UNIVERSITY OF CALIFORNIA AT LOS ANGELES GIFT OF CARNEGIE INSTITUTION OF WASHINGTON EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER By CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University WASHINGTON, D. C. Published by the Carnegie Institution of Washington 43 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER BY CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University WASHINGTON, D. C. Published by the Carnegie Institution of Washington 1915 CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No. 229 PREFACE. The present volume contains applications of the displacement interfer- ometer to subjects which suggested themselves from time to time. Unfor- tunately it was not possible, in the laboratory of Brown University, which is situated on a hill in the middle of a large city, to carry out any experiment to its final degree of rigor. Quiet surroundings, a location free from tremor, and irregular temperature variations would have been necessary. But the development of methods of the kind in question was nevertheless quite feasi- ble ; and without attempting to push them to a limit, the range of application could be fully investigated. Among the subjects selected for treatment was the horizontal pendulum. In the first part of Chapter I certain available forms of the pendulum, with and without a float, are considered and tested as to their discrepancies, through long lapses of time, by a reflection method. Among the interesting results obtained is the suggestion of an apparatus capable of measuring changes of elongation to the amount of even less than 4Xio~ 10 of the total length per vanishing interference ring. In the second part of the chapter the interferometer itself is used, a service- able method of application worked out, and the range of application studied through many months. With a relatively very wide scope (several seconds of arc) there should be no difficulty, under proper surroundings, of measuring changes of inclination as small as 3 X io~ 4 seconds of arc per interference ring, and it is probable that one could reach smaller angles by modifying parts of the pendulum. In Chapter II an attempt is made to use this interferential horizontal pen- dulum for the measurement of the gravitational attraction of two parallel disks. What was obtained, however, was a definite repulsion of the disks, decreasing with their distance apart and appreciable even within 1.5 mm. of this distance. As the method of measurement contemplates the viscosity of the film of air between the disks, and as the effect of any natural charge or potential would be insignificant in comparison with the forces observed, it is probable that the repulsion in question is attributable to the molecular atmospheres by which the disks are surrounded in air, supposing that such atmospheres of gas increase in density as the surface of the disk is approached. Chapter III is introduced as a severe test on the interference equation employed for the case of path differences resulting when glass columns as much as 10 inches long are inserted in one of the component beams of the displacement interferometer. It appears that the constants of any dispersion formula may be obtained directly from these observations. The equations for the relations of displacement and wave-length increments show, however, that the anticipation of great precision in the determination of refraction, iii 209210 iv PREFACE. by lengthening the column of glass, is not fulfilled. The ellipses become pro- portionately more sluggish in their motion as the path difference is increased. In Chapter IV a number of incidental experiments, on allied subjects, have been grouped together. In the first paper the possible bearing of certain disk colors of circular gratings on the somewhat similar phenomenon in coronas is discussed. The second paper deals with the performance of the easily available film grating to replace the ruled-glass grating, for purposes of dis- placement interferometry, from a practical standpoint. With the same end in view the third paper considers the use of the Nernst filament as an available illuminator, in the absence of the arc lamp or sunlight. In conclusion, an interesting case of regular reflection and refraction of scattered light, bearing on the X-ray phenomena recently discovered by Professor Bragg, is treated in the fourth paper. In Chapter V, finally, following the suggestive experiments made in an earlier report, the displacement interferometer is directly applied to the quadrant electrometer. In the several hundreds of adjustments made no serious difficulty was encountered in the optical parts of the experiments, and that was the question chiefly at issue. The sensitiveness obtained in this way should have been of the order of a millionth of a volt per vanishing interference ring; but owing to the uninterrupted commotions surrounding the laboratory already referred to, possibly also to difficulties residing in the electrometer, this limit could not be reached. The experiments, therefore, largely explore the available scope of the method. My thanks are due to Mrs. D. T. Knight and to Miss R. R. Snow for efficient assistance in connection with the preparation of the papers for the press. CARL BARUS. BROWN UNIVERSITY, September 25, 1915. CONTENTS. CHAPTER I. Elliptic Intcrferometry Applied to the Horizontal Pendulum. PART I. Direct Differential Reflection. PAOB 1. Introduction. Method. Fig. i I 2. Equations. Figs. 2,3,4 2 3. Observations. Fig. 5 6 4. New apparatus, without float. Figs. 6,7 7 5. Observations. Fig. 8 9 6. New apparatus, with float. Horizontal pivots II 7. Observations. Figs. 9, 100, lob 12 8. Second apparatus, with float. Jeweled bearings. Fig. na, nb, lie 14 9. Observations. Fig. 12 15 10. Observations, continued. Fig. 13 17 11. Effect of temperature on the float, etc 18 12. Further observations. Fig. 14 20 13. Effect of temperature on the scaffolding. Fig. 15 21 14. Inferences. Fig. 16 24 15. The precision measurement of elongations. Fig. 17 25 16. Improved pendulum 28 17. Observations with the new pendulum. Fig. 18 28 PART II. An Application of the Displacement Interferometer to the Horizontal Pendulum. 18. Introductory 30 19. Apparatus. Figs. 19, 20, 21 31 20. Equations 34 21 . Observations with a grating rotating on a fixed vertical axis 38 22. Observations with the interferometer. Fig. 22 39 23. Further observations. Film grating. Oil damper. Figs. 23A, 236, 24, 25, 26 41 24. Inferences 42 25. Improved aluminum pendulum. Observations. Fig. 27 45 CHAPTER II. The Repulsion of Two Metallic Disks, Nearly in Contact. 26. Apparatus. Figs. 28, 29 49 27. Equations 50 28. Equations for the vertical pendulum. Fig. 30 51 29. Observations with small plates. Figs. 3iA, 318, 3iC, 32A, 328 52 30. Observations. Plates of larger area. Tablet. Figs. 33A, 338, 33C, 330 53 31. The same, continued. Metallic contact. Table 2. Figs. 34A, 348 55 32. Retardation due to viscosity of air. Table 3. Fig. 35 57 33. Observations, continued. Presence and absence of electrical contact. Table 4. Fig. 36 60 34. Observations, continued. Change of distance apart. Table 5. Figs. 37, 38, 39A, 39B 61 35. Observations. Long periods and inversion. Table 6. Fig. 40 65 36. Plates electrically charged. Tables 7, 8. Figs. 4iA, 418 67 37. Conclusion 70 CHAPTER III. The Refraction of Long Glass Columns Measured by Displacement Interferometry. 38. Introductory 72 39. Glass columns. Fig. 42 72 40. Equations 74 41 . Equations. Sensitiveness in terms of displacement 75 42. Equations. Sensitiveness in terms of order 76 43. Observations. Green glass column. Table 9 77 44. Observations. Blue glass column. Table 10 78 45. Observations. Shorter column. Table 1 1 79 46. Summary. Fig. 43 80 v CONTENTS. CHAPTER IV. Miscellaneous Papers. PART I. Experiments Bearing on the Properties of Coronas. PAQ! 47. Introductory. Figs. 44, 450, 456, 45c 81 48. Experiments with a grating. Table 12. Figs. 46, 47 82 49. Experiments with small coronas 85 50. Experiments with large coronas, annular source. Coronas by reflection. Figs. 48, 49 86 51. Coronas from a point source. Figs. 50, 51 88 PART II. Displacement Interferometry with Film Grating. 52. Introductory. Figs. 52, 53 92 53. Films between glass plates 92 54. Continued. The groups i +4, I +5. Figs. 54, 55, 56, 57, 58, 59 93 55. Continued. The groups 2+4, 2+5 94 56. Continued. The groups 3+4, 3+5 95 57. Centers of ellipses 95 58. Film or ruling on one side of the glass plate. Ruled grating 96 59. Continued. Film grating not cemented to glass. Fig. 60 97 60. Single plate, film grating 97 PART III. Elliptic Interferometry with a Nernst Filament. 61. Introduction 98 62. The Nernst burner. Fig. 61 99 63. Remarks 99 PART IV. Scattering in the Case of Regular Reflection from a Transparent Grating, an Analogy to the Reflection of X-rays from Crystals. 64. The phenomenon. Fig. 62 100 65. Explanation 101 66. A further analogy to the reflection of X-rays 102 CHAPTER V. Displacement Interferometry Applied to the Quadrant Electrometer. 67. Apparatus. Fig. 63 103 68. Observations 104 69. Observations, continued. Fig. 64 105 70. Observations, continued. Figs. 6sA, 658 105 71. Observations, continued 106 72. Further observations. Table 13. Fig. 66 107 73. Observations, continued. Tables 14, 15, 16, 17, 18 109 74. Summary 112 CHAPTER I. ELLIPTIC INTERFEROMETRY APPLIED TO THE HORIZONTAL PENDULUM. PART I DIRECT DIFFERENTIAL REFLECTION. 1. Introduction. Method. Before using interferences, it seemed inter- esting to apply the interferometer adjustment to the case of simple reflection, the mutual displacement of the two direct images from the front and rear face of the mirror on the pendulum being used for the measurement of the angle of deviation of the pendulum. In such reflection from a glass plate there is necessarily considerable loss of light; but at radii of 20 and 30 meters, when the source of light is a slit closely in front of a Nernst filament, this difficulty is not prohibitive. It is necessary, however, that the lens of the collimator, as well as the plates of glass and mirror used, be of high opti- cal quality; otherwise it is impossible to obtain sharp condensation at a very dis- tant focus. One may also concentrate the slit images to a point by a cylindrical lens, placed with its linear element at right angles to the direc- tion of the slit. The interesting feature of the method is that it is inde- pendent of any zero-point, as the distance apart of the two images on the far screen at once measures the inclination of the pendulum axis, the normal position being that in which the two images coincide. If the two opaque mirrors are rigidly fixed, the direct or incident beam of light from the source and the subsidiary reflecting mirrors may shift without modifying the datum for the inclination. Further- more, the sensitiveness is twice that of the case of single reflection, other things being equal. The method is thus particularly adapted for the measurement of the inclination of the plumb-bob relatively to the earth. The annexed diagram, fig. i, will make the method clear. Here 5 is the fine slit in front of the Nernst filament/, and / the condensing doublet (about 60 cm. focal distance; for rough work an ordinary spectacle-glass answers very well) of the collimator. It is necessary that this lens be wide, weak, and good in order that a sharp focus and as little loss of light as practicable be obtained at the distant focus. 1 FIG. i. 2 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. The pencil from the collimator strikes the plate of glass G at the end of the horizontal pendulum, the greater part of this, d', being transmitted to the opaque mirror M , the remainder, d, reflected from the opaque mirror N. It is advantageous to have M and N equidistant from G, as nearly as prac- ticable (20 or 30 cm.), and far from the lamp/, to avoid the menace of tem- perature discrepancies. If the mounting is of gas-pipe, water circulation might be used, but this is not necessary. In the diagram the pencils d and d' are normally reflected at M and N. On returning d is transmitted and d' reflected, so that the beams reunite and proceed together to the far focus F, 20 or 30 meters distant, where they are caught on a paper or ground-glass screen, or directly observed with the lens. It is particularly necessary that the movable reflector at G be an excellent optical plate, i or more inches square. When the plate at G (which is at the extremity of the horizontal pendulum) rotates over a small angle $, the reflected rays d" and d'" now diverge in opposed directions from the center C on the face of the opaque mirror N, and pass to the distant foci F" and F'", where they are now at a distance x apart. If the rotation of the horizontal pendulum were 9, the positions of the beams would be exchanged (see F' and F iv ). In other words, if the pendulum vibrates, the two foci F" and F'" move in opposed directions, passing through each other, when the normal position is instantaneously assumed, irrespective of the amplitude of vibra- tion. It may be noted that a similar adjustment may often with advantage be attached to any ordinary pendulum. The mounting of the plate G and the mirrors M, N, etc., is identical with that of the interferometer described in the next section (the plate G is there replaced by a transparent grating) and need not be treated here. Necessarily the collimator and the mirrors M and N are attached to the same pier which carries the horizontal pendulum, to the end of which the mirror G is attached; but the horizontal pendulum and its case must otherwise be quite independent of the goniometric apparatus. 2. Equations. It will be convenient to suppose the foci F, F", F'" to lie in the plane of the mirror M, and the two mirrors M and N to be equidistant, so that d=d', taken from the plate G as an ideal plane. Let a be the angle between the normal to N and the direction of the incident pencil; i.e., let a be the angle between the mirrors, made as small and as convenient as prac- ticable. Then if the mirror at G rotates over any angle = b/2, the distance apart x of the foci F" and F'" will be (i) x =d s{n2b ( ' -f - M cos (a b) \cosa-f6 cos6/ This equation may be transformed to / 2 ) .j^__ L f cos 2 6 ,i cos b cos 2a+cos 26 2 cos (a 6) EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. Since the angles a and b are invariably small, the cosines may be expanded, so that (3) is nearly true, or (4) =2d tan b Since b = 2d is exceedingly small (but a few seconds), (5) Finally, if a is also sufficiently small, which will usually be the case, and D is 2d or zd' , so that the distances to the far screen, F" and F"' t are measured from the mirror N, (6) x=>4D6 One may note in passing that the distance over which the N ray travels from coincidence is cos (i cos a whereas the Af ray travels over xt=d' (tan (a+6) -tan a) where *=#i+# 2 . Hence, for small angles, the N ray travels over 3 times the distance db of the M ray, the total being ^db. Thus the angle of devia- tion 6 is measured by x, apart from any other consideration, except that the distance D is very large and therefore invariable and the sensitiveness is twice as large as in the case of single reflection. To test this result in its practical aspects, a millimeter micrometer was installed at the end of the pendulum, at a distance of 51.5 cm. from the axis. The two images traveled in opposite directions, in steps, from end to end of a 30 cm. scale, while the micrometer was moved forward i mm., successively, eight times, as follows: Micrometer. Mean. * Micrometer. Mean. X o.o cm. .1 ,2 3 4 1 6. i cm. 15-7 15-4 I5 'o 14.8 32.3 cm. 24.7 17-3 9-9 +2.5 0.5 cm. 7 .8 14.5 cm. 14.2 13-9 13-7 - 4-9 cm. -12.3 -19.7 -27.0 With the exception of the first and last readings the steps of x are 7.4 cm. and equidistant. Hence - 1/51. 5 -7.4/4^ EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. whence D is equal to 914 cm., which agreed with the direct measurement. The center of the images ("mean") holds pretty well to the scale, shifting but from 13.7 to 16.1, while the distance D is smaller and the total angle (0.8/51.5 radian, about i degree) larger than would usually be employed. In the experiments made below, the distance D was frequently above 2,000 cm. Since i second of arc is about 5X10"* radians, the deflection x corre- sponding to 6= i sec. would therefore be * = 4X20ooXsXio- fl =4Xio- l cm. or nearly half a millimeter. A sharp focus F", F" e is thus nevertheless needed if single seconds of 6 are to be read off visually. I frequently made use of what seemed to be the internal diffraction patterns of the slit, fine bright lines in each being used for measurement. The angle 6, denoting the deviation of the pendulum, is invariably very large as compared with the angle a, the corresponding change of inclination of the pier to the plumb-line. In fact, fig. 2 , cdg denotes the horizontal pendulum, with the grating at g, pivots at c and d, the center of gravity at G, at a distance h from the axis cd. The latter prolonged intersects the plumb-line through G at e, all in the plane of the diagram. The angle between the axis de and the vertical df is

