aa =O.O 1-5 2.2 4-5 4.8 6.6 7-4 8.1 cm. 9-3 showed a mean coefficient of AN/Aa = o.8j. Finally the prism was moved to the right, i. e., with its plane of symmetry on the other side of the axis. The results were io'AN=o.o 3.0 4.4 7.0 cm. io3Aa =0.0 3.0 4.4 6.7 giving a mean rate AA/"/Aa = 1.05. Thus the shifting of the prism right and left has made but little difference and can not account for the discrepancy. It is probable that the coefficients found are largely due to the half-silver glass mirror n (fig. 15), which rotates with the rail mn. To test this a com- 40 DISPLACEMENT INTERFEROMETRY BY pensator c of the same thickness and g]ass may be placed in the beam mP on the left. If c and n are parallel, both originally at an angle /3/2 to the beams traversing them, it is obvious that the compensation will not be de- stroyed by the rotation of the rail, provided c is fixed while n rotates. If c is effectively thicker than «, the part of the coefficient due to the compensator may become negative. This is apparently the case in the following experi- ments, in which a compensator was installed as in figure 15 at c: AATXio3= —o.o —.4 —1.4 — 2.8 —4.0 —5.2 cm. AaXio3= o.o 2.6 5.9 9.3 12.2 16.0 The rate here is AN/Aa = —0.31, so that the zero value is exceeded. How- ever, the path-difference in the compensator of thickness e at an angle of incidence i and refraction r, viz, e (cos i — sin i tan r) a, where 2 = 90° — jS/2, here becomes 0.77 (0.816 — 0.578X0. 6i8)a = o. 3530:, so that the whole difficulty is not explained away. Finally, a few experiments were made to compare the effect of displacing (A7V) the micrometer at n" (fig. 15) as compared with the effect of a microm- eter (AN') which displaces P in the direction of its plane of symmetry. The latter (AN1) is zero when a = o. Generally if e is the normal displacement of the prismatic faces, the path-difference is ze (cos (45° —a) — cos (45° -fa)) = 2AN sin a. since e = AN sin 45°. Hence, as AAf is a normal displacement AN /AN' sin a The results obtained were io3AiV' = o.o 50 100 150 200 225 cm. io3&N =0.0 1.5 3.4 5.4 7.2 9.2 cm. Thus the mean rate is a = AN /AN' = 0.036 or a is a little over 2°. To obtain these data the achromatic fringes were used as above. When the slit-images seen in the telescope are not quite parallel, they may be made so by slightly rotating the slit on a horizontal axis normal to its plane. The images rotate in opposite directions. A slight angle between the images is, however, of no consequence. 21. Second method. — In view of certain difficulties encountered in the use of reflecting prisms, in particular the loss of rays at the edge, the limitation to half -ellipses, etc., the method of figure 22 enlarged in figure 23 was devised. In this the prism is replaced by a half -silvered plate PP'. Hence the rays issuing at 5 and reflected by the opaque mirrors at m and n are thereafter respectively transmitted and reflected by the half -silvered plate at p, and then reach the spectro-telescope at T together. When the path-differences are sufficiently equal, elliptic interference fringes will be seen in the spectrum. When first found they are usually very fine straight lines; but they may be rectified by plate compensators in the beams d and d' or mp and np, though THE AID OF THE ACHROMATIC FRINGES. 41 the operation is not easy. Leaving these details for further consideration, the procedure for angular measurement may advantageously be treated here. For this purpose the half -silver P and one opaque mirror, n, for instance, are mounted on a rigid bar with an axis at P. The other mirror m is to remain fixed. If the bar is now rotated over a small angle a, figure 23, the mirror at n is displaced to n' and the ray Sn prolonged (intercept %) is now reflected from n' to q and thence along T into the spectro- telescope, parallel to its original direction or to the other ray mp. Hence the interferences remain intact, but many fringes pass during the transfer. The persistence of parallelism is easily seen, be- cause the angle between the incident and reflected ray at n is decreased by 20. when n passes to n', but is again increased by 2 a owing to the rotation of PP\ to PP' over the angle a. To control the fringes either the mirror at n (or at m} may be displaced on a micrometer screw normal to itself, or the half-silvered plate at P may be displaced parallel to itself. If the angle of incidence at n is i and the nor- mal displacement of n is e, the path-difference introduced will be 2e cos i. Similarly if the normal displacement of the plate P is e' and the angle of incidence i1 ', the path-difference will be ze' cos i'. As in the preceding experiment, the mirror at n may be a half -silver, so that the ray d' passes through it and may then be returned in its own path by a mirror at n" on a fixed standard. The displacement of this mirror over a distance e, parallel to itself, introduces the path-difference ze, so that the cosines are avoided. But a much more important result is the fact that the rays np or n'g now are stationary. The strips of light originally at p do not therefore travel over each other while one passes from p to q and the inter- ferences are kept at full intensity throughout. This is a great advantage. Moreover, the half -silver plate at n compensates the half -silver Pp, which is a further advantage, since both paths within are glass paths with high dis- persion coefficients. It is obvious that the path-excess nn" on the d' side, must be separately compensated on the d side. The method of doing this by an air compensator (fig. 21) will presently be considered, as a long glass compensator would not in general be desirable because of the sluggish motion of the small ellipses thus produced. 22. Equations. — The equations for this case are apparently very compli- cated. If in figure 23, m and n are in the same phase and Pp is symmetrical, there will be no path-difference at p. When Pn is rotated over an angle a. into Pn', the path on the right becomes nn'+n'q+qs while (ps wave-front) the path on the left remains mp as before. The path-difference is thus the differ- ence of these quantities, to which, however, the increased glass path at PP' would have to be deducted, and the surface PP'.must pass through the axis P, 42 DISPLACEMENT INTERFEROMETRY BY If the angle SnP is /3 and Pnp is 7, the values of the branch paths may be found to be (since nP = mP = b), if /3 — a = d and 7 — a = r, wp = w£ = 6/cos 7 wn' = b sin a/sin 5 nq =b sin /3/ sin 5 sin T _ ^ f sin a sin /3 sin r+sin2 7 sin 6 cos r 1 sin o cos T cos 7 [ — sin /3 sin 7 sin r cos r j Hence the path-difference is equivalent (after some reduction) to the equation % _ & [sin a (cos 7 cos r + sin fi sin r ) + sin /3 cos 7 sin 5 cos T cos 7 j — cos2 7 cos r sin 5 — sin /3 sin 7 sin r cos rj If a = o, then /3 = 5, 7 = ^, and the right-hand member becomes zero, as it should. I have not succeeded in putting this equation in a much more con- venient form. If a is very small, so that differential expressions may be introduced, the rigorous equation, to an approximation of the second order in a, may be reduced to / fr,\\ to \ cos 7(i-f-cos (p — 7); i+cos (p — y) n\ = ba — - = ba— sin 5 cos r sin j8 If /3 is nearly 90° and if the distance Pp is p, then n\ = ba (i-j-sin 7) =ba-\-pa cos 7 The same expression may be obtained geometrically by prolonging n'p and T'q to m'. The triangle n'qm' is isosceles. Hence if we draw the wave- front Pq', normal to pT, we may deduct the common length pq' from both rays, or add it to both. Hence the path on the right will be x+b' cos (y — a) -\-p sin 7, where bf is the line Pn' and on the left as before, 6/cos 7. Hence (if b = p/tan 7) the path-difference is /sin a b' i \ b I — ; h~cos (7 — a)+tan 7 sin 7 I \ sin 5 b cos 7/ which, as above, also reduces to ba. (i+cos (18-7)) / sin /3 There is a source of discrepancy which enters when the face PP' does not pass through the axis P, but is eccentric. In such a case, if e is the distance of the plate from the axis, a correction equivalent to e (i— cos a) cos 7 = e«27, nearly, will have to be supplied. Again, if there is lack of symmetry from such a cause, the base-lines will be b and b' and the angles 7 and 7', so that a modified equation is suggested. Finally, from all these expressions the changes of glass path on the left with a, if not compensated, must be deducted. As the method admits of a good achromatic phenomenon of reversed slit-images, it is theoretically interesting and I have given it some study. For the case where the ray Sn prolonged returns on itself, as from n" in fig- THE AID OF THE ACHROMATIC FRINGES. 43 ure 23, the mirror n being a half -silvered plate, the quantity nw" = 2&a/sin

.u== l-55> and 7 = 27.1°. A further discrepancy may be sought for in the fact that if the surface Pp (fig. 23) does not pass through the axis of rotation, this plate is both rotated and displaced. The error so introduced may be either positive or negative. If the displacement of plate is e' = e(i— cos a), the equation should read cos 7 2 AN COS = b (sin /3 + sin 7) where e is the distance of the plate from the axis. The following experi- ments were therefore made with some consideration to greater symmetry of apparatus. The constants were: /3 = 7o.5°, 7 = 27.1° and therefore i = 48.8°, or 2 cos 1 = 1.32. The data found were in different adjustments; AATXio3 AaXlO3 AA7/Aa 2 AN cos i/ Act 110.9 5-2) 203.7 9.6 21.2 28.0 246.5 II. 6J 102.9 208.4 4-8 \ 10. 