DISPLACEMENT INTERFEROMETRY APPLIED TO ACOUSTICS AND TO GRAVITATION By CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON WASHINGTON, 1921 v CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No. 310 PRINTED BY J. B. LIPPINCOTT COMPANY AT THE WASHINGTON SQUARE PRESS PHILADELPHIA, U. S. A. PREFACE. The experiments of the present volume are either direct applications of displacement interferometry or they embody the correlative work which has grown out of such applications, often at widely different times. In the ar- rangement of the chapters it has therefore been expedient to depart from chronological order in favor of an arrangement of subjects which belong together. In a former report* I had already used a U-tube in connection with the in- terferometer, but the design of the apparatus was limited in scope. In the present paper (Chapter I) the open mercury manometer is made directly available for pressure measurement, and as the attainable sensitiveness is easily a few hundred thousandths of a centimeter of mercury per fringe dis- placement, it is well worth while to see what can be gained by using it. The applications to air thermometry on a micrometric scale and an attempt to revive the old absolute electrometer in Chapter II are merely incidental, though in each case much more may be done than I have here attempted, as I hope to show at some other opportunity. The manometer also admits of further improvement in ways which can not be included in the present report. A more suitable field for testing the immediate capabilities of the mercury U-tube is detailed in Chapters III and IV, where the endeavor is made to give an account of the pressures and dilatations observable in a region vibrating acoustically. If this is quite closed or quite open to the atmosphere, the rec- ord of the U-tube within is without interest; but if the region is all but closed up — open, for instance, through a pin-hole less than 0.5 mm. in diameter — the gage shows pronounced fringe displacements as a rule and particularly at the frequency of the harmonics. If the sound generator is a telephone, the displacements are proportional to the effective currents actuating it and at the harmonics much more than 10,000 ohms may be put in circuit before the fringes cease to move appreciably. Similarly under low resistance, very small fractions of a semi-tone are registered and the ear becomes a poor ap- paratus for discrimination. The investigations are made along two lines, in the first of which the sound generator and the U-tube lie within the boundary carrying the pin-hole; in the second the sound generator lies without and is independent, so that the pin-hole valve carried on a long tube becomes an appropriate probe, or sonde, for the pressures within sounding pipes and cavi- ties. As all the harmonics are thus saliently registered, there should be no serious difficulty in exploring the acoustics of the mouth cavity uttering word sounds, for instance. A curious result of the survey of the distribution of pressure increments in relation to pitch is the replacement of pressures by dilatations in different orders of frequency. * Carnegie Inst. Wash. Pub., No. 249, II, Chap. V, 1917; ibid, Chap. VI, ? 72. iii iv PREFACE. In Chapter V, on the direct interferometry measurements of the compres- sion of a sound-wave, much of my work has been superfluous, as it was an- ticipated in an admirable paper by Raps, using the Jamin interferometer. I have therefore given only as much as is necessary for the coordination of the other chapters. My method, however, is, I think, superior, owing to its much greater flexibility and the ease with which fringes in any orientation may be produced and shortened to a string of silvery beads. The simple organ-pipe blower or adjustable embouchure much used in the chapter will, I think, be found serviceable for many purposes, both of research and instruction. As the telephone is an indispensable convenience throughout these chapters, it was thought necessary to begin an interferometer investigation on the vibrations of the plate of that remarkable instrument. What comes out definitely in the research, Chapter V, is the readiness of the plate to quiver in overtones. A small mirror at the center is not therefore displaced, as a rule, translationally, but rather rotationally, giving rise to very complicated wave-forms, difficult to analyze. In corroboration of this, it was found (in Chapters IV and V, for instance) that a telephone current may often be com- mutated. In the endeavor to place the Foucault mirror on the interferometer I have thus far, for incidental reasons, failed of achievement; but as different appara- tus useful in experiments of the present kind were tried out in the course of the work, I have given a brief account of it in Chapter VII. In Chapters VIII and IX, in deference to the wishes of Dr. R. S. Woodward, I have begun a search for methods of measuring the acceleration of gravity other than those classically in use. Such an inquiry necessarily consists in referring gravita- tional forces to forces generated in other mechanisms. An interferometer torsion-balance is first tested, but the results are found to contain relatively large and uncontrollable temperature coefficients, both of rigidity and vis- cosity, even if the ordinary effects of viscosity can be allowed for. The other (pneumatic) method for g, in which gravitational pull is referred to the pres- sure of a gas, has at the outset much to recommend it, for it admits of rough handling in spite of the otherwise surprising precision of results. The two errors which offer a serious menace to the accurate hydraulic weighing of the Cartesian diver, viz, the diffusion and solution discrepancies, though at first approach apparently insuperable, may not remain so indefinitely. At least, in experiments on the diffusion and convection of gases in narrow tubes, made in the lapse of years, coefficients of a negligibly small order of value were obtained. Though the work is very laborious, I think it will be worth while to carry it further. The remainder of the volume is largely concerned with work (Chapters XI and XII) bearing on the constant of gravitation. The object of these experi- ments was at the outset a mere endeavor to read the deflections of the gravi- tation needle by displacement interferometry. The plan succeeded at once, almost beyond my expectations; but on computing the Newtonian constant it came out actually several times too large. It was obvious that this could PREFACE. V be explained only by the presence of relatively large radiant forces. Noticing, in the course of the work, that the latter, in some mysterious way, almost al- ways acted in the same sense as gravitation, I became much interested in the endeavor to trace the radiant forces to their source and if possible to learn to control them. Much of Chapter XI is given to work of this kind, and among other things the attempt is made to refer the constant of gravitation to the viscosity of the medium in which the needle moves. The most curious re- sults in relation to the radiant forces were obtained by submerging the whole of the gravitational apparatus in a capacious water-bath well stirred, so that the temperature varied but a few tenths of a degree per day. Notwith- standing these apparently ideal conditions, the needle simply drifted and showed no response to gravitation whatever. The best method of reducing the radiation discrepancy in question, thus far found, consists in the exhaustion of the case containing the needle. Results so found came within i per cent of the normal value, but were still in excess. No doubt this is far from pre- cision, but it is a great advance from an error of several hundred per cent of excess. The further development of the method of attack I hope to complete this summer, by making the exhaustion as rigorous as possible. In the last chapter I have put together a number of incidental results, bear- ing (as in Chapter VIII) on the breakdown of molecular instabilities evidenced by the peculiar phenomena of residual viscosity. In a similar experiment, showing the magnetic set in iron produced by an electrical current passing through it, we have, as it were, an element of hysteresis. A useful method for the production of two groups of independent fringes, present in the same field, is also given. My thanks are due to Miss Rachel Tupper Easterbrooks for efficient as- sistance throughout the editorial work necessary for the preparation of the present report. CARL BARUS. BROWN UNIVERSITY, June 1921. CONTENTS. CHAPTER I. — The Open Mercury Manometer read by Displacement Inter ferometry. PAGE 1. Apparatus I 2. Experiments I 3. Equations and pressure observations 2 4. Air thermometer 3 5. Acoustic pressure, etc 4 CHAPTER II. — The Interferometer U-tube used as an Absolute Electrometer. 