6 Thus if de is a rigid stick pivoted at d and fe a flexible inextensible line, the motion is such as if the whole mass of the pendulum were concentrated at e, the diagram being the plane of the couple Mgh=MgH<-inch gas-pipe, usually screwed fast at one end and clamped at FIG. 7. the other. The position of the horizontal pendulum on a separate mounting is sketched in at kPh'P', the glass plate being at h and h', the pivots at p and p f . The two adjustable opaque mirrors are shown at M and N, the former being about 20 cm. to the rear, and held by clamps. 5 is the slit and L the lens of EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 9 the collimator, the lamp being at A, on a separate stand. In fig. 7, the tall standard A ABB of i-inch brass pipe, well braced (not shown), supports both the case abcf of tin plate and the pivot supports de of the horizontal pendulum HPH'P'. The rear and sides of the case are rigidly fixed, but the front may be removed as a whole. Similarly, the square boxes abcf, of which a and c are provided with glass plates, slide out horizontally or vertically, both in front and in the rear. The pendulum is thus easily accessible for adjustment. The pivots may be revolved around a horizontal axis, or moved fore and aft, right and left, or up and down. The fore-and-aft movement is provided with a screw adjustment, like the right-and-left movement. The method of attachment is much the same as that to be described below. Pendulum and case are quite independent of the truss (fig. 6) , a very essen- tial condition, as the truss must often be touched for optical adjustment. In a later adjustment the case also was mounted in complete independence of the pendulum. The present symmetrical horizontal pendulum was made of X~ mcn alumi- num tubing, the vertical brace PP of ^-inch aluminum tubing. The junc- tions are brass tubing and the cups and slots for pivots either jeweled or of glass-hard steel. There is a slot for the d pivot and a conical hollow for the e pivot. Inasmuch as the horizontal pendulum is invariably under tremor, with the consequent absence of static friction, the pivot e in the first experi- ment was adjusted vertically, though the means were at hand for adjusting it in any inclined direction, as will presently be shown. The horizontal brace gh is of tense brass wire, forming a rhombus when seen from above, the object being to enhance the lateral rigidity of the pendulum. Finally, the plane parallel glass plates, H, H', lie to the front of the pendulum, so that a beam of light may pass across, parallel to its plane, as called for in some of the interferometer measurements. They are below the line of horizontal symmetry and each is adjustable around a horizontal and a vertical axis. The case at / was specially adapted for the installation of a float K, which will be described presently and was not used in the first experiments. The vat C of the float must be supported on an entirely separate standard (not shown). 5. Observations. The total weight of the apparatus, including the mirrors, was M 740 grams, with the center of gravity at h= 13.9 cm. from the axis; the effective length of the arm was R = 59.5 cm. and the period T=2g seconds. The moment of inertia for an axis through the center of gravity was found to be 1.51 X io 6 . Since the mass was 740 grams, this is equivalent to a radius of gyration 4*0 = 45.2 cm. Hence, since the distance of the center of gravity from the pivotal axis is ^ = 13.9 cm., the radius of gyration for the same axis will be * = 47-3 cm. From this and the above period, 47T 2 * 2 (f>= * = 7 -7 1 Xio- 3 radians, 10 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. or about 0.44 degree. Hence, the change of inclination of the pendulum is nearly a = 0.00770. The force at a distance # = 59.5 cm. from the pivotal axis (place of the plate of glass or grating) is supposing the interferometer to be used and that AJV is the displacement of the micrometer. Thus F R here and in the preceding case, when referred to the same

it is independent of the presence or absence of the float, which therefore does not conduce to enhance the precision of the quantity a, except in dimin- ishing the friction of the pivots. 12 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. The force at the distance R = 6 1.5 cm. from the axis (this being also the position of the line of light passing through the plate) is, for like 6 and 0 = 0.0146 radian = 0.81* second of arc, since D = 9oocm., M =971 grams, M F= 169 grams, /? = 6i.$ cm., &=i3.icm. r=aosec. In fig. 9 the ordinate x is given in arbitrary units, which must be divided by 5.74 to reduce them to seconds of arc. ir The conical sockets for the horizontal pivots of the pendulum being of steel and not quite smooth, it is possible that the relatively enormous values of the changes of inclination, a, registered may have been due to displacements of the pivots in their sockets; but readjustments of the fiducial zero (before the observations marked n in the graph) were as frequently necessary when there EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 13 was no explosion (marked e in the graph) due to blasting under the hill, as after the occurrence of such a disturbance. In fact, the values of a range within 26 seconds and are often as large as that per day, whereas in paragraph 5, for the case of the apparatus without a float, the whole limit of variation was not above 1.5 seconds. In fact, the two slit images frequently separated to a distance exceeding # = 36 cm., or nearly as many seconds (compared with the isolated maximum of 8 cm. above), while the distance between mirror and scale was but .0 = 900 cm. (compared with 2,000 cm. above). The apparent change of inclination of the line determined by the pendulum pivots is thus, in the present case of the float, registered about 17 or 18 times larger than the above similar case without a float, whereas there should be no difference. At least, it is difficult to conceive how the float, which practically compensates the weight of the pendulum, can introduce any seriously variable torque around the vertical axis. In fig. ioa, let ABG be the horizontal projection of the lower pivot, the center of buoyancy, and the center of gravity of the pendu- lum. Then, with the correspond- ing notation, the forces involved will be A +B = G, the couples involved, Ah and Bh, very nearly, if h is the distance AG. The effective couple is thus Ch, the vector sum of Ah and Bh. If the angle between AG and BG is e, the couple Ch may be resolved into a normal couple N and a parallel couple H equal to Bhe nearly, whose axis is essentially horizontal. In fig. 1 06, where AZ PIG. 10. is the vertical and AF the line of pivots at an angle

(6'-d) with a corresponding value free from c for T'T. 14 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. There is another point of view from which the question may be approached : Any variation of buoyancy B, if B is eccentric, is virtually equivalent to a displacement of the center of gravity of the pendulum. This occurs when the temperature of the water, in which the float is submerged, changes. Let AG be the plane through the axis of the center of gravity of the pendulum when the float is not submerged, k the perpendicular distance of B from this plane. The center of gravity after the submergence of the float will be displaced laterally (if Vp is the mass of liquid displaced by the float) , f _Vpk_Yp ~ M ~ M** Since the center of gravity must lie in the same vertical plane with the line of pivots AF, the pendulum will have to rotate over an angle e' = k'/h=V P s/M The observed angle is thus to be divided by 0' to obtain the amount due to simultaneous changes of inclination only. Of course, e may be either positive or negative. Hence, the apparent change of inclination from a to a' is to be interpreted Before discussing the question, however, it is preferable to obtain data with a more perfect pivot adjustment; in other words, to use pivots inclined toward the center of gravity and provided with jeweled bearings. c XI * FIG. u. 8. Second apparatus, with float. Jeweled bearings. The anomalous results for a obtained in the last experiments were in the first place to be associated with the unsatisfactory pivots. Hence, these were readjusted so as to point toward the center of gravity of the pendulum. Moreover, the steel cup was inadequately smooth and could not be polished. It was there- fore replaced by a conical hollow of polished sapphire, placed so that its axis prolonged passed through the center of gravity of the horizontal pendulum. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 15 The manner in which the jewels were secured is shown in fig. na, and sectionally in fig. nb, turned at right angles around a vertical axis to the preceding figure. Here a is the lower end of the central vertical tube of the horizontal pendulum into which the stem b of the forked holder c fits very snugly. The two flat prongs c carry the brass screw with fine thread d, which is horizontal and is secured in any position by the lock-nut e. The conically hollowed jewel, black in figure, is firmly embedded in the small brass cylinder /, which in turn may be screwed centrally into the wider cylinder d and fixed by a lock-nut. P is the brass stem carrying the needle of glass-hard steel which dips in the sapphire socket. The cylinders P and / are coaxial, but may be given any inclination to the vertical and then locked. As the effective weight of the pendulum does not exceed 170 grams, the strain on the pin and jewel is not excessive, and the results appear to show that they rendered excellent service. The upper pivot played in a groove of glass-hard steel as before, and it did not seem necessary to modify this. The remainder of the horizontal pendulum was of the form already sketched in figs. 6 and 7 ; but precautions were subsequently taken to mount the water- bath for the float on a separate pillar, quite independent of the horizontal pendulum and its case. Later the case was also independently mounted. Trial was made of a water damper attached to the end of the beam (H ' in fig. 6) on the side opposite to the mirror H. This, however, was soon discarded because of the capillary forces introduced. As a rule, the damping obtained at the float is adequate. In order to set the zero of the pendulum at a given point, as well as to vary the inclination of the axis by the definite amount needed in the independent data of

being radian and T= 20 seconds. Hence, / 400 (I6.2) 5 = 0.0214 radian, or about 1.2 degrees, roughly. This is an unnecessarily large angle and is merely admitted as a first experiment, to be decreased in successive experi- ments. Hence, finally, a = < f >e = 6.oXio- 6 x radians =1.23% seconds of arc. The apparatus is thus relatively insensitive, seeing that 1.2 seconds go to a centimeter of deflection, x. ZOO 440 fcO 60 40 20 -20 -40 -60 -80 5 4 9 // FIG. 12. 60" ACT 30* These observations were carried on for some time in the midst of other work, during February and March of 1914, the method of reflection being used. The data are first given in fig. 1 2 . The ordinates of the latter are in arbitrary units and must be divided by 5.74 to reduce them to seconds of arc. In the continuous record from February 23 to March 13, the range of variation is almost as large as it was in fig. 9, showing that the use of jeweled bearings has EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 17 had little or no influence on the result. An interesting fact is the depression produced by the gale on March i (see g in fig. 12). Though the behavior of the apparatus, as such, apart from the anomalously large results, was throughout satisfactory, it was supposed that the attack of water on the soldered joints of the copper float was an objectionable feature. These were accordingly covered with resinous cement, with a removal of this trouble after March 13, the new zero being indicated at n in fig. 12 ; but the behavior thereafter was even more variable than before, showing that something not connected with the change of inclination is in question. 10. Observations, continued. The horizontal pendulum was now read- justed for greater sensitiveness and for a smaller vertical inclination , using, however, the micrometer method, instead of the more elaborate pendulum method. The following values of x were found for successive turns of 6 each of the micrometer screw: Turn of screw. X Mean Sx per 6. 29.0 cm. 18.5 cm. 6 11.9 12 - 7-5 18 -25-7 24 -44-5 The screw being a K'-inch screw with 20 threads to the inch, its pitch may be put 0.125 cm. If z is the displacement of the lower pivot for each partial turn of 6, y the distance apart of the pivots, and C the constant to be found, a = z/yCx radians, or, in seconds of arc, C 0-31 The value of C found for the pendulum used in the case above was C= 0.515. The difference is larger than was expected; but with the center of gravity but 12 cm. from the axis, the addition or removal of the weights at the end of the beam 60 or 70 cm. from the axis is of marked consequence. It is also surprising that the displacement method is so consistent in its results, as nothing more than an ordinary clock-dial with a pointer was used at the micrometer. These results could easily be much improved. In other words, the present direct displacement method for C=

gave a = o.^x seconds of arc. These (summer) data are given in fig. 14 and the temperatures are inserted in the same figure. The work was continued for about 6 weeks, not all of the data finding room in the figure, and the graph after July 3 had to be displaced, as shown. The new results still partake of the same tendency to enormous variations which characterize the older (winter) data. The essential error has, therefore, not been removed. On comparison with the detailed temperature curve above, however, the clue of the anomaly is obtained, for although the temperature variations are not quite contemporaneous with those of inclination, there can be no doubt of the immediate relation between them. The case is all the more cfcnp. '\ AV w -J FIG. 14. puzzling, however, as single degrees are in question, enormous changes of inclination being produced by 4 degrees. Under the circumstances, moreover, complete identity in the direction of variation of temperature and inclination graphs was not to be expected, for the temperatures given are those of the water in the float and will therefore vary more sluggishly than the tempera- ture of the metal parts. The air temperatures, again, which were also taken, would vary faster than those of the metal, evidence for which will presently be shown. It is therefore next in order to actually examine the structure of the standard of the horizontal pendulum. 13. Effect of temperature on the scaffolding. To give the columnar sup- port of the horizontal pendulum adequate steadiness, it was braced from be- hind as shown in fig. 15. It was not foreseen that any menace could lurk in such a system, such as was later detected. In fig. 15, ABC is a side-view of the brass vertical standard (in duplicate, as shown in fig. 7), the horizontal 22 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. pendulum being supported between A and B, while the heavy base CD rests on foot-screws on the cement subfloor of the basement of the laboratory. BD is the brace in question, extending almost half-way up the standard at an angle of about 0=14 to it. For ordinary quiet surroundings this truss seemed to be adequate, as the water-vat of the float was held on a separate standard, free from the pendulum and its case. The pendulum, in view of FIG. 17. the float, was therefore virtually very light. The difficulty encountered resides in the fact that even small differences in the coefficient of expansion of BC and BD will seriously tilt the axis AC. To express these relations let h, v, b, be the hypothenuse, the vertical, and the base of a right-angled triangle as shown in fig. 15 and idealized in fig. 16. Let =dh/h=db/b for the same temperature increment of i C. be the coefficient of expansion of the base and of the brace (for convenience), and fi=dv/v that of the brass post. Then it follows easily that for an increase of i C. of the environment, fe-l#4W-&0((-p=io~* would give rise to a deflection of dx = 2.7, nearly 3 cm. per degree of increase of tem- perature. In fact, this arrangement actually suggests itself as a remarkably sensitive method for measuring small elongations; for, since da= (<}> &) cot B, independent of all lengths, da increases as decreases without limit, and the question is merely one of experimental adjustment. If the hypothenuse h alone expands, the remaining temperatures being kept constant, EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 23 or for the above data quantities of the same order as the above. Thus if dx = i cm. (for the rela- tively small distance of scale from mirror, D = goo cm.), = o.3sXio 6 , and it should be easy to measure one-tenth of this expansion. The peculiar interest which attaches to this equation for da or any corre- sponding case is the absence of all need of length measurement in the combina- tion. In the right-angled triangle hvb, fig. 15, it is merely the angle 6 which must be given, all the quantities compared being numbers. Of course, the relation of x and a remains, into which the distance of the mirror from the scale will enter. In the complete equation (if da is replaced by a)