0 / 20.5 27.1 47.6 2.6' 80. i 4 I 170.4 8.5 20.7 27-3 223.6 10.7 ] 257.6 12.7 J 46 DISPLACEMENT INTERFEROMETRY BY These results agree much better with the theoretical equation than the former, and may be considered as coinciding with it. In the present case the attempt to get interference from rough surfaces was not at first successful. The slit-images are reversed, as indicated by the transverse arrows in figure 2 2 . Hence if the white slit-images are wide there can be coincidence only in a single vertical line. Fringes with white light will occur as a case of the interference of fine slit-images. To produce them it is first necessary to obtain the spectrum fringes with the ellipses, or else with horizontal fringes in the field. If now the spectroscope is removed and the white slit-images put out of focus, the phenomenon indicated in figure 24, where 5 is the superposed, washed slit-images wiJl usually appear, or may be found on cautiously moving the micrometer screw. Within the slit- image the fringes are coarse and colored, but they send out fine oblique stream- ers into the field of diffuse light or glare, on both sides of s. When the slit is widened these fringes are liable to vanish just 24 j//// £ -> as the spectrum fringes vanish, except perhaps at the edges of the images. These achromatic fringes climb up and down the slit-image with motion of the micrometers (A/V, Aa) with extreme rapidity and are easily lost, as there are not usually more than 10 or 20 of them. If the spectrum ellipses are huge, the white fringes are almost too coarse to be seen and too mobile to be controlled. I next removed the objective of the collimator. The fringes, though much changed in appearance, practically black and white, were not destroyed. In such a case the slit-image shrinks vertically. To obtain a long strip a highly illuminated ground-glass screen (sunlight and weak condenser) should be placed in front of the slit as a source of very diffuse light. In such a case this long white post (as it were) is covered from top to bottom with sharp blackish and usually oblique lines, which vanish at once, up or down, on moving the micrometer. No fringes are seen if the slit is in focus. When considerably out of focus (as in case of the diffraction patch in fig. 25,) strong, sharp-colored cross-markings are present, which would be quite available for measurement. However, in this experiment, when the slit was widened or removed, the fringes apparently vanished. The phe- nomena as a whole seem to me to be fringes of the two white slit-images, and seen either behind or in front of their focal plane, like the complemen- tary fringes described elsewhere. This is confirmed by experiments pres- ently to be described. 24. Reversed rays. — The apparatus was now adjusted for a reversal of rays by putting a half -silver plate at n and an opaque mirror on a micrometer (with the screw normal to its face) at some position n" (fig. 23) fixed inde- pendently of the rotation. In this case, therefore, the intercept nn' changes sign. Moreover, the angle of incidence at n" is o°. Hence the equation should be a = b (sin 0 — sin 7) THE AID OF THE ACHROMATIC FRINGES. 47 The constants of the apparatus were 6 = 20 cm. ,8=71. 3° Thus 7 = 34-9 — 0.57 2) = 7. 5 To obtain the ellipses a thick plate-glass compensator may be placed in the d ray to provide for the elongation 2% in d' '. About 14 cm. of glass col- umn were necessary. This makes it very easy to center the ellipses and to obtain them intensely black on a colored ground by rotating the compen- sator on a horizontal and vertical axis until the two strips of illumination at p quite coincide when the rays T, T' are parallel. But on the other hand, because of the thickness of glass used, the small ellipses obtained move rel- atively sluggishly with displacement of the micrometers. The sensitiveness decreases proportionately to the thickness of glass path. Experiments, of which the following data are examples, were made by alternately restoring the center of ellipses to the D lines of the solar spectrum first by the micrometer (A/V) and thereafter by the rotation of rail (Aa). The adjustments were very different. ANXio* AaXlo3 ANXio3 AaXlO8 o.ocm. o.orad. o.ocm. o.orad. 53-2 13-7 24-3 1-9 66.1 17.0 33-3 4.4 58.1 14.8 44.1 "•5 43-3 7-4 21. 1 5-2 57-4 10.7 68.8 14.0 Mean 2AA7'/A« = 7.6 Mean 2A]V/Aa = 6.8 The second series changed its rate enormously, almost one-half, owing to necessary intermediate adjustments (inserting a new zero). Otherwise the observations are as good as the apparatus permitted; but the computed 2 AA/YAa = 7.5 is above the observed value, usually, possibly owing to an ec- centric position of the plate Pp relative to the axis of rotation. To test this point of view the plate Pp was displaced eccentrically toward the left of the axis. The result should be a modified coefficient, but the following data obtained in the same way as before fail to bear this out, however: AAfXio3 = o. 19.9 5.9 40.2 12.2 56.8 cm. 16.7 rad. The rate, 2AA/"/Ao: = 6.6, does not differ essentially from the above. With these small ellipses there can not, of course, be an achromatic phenomenon. To obtain large ellipses the glass-path difference (i. e., the dispersion) must be abolished on both sides and an air-path difference introduced, preferably in a way which has been shown above in figure 21. As such experiments are so mach more trustworthy and sensitive, I did not pursue the glass- column work further. 48 DISPLACEMENT INTERFEROMETRY BY 25. Fringes from rough surfaces. — Experiments were now made with the use of the air-path compensator (fig. 21,) placed in the d rays when these rays were reversed. A magnificent set of achromatic fringes were here found, only about five in number, with the central members in black and white. Tests similar to the above showed that they are Fresnellian inter- ferences. To prove this the objective of the collimator was removed and even more brilliant fringes were found on placing the washed slit-images in contact. If these patches of light were slid over each other horizontally, by moving the adjustment screw for rotating the micrometer mirror on a vertical axis, the fringes rotated nearly 180°, passing from vertical hair-lines, through a maximum of coarseness for the horizontal fringes, back to hair- lines again. On focussing the telescope on the slit it was then found that the large horizontal fringes corresponded to coincident slit-images in focus, whereas for the very fine fringes the focussed slit-images are far apart. No fringes appear on the slit-images in focus, in any case. They lie in front of and behind the image plane. This is exactly the case found above, ex- cept that here the edges of the washed slit-images are exchanged. The endeavor to obtain the fringes without the slit was next tried. For this purpose a ground-glass screen illuminated by sunlight (a, fig. 17) was first placed in front of the slit 5 in the absence of the objective (L) of the col- limator. The fringes were still very prominent, though the light was darker. The slit 5 was now also removed. The fringes could then no longer be seen; but on narrowing down the illuminated ground-glass screen a to a vertical strip of light i to 2 mm. broad, they were unquestionably present. In such experiments, therefore, the chief function of the slit 5 is to cut off the light which does not interfere, so that the fringes are lost in the glare. In the absence of such excess of light the fringes are quite visible and therefore certainly always present. By aid of the offset air compensator huge achro- matic fringes may be easily produced ; but they are so sensitive as not to be manageable in an improvised apparatus. A number of measurements were now made with the achromatic fringes set at convenient (small) size by the air-compensator. In this work the plate PP' (fig. 23) was moved in three steps over about i cm. For each step a set of data was investigated. The results were in succession f N f AArXio3 = o.o 1.9 6.1 12.4 17.8 21. 9 cm. \ 2&N (*) 1 A V 1 J- (A«Xio3 = o.o .4 1.2 2.3 3.3 3. 9 radian] , . j A/VXio3 = o.o 7.1 12.2 19.2 26.6 32.6cm. 1 2A^ (2) 1 , . — = Q.6 [ Aa X ioj — .0.0 i.i 2.2 3.7 5.2 6. 3 radian J &a , . f A/V Xio3 = o.o 9.3 19.2 27.7 37. 7 cm. 1 2AA/" [ Aa X io3 = o.o 1.8 3.7 5.9 7.4radian j ±a The coefficients so obtained are practically identical ; and they agree as nearly THE AID OF THE ACHROMATIC FRINGES. 49 as may be expected with the equation, since the angle B and T are not easily specified with accuracy. They were 18 = 71.3° 7 = 28.4° 6 = 21 cm. so that theoretically o! = 21(0.947— 0.476) =9-9 26. Direct interferences without cleavage prism. — The next step in ad- vance was made by dispensing with the sharp prisms heretofore used for cleaving the rays issuing from a collimator (or the slit simply) in the endeavor to obtain two rays capable of interference. The assembly of apparatus is shown in figure 26, where S is the slit (to be replaced by a Nernst filament or a tungsten filament), m and n the opaque mirrors, pp' the half-silvered plate. The rays dd', diffracted at 5, pass after reflection into c and c' and may be observed by spectre-telescopes placed either at T or T1 '. In the first experiments the distance Spr was about 4 meters and the distance mn 10 cm. The mirrors « and pp' were on micrometers with the screws normal to their respective faces. The distance mn must be within the limits of the wedge of light from the slit and is therefore small, unless d is very large. Both pp' and n are on the rotating rail (as above), whereas m is fixed. The appa- ratus was also adjustable for reversed rays by attaching an auxiliary mirror, normal to the rays d' prolonged through n. S being distant, this slit must be long, as otherwise the spectrum band will be a mere horizontal line and the fringes difficult to detect. A doublet of lenses, each about 10 cm. in diameter and of the same focal power (1.60 cm.), but respectively convex and concave and having a combined focal distance of about 5 or 6 meters, is of advantage for focussing a large solar image, i to 2 inches in diameter, on the slit. The Nernst or tungsten filament gives the same advantages at once; but the former is too thick, at least for the initial experiments at shorter distances. The fringes are exceedingly difficult to find in spite of the brilliant spectra. It was not until after about three days of searching, in which (besides sunlight) the filaments as well as the methods of direct and of reversed rays were used, that the experiment ultimately succeeded with sunlight. The filaments are much less gracious. To obtain the fringes calls not only for very accurate adjustment for horizontal and vertical spectrum coincidence, but the fringes lie quite sharply in a definite focal plane, usually between that of the slit- image and the principal focal plane; the rays must interpenetrate at the plate and finally path-difference must be nearly annulled. And there are other conditions presently to be stated. After being found they are quite strong elliptic spectrum fringes, but when lost nevertheless difficult to rediscover. The slit may be broadened till the spectrum lines vanish, perhaps to more than a millimeter, before they disappear in a uniform spectrum band. The achromatics which coincide in adjustment with horizontal spectrum 4 50 DISPLACEMENT INTERFEROMETRY BY fringes and are seen with the slit-image out of focus are also difficult to find because of the short length of the slit-image. As first obtained they lacked brilliancy and were not easily observed. Similarly, experiments with filaments failed to show the fringes, although made in parallel with the successful result with sunlight. A considerable assistance in finding the fringes is an opaque screen with a vertical slit 2 mm. to 4 mm. wide placed just in front of the objective of the spectro-telescope, in the best position as to symmetry. This screen cuts out rays which do not interfere and makes the fringes stronger, even though the background is darker. Fringes are frequently found in this way when they are all but invisible in the full spectrum. In addition to the regular