6. Electrical condenser 5 7. Fringes from a free mercury surface 6 8. Equations 6 9. Specific inductive capacity 9 10. Allowance for the electrometer 10 11. Absolute values 1 1 12. Improvements and miscellaneous experiments 12 CHAPTER III. — Acoustic Pressures and Dilatations chiefly in Reservoirs. 13. Introductory. Apparatus 14 14. Observations. Closed and open resonators 15 15. Resonator all but closed 15 1 6. Pressure depending on the frequency and on the intensity of vibration 1 6 17. Fringe deflection varies as current intensity 16 18. Pin-hole sound-leaks 17 19. Effect of resonance 18 20. Inside and outside stimulation 20 21. Apparent removal of pressure decrements 21 22. Reversal of poles of telephones changes sign of fringe deflection 23 23. Change of volume of reservoir 24 24. U-tube used differentially 24 25. Conical vents reversible. Periodicity 26 26. Resonators of very large capacity 27 27. Resonators of very small capacity 28 CHAPTER IV. — The Pin-hole Probe for Sound Pressures. 28. The pin-hole sonde, or probe 31 29. Pressures in smooth, straight pipes 33 30. Symmetrical induction 35 31. Closed region with pipe 36 32. Closed organ-pipe 37 33. Open pipes and adjutages 37 34. Reversal of poles of telephone 38 35. Open pipe on the interferometer 4° 36. Helmholtz spherical resonator 41 37. Correlative experiments with the torsion balance 43 38. Conclusion 44 CHAPTER V. — The Compression of a Sound Wave in Diapason Pipes. 39. Introduction 46 40. Apparatus 46 41. Observations with crossed pipes 47 42. Deductions 47 43. Stroboscopic and other secondary phenomena 48 44. Observations with longitudinal pipes 49 45. Organ-pipe blower 50 46. Interference 51 47. Reed pipes, voice 52 48. General result 53 vi CONTENTS. Vll CHAPTER VI. — The Vibration of the Telephone Plate. PAGE 49. Phenomena 55 50. Interpretation. Lines. Oscillation about a vertical axis 56 51. Shadow waves. Oscillation about horizontal axes 58 52. Beating fringe-waves 58 53. Effect of temperature. Miscellaneous observations 59 54. Synchronism. Iron screw-core 61 55. Shattered fringes 62 56. Wave frequencies. Musical notes 63 CHAPTER VII. — Experiments Made in the Endeavor to Place the Revolving Mirror on the Interferometer. 57. Apparatus. Revolving mirror normal 65 58. Lens train 66 59. Estimate 66 60. The inclined revolving mirror and interferometer 67 61. The revolving telescope objective 68 62. Control fringes 69 63. Sensitiveness 69 64. Experiments with the rotating telescope. Fringes on washed images 70 65. The combined revolving mirror and telescope 71 CHAPTER VIII. — On the Torsional Measurement of Variations of Acceleration of Gravity, by Interferences. 66. Apparatus 74 67. Measurements 76 68. Thicker wire 77 69. Equations 80 70. Absolute viscosity of the wire 81 71 . Twist in one direction only 82 72. Observations on the permanently twisted wire 83 73. Further experiments with the preceding apparatus 84 CHAPTER IX. — A Pneumatic Method of Measuring Variations of Acceleration of Gravity. 74. Introductory 87 75. Apparatus 87 76. Equations 89 77. Observations 90 78. Sheathed or inclosed divers 93 79. The swimmer under pressure 95 80. Stems of small bore 97 81. The diffusion of air through water in the lapse of years 98 CHAPTER X. — Gravitational Experiments, chiefly with Reference to the Accompanying Radiant Forces. 82. Introductory 101 83. Apparatus 101 84. Long-period observations 102 85. Long-period observations in the lapse of time 103 86. Short-period observations 107 87. Further observations no 88. New apparatus 1 1 1 89. Trial observations. Radiation effect 112 90. Radiation effect observed on exhaustion 113 91. Tendency of needle to stick to glass window 1 16 92. Needle excursions under increasing pressure 1 16 93. Experiments with the exhausted case 118 94. Tentative estimate 1 19 95. Angular velocity at different pressures 120 96. Water-bath 120 97. Attraction in vacuo 123 98. Apparatus No. in. Brass and glass 124 99. Metallic case cooled by efflux 128 viii CONTENTS. CHAPTER XI.— Gravitational Experiments. PAGE 100. Slender needles 131 101. Experiments with slender needles 132 102. Torque exerted on the stem 135 103. Contemporaneous experiments 136 104. Filamentary needle 137 105. Observations 139 106. The residual radiation forces 139 CHAPTER XII. — Miscellaneous Experiments. I. Heavy Gravitational Systems. 107. Attractions in case of a heavy needle 141 108. Apparatus 141 109. Observations 142 II. The Torsional Magnetic Energy Absorption of an Iron Conductor. 1 10. Apparatus 144 in. Observations 144 112. Data 145 1 13. Longitudinal field 145 III. Liquid Refraction Near a Solid Surface. 114. Methods and first experiment 146 1 15. Second experiment 147 IV. Comparison of Two Independent Sets of Fringes. 1 16. Apparatus 148 CHAPTER I. THE OPEN MERCURY MANOMETER READ BY DISPLACEMENT INTERFEROMETRY. 1. Apparatus. — This is practically a U-tube, AmA1 ', figure i, with wide shanks A, A' connected by a channel m below. A and A' are cylindrical hollows, 2 to 3 cm. deep and about 5 cm. in diameter, cut in a rectangular block BBf, preferably of iron. The connection m must also often be large in section, so as to admit of rapid flow from A to A'. The U-tube is charged with mercury, MmM', M and M' being as shallow as possible to counteract the tendency to vibration. Thin plane parallel glass plates, gg' ', round disks of equal thickness and diameter, are floated on the mercury, which act as mir- rors for the interferometer beams L' and L" and also materially check the tendency of the pool of mercury to vibrate. It would be desirable to be able to use the mercury surfaces at M and M' directly without the intervention of the plate, but all attempts to do this, within the city limits, were at first fail- ures. Moreoever, the amplitude of vibration in the trough further from the source of light is much in excess. Later I found the fringes, but could not use them with advantage. The top of the iron block BB' is recessed as shown, to receive the plane par- allel glass plates G, G1 . These, like g, g', must be equally thick, otherwise the fringes will be multiplied and faint. The annular space cccc between G and B is filled with resinous cement, poured in the molten state. The air-space AA', shut off in this way, communicates with the atmosphere by two tubulures, t and f', in the front side. The ray parallelogram of the quadratic interferometer, of which L', L" are the interfering rays, should be vertical. The displacement of the achromatic fringes of white light are read off by a telescope with an ocular micrometer (scale-part o.oi cm.). The fringes parallel to the divisions of the micrometer are conveniently made a scale part in size. The block BB' should be mounted separately from the interferometer. If it is placed on the base of the latter, all manipulations there shake the mercury in BB' and it is necessary to wait for subsidence. This, however, occurs very soon, so the separate mounting is not absolutely necessary. Without manual interference the fringes are about as quiet as in a solid apparatus. 