be the inclination of the axis of the pendulum to the vertical and 6 an angular excursion of the pendulum, measured from its position of equilibrium. Let h be the normal distance of the center of gravity from the axis. The rise of the latter above its lowest position is w ;y=/t(i cos 5) sin ^= and the energy potentialized, if the total mass is M, will be which for small displacements corresponds to the torque Fh at the angle 6. This torque is (3) -~- = Mgh sin

= h/> nearly, or if the period T be introduced from (6) and 6 from (5) 4**? AAT (9) a= ng 1R It is in equation (8) that the condition of remarkable sensitiveness resides. Thus, if the interferometer is used, a = AN/2R, and, if A/V= 30X10-' and v? = io~ 2 (somewhat less than i of arc), R= in cm., as above, a = 13 X io- 10 radians = 00028" per vanishing interference ring. 36 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. If F, as before, is the force at the center of gravity, the corresponding force at the grating, a distance R from the center, is (10) F R - Mg is given by equation (6). This equation implicitly contains h, since * refers to an eccentric axis and i z =i\+h' i ; but i may be found directly. The deviation is given by (5). If, however, the device* of two parallel mirrors, equidistant (distance R) from the axis of the horizontal pendulum, be used, and if light impinges on either mirror at an angle of incidence I (the impinging and reflected beams being always parallel), AAT where AN 7 is the displacement of the micrometer. The horizontal pendulum is in this case constructed symmetrically to the vertical axis in the form of a balance beam, but somewhat heavier on one side. Finally, the compound pendulum may be supported on a cylindrical float, symmetrical or not to the vertical axis of the pendulum and submerged in water or some other liquid. In such a case, the mass of the compound pen- dulum may be reduced in any degree without serious difficulty from capillary forces, as will be shown below. If the center of buoyancy is in the vertical line passing through the center of gravity of the horizontal pendulum, the above equation needs but slight alteration. Let V be the volume of the float, so that Vpg is the buoyancy. Apart from the temperature conditions, p= i, and hence the equations take the successive forms, since (M V)g is sup- ported at the center of gravity, instead of Mg, (13) r-(Jlf The force at a distance R from the axis is, when the center of gravity is at a distance h, Hence the force has been reduced in the ratio of M/(M- V} for the same 6. One may also note that it is smaller, not only as

2 X io- 5 /3 1 = 6 X lo- 6 dynes, roughly. This would be equivalent to the attraction of two so-gram weights at i cm. of distance. *Barus: Am. Journ. Sci., xxxvii, pp. 83 et seq., 1914. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 37 Furthermore, whence, since 0=&N/2R f a x" M **** M ('6) ^-K-T^-tf all of which quantities are easily determined with accuracy. To find the radius of gyration *', for instance, a body of known moment of inertia may be suspended at the end of the horizontal pendulum and the periods T of the pendulum before and after suspension determined, with or without the float. Finally the change of vertical inclination a becomes If the pendulum is damped, which will usually be the case, it may be neces- sary to observe the logarithmic decrement, in order to compute the free period in the usual way. If the buoyant force due to the float does not pass through the center of gravity of the solid parts of the pendulum, but at a distance h' from the ver- tical or pivotal axis, the new distance of the center of gravity h" when the pendulum is partially floating will be i f _ M-V Hence, tih'=h, then h"=h, resulting in the equations just deduced. But if h' = o, i.e., if the buoyant force passes through the point of the lower pivot, Thus the equations deduced become identical with the original equations (2) ft seq. The float therefore adds nothing to the sensitiveness except in so far as it removes friction at the pivots and supplies a reliable damper for the pen- dulum. It is in this form that the float will be applied below. Since the torque equation is now again where all references are to solid parts of the pendulum, h may be accurately found by placing weight m at a distance / from the plane of the pendulum, or better, by placing weights alternately before and behind this plane, at a dis- tance / apart. The torque applied is then T=mgl, whence ml < I9) h= Mi This method will be used effectively in several experiments below. It is an excellent test on the reliability of the damper, since h can also be determined directly by the suspension of the solid beam of the pendulum. In the adjust- ment adopted, at a scale distance of 900 cm., m/=gramXcm. on the scale- pan, produced a deflection of about i mm. 20921O 38 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. A few other equations of minor importance may be added. If the indicated length is H and the horizontal pendulum be treated as a vertical pendulum of length L, the point of suspension being the intersection of the plumb-line through the center of gravity and the line determined by the points of the two pivots, the observed period is ( 20 ) r=WZ/g and P=LH where 7 is the corresponding radius of gyration. If the end of the horizontal pendulum is loaded with the weight m of a disk at a mean distance R from the axis for the measurement of gravitational attraction, since (M+m) h'=Mh+mR, the new force at R is When the end of the pendulum is similarly loaded for the determination of its radius of gyration, since (aa) the new period is r= v V Tj _ _ h Since T' and T are observed and m, M, R, h given, * may be computed. The horizontal pendulum itself thus supplies the value of i. If the lower pivot is provided with a strong micrometer screw, by which it may be moved over a small distance z to the front or rear of the plane of the pendulum, the computed value of a may be tested independently. Thus let y be the distance apart of the pivots and z the displacement of the lower, then when in the method of deflection x is the increase of the distance apart of the two images of the slit, at a distance D from the further mirror. Hence where

and h may be obtained inde- pendently, the torque T, etc., is given independently. This method will also be applied below. 21. Observations with a grating rotating on a fixed vertical axis. When the opaque mirrors M and N are identically concave and are put on the ordi- nary interferometer at a distance equal to their radius of curvature from the stationary grating, the latter may be rotated (without translation) as far as the breadth of the opaque mirror N permits, without readjustment. The ellipses are not lost. Inasmuch, however, as different thicknesses of glass are introduced into the rays when the grating is rotated, the ellipses travel hori- zontally through the spectrum from the red to the violet end or the reverse. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 39 They are about equally clear in all positions. A displacement at the mirror N of about 4 cm. per meter, i.e., 0.04 radian, equivalent to a rotation of 2.3 of the reflected ray, or a rotation of 1.15 for the grating, was within the scope of the interferometer and the tests were made within this limit. It is far in excess of anything required in the horizontal pendulum. No doubt if the mirror N had been wider, the ellipses could have been retained for larger angles of rotation of the grating, though they would in such a case travel several times through the spectrum. The micrometer at M would have to be used. If long columns of glass are to be inserted in either beam (GM or GN) the concave mirror is not available, since the direct slit images will then have different focal positions. The rays issue from the plane-parallel column, parallel to this focal direction, but from a virtual focus nearer the concave mirrors. Hence, if the column is placed in the beam GM, the beam GN will, as a rule, have to be correspondingly shortened. The algebraic relations are complicated. 22. Observations with the interferometer. The horizontal pendulum with which the following observations were made had the following constants, M being the total mass of the fixed parts, m the attached mass, h the distance of the center of gravity from the axis, R the distance of the vertical line of light on the grating (also mean distance of m and of FR) from the axis,

= a/6=o.QioBi radian=o.62 and H = 7,394 cm.; L=8,488cm.; #'=7,834 cm.; ,' = 8,853 cm. Since 6=AN/2R when AN is the mean displacement for the horizontal deflection (6) of the pendulum, a = i o- 5 X 4. 86 AAf radians. Thus, if AAf = 10 4 cm., a = io~ 3 second of arc, or the change of a per vanish- ing interference ring (AAf = 10-^X30) is 0.000310 second of arc. Since T may easily be increased over 3 times, this limit may be reduced to a = .000030" per ring. Similarly, the forces at distance R from the axis of the horizontal pendulum are Thus if AAT= io~ 4 cm., F'# = o.oo54 dyne or about 0.0016 dyne per vanishing interference ring, in case of the pendulom loaded with the disk m. In the graph which follows an example is given of a series of observations made for 6 and a, and no further explanation will be needed. Since a = X i 2 r x /!/ y V A ^ A / \j / ' s A L ^K s X s / L i / / v - 7 V ' J^ ^ I/ V W& 6 67 t 9 *> 11 & ft M 15 16 1T FIG. 22. As the observations were made in an unavoidably steam-heated room, it is probable that the flexure of the pier, etc., due to thermal causes, has been largely operative in modifying the trend of the curve; for on comparing the curve as a whole with the thermostat sheets (not shown) a retarded effect is possibly suggested, such as one would suspect if variations of surface tem- perature should penetrate massive masonry. It would then be possible for the curve to have different heights at the same temperature. Naturally such comparisons are very vague, and it is the range of values of a admissible in the apparatus which is here of paramount interest. Furthermore, as the hill on which the laboratory stands is, at present, being tunneled, so that the building is subject once or twice a day to the tremors resulting from the vigorous blasting underground, adequate conditions for the installation of an apparatus of the present kind are still remote. It is really surprising that interferometer observations could be made, without essential difficulty, under EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 41 these circumstances. During an explosion, of course, the ellipses vanish, to reappear, however, immediately afterward, sometimes with displacement, such, for instance, as is indicated by the doubled parts of the curve. The use of the water damper, moreover, which was necessary here, is objectionable, though it has not, probably, introduced any marked error into the observed curve (see doubled parts). Finally, the use of a steel horizontal pendulum with its plane in the magnetic meridian is inadmissible. I have not, therefore, endeav- ored to interpret the results, but they are given simply as an example of a systematic series of observations, extending over a month. I hope in the summer to resume the work in the absence of the annoyances referred to. I may add in conclusion that the experiments referred to above, for measur- ing the gravitational attraction of two identical brass disks, led to curious results. It is easily seen that for constant mass the attraction of nearly con- tiguous disks should increase roughly as the fourth power of their radius. For disks 20 cm. in diameter, however, the result is an invariable repulsion, several times as large as the estimated gravitational attraction, the position of equilibrium being reached gradually in the lapse of several minutes. The subject will be systematically discussed in Chapter II. 23. Further observations. Film grating. Oil damper. After the above experiment, the steel horizontal pendulum was used for other purposes and observations on the tilting of the pier were discontinued. Later, however, the apparatus was again available and a variety of experiments was made with it. In the first place, the water damper was replaced by an oil damper, as it seemed probable that the surface tension of illuminating oil and its slower evaporation would be an advantage. Under like conditions (though it proved sufficiently serviceable) it did not check the vibration as effectively as the water damper. The modification of chief interest, however, was the inser- tion of one of Mr. Ives's film gratings (in the usual double plate-glass pro- tection) in place of the plate-glass grating. The film grating in question had about 15,000 lines to the inch, so that the dispersion was excessive, the ellipses being large and diffuse and with a long horizontal axis. To obviate this diffi- culty a thick compensator was introduced into the component beam M passing to and from the micrometer. For this purpose three thick plates of glass were cemented together with Canada balsam to a combined thickness of something over 2 cm. The ellipses now became adequately sharp and almost circular in form. In consequence of the multiple reflections described in Chap- ter IV, Part II, the ellipses are not so strong as in case of the grating ruled on plate glass, and they are much harder to find; but they are nevertheless quite serviceable. The single-plate film grating of 60 was not at hand at the time. It is advisable to try out the double-plate film grating first on the fixed interferometer, in order to determine which lines of the individual images of the slits are to be placed in horizontal and vertical contact, together with the distance which corresponds to the different interferences on the micrometer. After this is done, the corresponding adjustment of the interferometer is easier. 42 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. It is also advisable to adjust the plate-glass grating in both cases for compar- ison. In figs. 23, 24, 25, 26 an example is given of these observations in the usual way. Data between January 15 and 24 (fig. 23, A) and March 3 and 6 (preceding fig. 23, B) exhibit the behavior of an oil damper with the ruled grating. In the observations after March 14, running as far as July 14, 1914, the ruled grating was replaced by the film grating. Inasmuch as no essential change was made at the steel horizontal pendulum, the constants may be taken to be the same as above, viz, M=i,2so grams; h = 8ocm.; R = m cm.; 7=18.48 sec.; * = 8s cm.; ^=0.0108 rad. = 0.62. Thus a = io- 6 X48.6 AN rad. = loAN", nearly. 