fringes, a much larger vague set seems to be present in another focal plane. They also rotate, etc., like the regular fringes, but the experiment led to no decision with regard to them, as they appeared in the field erratically and could not be produced at will. They may be shadow interferences of the principal set. An attempt was made to register slight lateral displacements of the slit in terms of the displacement of fringes, but as the slit-images are thrown out of coincidence when the slit moves, trustworthy numerical data can not be obtained. One may estimate that as a first approximation d^ x = —\ c if x is the lateral displacement of the slit and c the distance between mirrors m and n. Hence 400 x0 = - 6Xio~5 = o.oo24 cm. 10 should have been equivalent to the passage of one fringe in the given appa- ratus, or generally 2AAT cos (/3-f 7)/2 =cx/d If (j3+7)/2 = 70°, A7V=io~4 cm., and d/c = 4o, then . = 2.Xio~3 cm. could have been registered. Incidentally it appears that two vertical lines of the slit, xo/2 =0.0014 cm. apart, would wipe out each other's interferences; but this is not the case, as much greater slit-widths are admissible. To the right and left of the line of the slit capable of producing interferences the parallel lines either cease to produce parallel rays, or parallel rays come from symmetrical but different lines. After completing these experiments, the distance between slit S and the mirrors m and « was increased to about 9 meters. The same lens doublet, focussing a large solar image on the slit, was used as before. With the aid of the slotted screen in front of the telescope and the micrometer distances from the preceding experiment, the fringes were found without difficulty. THE AID OF THE ACHROMATIC FRINGES. 51 In fact, in view of the longer distance d, the slit could be opened to over a millimeter of breadth before the fringes quite vanished from the spectrum; but on using a somewhat stronger condensing system (concave lens of doublet preceding the convex lens) and consequently more oblique rays, a very fine slit was needed to show the fringes. They are thus more easily found when the rays are more nearly parallel. With artificial light, again, I obtained no results, even after long searching. Operating with two successive slits at about 9 meters from the interfer- ometer, one of which received the light through the other, I found that two independent sets of fringes very different in size and inclination could be put in the field together. The further investigation eventually showed that the size and inclination of the fringes is essentially dependent on the degree of parallelism of the two slit-images. When the images are parallel, the fringes are of maximum size and vertical. When the images are not quite parallel (they incline in opposite directions when the slit is slightly rotated in its own plane from the vertical), the fringes rapidly grow smaller and rotate. With parallel slit-images the spectrum ellipses are centered in the field; otherwise they are very far out of center. The adjustment for actual (not X-like, coincidence must therefore be made with precision if large fringes are wanted. Further work was also done with sunlight to obtain more pronounced achromatics. For this purpose a compensator was inserted to equalize the glass path in the half -silvered plate. Huge spectrum ellipses were obtained in this way and their centers were placed above the telescopic field, so that the fringes seen were large horizontal bars. On removing the spectroscope and placing the slit-images out of focus, brilliant achromatics were in fact obtained, of the concentric hyperbolic type, vividly colored and broad be- tween the apices, and diminishing to hair-lines laterally. With these it was possible to enlarge the slit to at least 3 mm., without destroying the fringes, though they became more vague. It is necessary that the slit-images, when in focus, should be quite parallel, otherwise any broadening of the slit will wipe out the achromatics. It was possible to place a plate of ground glass on the far side of the slit without destroying the fringes, but not on the side towards the interferometer. In other respects the behavior was as described in the case of achromatics in the earlier experiments with a cleav- age prism. Finally, the spectrum fringes and the corresponding achromatics were obtained with the light of a Nernst filament, at first by focussing an image of it with a strong condenser lens on the slit. The experiments, however, are very difficult. The spectrum fringes are often weak, out of focus, and extremely sensitive to small disadjustments in the horizontal and vertical coincidence of the slit-images. They require a fine slit. When well produced the achromatics are also obtainable on removing the spectroscope when the spectrum fringes are horizontal bars. The achromatics may also be obtained brilliantly without the condenser lens, but the adjustment must in such a case be made first with sunlight, as the spectrum from the Nernst filament 52 DISPLACEMENT INTERFEROMETRY. is too feeble for detecting fringes so elusive as the present. The achromatics, however, are strong and brilliant even here (Nernst filament) . An interesting result is obtained in case of the achromatic fringes by nar- rowing one of the beams, for instance that coming from the mirror m (fig. 26), by a screen with a vertical slit about 2 mm. wide. In such a case the slit-image (out of focus) is correspond- ingly narrowed. It may be passed from side to side of the broad washed slit-image coming from the mirror n, by moving its adjustment screws (vertical axis). The fringes then appear only in a particular position of the narrow image in the field of the broader ; but when they do appear they spread far beyond the margins of the nar- row image on both sides. Interference thus apparently occurs where but one beam is present. The phenom- enon is like those instances above (figs. 18, 24, Chapter II) and means, as I understand it, that the beams have met in some other focal plane, though one is tempted to conclude that interference is stimulated by resonance, in particular as it is often impossible to find a plane in which they have met. The achromatics may sometimes be seen before and behind the principal focal plane, but more frequently either in the one or in the other region only. 71' CHAPTER III. THE ELASTICS OF SMALL BODIES. 27. Introductory method. — At the request of Professor W. G. Cady, who was in need of Young's modulus in case of certain crystals used in experiments in which he is interested, the endeavor was made to adapt the above: inter- ferometer for measuring small angles with an auxiliary mirror for this pur- pose. The project seems feasible and apparently simple in execution when the method of end-thrust indicated in figure 27 is used. Here F is a rigid metallic bar subjected to the force couple P,P', carrying the coplanar mirrors m,m' and capable of ro- 3 Part of the sectional area of the short solid may in fact be quite free from strain. Hence the device shown in figure 3 2 was next tested, where a relatively thin rod of hard rubber r is surrounded by brass caps b and c with closed ends. The caps fit the rod loosely and room is left between them for compression. For hard-rubber rods of the dimensions L=i.7 cm., A =0.2 4 cm.2, and in three series of experiments the moduli io9E = 6.g, 7.5, 7.5 were computed, showing therefore no marked change from the preceding data, in spite of the greatly diminished sections. In the next experiments shaped rods like figure 33 were used directly. For the dimensions L = 2.2 cm., A =0.41 cm.2, the moduli were found to be EXio9 = 9.6, 9.6, 8.4, 8.4 in four series made after different periods of loading. 56 DISPLACEMENT INTERFEROMETRY BY In all this work the initial loads of i kg. each were not removed. The elon- gations between i kg. and 4 kg. were consistent, though naturally with definite evidence of apparent hysteresis. For instance, on the last series with brass caps the individual contractions were AP= 4 3 212 3 4 kg. io4A/V=i2o 94 55 o 40 78 141 cm. and in the last series with the mushroom-shaped solid, AP = 4 3 2 i 2 3 4 kg. 89 68 38 o 34 64 89 cm. As the values AAf are proportional to the decrements of length, the rod is shorter on unloading and longer on loading, c&t. par. In other words, the hard-rubber rod is influenced by its immediately antecedent history. The curved lines obtained may, however, also be influenced by gradual dislo- cation or decreased firmness in the seating of the rod. Thus the modulus in the last example decrease from about iogE = g between the two highest loads (3 to 4 kg.) to id9E = 6 between the two lowest loads (i to 2 kg.). Of the two the former (high loads) is unquestionably the more trustworthy and least influenced by imperfections of apparatus. Results of the same nature were obtained for brass, though here, from the much greater rigidity, the effect of dislocation at small loads is much more apparent. The first experiments were made with a relatively thick rod, of the dimension L = 2.2 cm., A =0.70 cm.2, for which E = 4.5Xio10 was ob- tained. This is enormously too low and the result is a mere indication of the yield of the apparatus or inequalities of stress. The rod was now turned down on the lathe to a mushroom-shaped solid (fig. 33), the dimensions L = 2.i cm., A=o.ii3 cm.2. With this the values EXion = i, 2, 5, were found, according as the rod was unloaded or loaded from 4 to 3 kilo- grams. Clearly this is a case of dislocation of parts of the adjustment. The rod was then further diminished to diameter of but 0.2 cm., so that L= 1.8 cm. and ^=0.035 cm.2 now obtained. With this, for the highest loads, values £=i.4Xion followed under like conditions. This is no improve- ment and shows that the apparatus yields seriously for moduli as large as that of brass (say io12). Thus in the series with largest values of E the in- dividual contractions were 4 3 2 i 2 3 4 kg. io4A7V=i.6 1.4 0.9 o .9 1.5 2.1 30. Rods in metallic sheath. — In the data thus far obtained the value of E is throughout too low, showing that the section of the rod has not been uniformly stressed. A final modification was therefore made as shown in figure 34, in which the rod r (to be tested) is rather loosely surrounded by a rigid metallic tube or sheath rigidly screwed into the bar FF of the inter- THE AID OF THE ACHROMATIC FRINGES. 