2. Experiments. — To test this apparatus the air-space A A1 was left with a plenum of air. With A' communicating with the atmosphere, A was joined through / and a filamentary capillary glass or metal tube to an apparatus by which slight pressure could be applied. In the first trials I attempted to use a water manometer controlled by a micrometer-screw ; but the vibrations of the meniscus were at once impressed on MM', so that the fringes were hard to keep at rest. I then devised the apparatus shown in figure 2, which is merely 1 2 DISPLACEMENT INTERFEROMETRY APPLIED TO an adaptation of the pin-valve of an oxygen tank, with a good micrometer- screw 5 and stuffing-box n. The head h of the screw is graduated. The barrel b is at right angles to the tube aa! ', which at a joins the capillary tube d, leading to I of figure i . At the end a' there is a cock C which shuts off com- munication with the atmosphere. Thus, when C is closed, pressure is applied directly at A, figure i, by rotating the head h in figure 2. This pressure is at once removed by opening C. The apparatus worked surprisingly well. When C is closed and h rotated, the fringes may be placed anywhere in the field about as conveniently as with the micrometer-screw at the mirror of the interferometer. There is, however, one difficulty which I have not thus far been able to remove. When the pressure increments exceed a certain small value, the plates g, g' no longer rise and fall in parallel. The coincidence of images is destroyed and the fringes vanish. There is here a conflict with the capillary forces present at the edges of the disk. I endeavored to improve this by using small plates g, g', anchored near the center of MM' by four loose threads, but the advantage was not marked. Fringes a scale-part in size will not usually be available for more than 50 scale-parts, being sharpest in the middle. This is about half the diameter of field of the ordinary telescope. 3. Equations and pressure observations. — If the cock C, figure 2, is closed and the temperature for brief intervals is considered constant, Boyle's law may be written (ignoring signs of increments) __ V " V = p "76 where V is the total volume inclosed, dv the increment at the micrometer-screw hs, and dV the corresponding decrement equivalent to the pressure decrement dp. If a is the area of the piston at g, dV = a dh/2, and if V = aH, H being the corrected depth of the air-space at A, equation (i) becomes , . dv dh _dh ~V~^H~^6 But dh on the interferometer is equivalent to n fringes of wave-length X, so that dh = n\/2. Hence finally (3) ^=y*(-^ + - 2 \76 2/f This equation gives a test of the trustworthiness of the gage. In the apparatus used the following constants were found by measurement : F=66.8 cm.3, 0 = 29.2 cm.2, H = z.2g cm. The pitch of the screw was 0.073 cm. and its mean diameter 0.51 cm. Hence, per turn, 6^ = 0.073X0.204 = 0.0149 cm.3, and ofo/V=io~4 X2.23 per turn. The mean wave-length being X = 6Xio-5cm., equation (3) reduces to (—(-^ + -jj\ = 0.0066+0.1092^ ACOUSTICS AND GRAVITATION. n- IO~4X2.23 = 32 fringes per turn of screw hs 6Xio-aXo.ii5S' In the experiments fringes of one-half scale-part were installed. In sep- arate experiments, immediately after closing the cock C, a half turn of the screw produced a displacement of 8.3, 8.0, 8.0, 8.5, 8.5 scale-parts; as the average, therefore, 16.4 scale-parts per turn or about 33 fringes per turn. This agrees as closely as may be expected with the number computed. The pressure increment per turn of screw is dp = n\/2 cm. of mercury or per turn of screw about io~3 cm. ; per fringe, therefore, 3 X io~6 cm. of mercury •as anticipated. A range of about 2 or 3 turns of screw was possible with each fringe, i.e., the range of pressure measurement should be from 3Xio~5 to 3 X io~3 cm. of mercury. Experiments of the same kind were made in great variety. There is no difficulty is using much larger fringes, so that 3 X icr6 cm. of mercury should be appreciable. By exhausting both sides of the U-tube the apparatus becomes a vacuum gage. I did not, however, attempt such work, as the present appara- tus was not well adapted for the purpose. \ . \ v "A. a _=_ • .^"^ • •• • C^/T~^ t» I 4. Air-thermometer. — If the cock C is permanently closed, the air-space A becomes the bulb of an air-thermometer of approximately constant volume. In this way the heat produced by the rays of light L' may be measured. In a variety of experiments of the kind, the mean result was about 10 scale-parts or 20 fringes in a lapse of 210 seconds. If r denotes absolute temperature, the intrinsic equation may now be written which reduces as above to dp , = ' V~~= T dh dh^dr 76 ~'' Thus, if T = 300°, dr =rn — ( — 7 H 7+ I = 2 \76 zHJ and for n=20 in 210 seconds, = 0.042° C. 4 DISPLACEMENT INTERFEROMETRY. or the heating produced was 2Xio~4° C. per second. Whether, supposing A A' to be filled with water, a pyrheliometer may be constructed on this prin- ciple I have yet to learn. Other interesting results of the same kind might be mentioned. Thus, if the screw stop-cock, figure 2, is closed quickly, there is always a decided increment of pressure. In other words, in consequence of the viscosity of air, the fine space at the plug is virtually a closure before the screw is checked by an actual cloture. 5. Acoustic pressure, etc. — If the glass plates G are removed and the air-space A partially closed with a pipe P (fig. 3) tapering to a neck, PAM becomes a closed organ-pipe with a bottom sensitive to pressure. The pipe may be blown with the adjustable embouchure / described in § 45, and suf- ficient free space remains for the component rays U L" of the interferometer to reach the mercury MM'. The neck should not much exceed an inch in diameter. It was hoped that when the pipe was sounded judiciously, acousticpressure, if any, would be shown at the bottom. A large number of experiments were made, at first with very weak pipe-notes corresponding to a very mild jet /. Displacements of 5 to 10 fringes nevertheless occurred in the direction of pressure, i. e., up to about 4 dynes/cm.2. In later experiments, however, suc- tions were quite as often obtained as pressures, so that what is registered here is the pressure effect of the nearly horizontal air-current from / across the top of the pipe. With strong notes the fringe displacement was much greater and the slit-images tended to separate, destroying the fringes. The results ob- tained, therefore, are merely those of a pipe sounding under slightly reduced or slightly increased pressure, and all attempts to eliminate the discrepancy failed.* Resonance was ineffective. The same effect is nicely shown with a thistle-tube t (fig. 4) , provided with a small-bore (o. i cm.) stem c, slightly raised as shown. If a jet from a pin-hole is blown in axially, as at a, the effect is pressure ; if blown in obliquely or tan- gentially, as at 6, the effect observed is suction. The displacements were 10 or more fringes, to the right or to the left. * The method which ultimately succeeded is described in Chapter III. CHAPTER II. THE INTERFEROMETER U -TUBE USED AS AN ABSOLUTE ELECTROMETER. 