24. Inferences. The curves in fig. 23, A, B, are independent so far as zero of measurement is concerned, but they already exhibit a tendency to decline in the direction of a decrease of a. This was pronounced in January and also in March. It is not a regular decrease, so that the cause can hardly be, or at least not wholly be, sought in the yield of the parts of the apparatus; for in such a case there would be no recovery (increase of a), a feature which is often marked. The continuous observations (i.e., with the same uninterrupted zero) are given in figs. 24, 25, 26. The same scale is used throughout, but on April 22 and May 15 it was necessary to displace the graph in order to accommo- date the observations on the sheet. The amount of displacement is shown. Here also there is a gradual and continuous decrease of the values of a. Begin- ning on April i with a about 2", the observations pass through a succession of oscillations to the lowest value of a recorded, about -4.2", on May 18. After this there is intermittent partial recovery, so that on June 28 a has EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 43 risen to -i". The decline, however, at once commences, and on July 8 a is about -3.3", from which it rises to -2.4" at the end of the work. The maxi- mum range of the tilting of the pier, a, between April and July is thus about from +2" to -4", or 6" of arc. -JL 7 9 ft & " >T~* B n FIG. 24. -2Q tr ' m 1 -4 rr \, /\ \ s* ^ ^ N -24 ^ ^ 7 J .A / n -3-0 j .J } rf 1 V % 1 f^ 1 *j \ I \l / to J -M I J 7 33 ? A u u t -4-0 i V ^ V ' / A v ^ 1 \J xj o Q 1 < 1 /j ' * 2swxrx94tW. 4 e ID /x ^ w FIG. 25. 44 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. Between the oscillations proper, which are as a rule sharply marked, both at the maxima and the minima, there are regions of relative constancy of inclination. Thus between April 9 and 21 the inclination is nearly constant and about 0.6"; between May 17 and 28 there is a slow ascent from about 4.1" to 3.6"; between June i and 12 a slow descent from 3.0" to - 3 ! 4 "; between June 16 and 25, relative constancy at about -2.8", etc. Regarding the observations as a whole, it is not impossible that there may have been some slow yielding (quiescent frictional forces probably yield vis- cously) of the mechanical and the optical parts of the apparatus. The main features of the diagram, however, are due to the pier itself, or the pendulum, in responding to actual forces. On these the former errors may have been ~/4 -44 -/9 -SO -2 -S-4 -M -2-8 ~zo -S-2 tfvj zr /' '"> $ - *f up \^_ -v_J k -'"\ s _ s S - s 3 cfe> / ^ y^ s*^ & ir J C ^ ~^^-^ IT 5 l fv li A J ^ 1 it ^ \ f, ft \ / t / V Til 1 I / i 1 L yf V 4 /v- f \ \ 'I V ie 18 ft 20 22 & 23 29 PIG. SOtfuhZ 4 6 8 A? ' /d 26. superimposed. Certain large drops on May 15 remain unexplained. The decline after April 3 is large and slow, so that it could be observed during its occurrence. This may therefore have been actual and not due to displace- ments of the pendulum resulting from subterranean shocks. The observations were continued into June and July, with the expectation that when the basement room was no longer heated, the variation of a would practically vanish; but this is not at all the case, as the play of a i.i June and July scarcely differs from the average run of values during the winter months. From the long-range point of view, inclination, a, decreases from about April i to about May 19, after which it increases intermittently again, recov- ering on June 29 and again on July 27 (not shown), about one-half the total decrement. It would not be difficult to arrange the minima in semi-monthly periods, if any reason for such large variations of a could be assigned. They are of course enormously above anything to be anticipated from tidal influ- EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 45 ences. So the maxima could be placed at March 17, April , May 9, June IS. July March 29, April 27, May 29, June 28, July 27, in which the monthly periods are at least marked. But here again any adequate cause for such behavior has not been found, nor in any case has it been possible to separate the true from the adventitious tilting record. As the pendulum was a thin steel tube and the direction north-south, one might infer changes of the earth's horizontal intensity. It is hardly probable, however, since whatever magnetization was present was induced by the earth, that forces of the required intensity could be present. The mechanical force at the grating for the displacement AAf would be, roughly, F=43AAf. Since i" of arc of a was about ioAAf, the mechanical force in question is thus 4.3 dynes per second of a. Such forces are not liable to be of magnetic origin. Finally, if we compare the run of air temperatures given after May 28, for instance (the thermostat sheets were not accurate enough), though there is no detailed resemblance in the two graphs, some relation is none the less apparent. Thus the fall of temperature up to June 10 and its rise through a maximum on June 14, to fall again to June 22, is followed by the pendulum graph with a lag. So also the next temperature maximum on June 27 is followed by a pendulum maximum. This lagging of the inclination of the massive pier is precisely what one should expect if the observed oscillations are of thermal origin. It would seem that the parts of the pier exposed to the light expand and contract on the more equally temperatured colder parts, as an axis, as it were. The result would be a pendulum mechanism, very similar to the trian- gular bracket which I have discussed above, 13 , and which is peculiarly sensi- tive to the elongation of its parts. The expansion of any side of a triangle produces relatively marked tilting of the axis when the instrument of detection is a horizontal pendulum. Taking the observations as a whole, there seems thus to be very little opportunity in the case of an ordinary massive pier of conducting observations, when fixity of inclination within i" of arc is in question, even for brief periods of time. Thus even after June 28, in case of the observed pier, there are changes of a amounting to 2" of arc in ten days, and 0.2" of arc per day must be looked upon as no unusual occurrence. 25. Improved aluminum pendulum. Observations. The outstanding ques- tion bearing on the above observations was the possibility of a magnetic influ- ence in case of the horizontal pendolum made of steel tubing, the pendulum being otherwise admirable because of its relative strength. A new pendulum, built entirely of aluminum tubing, with the exception of the brass clutch and the vertical hard-steel bearings for the pivots, was therefore installed. The aluminum tubes were screwed firmly together, the large triangle having the following dimensions and constants: Mass of pendulum, 554 grams; mass of grating holder and leveler, 456 grams; mass of (single-plate) film grating, 114 grams; mass of damper, 60 grams. This brings the total weight up to 1,124 46 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. grams, not differing much from the above. If lightness is an object (small torques being in question), the clutch and grating holder should also be made of aluminum and a lighter grating attached ; but this is a secondary considera- tion here, though the mass might easily be brought down to about 700 grams. The distance of the center of gravity from the axis was 11 = 93.1 cm.; dis- tance of line of light at grating from the axis, R=no cm. The period was about as above, Tig sec. The distance apart of the pivots, = 97.1 cm. Though, in general, the aluminum triangle was a copy of the steel triangle, some improvements in construction were introduced. Thus a micrometer attachment was added to the lower pivot, so that a direct value of , the incli- nation of the pendulum axis, could be obtained. Windows were put in the case and both pivots were now accessible without removing it. The micrometer did not work as well as was expected, for reasons which did not appear. In several series of experiments, the mean of the horizontal angle corresponding to 30 of rotation of the micrometer screw of 32 threads to the inch and a distance of 97.1 cm. between pivots was 6= j~ =0.00793 radian since the reflected spot of light traveled 6.5 cm., when the scale distance was 410 cm., for each step of 30 of the micrometer screw. The corresponding change of inclination of the pendulum axis would correspond to one-twelfth of the pitch of the screw and would be -I? ^=68.,X.o- radian Since a = \\\ FIG. 28. FIG. 29. The interior of the smaller disk d, (originally) about 10 cm. in diameter and 0.6 cm. thick, is suspended vertically by two fine wires / from the end of the arm of the horizontal pendulum, just below the grating. The disks D, d, D are coaxial, while d is relatively stationary; D or D may be brought as near to d as desirable by aid of the slide micrometer, the other disk being removed at the same time. 50 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. The method of attachment of the disk d to the horizontal pendulum is shown on a smaller scale in fig. 29. Here G is the grating, secured by three adjustment screws to the table T, the cylindrical shaft of which is grasped on a clamp (open form) of the horizontal pendulum P. To the bottom of the shaft in question, a cross-piece hgh is screwed and fastened with a lock-nut. The two fibers// which support the disk d are wound above around the pulley screws hh and thus adequate vertical adjustment of disk d is available. The slide micrometer is attached to the pier by a firm horizontal rail capable of adjustment forward and rearward. A strong clamp attaches the base of the slide micrometer to this rail, so that the whole instrument may also be adjusted to the right or left, roughly. The fine adjustment is completed on the slide micrometer itself. Finally a case is provided covering the disks D and d and part of the micro- meter, so that only the drumhead and scale projects. The apparatus was found to work satisfactorily. It is quite possible to reject the water damper at the end of the horizontal pendulum, above, and to rely solely on the effective air damping produced, when the disk d is very close to D or D'. In fact, the tin cover, in this case, was all but superfluous. D could be shifted from end to end of the course, without materially interfering with the visibility of the ellipses in the spectrum of the interferometer. The real interferences unfavor- able to the gravitational measurement were incidental, due either to the change in inclination of the pier, or to changes in the magnetic field (inasmuch as the pendulum was preliminarily constructed of steel tubing), or to the causes discussed in this chapter; for what was found was not an attraction at all, but a repulsion, much larger in absolute value than the attraction anticipated. 27. Equations. The chief equations to be used in the present work have already been given above. It is merely necessary to add those which bear upon the sensitiveness of the method. Since the disk of mass m is added, at the mean distance R, to the mass of the pendulum M, the force at R from the axis is now The gravitational attraction /' of the disks necessarily involves spherical harmonics, but may be written temporarily as where m' is the mass of the stationary disk at a mean distance d from m. Equating these forces and inserting the value of F R , the equation for &N, the displacement at the micrometer, becomes (3) EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 51 In the first place, therefore, m/M m ^f M so that the lightest available pendulum and the heaviest admissible disk is to be selected, although the increase of sensitiveness is not quite proportional to m/M, but diminishes as (i+mR/Mh)-*. This procedure, even when the float is used, is relatively inefficient and the value of AW can probably not be increased more than twice the above value (a difference of AW= 2X0.001 2 for the two extreme positions of the disk) by this means. 28. Equations for the vertical pendulum. A final word may be added with regard to the inclination a. This can be detected with such precision that a method based upon it deserves consideration. The apparatus in this case would take the form of fig. 30, where ABCD is the iron framework of the heavy, long, vertical pendulum, with the massive bob at D and knife-edges and tablets at e t so that the pendulum is capable of swinging normally to the plane of the diagram. The hori- zontal pendulum is attached by two pivots, f cj) a and 6, to the central rod CD of the ver- tical pendulum. It is to swing clear of it FlG> 3 - and to be in equilibrium in a parallel plane. The deflection of the horizontal pendulum is also normal to the plane of the diagram, and it measures the change of a of CD, as above, G being the grating, h the center of gravity. When gravitational attraction is to be observed, the bob D is one of the attracting bodies and of mass m', whereas the attracting mass m, with its center on the same level, is placed in front of or behind the plane of the diagram. If the mass m' at the end of the vertical pendulum is at the distance L from the horizontal axis, and the mass M' of the remainder of the pendulum virtually at a distance H (center of gravity) from the axis, where d is the mean distance apart of m and m'. Hence (6) AW' = ym2R/g Thus if d=io cm., p=io, with the other magnitudes as in the above inter- ferometer, AW' = io- 6 X7.5 cm., which increases but as the first power of the diameter of the spheres. Hence, in spite of the precision of a measurement, the method would not be available for the determination of 7. 52 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 29. Observations with small plates. The first experiments were made merely for the purpose of testing the method, using the same heavy horizontal pendulum as in the preceding section. There are two or three objections to this pendulum for the present purposes, of which the first is its weight M; the second is the water damper, which introduces inevitable discrepancies, due to such capillary forces as result from surface viscosity. The third objection is due to the fact that the pendulum is made of light steel tubing and points in the north-south direction. These tubes become weak magnets in the earth's field, and the angle 6 may change with the variations of this field. Finally the inclination a of the axis of the pendulum, due to terrestrial causes, is itself to be considered; this can only be eliminated if the time of observation is reduced. The two attracting plates of rolled brass were each 6 inches in diameter and 0.25 inch thick, weighing w' = i,o3S grams. The attracted disk d at- tached to the horizontal pendulum was 4 inches in diameter and 0.125 inch thick, weighing 227 grams. The distance between the large plates was 2 . 5 cm. on the micrometer, this being about the limit of the micrometer screw and sufficient for the diminution of the attraction in question to negligible values. The difference of AAT for the two extreme positions of the disks was esti- mated above as 0.0024 cm., or 5 drum-parts. It should have been easily detected, if not masked by the incidental disturbances referred to. The five series of observations are given in the curves, figs. 3iA, 316, 31 C, 32A and 326. They show both the release of the suspended disks from con- tact with the disk fixed on the micrometer, and the differential effect of the fixed disks on opposite sides of the suspended disk, but near it. Z\fl A 40 -20 30 40 -05 & & FIG. 31. In fig. 3 1 A, the abscissas are the successive excursions A* of the micrometer bearing the fixed plates, the ordinates are the corresponding excursions AN of the suspended plate. Beginning at a, the two plates are nearly in contact, and this contact is made more definite in the direction + x. Hence in the curve from a to d to 6, as shown by the arrows, 2A/V=A*, as it should be. After passing b toward c the suspended plate is released, but released in such a way as to suggest repulsion at 6, whereas the other four points nearer c EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 53 in their downward slope toward the left would be compatible with gravita- tion. Release should take place at the intersection of the two lines. Results of the same kind are shown in fig. 3iB, where the apparent repulsion is very definite. It is probable that in both cases the discrepancies observed are dis- torted by capillary forces, surface viscosity at the water damper, and by the inclination difficulties. This is borne out by fig. 3 iC, in which the fixed disks were alternately placed all but in contact with the suspended disk. The curve should have been zigzag, with the oscillations equal and in opposite directions; but it is quite irregular, due to extraneous causes. R and L indicate whether the fixed disk is on the right or left side of the suspended disk. A v B A 3 t 4 ft \ \ X 005 f 4 \ COO -005 w* T i \ -a FIG. 32. The water damper was now removed and the work repeated, relying on the air-damping at the disks only. No difficulty was experienced in obtaining the interferences; but the results fig. 32A show no evidence whatever of attrac- tion. Similarly in the alternations of fig. 326, the curve which should have been zigzag shows no regularity. Here again foreign disturbances have masked the effect sought, although the displacements themselves were ap- parently definite and satisfactory. It is therefore necessary to replace the disks by a larger set, as is done in the next section. 