57 ferometer. This tube is closed at t with a tightly -fitting screw-plug provided with a conical depression to receive the thrust P from the point q of the offset 5 s (fig. 30) . The rod r is thus compressed between q and the rigid abutment A of the apparatus, and projects but slightly (i or 2 mm.) beyond the tube b. Special sheaths b are provided, fitting neither too tightly nor too loosely, for different diameters of rod r ; or a number of coaxial tubes, b, may be tel- escoped for the purpose. This device gave more satisfactory results at once, and with such bodies as hard rubber showed the change of modulus with stress, the occurrence of hysteresis, and viscous deformation. It is particularly interesting, inas- much as it gives the apparent value of the modulus under each of these conditions. The successive contractions (AAr) and modulus values for hard rubber as found in several successsive series are exhibited in table 2 and figures 35 and 36. In the first series the relatively large micrometer displacements, A7V, are probably due to crushing or to fitting the unstressed rod to the abut- ments of the apparatus under increasing pressure. Thereafter these large contractions do not again occur. The moduli obtained from triplets of obser- vation between 3 and 4 kg. gradually increase to a fixed value. In the second series the rod which had been loaded for some time (see table) with about 40 kg. per centimeter shows the limiting value of moduli found, E = 4-4X io10. We may contrast with this the small modulus when the load is but i to 2 kg. In the third series, beginning with a rod but slightly stressed, a low value of the modulus at first appears, but it soon reaches the limiting values again. In figures 35 and 36 the contractions (the numerals show the loads) are given in succession. With the exception of the necessary break at the beginning of the second series, the work is continuous without modification of adjustment. As contractions are positive the rod is gradually becoming shorter and more viscous. The data are throughout consistent, much more so than was ex- pected. For instance, the effects of the removal of weights at b and c are practically identical. The triplets in figure 35 all show an upward slope or continuous viscous contraction of the rod under large end-thrust. In series 58 DISPLACEMENT INTERFEROMETRY BY TABLE 2. — Modulus of hard rubber. End-thrust apparatus with sheathed rods; .£, = 2.45 cm.; 2r=o.36y cm.; .4=o.io6 cm.2; 1=45°; 2^ = 10.3 cm.; 2R'=j.o cm. Permanent load I kg. each. Achromatic fringes. "b o "b 0 "b o p i— i X b M p M X fe b M p | b M "5t X *~H X ^^« X ^ K] "3 K) ^ K) kg. c?w. kg. CfW. kg. cm. ioh30m 0.5 -50 I2hom **2 105 4hiom 2 49 I.O o 4 128 4.41 3 87 I 2 90 3 108 4 124 •3-32 3 164 4 129 3 103 4 219 2-53 3 1 08 4.41 4 126 3 198 4 130 3 105 4-47 (0 34 228 206 3-70 (2) J 108 130 4-47 (3) I 127 105 *3 204 3 109 4 127 4-37 4 230 4.18 4 129 3 105 3 209 3 109 4 127 4 233 4T 1 4 129 3 1 06 4-43 3 2IO •AO 3 109 4 128 4 234 2 116 1 3 106 3 211 4-i3 I o 1.67 2 62 ioh45m 4 235 I2hl6m 2 48 J I o 1-77 (b) 4h30m 2 46 (c) 1 11.20- 4 240 (a) t2 47 5 167 ***4 149 3 85 ] 4 148 5.12 5 169 5-34 4 I2O [3-41 5 167 4 153 3 99 j 4 149 5 170 5 4 5 1 68 150 1 68 5.20 (7) I 4 153 171 153 5-43 3 58 94 (d) ] 4 150 5-25 5 170 5-81 \y) 5 165 3-63 5 1 68 4 153 4 146 4 151 3 I2O 2 73 * Slight adjustment. ** After adjustment for larger fringes, t Next day. J Two days later. *** 2 hours later. 6 and 7 a special collection of data was investigated two days later at the highest loads which the apparatus admitted, 4 to 5 kg. These are shown after d (fig. 35) and with a further lapse of time after e. It is interesting to note that whereas the contractions under 5 kg. are relatively stationary, the con- tractions at 4 kg. increase in the successive loadings and hence E also in- creases. Otherwise the behavior, allowing for the fact that the rod had been stressed for several days, is about the same, cast, par., as before. In figures 37 and 38 results1 with special reference to hysteresis are shown as a whole, indicating the three successive loops terminating at r, s, t, for loads between i and 3, i and 4, and i and 5 kg. on the 0.106 sq. cm. of section. It is difficult to interpret the individual data, because clearly their nature is 1 As the data are sufficiently reproduced by the curves, the numerical tables have been removed. THE AID OF THE ACHROMATIC FRINGES. 59 exceedingly complex. There may be some instrumental error; but it is interesting to note that in figure 37 the rod subjected to variations of loads "between i and 4 kg. shows gradual expansion at the lower pressures. Just as the loops are evidences of hysteresis, so the grad- ual upward trend of successive loops between the same end-loads is evidence of viscosity. Finally I collected the values of the modulus E, obtained under different loads P in the successive series of experiments. In each series the stable value was reached between given alternating end-pres- sures, gradually, as already explained. The mean values are Loads=i4.2 23.6 33.0 42. 4 kg/cm2. io-10E=i.8i 2.82 4.00 5.11 results which are exhibited in figure 39. They hap- pen to lie on a straight line. Although these mean values are unequally influenced by the character and number of the experiments made, the result as a whole disarms suspicion. It is improbable, in other words, that data which have shown such detailed consistency as appears in figures 35 to 39 should be seriously influenced by imperfections of apparatus or of method, though it is possible that at the higher pressures the sides of the hard-rubber rods may have expanded into and been sustained by the walls of the rigid sheath b (fig. 34). In the above estimates the rate R at which the modulus E increases per kg./cm.2 of pressure would be, nearly, 5.1- 1.8 40 :=io10- IO 10 42.4-14.2 The rate R is excessive as compared with subsequent values. 3 1 . Same. Thinner rods, hard rubber. — The suspicion left in the preceding experiments that the marked increase of E was possibly due to the lateral ex- pansion of the hard-rubber rod, so that it more or less filled the rigid sheath at the highest loads, induced me to repeat the work with slightly thinner rods. The former case would make E approach the bulk modulus. The rods last used were therefore turned down on the lathe until their dimensions were 2L = 4.9ocm. 2r = o.35cm. A =0.098 cm.2 They were then tested on the interferometer for cyclically increasing and decreasing loads P, as shown in table 3, where AW is the micrometer equiv- alent of elongation. A few extra triplets were added. The results are also given in figure 40 and show a beautiful case of hysteresis with gradually increasing loops. This hysteresis is in no way less accentuated when com- pared with the preceding cases. 60 DISPLACEMENT INTERFEROMETRY BY TABLE 3. — Hard rubber. Thinner rod. 2L=4.go cm.; 2^ = 0.353 cm.; .4=0.098 cm.2,- other data as above. Load kg. permanent. R/R' = 10.3/7.0 = 1.47. p A./Vxio6 Exio-10 P ANxio5 Exio-10 P AATxio8 Exio-H' kg. cm. kg. cm. kg. cw. I 0 2 725 3 1150 2 565 I 5 1. 60 4 1660 | 3 1103 2.15 2 585 5 2070 3.06* 2 670 3 H35 4 1800 [ I -20 >i 62 4 1620 ) 5 2225 2 575 i . ^ ' *_ 5 2035 3-18* 4 2590 3.76** 3 1125 4 1795 J 5 22OO 4 1605 2.52 3 1335 ™ 3 1260 2 730 4 2460 1 I 15 ) 3 292O 2-53 I o 1-58 4 2555 J 2 605 * Loads 4, 5, 4 kg. ** Loads 5, 4, 5 kg. The values of the modulus E were then computed from the triplets in suc- cession. One may note that the succession 5, 4, 5 kg. gives a much larger value of E than the contrasting succession of loads 4, 5, 4, kg. One can not, there- fore, expect a smooth march of values unless this difference is systematically included, while it would be exceedingly difficult to even conjecture a rational method of quantitative interpretation. With this conceded, the mean values Were Load= 15 25 35 is l 42 2 — I — 2 455 2.1 Id 5-4-5 165 5-9 j I —2 — I 465 2.1 j 37. Apparent yield within the apparatus. — It remains to determine how far the above apparatus may be made rigid. With such an end in view, steel rods of the dimensions 5.o3 cm. 2r = o.37 ^=0.107 cm.2 THE AID OF THE ACHROMATIC FRINGES. 67 were put in the sheath. In this case (P in kg.), if C = 98 is the constant of the apparatus, i o- 6E = C/ (AAT/AP) or A7V/AP = 49 X i o- 6 about one-half the smallest division of the micrometer, or a little over one fringe (1.2), is all that may be looked for. The results with the fixed offset were larger, 3 to 4 fringes ; or at the micrometer P = 4 — 5 — 4kg. io5AJV/AP=io to 30 io-12£ = o.5 to i.o Thus E is about one-quarter of the true value and three-quarters of the displacement is apparently in the apparatus. Under lower loads the con- ditions are not essentially different. I obtained in the triplets, for instance, P=i — 2 — i kg. io5A./V/AP = 33 to 50 io-l2E = o.2 to 0.3 largely influenced by errors in setting the micrometer, and the increased uncertainty of the seat of the rod. The fixed offsets (ss, figs. 45, 46) were now removed and replaced by the loose offsets with conical plugs and sockets (figs. 28, 34). Observations were made in triplets as before, showing P = 4 — 5 — 4kg. io5A./V/AP = 33 to 34 io-1<2E — 0.23 to 0.30 i — 2 — i 27 to 40 0.24 to 0.36 These results are identical and substantially the same as the preceding set. In a series of decreasing pressures I found P= 5 kg. 4 kg. 3 kg. 2 kg. i kg. io5A/V/AP=i95 iS5 125 75 o IO~l~E = O.24 O.32 O.2O O.I3 so that it is not until all but 2 kg. are removed that the apparatus registered a yield or dislocation of seat. In case of the loose pulley there was no flexure of the bar F; the slit-images remained in coincidence, and the fringes strong and clear at all loads, without changing their inclination. Hence, since the above mean AA/"/AP = 0.0003 7 cm., and as x = EAN/iol* = 0.000098 cm. is to be ascribed to the steel, 0.00027 cm. per kg. may be considered as yielding coefficient of the apparatus; i. e., about 0.0003 cm. should be deducted from all deflections per kilogram of load. The endeavor was now made to ascertain where this yield is to be located. It was found that by loading the pulley standards b,b' (figs. 28 and 29) no appreciable displacement of fringes was produced, provided the load was the same on both sides b and b' '. This is the case in the above experiments. On loading b or b' alone, however, the yield was quite marked, showing io5AA/YAP = 25 in case of loads of i and 2 kg. But this non-symmetrical method of loading is never used. In a further examination of the apparatus it was found that vertical or longitudinal slight pushes on the bed-plate (tripod) were ineffective, but that a cross-push, even if a mere touch, as this directly tends to change the angle a of the bar F, was very appreciable. Somehow, through the inter- 68 DISPLACEMENT INTERFEROMETRY BY action of the bifilar and the load, stress or spurious torque about the vertical (hence Aa) is introduced; but it is extremely difficult to state just how when all loads are symmetrically vertical. To obviate this the permanent parts of the apparatus should be cast in one piece. An interesting corroboration of these observations is given by the pendu- lum oscillations of the loads. These produce (even for loads as small as i kg.) marked vibration of fringes if the load vibrations are transverse to the rays, while vibrations are ineffective if longitudinal (in direction of the rays). Noticing the behavior of fringes for the first time, I supposed that the centrifugal accelerations introduced by the vibration might be the cause. In fact, if the length of the compound pendulum (load w, figs. 28 and 29) treated as a simple pendulum is L and its mass M, the centrifugal force at any displacement 5, corresponding to the displacement velocity v, is Mvz ly __ _ _ __ But if s = A sin ut, then v = Aucos o>/, and thus since co2 = g/L. This force is a maximum at / = o sec., or FQ = Mg. AZ/L2. If A = i cm. and L=is cm., F0 is but ^ of the weight of the load Mg, say a maximum not exceeding 2 5 grams of weight. Since the whole displace- ment is but a few fringes, this small fraction could not be discernible with a body like the steel rod above. Thus the alternating transverse stress pro- duced by transverse swinging alone can account for the observed effect. 