6. Electrical condenser. — Adjusting the U-tube as in figure 5, with the top plates removed so as to admit a metallic disk C, above the mercury M (earthed at E, electrode at a) and parallel to it, CM becomes an absolute electrometer. The disk C is perforated at c, so the component rays L', L" may reach the mercury. Unfortunately this instrument is not very sensitive in any case and is chiefly useful in measuring electrostatic potentials. If p is the electric pressure below the disk (7, charged at potential difference V, and h is the head of mercury resulting, (A) V = d\/%Trp = d \/&irhpg = d where d is the distance of C from M, p the density of mercury, and X the wave- length of light when n fringes correspond to V. Hence, if d = o.i cm., 13.6X981 Xw = 0.3 i7\/w els,. units ; or V= 95\/wVolts. In the experiments d was made slightly over 0.38 cm.; but of this, 0.23 cm. were in glass and 0.15 cm. in air. The usual experiments with the electroscope were carried out about as con- veniently as with that instrument, with all the data in absolute units. Thus a charged hard-rubber rod near a metallic plate connected with C gave meas- urably up to 40 fringes. With larger potentials the images were apt to separate and the fringes van- ish. The range may be varied (d), and steady or alternating potentials are both within reach of the instrument. To avoid the effect of specific inductive capacity, a small thin mirror, say i cm. in diameter, may be anchored with loose silk fibers below the hole in the center of the electrode. Otherwise, in the presence of glass plates it would seem that the above equation should be modified to read (B) and if KK is the specific inductive capacity of the glass, d& and ds the thickness of the layers of air and glass respectively so that d = i/K = (da+dg/K)/d Thus, if Kf — 6, dg = o.23 cm., da = o.i5 cm.,i//C = o. volts where n is the number of fringes. It is not, however, obvious that either equation (A) or equation (B) are at once applicable. An investigation of this will be made in § 10, showing that equation (A) suffices and that d is to be measured from the top face of the glass plate g to the lower face of the parallel electrode C. 5 6 DISPLACEMENT INTERFEROMETRY APPLIED TO To test the electrometer, figure 5, a screw electrophorus was constructed as. shown in figure 6, where p and p' are the two plates and r the insulating layer of hard rubber. The screw-post s, rising from the center of the plate p', passes through the screw-sleeve of brass b, embedded in the hard-rubber handle h, which in turn is firmly screwed into and supports the plate p. By rotating the handle h, the plate p is raised or lowered by an amount shown on the spring rod a, which is also the upper electrode. Fractions of a turn are read off on the graduated top face of p. The lower electrode is at E. As the insulation sbh appeared inadequate, the construction was modified by removing s, raising b, and allowing a supporting screw, similar to s, to enter the handle k from above, retaining the feature of rotation. The new form is shown in figure 7, k being the graduated head referred to the stem C. The plate d is clutched by the same standard which supports p'. This left the space between r and p clear for the insertion of dielectrics of different specific inductive capacity. I =4>4 a ; 5 1 c - _TL J CM. 'JH c^.'- c^ 6 P u \t r a £ — qf TL , fe 7 r cf - -? 4 flf a g V ! -, In figures 6 and 7 , a and e are to be joined with a and e of figure 5 . It would have been desirable, of course, to provide the condenser CM in figure 5 with a guard-ring. If B is an insulator, this would apparently occasion no serious difficulty; for a fixed metallic ring or short cylinder coaxial with M and partially submerged in it would suffice. The upper face of the ring should be flush with M. The apparatus used, however, was not well adapted for this -purpose. 7. Fringes from a free mercury surface. — By making the troughs of the U-tube very shallow (a few millimeters), fringes were obtained rather easily. They were entirely too mobile to be used with convenience. The curious fea- ture was observed, however, that even with pools 6 cm. in diameter the rise and fall of the mercury faces on the two sides was not rigorously in parallel. In fact, the slit-images separated and the fringes vanished about as soon as when floating glass plates were used. This discrepancy is to be referred to some type of surface viscosity, which, if the mercury had been quite chemically pure, would possibly have disappeared, rather than to an inequality of the electric field at the surface. The experiments with the guard-ring alluded to were also abandoned because of the mobility of the mercury surface even when using thin glass plates i to 2 cm. in diameter. 8. Equations. — If we treat the case of the electrophorus as a closed cylin- drical field of cross-section A, and if V0 is the potential of the charged hard- rubber surface, we may write ACOUSTICS AND GRAVITATION. (0 where Q' V is the positive charge and potential in the top plate, at a distance d' from the rubber surface at potential V0 and K' are the specific inductive ca- pacity of the dielectric medium. Similarly, if Q"V" is the charge and potential of the lower plate at a distance d" from V0, a layer of specific inductive ca- pacity K" lying between, w But (3) the fixed charge on the rubber surface. If the two plates are put in contact, (4) V'=V" so that on combination Vti" n' — n A a (5) ~^K" Furthermore, if the two plates thus charged are kept insulated and the top plate is moved normally towards the lower a distance of y (figure 6), (6) AF being the potential difference thus produced and measured at the U-tube electrometer taken as small in capacity in comparison with the electrophorus. Hence, on inserting equations (5) and reducing, _ _ A(K"d'+K'd") It follows that yi= (constant) n; or the locus is a parabola. Among the experiments made to test this equation, it suffices to give the data in figure 8, in which the displacement of fringes, n, is laid off vertically downward, and the turns of the screw s in the apparatus figure 6 (pitch one-twentieth inch) horizontally, the plates having been put m contact at 4 turns above the hard -rubber face (^' = 0.51 cm., d" = o.i6 cm., turn = 0.127 cm.). The zero of « is heie arbitrary. When the plate moves down the para- bolic form of curve is apparent. When it moves up from 4 turns, however, the character of the curve soon changes. In other words, lines stray in the latter case, whereas the stray field is caught to some degree in the former. Furthermore, the capacity of the U-tube has been disregarded. Finally, the unfortunate leak in the apparatus (fig. 6) is shown by a marked hysteresis- like difference in the outgoing and return series, as indicated by the arrows. 8 DISPLACEMENT INTERFEROMETRY APPLIED TO The second modification (fig. 7) of the electrophorus was then used with the plates quite separated and the micrometer-screw above. The insulation was much better, the loss amounting to not more than 2 fringes in 10 minutes at full charge. The pitch of the micrometer-screw being now o.i cm., the upper plate was conveniently discharged when d' = i cm. above the hard-rubber ID SO surface. Large fringes (about 1.5 scale-parts) were installed. The results (fringe-readings « in terms of displacement y) obtained in the same way as the preceding are shown in figure 9. The outgoing and incoming series practically coincide. Figure 10 shows a series of results in which the plates were discharged at different distances d' = i.o, 0.8, 0.6, 0.4 (clear distance between rubber and metal faces) apart. If we write y* = Cx, x= i.$n (x in scale-parts, about two- thirds fringe), the relations are = i.o io3C=i8.o 0.8 7.1 90 0.6 2.9 124 0.4 cm. 0.8 200 These parabolas are only approximate ; in each case except the last, C decreases appreciably as x increases. The frinpe displacement is in excess of the equa- tion yz = Cx, owing, as I take it, chiefly to the escape of negative charge to the electrometer as d' decreases. The relation of C and d1 (fig. n) is again apparently parabolic in shape, but remotely so; for if we write Gi = cd> ', the values of c (fig. n) rapidly decrease as d increases, probably for the reason just indicated. ACOUSTICS AND GRAVITATION. 