30. Observations. Plates of larger area. The brass plates were now re- placed by a set larger in area but thinner, this being in the direction of the improvement of method indicated. The same unnecessarily heavy steel pendulum had, however, to be used, so that M=i,2so grams, h = So cm., R = 111.3 cm., ^ = 0.01081 radian, ^ = 42.9 AAf. The new brass plates were identical in size, the mass being m = 468 grams each, the diameter zr = 20.3 cm., and the thickness 0.17 cm. In place of gravitational attraction an apparent repulsion, equivalent on the average to 0.0338 cm., or about 68 drum-parts, was observed. The observations are given in table i and in figs. 33A, 336, 330, the arrows showing the direction of successive observations. The abscissas denote the positions A* of the attracting "fixed " plate on the micrometer, the ordinates the corresponding value of the displacement AN of the plate suspended from 54 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. the horizontal pendulum, read off on the micrometer of the interferometer. The plan was to begin with the plates more than in contact, so that the mov- able disk is carried by the fixed disks until released. In fig. 3aA, at b, the plates adhere until at a a release or fall suddenly takes place, the plates being now over i mm. apart. The figure shows that release should have ^ occurred at the position 2.09 cm. In fig. 33B, corresponding to the other ("small") side of the Fraunhofer carriage, the plate is released at the position 0.51 cm. and there is no adhesion. In fig. 33 C, on the original ("large ") side, the plate is passed into cohesion and then released with a smaller fall at a. TABLE I Large brass disks, not in metallic contact. Steel horizontal pendulum, m =468 grams; r = 10.2 cm.; *=o.i7 cm.; ^ = 0.0108; M= 1,250 grams; h = 80 cm.; = 111.3 cm.; F = Fixed plate at Ax Movable plate at io*AN Fixed plate at A* Movable plate at io 4 AW Fig. 33A. 2.15 cm. .10 05 .00 1-95 .90 - 955 + 37 1030 2015 2875 283 Loosening. 2.05 cm. 2.05 2.00 2.05 +650 185 295 215 Fig. 338. 0.45 .40 50 :io 65 2493 3493 1605 625 495 495 Fig.33D. 2.00 /.6o Us 2.00 65 2.00 65 2.0O 65 302 580 517 187 565 ISO 535 177 550 Fig. 3 3C. 0.65 1.90 1-95 2.00 2.05 2.10 495 285 320 320 + 175 - 285 * Mean values. Thus the positions 2.00 and 0.65 are guaranteed as free, the space between the reacting plates being over i mm., as compared with the distance 1.4 cm. between the fixed plates. The effects of alternately approaching the opposed fixed plates to the movable disk are shown in fig. 330. They are quite definite, larger in order of value than would be anticipated and constitute repulsions instead of attractions. In fact, figs. 33A and 336 show that in case of attrac- tion or of cohesion, AJV should be too large on the "large " side, and too small on the " small " side of the stationary disk. In fig. 330 the reverse is the case. To explain this repulsion a number of facts have to be taken into ac- count. Both the fixed disks are separate metallic systems, but ultimately anchored into the pier with iron bolts, so that a volta contact force, iron-brass, would be inevitable. The disks are thus carrying charges, depending on the nature of the anchorage in the pier, whether this is moist or quite dry. It seems probable, as will be shown below, that these small potentials are negligible. Again, with the small forces per square centimeter of area in question, the EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 55 viscosity of air is sufficient to necessitate the lapse of considerable time before a position of equilibrium is assured, even with the disks a millimeter or more apart. In the above experiment sufficient time was allowed until the motion became vibratory, after which the reading was taken; but it is difficult to assert that, even after indefinite waiting, further subsidence would not have taken place. Finally the flexure or tipping of the pier, where long intervals of time are in question, can not be eliminated. There would inevitably be some error on this account. It seems improbable, therefore, that the actual gravitational attraction of metallic disks will be determinable, while a non- metallic system is liable to introduce even greater errors. D \ ftO 20 fcf 4 -5-6 D FIG. 33. 1-95 \^, 31. The same, continued. Metallic contact. The next advance consisted in placing the disks in electrical (metallic) contact, which was easily done by joining the pivots of the horizontal pendulum with the slide of the micrometer bearing the fixed disks by a copper wire. Moreover, since the position of equilibrium is gradually reached in the lapse of minutes, the time of the observations is taken in minutes. These results are given in table 2, and are inscribed in figs. 34A and 346. The figures on the curve show the series in question and the plate ("large" or "small" side of the plate micrometer), which is actively repelling. In fig. 34A the alternations are found after long- waiting; in fig. 343, however, in time series. When equilibrium is reached, and this is always relatively quickly, the disk oscillates due to incidental causes. It makes no difference from which side the position of equilibrium is approached (series 2, 8, 15). The presence of radium on the plates has no effect, other than the mechanical disturbance given by placing it there; the same position of equilibrium again results. When the disks are jolted by contact (case between series 7 and 8), the equilibrium position may be tem- 56 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. iO 10 a 06 15 04- 02 00 -OS, /\ 10 B FIG. 34. 50 T TABLE 2. Large brass disks in metallic contact. Constants as in table I. Ax lO'AW Time. Ax IO*Atf Time. Ax itfAAT Time. cm. 0.65 2.00 65 2.00 2.05 2.10 cm. 3 43 29 29 26 min. cm. Vi 5 cm. 68 64 68 min. 2 4 cm. 2 cm. % 49 45 min. i 2 4 VI? 28 49 55 *50 53 53 i 2 4 5 6 & 90 68 63 60 2 4 2.20 I 29 36 4i 46 46 47 2 4 6 8 10 12 0-75 XIII Radium on 69 65 62 62 6 7 8 9 0.65 144 0-75 VIII - 7 21 tS3 *62 66 3 5 7 2 i'i 5 132 103 t-50 2 10 xft Radium off 62 62 12 14 0.80 III 7i 68 ?59 70 O 2 4 6 2 rJ? 36 50 * 4 8 48 48 48 o 2 3 4 5 53 xv 5 30 48 50 52 I 2 4 # 66 67 65 o 2 4 2.15 Radium on 59 54 T 88 68 60 61 60 i 2 4 6 '$' 48 56 55 2 4 Vibrating. t Fall. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 57 porarily disturbed (series 8). Apart from flexure of the pier, long waiting (50 minutes in series 9) does not further change the position of equilibrium, if the slight swinging is taken into account. The cause of the gradual motion may reside in the viscosity of air, as indicated in the next paragraph. If we compare the results of fig. 330 (system not in metallic contact) with the present (system metallically connected) the results appear as follows: System not in metallic contact System metallically connected. 2 AW = 0.02 1 22 2AJV=o.oi6 H 38 38 37 015 12 13 10 II Mean 2 &N= 0.038 Mean 2AJV =0.013 The disks were usually about a millimeter apart. Metallic contact has thus apparently made the repulsion smaller; but it is not certain that the distance apart of the plates is quite identical. Moreover, data obtained at different times vary considerably. In the present case the repulsion observed for the disks 20 cm. in diameter is 2^ = 65.2^=65. 2X0. 013 =0.85 dyne, at about d = i mm. of air-space. 32. Retardation due to viscosity of air. It will next be necessary to ex- amine the above suggestion, that the very gradual approach of the suspended disk to its position of equilibrium may be due to the viscosity of the interposed film of air, in view of the small forces and small displacements involved. The case may perhaps be treated in terms of Poiseuille's law, assuming that the flow is from the center of the two nearly contiguous parallel disks radially toward the circumference. Let y be the initial distance apart of the disks, and the time t = o second, measured from the fixed toward the movable disk. Let y' be the final position of equilibrium of the movable disk, so that its excursion is y y'. Let a small impulsive force P act normally on the outside of the movable disk, by which it is put into the position y. The pressure generated will cause a flow radially outward, and if p is the pressure in the fluid at a distance r from the center, Poiseuille's law may be written d) -*v= fr = Cwy)' = 0.10 cm.; >j = i9oXio-, so that roughly Table 3 contains some corresponding values of y and P computed in this way. TABLE 3. Motion of movable brass disk retarded by viscosity of air film. io 4 X y=AN/2 > io* X t cm. sec. cm. sec. IOOO 0.00 680 548 900 850 39 .70 670 668 8.47 10.47 800 1.16 666.8 19.41 750 i-93 y' = 1/15 oo 700 3-68 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 59 These values are reproduced in fig. 35, which with table 3 shows that the position of equilibrium is reached to AN=io~ 4 , the smallest quantity easily measurable, in about 25 seconds. 8 10 FIG. 35. /4- 1 6 The fluid has been treated as incompressible. If this is not done, the results apparently become unavailable. A further step may, however, be made: Poiseuille's equation (i) if the condition VP = Vo PO is introduced, leads on integration to the form (n) P'-p'o where P is the pressure and Vo the volume issuing at the edge, per second at the normal pressure PQ. In endeavoring to use (n) directly, I have not succeeded in producing a practical form of equation. Equation (9) may be put in a different form suitable for computing in the ultimate times of very close approach to equilibrium. For this purpose, let y'/yo = a and yy'=b where 6 is to be very small, so that y=ayo+b. Equation (9) then reduces nearly to ay Usually b/ay may be neglected compared with i a. Thus if 6 = io~ 4 cm., t = 1 1 . i sec. , with the other constants as above, y = o. i cm. ; a = 2/3 . For the same case, 6 = io~ 4 cm., if 70 = 0.05 cm., = 2/3, ^ = 38.3 sec., are needed to approach within io~ 4 cm. of the position of equilibrium, etc. In case of repul- sion, a> i and 6 is negative. Thus for 0=3/2 cm., 6= io~ 4 cm., y Q = 1/15 cm., * = 6.53 sec. For 70 = 2/45 cm., y' = 1/15 cm., / = 13.8 sec., etc. The intervals so computed are small as compared with the times actually observed, where many minutes have to elapse before equilibrium is obtained. It seems diffi- 60 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. cult to interpret this excess by supposing that the method is inadequate; for the effect of gravitational attraction between the disks, which has been ignored, would be a virtual increase of P . Since, on the average, pressure excess, p po, is very small as compared with p Q in the actual case, equation (n), seeing that p*-p*o = 2p<> (p-po) becomes and which is identical in interpretation with equation (2) above, where p = therefore leads to the same conclusion. 33. Observations, continued. Presence and absence of electrical contact. Notwithstanding the improbability of electrical effects, it was thought neces- sary to test the case directly. Accordingly, in table 4 and fig. 36, series i to 8, experiments are recorded with the plates not in metallic contact, series i to 5, and with the plates in metallic contact, series 6 to 8, respectively. The behavior in both cases is virtually the same, when the shift of zero is taken into account. Observations are plotted in time series, with the last observa- tion marked by a circle, and they are in each case continued until the motion of the plate is retrograde, whereupon the real oscillation of the plate begins. To throw further light on the subject, a Leclanche' cell was introduced in series 9 and 10 and removed in series 1 1. The lighting circuit of the room was placed more remote in series 12 and the system earthed in series 13 and 14 for both fixed disks. iO 08 06 04 02 00 ( 1 * r I 5 N* 8 b 8 I ?1 ' a ft 5 10 70 70 70 I / 10 L f 2 - 2/6 1 r fs d / r I C \ A t /a y $ I & me, * ' K - ty 05 2 61 ocn 0-75 & .80 64 19 50 II 9 0.036 (Dist. .004) 101 .075 (Dist. .050) Swinging. 05 03 01 -41 ~G3 -05 -47 < "ft Iff i 9 < 1 <# >>& > 6 no \ 4 h M f 1 W \ /> \ A <2-5t> ' S J 1 v* J-6U \ 4 i i %) -n c, 8< - to ft i 8 i 1 i / 3 t +\ i // s \ -> s. J 1 "" 20 40 60 80 100 #0 O 07 * -05 -10 & FIG. 41. the pier is liable to warp during the time in which the observations are made. Nevertheless the corroboration obtained is of great value. 68 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. The observations were made in time series, when the plates were close together. For plates farther apart this is not essential, but in the absence of other than the air-damping the suspended plate oscillates, so that mean values have to be taken. It will be necessary to await the summer in order that all observations may be made in a room free from artificial heat. In table 7 and fig. 41 A I have inscribed the first observations from plates close together. In table 8, the summary of all the observations is given. In series i to 4, table 7, the plates are probably not far enough apart, though the contact position is still beyond the equilibrium position by about 0.014 cm. It was not necessary to wait for the uncharged position of the plates, as this remained pretty constant long before and after the experiments. In series 5, 6, 7, and 8, the distance apart is increased about 0.042 cm., and there is no danger of actual contact (free space 0.056 cm. when charged), so that actual repulsion is in question. TABLE 8. Summary. d Atf d' Vots. V'comp. FJW,. Fltfomf. Correc- tion AN FM F'R * y / (I) cm. (2) cm. (3) cm. volts (5) volts. (6) dynes. (7) dynes. (8) (9) (10) cm. (ii) cm. (12) dynes. 0.056 0.039 0.075 12-5 7-43 7-i 2-54 O.IIO 2.8 0.036 + .020 4-7 .044 .014 .051 6.2 3-56 2.85 94 .044 3-o .0146 +.029 1.90 .102 .021 .112 12.5 9-93 2-15 1-37 033 1-57 .003 +.096 .46 133 .Ol6 .141 12.5 i 1-3 1.27 1.04 .019 1.22 .001 +.132 19 .014 045 .036 12.5 19.0 114.0 2-93 1.75 39- 85 -.84 no. Table 8 shows the essential data of these and subsequent experiments in which the distance d, in centimeters, between the plates is gradually increased. AAf shows the difference of displacement (in centimeters) observed, when plates were respectively charged and uncharged. AAf thus measures the displace- ment of the suspended plate in half centimeters. From &N the approximate electric force (ignoring the repulsion of plates) may be computed as above, F' R = 6$.2hN. This is given under F' R computed, in the seventh column. Plates were charged by aid of a storage battery to the potential shown under V<*s (volts), in column 4. From V and d, the attraction of plates F' R may be computed, since A = irR i being the area of each plate. The results are given in column 6 and the ratio F/F' in column 9. Furthermore, the potential V may also be computed from F R , since V.I R and this is inserted in the fifth column. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 69 From the computed values of F' R , since F R is equal to 6s.2XAAf, the true value of the displacement in the absence of repulsion, AAT.may be computed as 65.2 the result being given in the eighth column in centimeters. It shows the corresponding displacement of the suspended plate in half centimeters. The two values of A/V and AA/ 7 now give us the displacement due to repul- sion in centimeters, as shown in column 10. Again, the distance apart (in centimeters) of the uncharged plates d' is given in the third column, being and found from observation directly. Finally, the residual distance apart of the plates, y, if the suspended plate had taken its true displacement A/V' (in the absence of repulsion), is given in the eleventh column, since In every case, except the first, in which y is negative, the plates when charged at a distance d apart were not under forces sufficient to put them in contact. One must observe, however, that for a distance apart y when d 0.13 cm., the ratio is nearly i ; i.e., the repulsion of plates nearly vanishes when their dis- tance apart markedly exceeds i mm. Just how large d would have to be in order that F' R /F R =i, I did not endeavor to find, since the suspended plate vibrates annoyingly for large distances apart. In other words, definite ex- periments of this kind must be left for the summer months. The constants of the pendulum should then also be determined. Moreover, in a lighter pen- dulum, the sensitiveness may be indefinitely increased, particularly when the pendulum is provided with a float, while the error due to the inclination of the pier does not simultaneously increase, an obvious advantage. It seemed wise, therefore, to stop the work for the present at the point of progress reached. 