38. Ocular micrometer. Collimator micrometer. — These methods of meas- uring the displacement of fringes have been discussed in Chapter I, § 4, 5. It was of interest to test them here. The scale on the ocular plate inserted divided the field width into 100 parts, the division being in o.i mm. The dis- tance apart of the achromatic fringes was a little more than this (1.4 cm.) The correspondence could be made exact by rotating the auxiliary plate (bar F, fig. 28) about a vertical axis slightly, but the adjustment was not necessary here. These excessive tenth millimeters thus correspond roughly to an elongation zAl of both rods, equal to the mean wave-length of light, or more accurately, _, 2A*=-X where zR is the distance apart of the interfering rays ab (fig. 27) and 2R' the distance apart of the rods rr'. If the size of fringes is known on the ocular micrometer, they may be counted by their displacement along it, since the central achromatic fringe is always distinguishable and serves as an index. But the width of fringes, if the laboratory is not quiet, is hard to measure, for they quiver or vibrate. It is easier to express the displacement A# of fringes in the ocular in terms of AN, the corresponding displacement of the micrometer of the interferometer. THE AID OF THE ACHROMATIC FRINGES. 69 By direct comparison the following values were found: 480 405 460 cm. io5AAf=i45 165 125 i55 cm. io4Xratio= 32 34 31 34 The mean value is AAV Ae = 0.003 2 8, or A is thus over 300 times larger than AN. The cyclic experiments with the steel rods previously used showed a lack of coincidence of the ascending and descending graphs which must here be spurious. I obtained, for instance, AP=i 2 3 4 5 4 3 2 i kg. io3Ae = o 90 170 250 330 275 190 no o cm. As the ascending graph is fairly regular, the apparent modulus may be found from it. Three such series gave mean values of A If we take the first case (/ = o.i, Z' = o.5, L = 5o) for the bifilar ju = 73 74 DISPLACEMENT INTERFEROMETRY BY X io~3 (since the load of the bifilar is 2m if m is the mass of each ball) and put M=io3 grams, w = i gram, 2R= 10 cm., /" ' = 30 cm., 7=- -2— AAT = 4.8Xio-6Xd2XAW, nearly 3oXio3Xi 10 or If d is estimated as 3 cm., then AN = 0.0016 cm. With the given interfer- ometer and reasonable estimates as to the other magnitudes, one should therefore obtain nearly 40 achromatic fringes (even with the bifilar as stated) for the attractions of i kg. There would be no gain, in case of the bifilar, by increasing the mass m at the ends of the needle; for the modulus of the bifilar increases as m, which would therefore cancel the m in the denominator of the expression for 7. But if the system is floated in water, m may be increased with advantage indefinitely, while the load of the bifilar is kept constant by providing a corresponding float. If this bifilar load is w', we therefore have d- II' cos * ~ — ni s Mml" gL •v = — ~ — ni s -- AA " g Inserting the data of the apparatus of the next section, M=iog. w = 3og. m' = 2 cm.2- L = $o cm. 2 = 45° 2.R=iocm. 3oXio3X3o 50 10 or AW = 0.01 7 cm. per attracting kilogram. With a reasonable size of float there is no difficulty in increasing the attracted mass m to over 60 grams and the attracting mass (with a slight increase in d) to 10 kg., so that values of AN of the order of a millimeter are not out of the question. The difficulty with the method lies in the simultaneous increase of the period of vibration of the needle, and this seems fatal; but I thought it worth while, nevertheless, to give the method a trial. 42. Observations. Floating system. — Figures 52 and 53 represent the floating needle submerged in the narrow trough AB provided with two win- dows of plate glass w w', through which the interfering beams enter and leave, nearly at right angles to the co planar mirrors n and n' attached to the needle. This consists of a tube of aluminum, about 30 cm. long and 6 mm. in diameter, with balls at the ends m and m', pairs of 30 grams to 60 grams each being admissible. The two hooks h and i carry the floats F, F', test tubes as much THE AID OF THE ACHROMATIC FRINGES. 75 •as 15 cm. long and 2.5 cm. in diameter, but to be changed in capacity with the balls m,m' used. The stems of the hooks, screwed and sealed into the tube rr, carry the mirrors n, n' . The needle is suspended from a torsion-head by the bifilar of silk fiber kl, also com- pletely submerged in the water-bath. The hooks k and I are provided with a thin flat sheet-metal link, by which the bifilar may be appropriately spaced. The needle system is so adjusted as just to float. It is then weighted by i or 2 grams to sink it. The weight in water may be measured at K. In view of the weight of needle, the mirrors n,n' may be rigidly connected by a very nar- row strip of this plate glass, which facili- tates adjustment. The attracting weights M and M' up to 5 kg. were used in pairs M,Mi and M'.M'i, on each side suspended by a steel belt from a pulley overhead in such a way that when MM' are in Place M'iMi are raised out of effective reach. The chief difficulty was encountered in floating the needle. When this was done the whole tank was fastened in place on the interferometer, as the torsion-head at k is attached adjustably to the tank. The fringes were found without much difficulty ; but they were in incessant motion, owing no doubt to eddy currents produced by temperature differ- ences. After many trials I concluded that measurements would be untrust- worthy and further trials were therefore abandoned. The experiment is in fact too difficult for a single observer and would be feasible only in an environment of perfect quiet and constant temperature. 43. Expeditious fringe detection. — In work like the present it is necessary to find the fringes quickly under considerable disadjustment of parts. In this case, if the auxiliary mirror m (fig. 4, Chapter I) can be manipulated, the two images in the telescope T are first made coincident by adjusting M', in which case the rays are parallel as they enter T. The mirror m is then rotated around the vertical and the horizontal axis, until, to the eye, the spots of light coincide locally on the face of M'. Fringes when found by moving the micrometer here are then strong. If. mm can not be interfered with, the rays are first made parallel as before by the coincidence of images in T. Thereafter the mirrors M and M' are rotated around a horizontal and a vertical axis in parallel (i. e., successively or alternately retaining their parallelism) until the spots of light on M' again coincide, locally, to the eye, or better, when caught objectively on a screen. It is clear from figure 2, Chapter I, that the remoter ray from M is displaced on the screen more 76 DISPLACEMENT INTERFEROMETRY BY rapidly than the nearer ray from M' and therefore parallelism with local coincidence of rays on M' is generally possible. If the parallel rays T\ and T2 which coincide in T are too far apart, no fringes can be found, even when the path-difference is annulled. 44. Heavy needle in air. — The needle was now deprived of its floats and other appurtenances, provided with somewhat lighter balls (w=i8 grams each), and suspended in the same float-vessel in air. Since in this case AAf is independent of m, and as the dimensions were about of the same order given in the example above (//' = o.c>5 cm., L=5o cm., M=io3 g., 2R=io cm., /" = 3o cm.) with a somewhat larger d = 6 cm., AN = 0.0004 cm. per kilogram of attracting load was to be expected; i. e., about 10 achromatic fringes. The adjustment is quite difficult, since the small mirrors must not only be approximately parallel, but must be so spaced and inclined as to receive the component beams and to reflect them through the mirrors of the inter- ferometer. This was accomplished with some patience and the achromatic fringes were found. But here again they proved to be in incessant and rel- atively rapid motion, so that they swept continually through the field. Though observed for some time, on different days, they were never found .sufficiently quiet to admit of the above measurements. It was impossible, in other words, to eliminate the air-currents within the shallow envelope sufficiently to warrant counting at the rate of 10 fringes per attracting kilogram. In the summer I think this would have been feasible, as there was no other drawback militating against the completion of the experiment. 45. Light needle in air. — In view of the failure of the heavy needle, I went to the opposite case of a relatively light needle, weighing when loaded but 1.49 grams. This was made of a rigid shaft of straw (mm', fig. 54) 25 cm. long, the ends being slit slightly into four symmetrical segments each, which receive the two shots m and m', additionally secured with a little wax. The two light mirrors b and c were differently mounted; b on a fine pin d, snugly fitting corresponding perforations in the straw, was thus capable of rotation around the vertical axis (d) and moving up and down slightly. The mirror c, however, was mounted on a thin elastic strip of alu- minum, clasping the shaft as shown in section at e'c', and thus capable not only of rotating on a horizontal axis, but of being placed at different distances (moving right and left) from d to accommodate the rays of the inter- ferometer, to which b and c are to be normal. The needle is swung from a quartz fiber q and a strip or hanger of elastic aluminum a (see a',q'), which clasps the shaft. Hence the latter may also be moved endwise to be balanced and rotated around a horizontal axis till b and c meet the rays normally. These operations are completed by trial before the needle is definitely hung, preferably in a broad beam of sunlight. No great accuracy is required. The shallow case is made of two plates A, A' (figs. 55, 56) of Ys-inch plate THE AID OF THE ACHROMATIC FRINGES. 