9 Similar experiments were made with small induction coils. The fringes here were always in motion, with relatively very large pulses recurring at intervals. 9. Specific inductive capacity. — In equation 7, if the space d' is filled with air, K' = i. On the other hand, if a plate of some insulator like glass is inserted of thickness d'K (8) d' = d where d'& is the thickness of the air-layer. Moreover, if Kg is the specific inductive capacity of the insulator, If, therefore, in the absence of the insulator, y is the downward displacement of the upper plate which gives the same fringe displacement «, and hence the same AV as the insertion of the insulator-plate, the two equations of the form (6) (in the second of which y = Q, but K' as in equation (9)) are equal. Hence -^Qd"y _4*Qd'd"f i ___ i \ A(K"d'+d") A \K'(K"d'+d" ~ K"d'+d" ) or inserting (9) or (ix) K* = d To determine the specific inductive capacity of a given insulating plate, the electrophorus is discharged at a convenient distance d' between plate and hard-rubber face. The insulator (Kg) is then inserted (noting the fringe displacement n) and withdrawn. The fringes must then return to zero, show- ing that no charge has been imparted by the friction of the insulator. The upper plate is now depressed (y) on the micrometer-screw until the same fringe displacement n is obtained. Equation (n) is thus applicable. The operation should be quite rapid. The following are examples of this method, among many experiments made, most of which proved quite disappointing. Thus, in the case of different plates of glass, d' de y x K 1.2 cm. 0.81 cm. 0.70 cm. 19 s. p. 7 1.2 .81 .72 22 9 I.O •52 .48 16 13 .8 •52 •44 39 6.5 .8 •52 •43 37 6 .8 .40 •34 23 7 .8 .40 •35 24 8 •7 • 19 • 157 18 6 •7 • 19 .160 19 6 10 DISPLACEMENT INTERFEROMETRY APPLIED TO Results are apt to be even larger than these, much depending on the time during which the glass plate is left within the electrophorus, or on the quick- ness within which x can be read with the subsidence of motion of fringes. The chief causes of discrepancy, however, are the large values of d' ' ', since in this case the dielectric plate operates in strong fields and the descending metallic plate of the electrophorus in weaker fields. Thus the results for d' ' = 0.7 cm. and below are the most uniform. In other words, the difference of d and y is so small that any slight charge on the plate has a rela- tively large effect onde — y and vitiates the result. The glass plates virtually conduct. If they happened to touch the upper plate a throw of the fringes resulted. The same difficulties are present in a less degree even with hard rubber, of which the following data are examples : d' de y x K 0.8 cm. 0.32 cm. 0.23 cm. 5.5 s. p. 3.4 i.i 0.64 0.45 6.0 3.4 0.7 0.156 o.ii 5.5 3.4 results again too large, but quite apt to be larger still. Baeckelite usually leaves a charge on withdrawing the plates and the method quite fails. Similarly, in case of hard rubber flamed to clean its surface, as the temper- ature rises in successive trials, K rapidly increases. Thus in the above case the effective K rose from 3 to 4, 6, 10, etc., and finally, when the rubber was purposely well warmed, became practically infinite. This recalls the rapid de- crease of viscosity under the same circumstances. Again, if one side of a cold hard rubber is charged and placed contiguously with the charged surface of the electrophorus, the electrometer, initially at zero, shows no appreciable effect; but if the two charged surfaces are both up, the deflection is enormous. In such a case the charge has virtually passed from bottom to top of the rubber plate, as it probably actually does in small part in case of hot rubber and glass. On the other hand, if the rubber is cooled to freezing or below, K decreases to a limit. 10. Allowance for the electrometer. — If the latter has considerable rela- tive capacity, the equations in the same notation as above change to , , where q is the charge which has gathered in the electrometer of area a, specific inductive capacity of dielectric (glass) K, and distance between plate and mercury d. Treated in the way given these equations reduce to (13) ACOUSTICS AND GRAVITATION. 11 and If the values of AF in (13) and (14) are the same, since by equation (9), it follows that equation (i i) again results. 11. Absolute values. — It is next to be determined whether the equations (A) or (B) of § 6 are to be used ; in other words, the value of the condenser space in the electrometer is to be found. For this purpose it was convenient to com- pare the data of the latter with the corresponding results of the Elster and Geitl electroscope. There happened to be three of these in the laboratory, all standarized by the aid of storage-cells compared with the Clarke cell, the range of available voltages being of the right order. To obtain corresponding read- ings, it was merely necessary to join the U-tube electrometer and the electro- scope in parallel with the electrophorus, figure 7, and gradually depress the plate p on the micrometer-screw s. The results of these comparisons are shown in figure 12, the U-tube reading being reduced to volts by equation (B), § 6, and the electroscope readings by the charts. Not much accuracy is to be expected in the individual readings, but the mean results of different instruments are quite trustworthy. The ratio of data (the U-tube numbers being in excess) is for the case of electro- scope No. 1499, 4.3 and 4.2; for No. 1727 it is 4.3, and for No. 1277 (this was the Ebert apparatus for testing atmospheric ionization, not well adapted for the present purposes) about 3.9. The mean of these results is 4.2, so that the data of the U-tube electrometer computed by equation (B) are about 4 times too large. This equation is thus inapplicable and the simpler equation (A) is to be taken. In other words, the thin glass plate floating on and in intimate contact with the mercury acts like a conductor and carries the charge practi- cally at once to its upper face. Thus an equation of the type (A), § 6, should have been used, and this would make the distance between the lower face of the upper electrode and the upper glass face 60/95 = 0.63 mm- Examining the apparatus, it was found that in the long usage to which it had been put the levels had slightly shifted. The distance between the electrode surfaces is difficult to determine without special apparatus. I obtained an estimate by rolling iron wires of different diameters between the faces. In this way, with some patience, 0.07 cm. was obtained. It is suf- ficiently near the datum from electroscope comparisons to prove the case. The equation should, therefore, read n volts 12 DISPLACEMENT INTERFEROMETRY APPLIED TO In the case of an experimental apparatus improvised for the purpose, and without a guard-ring, and in which the rigorous parallelism between electrodes could not be guaranteed, a closer approach was not to be looked for. The equation may be taken as trustworthy within its range. 