70 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. Table 8, however, admits of a number of preliminary estimates of the decrease of repulsion (f) with d, the distance apart of plates, for we may write, /= 65.2 X2X= 130* dynes, nearly. These values are given in column 12 of table 8 and in fig. 416, with the ex- ception of the first, which is liable to be anomalous from actual contact. The second observation also seems to be in error for some reason not detected. The others make a compatible series. The forces found in the above work (paragraph 10) lay between 1.3 and 0.4 dynes, for distances of the same order of value, but which were not quite the same in the two positions of the fixed disks. If we take the results in table 6, which are probably the best, AAT= (0.0490.014)72=0.018 cm.; d= (0.031+0.097)72 =0.064 cm.; /=6s.2X 0.018=1.17 dynes, the results of / and d, as shown by the cross in fig. 4iB, fit in very well with the present data obtained from electric attraction. The repulsion therefore has throughout been found of the same order of magnitude. The pressure corresponding to the above thrust / is found (as above) by dividing by the area A of the disks, whence We may then compute the attraction of the disks per gram of air film, at a distance h from the disk, similarly to the ordinary case of the barometric formula, or - an Thus, if one can detect the variation of / with h, the molecular attraction of the disk per gram of air should be discernible. 37. Conclusion. By the application of displacement interferometry to the deviations of the horizontal pendulum, I find that two parallel rigid plates whose distance apart is of the order of i mm. and less repel each other, in air, with a force far in excess of their gravitational attraction. This force in- creases rapidly (certainly as fast as the inverse square) as the distance of the plates decreases, and vice versa, but can be recognized beyond a millimeter of distance. For brass plates 20 cm. in diameter and i mm. apart, the repulsion in question is of the order of 0.5 dyne and therefore equivalent to a pressure of 0.0015 dyne-cm, or roughly io~ 9 atmosphere. It is in excess of any electric repulsion due to the absolute voltaic potential of the disks. The suspended plate reaches its position of equilibrium gradually, the motion progressing at a retarded rate through infinite time, in a way characteristic of the viscosity of the film of air between the plates. I have estimated the intensity of the force both from the repulsions of a vertical plate suspended from the horizontal pendulum on opposite sides of a fixed parallel identical plate; also by charging pairs of plates to a given differ- ence of potential for a given distance apart. So far as can be seen, the repul- sion is caused by the condensation of air on the surface of the plates by molec- EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 71 ular and not by gravitational force (which is too small). Hence, the method employed should enable the observer to find the density of the concentration in terms of the distance from the plate and the law of attraction of the plate in terms of distance within the small distances in question. In other words, a method for direct investigation of molecular force is here apparently given. Correction: The effect of gravity in Chapter II was overestimated and the data have been withdrawn. To make an estimate of the gravitational attraction between the largest plates, it is sufficient here to consider the attraction of contiguous disks under the approximate form m* w? F = where ^ = 6.7X10-* is Newton's constant, a the density, ^ = 468 g. the mass, A the area, r = 10 cm. the radius of the disks. We may then write ,# m M _ ~ 2r r 2 g . 43. Observations. Green glass column. In spite of the clearness of the column, the light absorbed at the ends of the spectrum makes it nearly im- possible to recognize the small, sluggishly moving ellipses. The observations, therefore, are reasonably good only between the D and E lines. In some cases, moreover, it is easy to mistake the lines, from the coincidence of the direct TABLE 9. Green column. = 22.87 cm.; = o.68 cm.; B =4.6X10-"; 1 = 15; =9.7: =sin 7/sin-R; (3-E+ cos R +2e/cos R) =70.66; M = i-53- Lines. C-D. D-E. E-b. 6-? b-F. 6N, 0.185 0.235 0.046 O.I 12 .166 233 50 .122 .... .168 175 .236 237 48 40 .06 5 67 .... .181 .232 045 075 .181 .175 175 .... 48 43 45 .084 70 67 '.'.'.'. .174 43 I7d Mean SN f observed 1754 .2346 .0453 SNf computed .i8n .2347 .0442 0.1610 78 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. spectra of higher orders, with the two interfering spectra. For work of this kind it would have been preferable to make I (angle of incidence) about 45. The individual observations are given in table 9. Between D and E the error is about 0.002 on a length of i2.6=AA/" e for the total displacement; i.e., about 1.5X10-* of M * I Q computing 8N C from equation (9), the value of B computed for light crown glass from Kohlrausch's tables was accepted, as a special measurement for so many plates of glass seemed out of the ques- tion. The films of Canada balsam are negligible. The quantity AAT C was roughly measured. To find this accurately it would have been necessary to make special adjustments, as it is large (AA/" C = 1 2. 6 cm., quite above the range of the micrometer), and as a readjustment of the mirror must be made in the presence and absence of the column, for which it is difficult to make an allowance. The end faces were not quite plane parallel. Using equation (6) the value of the correction zEB/\* is 0.605 cm., whence 2/# = 1.525 at the D line. The value found directly from the total reflection for a similar glass was 1.521. As without the correction ^=1.551, the corroboration of the equation is adequate. One may note that zB/\* = 0.0265, a $ a correc- tion of n, is independent of E, the length of the column. But, for purposes like the present, a small thickness of glass (E about i cm. and within the range of the micrometer screw) is preferable, even if the accuracy could be enhanced by using a stronger telescope. Table 9 shows that the computed values of SN C happen to coincide with the observed values between the D and E lines. Between D and C, D and 6, the results are quite within the errors of observation and satisfactory. The F line was obviously not observed, some other line in this dark part of the spectrum being mistaken for it. Thus the line X = 49. 5 8 would give dN c = o.ioS, the line X = 50.41 would give 0.064, each coming close to some of the observa- tions. The results as a whole therefore attest the accuracy of the equation used, as the computed lines are clearly better than the observed lines. 44. Observations. Blue glass column. Although this column was more colored than the other, the observations were apparently not inferior. Table 10 contains the results. It was just possible to reach the F line, visually. As before, these data reduce largely to the shift from D to E. The column was too long to be compassed by the contact lever used, and the length E given is therefore approximate. To compute M from AAr e =i 4 .o cm., observed, the equation is as before where the last term is 0.0265 for the same B and X; hence /i = 1+0.5503 -0.0265 = 1.5248 agreeing sufficiently with the experimental result. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 79 The observed and computed values of 8N C show the same agreement within the errors of observation as before. Differences are due to the value of B used, which is naturally not quite the same as in the preceding case and should here have been about 2 per cent, smaller. Between b and F, io 6 X = 50.41 would give 8N C = 0.0709, which was probably observed. TABLE 10. Blue column. =25.4 cm.; =o.68cm.; 5=4.6X10-"; 7 = 15; -=9.7: Lines. C-D. D-E. E-b. 6-?. b-F. SNc 0.178 0.2=^2 O.O5O 0.063 O 157 .218 .196 254 255 257 .256 254 .256 53 47 49 47 50 48 8 .191 Mean 6N e observed .197 .2549 .0491 .174 SNf comouted . . . .200S .2<%QQ .0400 .1782 45 Observations. Shorter column. The results for this column are given in table n. It was less than one-third as long as the other columns, but, absorbing less light, all the lines were seen. The ellipses being more mobile, sharper adjustment is implied; but the F line could not be recognized with certainty and there was difficulty at the C line. In this case, ^N c = ^.g6i cm. lay within the compass of the micrometer. The only error therefore is the intermediate readjustment of mirror in presence and in -absence of the column. The index of refraction of the D line is thus M=I+ 3.961 7-1675 which is of the same order as before. 1.5261 TABLE n. Short column. = 7.1675 cm.; = o.68 cm.; B=4.6Xio- 15; -8=9.7; Lines. C-D. D-E. E-b. fr-?. b-F. &N C 0.0588 0.0795 0.0147 0.0432 0.0560 589 790 158 353 558 591 576 792 785 784 153 155 ISO 0305 319 323 560 571 548 .0224 230 213 331 Mean SNe observed .0586 .0789 .0153 .0559 SN f computed .06035 .07821 .01473 .... 05363 The results for this series are scarcely as good as the preceding, relatively, since finer micrometric measurement was required; but, absolutely considered, 80 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. they are better. Thus between D and E the agreement is within 7 X io 4 cm. It is obvious that between b and F different lines were sighted, some of them possibly due to superimposed direct spectra. Thus for ioX= 50.41; = 49.58 SN e = 0.02 13 1 = 0.0359; etc. Both these and other lines seem to have been used. 46. Summary. The results contained in tables 9, io, u, reproduced in fig. 43, show that equation (i) above, or any of its derivatives (4) and (8), gives an accurate account of the motion of the center of ellipses throughout the spectrum, even in case of such extreme conditions as are introduced by glass columns io inches or more long. The constants of Cauchy's or any simi- lar dispersion equation may therefore be obtained directly from observations of this character. In such a case a linear interferometer, i.e., one in which IR=Q approximately, would be specially convenient. If 8 refers to the difference of the variables for 2 lines X and X' which in case of the simple dispersion equation gives $B(E+ e )&-jp- As this linear interferometer will have other interesting properties, it has been thought worth while to construct it in connection with the present work. 4 s> 2 -4 4 -2 ^ ^ 4 I. T^ K -4 ^ X ^^ ---. ^\ S3 ,t ^^; ^ ^fe ^ 1 - f TdxKf- * -< *^! ^ 9 X^ 4850&H54-56&8606264- FIG. 43- The expectation of reaching great sensitiveness by using long columns was not fulfilled in view of equation (17), which shows that the ellipses become more and more sluggish in their motion through the spectrum, as the column CHAPTER IV. PART I EXPERIMENTS BEARING ON THE PROPERTIES OF CORONAS. 47. Introductory. There are a number of obscure points in the theory of coronas when the particles producing them range in size from about io~ 3 cm. to 10 ~ 4 cm. in diameter. These relate chiefly to the colored central disks and to the color which for very fine particles spreads uniformly over the white source of light. In the latter case the colors are strictly axial and they suggest the in- terferences due to thin plates. At least a tentative explanation along these lines seems available.* Light, moreover, is abundantly reflected by the par- ticles, as may be tested by using a Nicols prism. It seems reasonable, there- fore, to assume that in spite of their small size the light is also transmitted and that the effect is appreciable when the column of fog is long enough in the direction of the impinging light. All of this is in accordance with the condi- tions under which axial colors are produced. If they were regarded as dif- fractions within the geometric shadows of the droplets whose diameter d is decidedly smaller than io~ 3 cm., the axial distance 6 in front of the droplet corresponding to the color X would be b=d z /n\ nearly, for the fringe of the nth order. Hence, even in case of n= i, b would be much less than 0.2 mm., whereas the axial colors are seen for all values of b; i.e., they do not vary with 6, however large it may be taken. The disk colors, however, belong to the phenomenon itself. If the element- ary equation for a single particle were true, i.e., if where 9 is the angle of diffraction for the wave-length X and the diameter of particle d, C the constant given by Airy's series, and s/R the aperture of the corona shown by the goniometer, the disks should invariably be white and red edged, as is the case of relatively large particles and small coronas. Actually, however, the white disk is more and more evanescent as d is smaller, the color being particularly vivid in case of the green coronas, where the disk is almost quite green. The disk and annuli thus recall the appearance of the rotary polarization of a quartz crystal cut normal to the axis, though of course all polarization is strictly absent in the colored diffraction phenomenon. I have in fact endeavored to identify the colors by the aid of a rotary polariscope, fig. 44, B and A being the polarizer and analyzer, Q' the quartz rouge, FC the fog-chamber at a distance from Q', Q" a quartz column sufficiently long to give a white field. Hence the coronas could be seen directly through Q", * Barus, Am. Journal of Sci., xxv, 1908, pp. 224-226. 81 82 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. whereas the color from Q' appeared contiguously on the left, the apparatus being used like a half-shade. But the attempt was not of practical value for incidental reasons. . 6 , \ 6 inch) laterally. For this purpose a screen with an appropriately wide vertical slit is placed either at F or on the grating table at b. When this is done two slit images may usually be seen at the mirror N and the brownish one is screened off there. To obtain the soli- tary ellipses, the ruled side g of the grating should face the source of light. As the grating is of ordinary glass plate and therefore wedge-shaped, the top and bottom of the grating should be selected so that the wedge and the thick- ness effect act in concert, to separate the two slit images at N, referred to. In this case the undesirable image may be more easily screened off. 63. Remarks. It was my expectation that the Nernst filament might itself be used as a slit without further appliances than the screen tube c. But this is not adequately the case, as the filament is a little too thick. Without the slit and the tube only, the ellipses are just suggested. Possibly if the white porcelain surface of the Nernst burner were black instead of white porcelain, clearness would be enhanced. But the ellipses would not be useful for measurement. Without the slit, but with the slotted screen T or 6, the ellipses are strong but somewhat washed, so that the fine lines to right and left soon vanish. The rings could actually be used for measurement, for 100 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. the centers are well indicated and the motion of rings adequately clear. With the slit and tube, or slit tube and screen, the ellipses become sharp and the fine lines indefinitely visible. The slit need not be very fine, but as it is finer the velvety black lines on the colored spectrum become more marked. The interference pattern is now quite as good as with the arc lamp. Naturally when the filament is so near the slit, the rays on leaving the collimator diverge strongly in the vertical plane. Hence the illuminated parts of the mirrors M and N may be 2 or 3 inches long. If these mirrors are of ordinary plate glass it is not liable to be adequately perfect over the whole length, and the ellipses will be imperfect in form. But this is not a serious disadvantage. At wider ranges (g to M and N i to 2 meters), the arrangement is not very satisfactory for photography, because the light passing through the telescope, unless the objective is very large (a ^-inch objective was used), is only a small part of that passing through the slit. Hence the light camera at the end of the telescope is insufficiently illuminated. For photographic purposes it would then seem to be better to place the Nernst filament at a distance from the slit and to use a condenser; but I was unable to obtain marked ad- vantages in this way, while the condenser is an annoyance. Hence for photo- graphic purposes it is better to replace the plane mirrors M and N by identical concave mirrors in which the light is appropriately condensed. This is done in the inclination apparatus in Chapter I, 21, and further reference has been made there. In any case, however, greater steadiness and freedom from tremor than the laboratory affords would be desirable for photography, and though it is not difficult to obtain families of ellipses in the way given on the ground-glass screen of the camera, few experiments in actual photography were made. The spectrum of the Nernst filament is free from the Fraunhofer lines. It is, however, easy to obtain the reversed D lines, by using an ordinary sodium flame placed either in front of the slit or (contrary to expectations) even be- hind it within the collimator. One would have expected the latter method to interfere with the definition, but it does not seem to do so. When the sodium lines have once been indicated, the cross-hairs of the telescope may be placed in coincidence with them and the desirable fiducial lines of the spec- trum thus obtained. PART IV.