77 54 6 a' P (>- >i .n )! 11 --30 glass about 32 cm. broad and 45 cm. high, spaced by two strips if, 3 cm. wide, of thick (9 mm.) plate glass, reaching not quite from top to bottom. Six steel clips h,h,h,k,k,k, (such as are used for binding pamphlets) held these strips in place and also clasped securely on the outside two wooden strips of about the same width and one-half inch thick. At the top of the wooden strips, two nipples, s and t, of >^-inch gas-pipe, projecting normally to the strips, served for the hanging of the case and needle from a firm wall- bracket. It would have been preferable to use the wood strips (without the glass strips) clasped between the glass plates A, A' as spacers and hangers at once, and this was eventually done. The bottom of the case is subsequently to be closed from below by a strip of felting gg. To diminish the space within the case two thin cloth-covered wooden boards w,w' are inserted from the top. The quartz fiber q, to carry the needle mm, hangs from a long Y8-inch brass rod d, which may be raised or lowered in view of the sleeve e, held by a separate ad- justable arm without. The rod d must fit the perforation in the cork nicely, so that the former may be smoothly raised or lowered and held in any position by virtue of friction. To swing the needle this is first placed on cork Y's below the opening of the case A, A', the felting gg having been removed. The quartz fiber is then lowered on the long stem d, until the lower hook on the fiber is in position to grasp the clasp on the needle mm, still below the case. The needle is then cautiously lifted by raising d until it has the required position relative to the interferometer, about as shown in the figure. After this the felt strip gg is inserted to close the case, and the necessary adjustment made at e and d to swing the needle freely in the restricted space provided for it. Observations were made with the needle at some length. The quartz fiber used was L=iy cm. long; the distance d apart of attracting weights M and attracted weights m was 6 cm. Hence for M= 10 gr. ^ ' T jo1 rf r^f- I//" ] 55 c [ b c _t*d a _ tn! 7 = 23Xio3X75 2X17 10 Thus, if r= 10 2 cm. r= io~3cm. AA/"=i.iXio 6 cm. per kg. of M 7 = 4.6 sec., AAT=i.iXio~2 cm. per kg. of M T = 465 sec. The first case would then correspond to but one-fortieth of a fringe; in the second case there should be 250 fringes per attracting kilogram. If the tenacity of quartz be taken as i.sXio3 kg. per sq. cm. the latter 78 DISPLACEMENT INTERFEROMETRY BY filament (r=io~3 cm.) should still hold 4.5 grams, or much more than the weight of the above needle (within 2 grams.) As the moment of inertia of the needle is2Xo.75Xi32 = 253, and if the mod- ulus of torsion be computed from the slide modulus w = 5Xion, the periods would be as given above, the second being nearly 8 minutes. The period of the above needle was estimated at about 2 minutes. This would give a modulus of torsion about 0.7 and make AN = 6fXio~5 cm. per attracting kilogram; i. e., about 17 achromatic fringes per kilogram were to be expected. Having mounted the needle as stated, the fringes were found without much difficulty and the image of the wide slit (i. e., the reflected beam seen in the telescope) was almost quite stationary, the light needle being thus adequately damped. But within this virtually stationary slit-image, the fringes (preferably made horizontal) continually wandered up and down, showing that micrometric vibration had not been eliminated. The experi- ment is a very impressive one, but as the drift is still much larger than the 17 fringes per kilogram to be anticipated, any attempt at measurement is again idle. Whether this drift is due to temperature or to the tremors of the laboratory would be difficult to state. Cutting down the intensity of the interfering rays (which need not be strong) made no difference. When the case is open above, there is no difficulty in finding the position of equilibrum of the needle symmetrically to the glass walls of the case. This position is assured by the parallelism of the images of the needle in both faces with the needle itself, and their distance from it. When, however, the wooden boards are inserted, the needle does not seem to be in stable equi- librium in the symmetrical position. It tends to move either into one or the other extreme of oblique positions at which the balls touch the plates of the case. Believing the phenomenon to be electrical, I placed a radium tube in the vicinity of the case for some time, but this made no difference. Damp cloth did not change the result. This was particularly true in the earlier experiment, where >^-inch plates were used for the case in the absence of thin plates, and in which one plate was thicker than the other. One would thus be inclined to interpret the instability as possibly due to the gravitational attraction of the residual disk. Considering the case as that of a point mass confronting an infinite disk, the potential would be y(Cd=x) 27ro- and the force 2^70- per gram attracted, which is a little above the mass of the ball. Thus, if . (0=1.7X10 6/o. 024 = 7.1X10 8 rad.) by the glass plates; and yet this tendency is marked. One may suppose, therefore, that each end of the needle is attracted by the nearer glass plate independently. This will give a superior limit enormously larger than the preceding estimate ; for the force is now , if (7 = 3X0.6 = 1.8 the mean thickness of plate bearing 0.6 cm. Hence the torque is since the length of needle is 26 cm., and finally 0 = 4.8Xio~5/o.024 = 2Xio~3 radian Even this excessive estimate, however, fails to account for the result; for the deflection is but a little over a half degree. Moreover, the short brass cylinder in question (length 3 cm.) showed a similar tendency to take oblique positions not corresponding to the torsion-head, as the long needle. 46. Summer experiments. — It is obvious that a fair trial of the apparatus can not be made in an artificially heated room. For this reason experiments in a semi-subterrranean room of the laboratory were reserved for midsummer. In the summer installation in a subcellar at constant temperature, with a few improvements of apparatus (the mirrors being readjusted, etc.), the needle was without difficulty made to take a stable position midway between the glass plates, subject to the torsion of the fiber. Some trouble was expe- rienced in finding the fringes, owing to incidental causes. The adjustment is made difficult in view of the definite distance apart of the small mirrors to which the breadth of the ray parallelogram must conform. With the microm- eter at 45° the latter is very limited in its displacement. I later attached a special micrometer with three identical pairs of parallel V-mirrors (the lat- ter at 90°) similar to the design shown in figure 2 1 . The middle V-mirror is movable in a micrometer. This, when the mirrors are parallel, has the ad- ditional advantage of being independent of slight changes of inclination 80 DISPLACEMENT INTERFEROMETRY BY in the micrometer. The displacement of mirrors is now virtually parallel to the rays and no difficulty in finding the fringes need occur. Naturally the mirrors must be good, there being now four additional reflections in each ray; and this V-micrometer must be accurately adjusted for parallelism of mirrors. With the fringes found, there is now no difficulty in showing the attraction of gravitation. In fact, an iron brick moved on a small truck, near the shot at one end of the needle, grips these balls very much like a magnet acting on the pole of a magnetic needle. By approaching and withdrawing the iron mass on one side, the fringes could be put in regular and uniform vibration over enormous (relatively speaking) arcs as measured by fringes. Thus in an incidental experiment the micrometer reading was 0.255 cm- with an iron mass near and slow vibration and 0.026 cm. (eventually) with iron mass remote. Throughout the whole of the experiment the fringes were under the perfect control of the micrometer. A more systematic experiment was then made by testing the attraction of a lead ball 5.43 cm. in diameter and weighing about M = Q5o grams for the shot (at the end of the needle) weighing w = o.6i gram. M was moved on a circular track with stops to a distance of ^ = 4.24 cm. (between centers of balls) from the ball of the needle, alternately. The position of the large ball M was reversed every 10 minutes, but the period of the air- damped needle can not have been less than 18 minutes. The case is, then, that of a forced vibration under constant force and a large logarithmic decre- ment. The observations are given in table 4, the reading being made every minute, beginning with the equilibrium position (M in the neu- tral position). If these data of the displacement x of the mass m are con- structed graphically it will be seen that the motion of the needle is nearly dead-beat. The successive arcs of vibration increase, and from the limiting distance between elongations the attracting force could be computed, if the torsion coefficient of the quartz fiber and the logarithmic decrement were known. The limiting arc was not reached, owing to incidental reasons. From static experiments made during hour intervals this elongation was found to be about 0.116 cm., or a departure of the shot in at the end of the needle from its position of equilibrium of 0.058 cm. in response to the attrac- tion of M. If I is the semi-length of the needle (between centers of shots), the micrometer displacement A/V and the displacement A# of the mass m are given by the equation cos i/b = 0.89 AAT where b is the breadth of the ray parallelogram and * = 45° the angle of in- cidence of the interferometer. Thus the micrometer displacement is of the same order as the displacement of m, and if the latter is 0.116 cm., we should have o.i cm. THE AID OF THE ACHROMATIC FRINGES. 