12. Improvements and miscellaneous experiments. — The electrometer eventually took the form shown in figure 13, which gives the apparatus in connection with the electrophorus and a commutating key similar to Mas- cart's. To put the mercury M to earth, a steel screw S, which also carries a flat clamp for fastening one end of the earthed wire, has been inserted. This screw 5 has the further important purpose of damping the oscillations of the mercury M or M't by adjustably closing the channel m. The deflections can thus be made quite dead-beat, which is an advantage. To level the electrodes C, C1 (using a small spirit-level placed on them), each has a connecting-rod d, which carries a clamp at one end, allowing the rod aa' to slide up and down, rotate around a, a' and d, df, and admitting of small displacements along d. At the other end of each rod is a flat vertical plate, which is received in a fissure at the top of the corresponding hard-rubber post k, k' and clamped. This gives a horizontal axis at right angles to the preceding. The lower ends of the posts k, k' are suitably clamped to the tubes t, tf, attached to the body B of the electrometer. Here further motion along the tubes t, t' and rotation around them is possible. In this way it is not difficult to place C, C' symmetrically above the mercury pools (which must be shallow) and parallel to their upper faces, for experimental purposes. It is not sufficient, however, if precision is required, nor well adapted for measuring the distance between glass face and electrode. In such a case micrometers at the inner ends of d, d' would be needed with a more efficient method of leveling. I made the endeavor to change the position of surfaces of the mercury, by putting M, M' in connection with a third vessel on the outside, which could be raised or lowered by a micro- meter-screw; but the method was not satisfactory in the long run, as the apparatus shifted its zero-point and sensitiveness. To control the latter in this way proved to be inadvisable and was eventually abandoned. Figure 13 shows the Mascart key below on the right, which consists of the elastic brass strips /, I', the earthed cross-bar n above them, and the cross- bar q below. All metal parts are carried on high, insulating hard-rubber posts / and /' at their right ends and they are both in contact with n if not depressed by the insulating knobs at their left ends. The figure shows that / and /' are electrically connected with C and C' by insulated wires, while 5 is con- nected with n. Thus the whole U-tube is earthed when not in use. The bar q is connected by wires with the brush A of the electrophorus shown on the left, so that when / or /' are depressed into contact with q, C or C' ACOUSTICS AND GRAVITATION. 13 respectively receives a positive charge while the other electrode and the mer- cury is earthed. This affords a satisfactory means of commutation; for since AV=C\/n = C'\/rn' if the electrodes are so adjusted that C and C' are nearly equal, A V = Tests were made with this apparatus and a known AF=i73 volts. For example, the scale-readings in the ocular on commutation were # = 34, #' = 17, so that (x— x')=o.7 (w+n'), as the fringe breadth was 0.7 scale-parts. Thus AV = A\/24^ or A =35. 3 volts per fringe, initially. With large fringes and under quiet surroundings 3 or 4 volts could have been detected. To endeavor to carry the sensitiveness beyond this, either by the outside mercury receiver or by cautious manual adjustment of C and C', did not suc- ceed. This would have to be done by special mechanism. Similarly the experiments made to put in a guard-ring led to no results of consequence, as the sensitiveness decreases rapidly if only the central parts of the mercury surfaces are utilized. Conversely, the central perforation of the electrodes may be enlarged to nearly 2 cm. in case of electrodes nearly fitting the air-space, without appreciable fall of sensitiveness. If wire gauze is placed over these holes the interferences are not destroyed (though they become weaker), but the electrical advantage is negligible. CHAPTER III. ON ACOUSTIC PRESSURES AND ACOUSTIC DILATATIONS, CHIEFLY IN RESERVOIRS. 13. Introductory. Apparatus. — On a number of occasions heretofore,* I have endeavored to use the interferometer for the measurement of Mayer and Dvorak's f phenomenon, but though the experiments seemed to be well designed and were made with care they invariably resulted in failures. The present method, however, has been successful and has led to a variety of sur- prising results, even if the term acoustic pressure is not directly applicable. The apparatus is shown in figure 14, where B is a mercury manometer described in Chapter I, the displacements being read off by the component rays L, L' of the vertical interferometer. The mercury of the U -tube is shown at mnm', above which are the glass plates g, g', the former being hermetically sealed, the latter loose, so that the air has free access. The closed air-chamber R, above m, receives the air-waves from the plate of the telephone T, by means of the quill- tubes t, hermetically sealed into the mouth-piece of the telephone, and t' sealed into the manometer. Finally, t" is a branch tube ending in a small stop-cock C, or similar device at one end, while the other communicates with it'. Flexible-rubber tube connectors may be used at pleasure, so long as the space bounded by the outer face of the telephone plate, the mercury surface m, and the stop-cock C are quite free from leaks. The cock C will eventually be replaced by the glass quill-tubes c and c' (enlarged) perforated with minute orifices at 0. The telephone is energized by two storage-cells and a small inductor, with a mercury or other break. Large resistances are to be put in the telephone cir- cuit, so that the inductances are of secondary importance. The bore of the tubes t, t', c, c' need not exceed 5 mm. Thus the R chamber in B, about 6 cm. in diameter and 2 cm. deep, is the resonator (capacity with appurtenances, 57 cm.8) of the apparatus. *Cf. Carnegie Inst. Wash. Pub., No. 149, Part III, § 122. 1914. t This and the Konig resonators are adequately described in most large text-books (cf. Chwolson, Acoustic g 5, Vieweg edition). Reference should be made to Bjerknes's famous treatise of hydrodynamic forces in pulsating media (Earth, 1902); to the papers of W. Konig, Wied. Ann. xlii, xliii, 1891, and others. Recently much work along similar lines has been done by Prof. A. G. Webster and his students. Cf. A. T. Jones, Thesis, Clark Univ., 1913, on the acoustic repulsion of gas-jets; and Phys. Review, xvi, p. 247, 1920, on the tone of bells; H. F. Stimson, Clark Univ., 1914, on interferometer methods in acoustics. 14 ACOUSTICS AND GRAVITATION. 15 The displacements of the achromatic fringes corresponding to the head of mercury in B may be read off by a telescope provided with an ocular o.i mm. micrometer. It is, however, advantageous to place the micrometer in the wide slit of the collimator, the fringes being parallel to the scale-parts. To obviate the need of adjusting the inclination of the fringes (as this frequently changes), the slit-holder should be revolvable around the axis of the collima- tor, the scale being parallel to the length of the slit and the fringes moving in the same direction across the white, ribbon-like field. Fringes equal to a scale -part in breadth are most convenient. 14. Observations. Closed and open resonators. — Spring interrupters dip- ping in mercury were first used, having frequencies of n = i2 and 100 per second, respectively. When the cock C is closed, there is no appreciable effect until the telephone resounds harshly. In such a case there is marked dilatation in the resonator R, increasing with the intensity of vibration. The successive readings (sr fringes) are liable to be fluctuating; for instance, Current off 20 25 22 21 26 21 24 16 Current on 30 34 30 30 31 34 26 27 but the sense and mean value are definite. Since the head is (sf— $0)_ = 2 s\/2 cm. (the displacement being 5 fringes of wave-length X) , this mean value s of the many experiments for the given intensity of vibration is at once equiva- lent to the pressure A£ = 2Xicf4 cm. of mercury, or about jXicT8 atmos- phere. If but 500 ohms are put into the telephone circuit, however, appreciable deflection here ceases. One might suppose (particularly in view of the irregular readings) that the equilibrium position of the telephone-plate differs slightly when in vibration from the position when at rest, and refer the above deviation to a cause of this nature ; but in view of the results with pin-hole leaks presently to be described and in which an effect of the supposed shift of the plate would be untenable, the dilations are probably real. In the closed space R in B, temperature and other discrepancies would inevitably be detrimental to a smooth record. Again, if the stop-cock Cis completely open, no effect whatever is obtained. The bore of the small stop-cock in this case need not exceed 2 or 3 mm. All the negative results which I obtained by other methods heretofore are thus explained. 