-SCATTERING IN THE CASE OF REGULAR REFLECTION FROM A TRANSPARENT GRATING, AN ANALOGY TO THE REFLECTION OF X-RAYS FROM CRYSTALS. 64. The phenomenon. No doubt the following phenomenon has been noticed before, but I have seen no description of it. If a vertical sheet of white light L, from a collimator, is reflected from the two faces of a plate- glass grating, having about 10,000 or more lines to the inch, g being the ruled face, the two beams 6 and y going to the opaque mirror N are respectively vividly blue and brownish yellow. In other words, more blue light is regu- EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 101 larly reflected from the ruled surface than is transmitted, and more reddish light transmitted than is reflected. Since the plate grating is not quite plane parallel, two of the four rays, b' and y', are seen in the same colors in the telescope. This is a great convenience in adjusting the displacement inter- ferometer, where the spectra from b alone are wanted, and the y ray may be screened off at N, while the other y 1 has no spectrum. The transmitted rays t after reflection show very little difference, the one reflected at g being perhaps slightly yellowish as compared with the other. The spectra from 6 and y, if compared one above the other, are practically identical. The difference is not sufficiently marked to be discerned by the eye. Multiple reflection from the two faces gave no further results. Finally, to be colored blue, the beam must be reflected from the air side and not from the glass side, where but little appreciable effect is produced. If the grating is turned 1 80, both the 6 and y rays are nearly white, while the t rays now correspond to the b and y rays and are vividly colored. Outside the ruled surface and with any or- dinary unruled plate of glass, all images are XT' of course white. I mention this merely since c4{ > one might suppose the absorption or color of the glass to have something to do with the experiment. The film grating, where sharp reflection takes place from the glass and not appreciably from the film, does not ordinarily show the phenomenon; but in case of the single-plate film grating of paragraph 60, it is astonishingly strong in the refracted slit images seen in the telescope. These are, in fact, azure blue when coming from the mirror N and reflected from the front side (toward the lamp) of the film; deep brown when reflected from the rear side, after having passed through the film. The two images may be superposed by rotating M with the production of nearly white light. Moreover, the marginal light (otherwise identical but not passing through the film) is white. The images in question are sharp, but it is possible that the material of the film may somewhat contribute to the color. 65. Explanation. Scattering is usually and perhaps essentially associated with diffuse reflection. The present phenomenon, however, is strictly regular reflection; i.e., there is a wave-front, for the blue and yellow slit images are absolutely sharp in the telescope. This is the interesting feature of the phe- nomenon, which associates it at once with the recent famous discovery of Friedrich, Knipping, and Laue relative to the reflection of X-rays from the molecular planes of crystals, and it is for this reason that I call attention to it. FIG. 62. 102 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. In case of the grating the sources of scattered light-waves are not only identical as to phase, but these sources are at the same time equidistant. Hence collectively they must determine a wave-front of somewhat inferior intensity but otherwise identical with the wave-front of normally reflected or diffracted light; i.e., the wave-fronts of regularly reflected and scattered light are superposed. Moreover, if the grating is turned in azimuth even as much as 45 on either side of the impinging beam (after which the many reflections and diffractions seriously overlap) the blue and brown colorations are distinctly intensified. This also is in accordance with anticipations, for the number of lines which are comprehended within the lateral extent 5 of the narrow beam L as the angle of incidence * is varied, increases as s sec i; whereas the lateral extent of the reflected beam is no larger than that of the impinging beam. Hence there should be increased intensity of scattered light in the ratio of sec i or increasing markedly with * from i for i = o, to oo for * = 90. In other words, the scattering lines of the grating are virtually more densely disseminated when * increases. For the light reflected from the inside of the glass plate the evidence to be obtained from color in case of the ruled grating is too vague to admit of definite statements. I have not, therefore, attempted it. 66. A further analogy to the reflection of X-rays. With regard to the recent experiments (I.e.) on the reflection of X-rays from crystals, it may further be interesting to recall my experiments (Phil. Mag., xxu, p. 12 1, 1911) on the interferences produced by two identical but separate halves of a reflecting grating, with the rulings parallel and originally in the same plane. The interferences observed are brought out by moving one of the half gratings micrometrically parallel to itself, to the front or to the rear of the other half, and are here necessarily linear and parallel to the rulings. If i (angle of inci- dence) =6 (angle of diffraction) and d is the normal distance apart of the grat- ings, the same equation n\=zd cos 6 holds. In other words, two identical spectra originating in parallel planes, at a distance apart commensurate with the wave-length of light, are superimposed throughout their extent and pro- duce interferences. I pointed out the bearing of this phenomenon on the theory of the coronas of cloudy condensation (I. c., p. 129), where the compound diffraction spectra, due to successive, parallel, equidistant layers of fog- particles (a sort of space lattice), are superimposed and interfere in a manner evidenced by the disk colors of coronas. In the actual case of distribution, however, the fog-particles (as I also pointed out) are too far apart to admit of the immediate application of the direct theory in question. Some extension of this point of view must therefore be forthcoming if the experiment with halved gratings one behind the other is to be reconciled with the circumstances under which coronal phenomena appear. CHAPTER V. DISPLACEMENT INTERFEROMETRY APPLIED TO THE QUADRANT ELECTROMETER. 67. Apparatus. In an earlier report experiments were given showing the adaptation of the quadrant electrometer for the measurement of very small potential differences, when the needle is provided with two symmetrical, light, plane mirrors, in parallel. The excursions of the needle may be read off, for small angular deviation, on the displacement interferometer. If 5 = AN is the displacement of the mirror of the micrometer of this instrument, and i the angle of incidence of the ray impinging on either of the small parallel mirrors on the needle, 5 = 20 cos * dS/di= 20 sin t where a is the normal distance apart of the parallel mirrors. If degrees of arc are used the ratio is 0.035 a SU1 * an( i * is usually about 45. It should be pos- sible with such an arrangement to obtain a sensitiveness of a few millionths volts per vanishing interference ring, and the following paper is a further attempt to reach this result, practically. The main, if not insuperable, difficulty encountered in such an apparatus is the continual and often irregular drift of the needle, when the condition of rest is so sharply determined. A special environment, without city tremors and at constant temperature, seems to be the only means of obviating these annoyances. The apparatus used is shown in fig. 63 in vertical section. AA is the per- forated base of a massive brass plate, i cm. thick, securely fastened by a large clamp to one arm of the interferometer, capable of some rotation around the vertical and horizontal axes for leveling the whole apparatus, etc. To this the quadrants a, b are firmly attached, by aid of screws i, j, but in such a way as to be quite insulated from the brass plate, in view of the perforated columns g, h and nuts u, v of hard rubber and of the form shown. The clamp-screws k, I are in metallic contact with i, /, and carry charges to the quadrants. There are about 2 inches of free space below the plate A A, available for the connec- tions and, if necessary, for a liquid damper, w. The needle consists of two 8 -shaped leaves, c and d (biplanes), symmetri- cally fastened to the stem st, on which the needle is bifilarly suspended from silk fibers. The two small parallel mirrors, e and /, are adjustably attached to a fine metallic wire at right angles to st and in contact with d. Each mirror has a light vertical and horizontal axis in a bit of cork (not shown). The mirrors are first made parallel by using sunlight and then fixed with melted wax, after which the aluminum foils c and d are centered in place, the eyelets at s and t having not as yet been bent. Light reaches the mirrors e, f through two corresponding holes cut in the vertical walls of the quadrants. The 103 104 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. weight of such a needle is easily kept within 0.75 gram and the air-damping is quite sufficient. Unfortunately its period is large, being about i minute, and it is apt to vibrate as a pendulum. Hence it is often convenient to hook on a wire at t, bent like an inverted V, with the free ends submerged in water to secure greater steadiness on the interferometer; or a mica vane may be added, as at w. The bifilar suspension, 13.5 to 20.2 cm. long in the different experiments, terminating above in the hooked brass rod r, is adjustably fixed in the brass cylinder p, which in turn is secured in the hard-rubber insulator n, attached at right angles to the brass standard GG, the lower end of which is screwed to the a FIG. 63. brass plate A A. This rod can be lengthened telescopically (not shown) ad- mitting of different lengths of bifilar suspension. The hard-rubber lever enables the observer to twist the bifilar. The charge, from a Zamboni cell or the lighting circuit (250 volts), is conveyed to the needle through the hard- rubber insulator at m and the clamp-screw at q (which in turn secures the plug p), through the moistened bifilar wires, as in Dolezalek's apparatus; or it may be admitted through the insulated damper below t. Finally, the lower part of the case CD envelops the quadrants more or less permanently and is provided with wide plate-glass windows for observa- tion. The upper part EF of the case may be taken off like a hat. 68, Observations. Experiments were made with this apparatus at con- siderable length, but they were not sufficiently definite to lead to any quanti- tative statement. Great difficulty was experienced, in addition to the drift of the needle, in securing an adjustment of the mirrors such that the beam of EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 105 light might pass from mirror to mirror and return through the inside of the quadrants. For since the mirrors are quite inclosed in the latter, the path of the beam of light can not easily be seen, and it is troublesome to obtain the several reflections to the best advantage. As the needle fits the quadrants with but one-eighth inch of clear space, it is very liable to be unstable if the parts of it are but slightly out of true. 69. Observations, continued. For this reason it was thought preferable to conduct the experiments by using a needle provided with mirrors attached on the outside of the quadrants. Such a needle (No. i) is shown in fig. 64 (of the wedge type) at cd, the two 8 -shaped leaves meeting on the outside, in a horizontal circular arc xy. The mirrors with the axles in cork are shown at ef t and should be several centimeters above the quadrants ab. The adjust- ment here is comparatively easy, as the mirrors and the path of light are all quite visible. The needle, being sharp-edged, may be charged to a potential of several hundred volts, with- out instability. The period, however, is still large. In the first series of experiments the needle was charged with a Zamboni cell to about 150 volts, and the voltage measured at the quadrants was about 0.04 volt. The ellipses showed continual drift, the needle moving as if a force acted in one direction for a time large as compared with the period of the {, needle. The mirrors were slightly curved, so that in FlG - 6 4- place of ellipses the interference figures were lemniscates. In spite of the diffi- culties, the two series of experiments show sensitiveness of 0.5 and 0.4 cm. per volt, respectively, which is equivalent to about 7 X 10-* volt per vanishing interference ring. Using the same needle, the voltage was now presumably doubled by using two Zamboni piles. The sensitiveness, however, not only was not enhanced, but showed a decrease, 0.02 volt being measured. In other experiments the sensitiveness was successively 0.5, 0.4, 0.4 cm. per volt, respectively. 70. Observations, continued. The sensitiveness was now increased by using a new needle (II) of the form given in figs. 6$A and 656. The two 8- shaped leaves or biplanes, c and d, of the needle are parallel and the circular edges at x and y closed with parts of cylindrical shells of aluminum foil. It is presumable from the elementary theory of the instrument that these walls x, y must contribute essentially to its sensitiveness. In the present case the capsule of the quadrants (II) was about 10 cm. in diameter and about 2 cm. in vertical height, within, with a length of the needle, xy, of about 9 cm. and a distance apart of the biplanes c and d about 0.8 cm. This gave about 0.5 cm. of dear space at the ends and about 0.6 cm. of clear space above and below the needle, as an allowance for stability. The needle swung freely and was inserted without difficulty. The mirrors were about 8 cm. apart. To secure greater steadiness a water damper was installed below, though it would not 106 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. otherwise have been necessary. The drift was at first marked, but finally subsided to a reasonably small value. The sensitiveness (centimeters of dis- placement per volt) in successive groups of observations was (AF potential increment, A/V micrometer displacement) AA/ r /AF=i.o, i.i, i.i, 1.6, 0.8, i.o, i.i, 1.2 ; therefore about i.i cm. per volt, equivalent to 3 X lo" 5 volt per vanish- ing interference ring. In these data some extraneous oscillation of the needle is manifest, which vanishes in consecutive groups of results. The experiment was then repeated with two Zamboni cells charging the needle. This, however, again actually reduced the sensitiveness to AA/"/AF= 0.8, 0.7, 0.6, 0.7, 0.7 cm. per volt, in successive groups of observations, a result equivalent to 4 X iQ- 6 volt per vanishing ring. The data remained of the same order, whether the needle was charged from above or below, so that it is inherent in the theory of the instrument. On returning to the single Zamboni charging cell, sensitiveness again increased to A/V/AF= 1.15, or to about 26 X io~ 6 volt per ring. The Zamboni cells were now removed and the needle charged from above with the electric- lighting circuit at 250 volts. To obviate the effect of drift, which is liable to be persistently in one direction, observations were taken every 1.5 min- utes. The sensitiveness in two groups of experi- ments of about 5 observations each was then i.i and 1.4 cm. per volt, or on the average 24Xio- volt per vanishing ring. Many other experiments were made with similar results. The annoyance of a drifting needle, which occurs throughout the above results and which at first seemed to have a definite direction from the dark to the light side of the parallel mirrors, was also FIG. 65 made the subject of considerable study, sunlight being used to avoid the radia- tions from the body of the electric lamp. In these cases the displacement of about 0.07 cm. within a half hour was usually reversed in the course of this time, so as to bring the needle nearly back to its original position. As the experiments were made with the apparatus uncharged, the only reason for this drift seemed therefore to be the occurrence of steady air-currents, in spite of the protection >t the case and the rapid subsidence of the pendulum vibrations of the damped needle. The attempts made to obviate these difficulties were all futile, i 7 /; bser j ations ' continued.