81 TABLE 4. — Gravitational attraction. M = 950 grams, m = o.6l gram. R = 4.2cm. Period (damped) about 18 min. Length of needle, 25.2 cm.; weight (total), 1.9 grams. Adjustment Time. xXio3 Adjustment Time. xXio3 Adjustment Time. *Xio3 min. cm. min. cm. min. cm. Ball M o 26 Ball M 21 50 Ball M 4i 94 right i 3i right 22 60 right 42 101 (from equi- 2 35 23 69 43 in librium) 3 40 24 79 44 126 4 46 25 90 *45 139 5 54 26 98 6 58 27 105 7 64 28 no 8 68 29 117 9 72 Ball M Ball M turned 3« 124 turned 10 77 Ball M 31 124 Ball M left 32 123 left ii 78 33 I2O 12 77 34 H7 13 74 35 H3 14 69 36 1 08 15 64 37 105 16 61 38 IOO 17 56 39 96 18 53 Ball M 19 49 turned 40 93 20 48 * Out of field of telescope. As the micrometer reads to io~4 cm., 1/1300 part of the attraction between M and m = o.6i gram could therefore be detected; i.e., the attraction of 950/1300 = 0.73 gram, or per interference fringe well within one-third of this, for the given quartz fiber, which was not specially selected, and distance R. This is equivalent to the attraction of two tenth-gram masses per centi- meter of distance per fringe. Apart from the measurement of the torsion coefficient of the fiber, there is, however, a real difficulty involved, and that is the occurrence of marked drift in the needle. It is only incidentally that the fringes are found at rest. The chief contributory cause of this is no doubt the occurrence of motion of air around the needle provoked by small differences of temperature, result- ing (for instance) from illumination. If the possible accuracy of deflection measurement is to be of any value, therefore, the apparatus must be kept in the dark, except during observation . Fortunately, the achromatic fringes require little light. Even then a closed case which can be exhausted of air is essen- tial, for such radiometer forces as may enter would in any event be differential, seeing that the mirrors are symmetric and the illumination is not subject to alternations like those in the table. It is probable that in case of the above regular method a thicker quartz fiber and a greater distance R would con- duce to the best results, since A N is the least difficult quantity to determine. Measurements could not be attempted in time for the present report, but the question may be asked whether it is not possible in the present case to 6 82 DISPLACEMENT INTERFEROMETRY BY determine the attractions in terms of the mere acceleration of balls resulting. With an ocular micrometer this would not be difficult, as the fringes move slowly enough so that the position can be sharply specified; but with a needle of long period in vacuum, the screw micrometer would also be available. If there is no damping we may write yMm/R2 —ax = zma where a is the acceleration, x the displacement of m, and a the torsion coeffi- cient, referred to the displacement of m, t the time, supposing the needle starts from rest, and the gravitational force is applied at t = o. If at the out- set we may put x — vt/2 = atz/2 an equation whose interesting feature is that if t is kept very small (which should be possible with an ocular micrometer and the achromatic fringes, a fine quartz fiber presupposed), the term involving t may be neglected and the experiment interpreted as a case of uniformly varied motion, in which For instance, if R = 5 cm., M =io3 grams, and 7 = 6.7 Xio~8, and if* = sec. is admissible, a=i.3Xio~7 cm./sec.2 and the distance traversed in 100 sec. would be 0.0065 cm-» weU measurable on the interferometer, quite so if the work is done reciprocally and the interference fringes are used individually. The theoretical error will be a minimum if m is as large as the fiber can safely carry and t as small as possible. On the other hand, x is independent of m, and if t is to be kept small, the result may be compensated in a large M/R2. The measurement is thus to consist in keeping the fringes at zero by moving the micrometer screw for the small interval t during which the weight M acts. The constant would then follow from the micrometer reading M and R only, all other quantities entering secondarily as corrections. The ex- periment seems well worth while. THE AID OF THE ACHROMATIC FRINGES. 83 II. USE OF THE RECTANGULAR INTERFEROMETER IN CONNECTION WITH THE HORIZONTAL PENDULUM. 47. Introductory. — In 1915 and in the reports of the Carnegie Institution of Washington, No. 229, Chapter I, part 2, pp. 30 et seq., I adduced a method for the application of the displacement interferometer to the horizontal pendulum with a graphic exhibit of the results obtained during a series of months. The concave-mirror design by which the spectrum interference ellipses were made available showed a very satisfactory performance, in spite of the fact that deformations of the pier to which the pendulum was attached were local disturbances and excessive in amount. The attainable accuracy was such that for moderate constants in the installation of the pendulum, an inclination of 3Xio-4 second of arc should have been registered per vanishing interfer- ence fringe (ellipse), or about icr3 second per io~4 cm. of displacement of the micrometer. The inclination of the line of suspending pivots was here about i° to the vertical. A smaller angle would have correspondingly increased the sensitiveness. The apparatus, however, required a space about 2 meters long between the extreme mirrors for its installation. This is in a measure a disadvantage, since small changes of temperature in the brackets and supports, as well as in the pier, would interfere with the full realization of the precision of the method. In this respect the rectangular interferometer with an auxiliary mirror is to be preferred ; for here all the necessary parts may easily be placed within a distance of i foot from the wall of the pier carrying the horizontal pendulum. If the achromatic fringes are used, these are straight and in- tense, so that photographic methods are available, while for visual observation a gas flame would give sufficient light. The sensitiveness under similar con- ditions would be slightly smaller, but not enough to cancel the advantages specified. 48. Apparatus. — The old horizontal pendulum formerly described was again used. It was made of thin steel tubing, and in this respect, since its plane was nearly in the meridian, may be subject to change of the earth's magnetic field; but as my object here is merely the trial of a method, these annoyances are of slight consequence. Figure 57 gives a sectional plan of the pendulum installation and figure 58 a front view of the pendulum HH alone, on a somewhat reduced scale. Its general shape is that of an isosceles triangle and the distance from the line of pivots tt' to apex B about no cm., while the distance between pivots was 97 cm. The pivot supports 55' are fine screws ending in hard-steel points, which enter a glass-hard steel socket (below) and a steel groove 84 DISPLACEMENT INTERFEROMETRY BY (above) . The line / 1 ' of the horizontal pendulum can thus be given any incli- nation to the vertical, while the rods p,p' which receive the screws 5,5' may be moved mormally to the wall of the pier PP', inward or outward, and clamped to secure parallelism between the pier PP' and pendulum HH'. The apex B of the pendulum is also provided with a clamp, holding vane D submerged in an oil- vat v for damping. The whole pendulum is inclosed by a flat case CC' of tin plate provided with a plate-glass window at g, through which the auxiliary mirror m of the interferometer may be seen. This is attached to one or two vertical tubes h,h' of the pendulum, adjustably, so that it can be moved up or down, and rotated slightly above a vertical and a horizontal axis. The interferometer consists essen- tially of the 4 plate-glass mirrors M,M', N, N', all but M being half -silver, the collimator (beyond L) and the tele- scope at T or Tr, t" being a telescope support. The collimated white beam L is thus separated into the component rays LNmadt and LbN'mcT, to be observed at either T or T'. M' is on a micrometer slide with the screw normal to the face of the mirror. All mir- rors must be capable of slight rotation about horizontal and vertical axes and the silvered faces all lie towards m for compensation of glass paths. The rays leaving M' for T must not only be accurately parallel, but locally (visible as spots of light) nearly coincident, as specified above (Chapter IV, § 43). Otherwise the fringes will be weak or invisible. The telescope T should be provided with an ocular micrometer (centimeter divided in tenth millimeters) standardized by aid of the sliding micrometer at M', since the main purpose here is the measurement of small angles. More- over, the image of the wide slit of the collimator adapted to the use of the achromatic fringes should be placed at right angles to them, with the ocular micrometer so placed as to read from end to end of the slit-image. A very fine wire beam across the slit gives the fiducial line relative to the ocular micrometer. Figure 59 shows the general arrangement, 55' being the wide, oblique slit-image, //' the achromatic fringes, w the image of the fiducial wire across the slit, and ssr the ocular scale. Of course the fringes may be made horizontal or vertical ; but this requires much adjustment or else compensa- tion, and is therefore an unnecessary complication of the preliminary work. With this fiducial mark at the collimator (which is permanently out of reach) , if the telescope is accidentally shifted, or temporarily removed, it maybe THE AID OF THE ACHROMATIC FRINGES. 85 replaced without difficulty. It is the telescope, however, which contains the ultimately fiducial scale, and like the collimator it should be held on a standard /" suitably attached to the pier. Similarly the mirrors M and M' ', N and N', fixed in pairs to slides or carriages F,F't are clamped to two parallel horizontal tubes E,E' (/s-inch gas-pipe smoothed, for instance) anchored in the pier. The highest attainable rigidity in the placement of the mirrors M, M', N, N' and of the telescope is essential. At the outset of the work the viscous yielding of standards and braces is quite apparent. When, as in the present paper, the observations are made at T' and not at T, the telescope is conveniently attached to the slide rods E,Ef joined in front at t". 49. Equations. — In figure 60 let pBd denote the horizontal pendulum in the plane of the diagram and dpe the line of pivots prolonged, terminating in e vertically above the center of gravity G. Let the incli- nation of de to the vertical be was directly measured, as the inclination of the line joining the points of the pivots to the plumb-line. Under these circumstances the constants given at the head of the table suffice. Since Aa = o.9 Ae seconds, roughly, the tenth millimeters of the ocular scale are about o.oi second of arc in relation to Aa and the fringes were of about the same size. There would have been no difficulty in making them much larger and therefore more sensitive, as they were clear and strong. As it was, there should have been no difficulty of estimating within io~2 second. The end of the compound pendulum was damped in lubricating oil. This is probably too viscous for refined work, but the purpose here is merely to try out the method. THE AID OF THE ACHROMATIC FRINGES. 