15. Resonator all but closed. — If now, from the open position, the plug of the cock C is rotated gradually until the opening is reduced to the merest crevice, the fringe deflection 5 will, on further slow rotation, be found to in- crease from zero, with great rapidity, to a positive maximum. The deflec- tion then falls off with similar rapidity through zero to the negative value when the cock is again quite closed. I have indicated this result graphically in figure 15, in which the abscissas show the degree to which the cock has been 16 DISPLACEMENT INTERFEROMETRY APPLIED TO opened and the ordinates show the fringe deflections s obtained. The max- imum or pressure is yery much larger than the minimum or dilation. The experiment may be repeated at pleasure in either direction with the same results. A loose plug moving easily with a minimum of oil is essential; but many trials to and fro are necessary, nevertheless, to set the plug at the optimum. The maximum pressure obtained in these initial experiments was the equivalent of about 50 fringes; i. e., A£= 1.5 X io~8 cm. or about 2 X io~B atmosphere, for a frequency of about 12 per second. At higher frequencies this datum is much increased. These pressures are real ; for on suddenly closing the cock at the maximum and breaking the current, they are retained until discharged on opening the cock. There has thus been an actual influx of air into the resonator R in B through the crevice in the cock C, acting like a valve while the telephone is sounding. Screw-cocks are not at once adapted for these experiments, since (on closing) a pressure is introduced owing to the viscosity of air. On open- ing, the minimum is accentuated. 16. Pressure depending on the frequency and on the intensity of vibra- tion.— The maxima are observable for very considerable reductions of the intensity of vibration. In figure 16 I have given the observed fringe location (here incidentally decreasing when pressure increases) when different resist- ances r, given by the abscissas in io3 ohms, are put in the telephone circuit. The frequency is n = 1 2 per second. In curves i and 2 the rift in the cock was either slightly too large or too small; in curve 3 it was nearer the optimum. In case of i and 2 the curious result of an initial increase of pressure with the first few hundred ohms inserted shows itself. This disappears in curve 3. Figure 17 contains similar results 5' when the frequency is n = ioo per second. The sensitiveness has obviously greatly increased and in a general way this is the case for higher frequencies. Anticipating results obtained in a better adjustment below, the following data may be cited. There were 2,000 ohms in circuit. The zero-point is at s°. 100 SQ-S'=II 300 per sec. 29 fringes 250 IS 23 It will be shown, however, that the magnitude of the phenomenon depends essentially on resonance, so that these data have but a general bearing. 17. Fringe deflection varies as current intensity. — If the readings in figures 16 and 17 be changed to actual deflections 50 — s' — s, the curves (3) and (5) take the form given in figure 18. They are roughly hyperbolic, so that thfe equation rs — C(r being the high resistance inserted into the telephone circuit) may be taken to apply within the errors of observation for resistances exceed- ing 1,000 ohms. So computed for convenience, rs is 24X10* in series 3 ACOUSTICS AND GRAVITATION. 17 and 36Xio3 in series 5. Hence, at r=ioo ohms (taken as a standard), the pressure would have been yXio~3 and i.iXio~2 cm. of mercury. The in- strument considered as a dynamometer is thus noteworthy, since its deflections would vary as the first power of the effective current or i = i0s. It is of interest, therefore, to ascertain how far the sensitiveness, which can not here be esti- mated as above io~4 amperes per fringe, may be increased. 18. Pin-hole sound-leaks. — Pin-holes less than a millimeter in diameter seem more like a provision for light-waves than for sound-waves often several feet long; but one may recall the phenomenon of sensitive flames. It is so difficult to make the fine adjustment for maximum conditions with stop-cocks that their replacement by the devices given in c and c', figure 14, is far preferable. Here c is a quill-tube, to one end of which a small sheet of very thin copper foil has been fastened with cement. The sound-leak at 0 is then punctured with the finest cambric needle. The other end (somewhat reduced) is thrust into a connector of rubber tubing at t". In case of c' the tube has been drawn out to a very fine point. This is then broken or ground off, until the critical diameter (0.04 cm.) is reached. Both methods worked about equally well, but in the case c several holes side by side, or holes of different sizes, may be tried out. Such results are shown in figure 19, which exhibits the deflection of fringes (ordinates s) for different diameters of hole in millimeters (abscissas), when 1,000 ohms were put in the telephone circuit. It will be seen that here the optimum 0.4 mm. in diameter is quite sharp. The finest size of needle is needed. The results obtained with the sound-leak c (when different resistances r are in circuit) are given in figure 20 for three diameters. As a whole the sensi- tiveness has been much increased, which is additionally shown by the com- parison in figure 18, series 8. The value of rs, viz, 5 = 51 25 1 6 12 10 5 fringes io~3r= i 23 4 5 10 ohms io~3rs = $i 50 48 48 50 50 is also much more constant than hitherto and reaches 50 X1^3- Hence at 100 ohms the pressure increment should be Ap — 1.5 X io~2 cm. of mercury. Figure 21, finally, indicates that the multiplication of pin-holes, all of the same diameter (0.04 cm.), is similarly disadvantageous. The deflection for four holes is scarcely half as large as for one. 18 DISPLACEMENT INTERFEROMETRY APPLIED TO Capillary tubes were also tested, but with less satisfactory results. Energy seemed to be uselessly frittered away in the channel. Thermometer capillaries act like closed tubes. Capillaries o.i cm. in diameter and 0.7 cm. long gave but moderate results, which were not improved by partially closing the tube with thin wires. An ordinary small one-eighth inch gas stop-cock of brass, with a loose plug, is preferable to a glass cock, when adjustable channels are wanted. The sharp edge seems to be advantageous, though I later adapted fine screw stop-cocks with good results. In case of the critically small leak, pressure decrements frequently precede the pressure increments when the telephone circuit is closed. If, in place of the quill-tubes c, the copper foil with an 0.04 cm. perforation is cemented to the mouth of a thistle- tube, the deflections on closing the cir- cuit are liable to be either absent or very small. The conditions for the valve - like action of the pin-hole are thus quite involved; but with a given tube c (fig. 14) the data are surprisingly constant. With the current on, if a drop of water is placed on the hole 0 in c, figure 14, the pressure is long retained, whether the current is thereafter broken or not. Pressure is gradually dissipated, however, as the joint at the telephone plate is rarely quite tight; and when the telephone sounds pressure tends toward the negative, as above. If most of the water is removed by blotting paper, a moderate, fairly constant pressure is usually observed for some time, until (doubtless with the breaking of the film across the hole) the maximum is sud- denly again attained. Thus from 60 fringes the deflection slowly fell to — 10 fringes; thereafter rose to -f 10 fringes, from which it eventually jumped to 60 fringes again. The result may be interesting, inasmuch as some pressure action through the film of water seems to be in evidence. 