-Another biplane needle (III) in place of the last (ngs. 6 5 A and 6 S B) was now installed. The blades of the needle were 1.2 cm part and to gi ve it stiffness a vertical partition running symmetrically rom end to end was fixed within, the whole being of aluminum foil 0.002 cm. thick and the frame, as before, of steel wire 0.044 cm. in diameter. The weight e needle with mirrors adjusted was about 1.2 grams, the bifilar suspension =m. long and the threads about 0.05 cm. apart. Unfortunately the damp- EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 107 ing, even with the presence of a water well, proved insufficient and the period too long in the case of the suspension used. An example of the six groups of results obtained successively, in the measure- ment of one-fortieth volt, with the needle charged to about 150 volts may be omitted, the mean being about AAf= 0.028 cm. or A/V/AF=i.n cm. of micrometer displacement per volt, equivalent to 27X10- volt per ring. In spite of the vertically more extended needle, there has therefore been no advantage in sensitiveness over the preceding case. The following experiments were made at different voltages, showing better agreement for the larger voltages, in which the drift is less significant, rela- A7 AJV AJV/AF Needle. Quadrant. volt 0.0125 0375 .0625 cm. 0.0135 045 .077 cm. i. 08 1.20 1.23 // II tively. Further experiments with this needle led to no new results. In partic- ular the endeavor to replace the silk suspension used by a bifilar the threads of which were single fibers of silk, proved a failure owing to the instability of the needle. 72. Further observations. The same needle was now placed within large quadrants (III), n cm. in diameter and 2.3 cm. high within, to obviate the difficulty from instability in a needle carrying 250 volts. While this was accomplished, the drift now became excessively large. The mean results were AF = o.oi25 v ^; AA/" = 0.076 cm.; AA/"/AV=6.i cm.; or about 5X10-' volt per vanishing ring. Unfortunately this large sensitiveness, the largest obtained, could not be controlled. It appears from these results that in the above cases the actual restoring torque could not have been the torsion of the bifilar, but rather a directed residual electrical attraction between the needle and the quadrants, the torque of the fiber being operative merely in placing the needle in the fiducial position, symmetrically with respect to the division line between the quadrants. In other words, the displacement of the needle is not to be estimated in terms of the rate at which the bifilar torque changes per degree, but in terms of a very much larger coefficient of the electrical forces in question, so that apart from giving position to the needle as specified, the bifilar acts not very differently from a unifilar suspension. The instrument is thus much less sensitive than would be inferred from the dimensions of the bifilar. Hence it appeared de- sirable to return to the needle and quadrants in 70, with the object of ascer- taining whether the sensitiveness might not be actually increased by decreasing the potential of the needle until a deflection fully corresponding to the torque of the bifilar should show itself. The present point of view also indicates why nothing was gained by the use of a larger needle in 71, seeing that in such a case the electrical restoring forces increase at the same rate as the 108 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. deflecting forces arising in the difference of potential of the quadrants. In the same way the negative effect, as regards sensitiveness, of an increase of the potential of the needle above a certain value is accounted for. Table 13 contains data bearing on this inference. TABLE 13. Needle II; quadrants II; steel frame. Needle at AF AN AJV/A7 I 126 volts volt o.os cm. 0.032 cm. 0.64 II 150 volts .os .040 .80 III. 183 volts .025 .009 .36 .05 075 .100 .025 .047 .062 50 63 .62 Thus the sensitiveness rapidly reaches a maximum when the potential of the needle is about 150 volts, after which it more gradually diminishes (see fig. 66, curve 6). Furthermore, in Series III, where the needle is at the highest potential applied, the sensitiveness seems to increase with the voltage measured. This, however, is merely the result of the fact that there is apparently a small fixed voltaic potential difference between the quadrants, even if they are nominally identical or in the connections. Thus in fig. 66, curve a, AF and A7V are pro- FIG. 66. portional within the inevitable errors; but the deflections begin with a differ- ence of potential of about 0.012 volt. In measuring such small voltages electrostatically these voltaic differences become of serious moment. The maximum sensitiveness obtained is not as large as above, being but 4oXio- volt per vanishing ring. Finally, the water damper was removed, ' that the needle was subject to air damping only. After a long trial it was necessary to abandon the work, as, in consequence of the excessive drift, measurement was out of the question. Most of this drift is probably intro- duced by the steel frame. EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 109 73. Observations, continued. The framework of the needles above was of steel. It was supposed that even if not originally magnetic, such a needle might be subject to variations of the earth's field, through which it becomes temporarily magnetic by induction. Accordingly a needle of the same dimen- sions as the preceding was constructed on a frame of thin copper wire and tested, with the results of table 14. TABLE 14. Needle II, copper frame. Quadrants II. Needle at AF AW AW/A 7 140 volts volt 0.0125 .025 .050 075 .100 cm. 0.008 .012 .020 035 .044 cm. 0.64 .48 .40 47 44 The sensitiveness is on the average 70 microvolts per ring, a smaller value than in the last experiments, owing to a somewhat greater weight of the needle. The voltage was now increased further and the following experiments made: TABLE 14. Continued. Needle at AF AW AAVAF 1 80 o 025 O OI7 O 7O 2 SO 050 075 .100 125 .187 OTO .025 .046 053 .071 .098 025 50 .61 53 57 52 CQ 075 .100 033 .041 44 .41 As before, the sensitiveness passes through a maximum when the voltage of the needle is about 180, and is about as great for the voltage of 140 volts as for 250 volts (see curve /, fig. 66). The maximum sensitiveness is 53 microvolts per ring. Though the drift was not quite removed, the stability of the needle under any given circumstances proved in fact to be greater than before, indicating a marked improvement as the result of replacing the steel frame by one of copper. The curves c, d (raised 0.05 cm.), e (raised o.i cm.), show that AF and AN are proportional within the limits of error. The latter, e, seems to begin with an initial potential which would mean that the sensi- tiveness is even lower at 250 volts than at 140 volts. It therefore seemed necessary to replace the needle in some other of the above experiments by structures not containing steel. Thus the needle and quadrants used in 71 with this improvement gave the results shown in table 15. In view of the low voltage of the needle, the sensitiveness attained is, as above, exceptionally high. The displacement AJV is proportional to AF (see fig- 66, g), but begins with a permanent potential of 0.008 volt. Something similar to this occurs in some of the above results on a smaller scale, so that 110 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. it is not impossible that voltaic potential differences in the quadrants (or connections) may be in question. These were of brass and nominally identical ; but a difference of o.oi volt is not out of the question. Allowing for the initial potential difference, the sensitiveness is about AW/A V= 2.0 cm. per volt or 50 microvolts per vanishing interference ring. The needle was far more steady than in the above cases, so that measurements could be made with reasonable assurance. A curious result is thus attained: on widening the quadrants, so that the distance between the quadrants and the needle is increased within limits, greater sensitiveness is secured. The reason has been suggested, that inasmuch as the electric forces which place the needle are now small, the latter is subject to the force of the bifilar suspension only. TABLE 15. Large needle III in large quadrants III. Copper frame. Needle at AF AJV AJV/A7 185 volts volt 0.009 .018 .036 cm. 0-035 055 093 3-8 3-0 2.6 Further work was done with the needle at 250 volts; but an adequately stable condition of the needle could not be obtained, as it gradually crept beyond the range of the interferometer. Finally, experiments were made with a needle of the ordinary form (I, 69), inclosed in the intermediate quadrants (II). The relatively sharp edges of the needle should reduce the electric torque. TABLE 1 6. Needle I. Quadrants II. Copper frame. Needle at A7 AN AJV/A7 145 volts volt 0.018 .072 cm. O.OIO 053 0.56 74 The sensitiveness here is not inferior to the usual cases above, being on the average 46 microvolts per ring, and this in conformity with the relatively low potential of the needle. The following results were obtained at higher poten- tials with the same needle: TABLE 16. Continued. Needle at A7 AAT Atf/AF 240 o 018 036 .072 .108 145 .019 035 051 unstable. -53 50 47 In spite of the much larger potential of the needle in the last series, the average sensitiveness is again less, showing the same behavior as the above ses. The relation of potential and displacement is linear (fig. 66, curve fc), EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. Ill but the displacements begin with a potential of 0.009 volt. Allowing for this, the sensitiveness is 66 microvolts per vanishing interference ring. The steadi- ness of the needle in both series of experiments was exceptional for this labora- tory, so that the motion of single rings could at times have been counted. Nevertheless the relative values of successive displacements for the same potential increment were not superior to the above. Since di=dd/2a sin i, the electrometer itself in the most sensitive detailed case (needle III, quadrant III) was only moderately sensitive, for if 1 = 45, = 4.5 cm., dd/AV=&N/AV=2 or A*/A7=o.3a radian per volt = i8.4 per volt. Hence, the reflected ray in the ordinary mirror and scale adjustment, at i meter distance of scale from mirror, would move over about 64 cm. per volt. In one of the incidental cases above, it is true, about three times this value was reached. In the other cases it was proportionately less sensitive. Thus for AAT/A V = o. 5 , the deflection would be but 1 6 cm. per volt. No doubt much could be accomplished by making the electrometer itself more sensitive; but this improvement was not the immediate purpose of the present article. Other comparative experiments with copper-framed needles were now made. The sharp-edged needle, I, placed in the large quadrants III gave the results of table 17. TABLE 17. Needle I. Quadrants III. Copper frame. Needle at A7 AN Atf/A7 2 SO volt 0.018 cm. O.OIO O S6 .036 .072 .108 .144 .016 .040 .081 .115 11 fi In this case the displacements are not proportional to the voltages, but increase at an accelerated rate. Neither do they seem to begin at the origin. The sensitiveness accordingly increases rapidly with increased deflection, but its mean value is of the ordinary magnitude. This behavior of a thin needle, in relatively wide quadrants, where stability should have been insured, was unexpected; but by raising the needle the following results were found, showing that the needle above was inadequately centered: TABLE 17. Continued. Needle at AF AJV AW/AF 2 CO 0.018 0.023 1.3 .036 .072 .108 .038 075 .in i. 06 1.04 1.03 This result is a great improvement; for not only is the potential proportional to the displacement (see curve i, fig. 66), but the sensitiveness is much larger than heretofore, for the same needle, being about 30 microvolts per vanishing 112 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. ring. The advantage of wide quadrants is thus again sustained. The sensi- tiveness, however, lags behind the result for the large biplane needle III (curve g) under the same circumstances. The intermediate biplane needle II in the quadrants III shows the results of table 18. TABLE 18. Needle II. Quadrants III. Copper frame. Needle at A7 AN &N/&V volt cm. 0.018 0.007 0.39 036 .022 .61 .072 .050 .70 .108 .144 .068 095 A The sensitiveness is low, owing again, no doubt, to the position of the needle. Otherwise the observations are good. The needle was then raised, with the following results: TABLE 18. Continued. Needle at A7 Atf AW/A 7 250 0.036 .072 .108 .144 .018 0.023 .050 .078 .103 .015 0.64 .69 .72 The sensitiveness has been slightly increased. Moreover, it grows larger in the course of the work, as if some surface viscosity in the liquid of the damper were gradually overcome. The instrument in general behaved admirably, barring alone the presence of drift which can not in the present laboratory be quite overcome. As the fibers were but 13.5 cm. long as compared with 23.0 cm. above, the mean reduced sensitiveness was 25 microvolts per ring. It is thus inferior both to the large biplane and to the wedge, for reasons which do not appear. The sensitiveness should have been intermediate. Finally, the intermediate quadrants II were again mounted with the same needle, the quadrants being specially smoothed inside, so as possibly to elimi- nate electric restoring forces. The results, however, were not essentially different from the above. 74. Summary. The results of this long and excessively laborious paper may be given in a few words. By providing the needle of the quadrant elec- trometer with a pair of mirrors, in parallel, and observing displacements on the interferometer, voltages as small as 10 microvolts may be detected per vanishing interference ring, so that a single microvolt should be reached by estimation. In the above experiments this could not be done, because the needle was never confined to a fixed position of equilibrium, to an extent com- patible with the use of light-waves. The causes of this drift are incidental, EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 113 probably attributable to air-currents, convection currents due to temperature differences and pendulum motion of the needle resulting from tremors. Steel must always be excluded from the framework of the needle. The sensitiveness as is otherwise known, theoretically, does not in any case increase with the potential of the needle, but passes through a maximum (in the above designs) usually at about 150 volts. This is the case both with the sharp-edged and the cylindrically-faced biplane needles. The directing force in the case of such needles is essentially electric; i.e., they are set in a position of equilibrium relatively to the quadrants by electric stress large in comparison with the torque of the bifilar. As soon as these forces increase at the same rate as the potential of the needle, the further increase of the latter is no longer serviceable. Hence the biplane needle, set in relatively wide quadrants, was found to offer the best conditions of sensitiveness, and it is in the case of needles and quadrants of this design that the best results were obtained. In other words, the sensitiveness also passes through a maximum as the mean distance between the outside contours of the needle and the inside contours of the quadrants increases. After preliminary experiments, the optics of the instrument offered no serious difficulty. It is merely necessary to follow the reflected light by placing white screens behind each mirror in the direction of the impinging rays. Since the rays are reflected at the grating, the returning ray also necessarily passes through the grating, and this part of the adjustment is therefore automatic. With a copper-framed needle, the water damper will probably not be essential, in which case the discrepancies due to surface viscosity will also disappear. 43 University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. 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