87 The earlier observations were discarded, but even after January 14, after which time the apparatus worked comparatively smoothly, instances of displacement within the apparatus required readjustment. These betray themselves in a lack of coincidence of the two wide slit-images (fig. 59) or of the cross-wire w (the slit is really superfluous except as the collimator lens may be so placed as to widen the illuminated field). This is probably referable to the two supports E,E,f which change their parallelism with marked changes of temperature in the room ; or it may have been within the apparatus at M, N, N,'M'. It is difficult to allow for it, and a reconstruction of apparatus is the only resort. The illuminant was an electric arc at a distance of about a meter from the interferometer. This was chosen for convenience solely, as the achromatic fringes can be adequately seen with a Welsbach lamp closer at hand. The observations will for convenience be given graphically. I merely recall that the breadth of the interferometer rectangle was .R=io cm.; the inclination of the pivots of the horizontal pendulum about <£> = o.oi radian; the angle of incidence of rays 1 = 45°. Hence the change Aa of inclination a of the pier will be (if AAf is the displacement of the mirror micrometer and Ae of the ocular micrometer) Aa = ^AAr cos i/2R = 5-gX io~3 a 1 1 ; y~ |[ m 'j 1 t $a 69 / d 1 ^ i =fta 94 DISPLACEMENT INTERFEROMETRY BY The telephonic system was now put in place of the galvanometer of a Wheatstone bridge, the small induction-coil being used as a source of cur- rent. It was then found that the adjusted resistances of the bridge could be changed by no more than a few tenths of i per cent before the even band of fringes changed appreciably to the wave-form aa (fig. 68). But as a dyna- mometer the instrument was still much inferior to the audible telephone. Cur- rents of an order less than an average icT4 ampere would be difficult to detect. 55. Bifilar systems. — To utilize such a system as figure 69 to full advantage it would be necessary to attune the springs ab of the two telephones to the same period, which should then be as nearly as possible identical with the period of the source of intermittent or alternating current. As an earth inductor or a small magneto inductor (single magnet rotating in a flat coil) would have to be used in the latter case, it seemed best to convert the appa- ratus into a bifilar vibrator as shown in figures 70 and 71 in elevation and plan. Here mm' is a strip of thin mirror plate- glass, about 32 cm. long, i cm. broad, and 2 mm. thick, horizontal and in a position to receive the rays NN' of the interferometer (compare fig. 66). Motion of mm, parallel to itself, fore and aft, will therefore produce no effect on the fringes ; but any rotation around a vertical axis will be immedi- ately apparent, as indicated in the above methods for small angles. This strip of glass is supported by the bifilar system ee, e'e', made of a single thin wire of brass, 0.2 mm. in diameter. The ends of ee' are wound around the horizontal screws a, b, which rotate with friction and are supplied with an index and scale n,n' , so that any tension may be imparted to the wire. This passes below under the pulleys c,d, as nearly free from friction as possible, with the object of secur- ing the same tension throughout ee'. Flat clamps//', of fiber and screws,, attach the strip mm to the wire at any height, but necessarily near the middle of the vertical threads, where it receives the rays NN'. The telephonic system consists of the soft-iron horizontal screws hh', similarly attached to m m' by the flat fiber clamps g, g' and the telephones /,/' (omitted in fig. 70). These were made of large flat files, each provided near its end with an appropriate bobbin, kk' , of fine telephone wire, the ends of which are attached to clamps, as already shown in figure 66, with one of the telephones provided with a commutator for reversing its current. The re- sistance of each bobbin was about 140 ohms. The screws //' are used to ap- THE AID OF THE ACHROMATIC FRINGES. 95 proach the telephone magnets tt' as near the soft-iron armatures hh' as possible without overstepping the unstable position, in view of the tension •of the tense wire e, e'. To prevent the sticking of h to t, which is very annoy- ing, small rubber buffers may be placed between. It is not usually practi- cable t approach h to t by more than i or 2 mm. and keep h perfectly free. To give the vibrating system adequate damping, thin wires qq', less than a millimeter thick, bent, and dipping into lubricating oil in small vats p,p', suffice. To change the damping the latter may be lowered or even removed. The fibers e,e' were about 45 cm. long and their distance apart about 29 cm. Their period and that of the vibrating telescope were made about the same, on the average about 0.2 sec., and this was for convenience nearly the same .as the period of the vibrating telescope and of the induced alternating current. It is convenient to insert an extra telephone (resistance about 100 ohms) in circuit, in order to insure against breaks of contact or other discrepancy, when the perturbation of fringes ceases. As a generator an earth inductor with a coil of wire 60 cm. in diameter was at first used. It was turned by a small motor, and by putting a sliding rheostat in circuit the period could be varied from about 0.19 to 0.26 second. To measure the average intensity of current a Siemens precision dynamometer was installed ; but though indicating currents as low as even within one-tenth of an average milliampere, it was not influenced by the earth inductor, though at maximum speed. This was therefore replaced by a small magneto con- sisting of a bar-magnet about 6 inches long, rotating in an oblong coil. By aid of the rheostat and appropriate pulley-wheels large differences of speed could be obtained and maintained at any value, so that periods from about o.i to 0.3 sec. were available. With a period of 0.2 sec., moreover, it showed a deflection on the dynamometer and, being in general lighter and more easily controlled, was preferable to the earlier instruments. The three vibrating systems (mirror, telescope, alternator) thus all admit of an adjustment of their periods, and these should be nearly the same if the elliptic system of Lissajous curves are to be obtained, which is the prefer- able case. A change of the tension of the wires ee' in figures 70 and 71, or any adjustments at the telephones, calls for a fresh search for fringes; but this is not difficult if the spectro-telescope is first used and the admonitions relative to the objective coincidence of pencils entering the telescope, as well as their parallelism, as above explained, are given consideration. These difficulties do not enter when the mirrors can be displaced normally to the incident rays. In addition to the telephone it' in figure 71, coils in great variety were used. The telephones were also placed within and without the rectangle of wires e,e' and in the same or on opposite (as in fig. 71) sides of mm'. But the phenomena in such cases were not dissimilar nor advantageous. For general purposes the mass of the vibrating system mm' should be diminished, as it could easily be ; but the straight blade of glass (in preference to two small mirrors) is a convenience in adjustment and here suffices. 96 DISPLACEMENT INTERFEROMETRY BY 56. Further observations. — Without synchronism in the two vibrating sys- tems (current and telephone), the motion of fringes obtained is practically inappreciable when average currents within the order of milliamperes are treated. This is a curious and at first a disappointing result. As soon, however, as approximate synchronism is established, the sensitiveness of the apparatus increases enormously. It is best for this purpose to vary the period of the motor of the alternate-current generator by the slide rheostat. If the fringes are horizontal and the objective therefore vibrating horizontally across the vertical slit-image, the motion of the fringes is vertical. Hence the hori- zontal band a (fig. 72), in the absence of current, at once takes the form b,c,d,c,b, in succession, with the opposition of rotation in c quite visible. The continuous change of these may be most conveniently accelerated or retarded by controlling the motor of the alternator with the slide rheostat. Since the lines b and d are of different inclination, they will usually show a dif- ference of breadth. Circles appear when the ampli- tude of the objective has sufficiently decreased, and it is advantageous, as a rule, to keep this ampli- tude small, for the phenomenon is then more luminous and brilliant. Very large fringes are not usually desirable, as they are too mobile and may pass out of the field. The smaller fringes are quite satisfactory and more easily obtained. When the tension of the fibers e,e' (fig. 70) is too small the fringes drift, showing that with the varying magnetization there is no persistent position of equilibrium. It is annoying if they leave the field while executing their gyrations, though they may always be restored by moving the micrometer. For mean tensions the higher Lissajous curves 2 13, 3 14 may be obtained, both for the alternator moving at smaller and at larger periods than the vibrating mirror. To obtain them the motor running at maximum speed is gradually slowed down by means of the rheostat, when the forms appear in succession, passing through the elliptic series at mean speeds. This in fact is the best tension for practical purposes. The size of the curves and the brilliancy of the whole display is increased by decreasing the damping or lowering the cups p, in figure 70. A beautiful phenomenon is observed when the magnets t,t' hold the arma- tures h,h' to the intervening rubber cushions and the fringes are fairly large. The slightest vibration anywhere in the vicinity will then cause the even band a (fig. 72) to change to magnificent large roof-shaped or violin waves. This loose contact device could, no doubt, be made useful for observational purposes. Without the vibration telescope such fringes would not be visible, as they overlap during vibration. The Lissajous curves continue to be very marked when additional resist- ances as high as i ,000 ohms are put into the circuit of the alternator. Indeed, they do not vanish appreciably, even for an additional 10,000 ohms, if well / o/t 73 THE AID OF THE ACHROMATIC FRINGES. 97 produced. They do not, however, increase in size in proportion to the de- crease of variable resistance, a result attributable to the amount of resistance and inductance necessarily in circuit, the effect of which is relatively large when the additional resistance is small. To obtain some idea of the smallest average current appreciable, the Siemens dynamometer may be put in circuit, though unfortunately it has a resistance as high as about 1,000 ohms. With the magneto inductor running at a speed of r = o.i7 sec., the Siemens showed a deflection of but 0.02 cm. on the given scale, owing to this resistance. Estimating the average current i asi = C V