1 9. Effect of resonance. — While a relation of the maximum pressure parallel to 1he frequency of the telephone note has been shown to exist, it is obvious that the best conditions for high maxima will occur under conditions of resonance between the natural R vibrations and the T vibrations (fig. 14) or their harmonics. I therefore used the same small induction coil with two storage cells, but with a commutator-like current-breaker, controlled by a small electric motor with a variable resistance in circuit (electric siren). By gradually increasing this resistance all chromatic intervals between about f' and a" were obtainable. The speed of the motor is, however, liable to fluc- tuate slightly in any given adjustment, while intervals within a semitone often produce large pressure differences. Thus the determination of the intervals of a somewhat flickering pitch in all chromatics is here quite difficult, even for a musical ear. A series of organ-pipes within the given range seemed to offer the best standards of comparison , as it was necessary to turn rapidly from one series of observations to another. In this way the graphs given in figures 22, 23, 24 were worked out, the curves showing the fringe displacement s to the logarithmic frequency n of the tele- phone. In figure 22, to limit the deflections within the range of the ocular, ACOUSTICS AND GRAVITATION. 19 about 2,000 ohms were put in circuit. Three maxima and three minima (one positive and two negative) are indicated. The maximum below f' could not be reached. The strong one at c" was well marked and approachable from both sides. The small one near g", though easily observed by continuously changing the pitch, was difficult to record. The latter, however, is particularly interesting, as it introduces the strong decrement at e" and a". I therefore re-examined it in figure 23 with less re- sistance (1,000 ohms) in circuit, and the results came out more clearly. The deep minimum at a" deserves further investigation, as it precedes a probably very high maximum at the near c'". At least, this may be inferred from the stimulation produced by an organ-pipe used on the outside of the apparatus. In figure 24 I repeated the work with a finer micrometer scale, placed in the collimator to secure a larger range. The curves are far from smooth, owing to the mental confusion produced by the wealth of chromatic intervals, to which I have adverted. Something better than the electrical siren used will have to be devised; but apart from this the results are very definite. i 25 3a f a c? aTe,' yaT cf e' e' g: a1 e,' cL' e'(jf a? Adjusting the siren for the maximum c", the sensitiveness with different resistances in circuit (2,000 to 9,000 ohms) was determined. The curve is shown in series 9 , figure 1 8, and is the highest thus far obtained. The equation rs = constant does not fit so well here, a result inseparable from the slightly fluctuating note; for this makes much difference in the maximum. The mean value is about rs = 80 X io3. Referred to a circuit resistance of 100 ohms, this is equivalent to a deflection of 800 fringes and a pressure of Ap = o.o24 cm. of mercury. In the course of the work this was at least doubled. Treated in the same way, the minimum at a", if the resistance in circuit were but 100 ohms, would produce negative deflection of 400 fringes, equiv- alent to a negative pressure increment of A£= —0.012 cm. of mercury. This also was later much increased. These are striking results deserving more careful investigation than I have thus far been able to give them. The pres- sure, decrements also, are real, as may be shown by inserting a stop-cock and 20 DISPLACEMENT INTERFEROMETRY APPLIED TO closing it suddenly. There has been an efflux of air corresponding to the influx above; *. e., the acoustic pin-hole valve may be reversed. An auxiliary telephone placed in circuit with that of T, figure 14, affords no suggestion of these occurrences. Its notes rather increase in strength regularly with the pitch. Yet if the notes should happen to be near e", the other telephone would show no fringe displacement. 20. Inside and outside stimulation. — I have as yet given but incidental at- tention to this development. If, for instance, the branch tube and cock t"Ct in figure 14, is removed and the pin-hole tube c' is put within the rubber connectors t, t', the apparatus is now completely closed. It functions, however, though in a way totally different from the original adjustment, figure 14, as I shall presently show. In this case, if the telephone connector t is even slightly detached, no reaction is obtained, at least for the size of hole and pitch tested. When the tube c' is inserted in the absence of vents, there is much undesir- able pressure disturbance at the outset, which is but very slowly dissipated. Moreover, the closed space can not be opened again at pleasure without similar commotion. I therefore used the apparatus, figure 25, in preference, in which the pin-hole tube c' is provided with a branch tube t" and cock C. The rubber tube t leads to the telephone (beyond T) and the tube tr to the mercury U-tube (beyond £/)• K C is open, c' may be inserted or withdrawn with fa- cility. If C is closed, the resonator R is closed, as in the above case. The pin-hole O may point either toward U or toward T, without modifying the results of this experiment. Using the mercury interrupter (frequency c#) with 2,000 ohms in circuit , the deflection of the closed region was invariably negative. The deflection was peculiar, moreover, inasmuch as it is a slow growth, within a minute or more, to a maximum. On breaking the current the deflection died off in the same slow, fluctuating manner, with a kind of loitering on the way, as if something were removed in the first case and again supplied in the second. If the cock C is opened, the zero is instantaneously recovered . In other words, the dilatation is due to a loss of gas within the closed region, which loss is but slowly re- covered after the telephone ceases to vibrate. True, the closed region is an air-thermometer, whose fringe displacements also increase in this fluctuating manner; but these are pressures, resulting from the very small increase of temperature in presence of the interferometer ray, L'. They act against the dilatations just described. If the cock C, figure 25, is opened at the critical point, or if it is replaced by the tube c, the deflection is again positive (about 14 fringes at 2,000 ohms when c above would give about 40) . The action of c thus exceeds that of c't probably because the pin-hole in c happens to be nearer the critical size than in c'. The question next at issue is the influence of pitch upon the dilatation of the closed resonator R. The electric siren was here used with 2,000 ohms in the ACOUSTICS AND GRAVITATION. 21 telephone circuit. The results were most striking and appeared about as follows : g'tob' c" d'toef f g" pitch max. estimated —5 —25 at -200 —35 o fringes There is thus an enormous maximum dilatation somewhere in the range of frequency d" to e" which, from the hovering character of the deflection, is not further determinable. This amounts to a pressure decrement of A£ = — 6 X io~* cm. of mercury with 2,000 ohms in the telephone circuit. At 100 ohms it would have been about a millimeter of mercury. Thus the introduction of the pin-hole valve has enormously accentuated the simple phenomenon in §14 and guarantees its trustworthy character. It remains to place the two shanks of the U-tube, figure 14, in communi- cation, respectively, with the two sides of the pin-hole O in figure 25; i. e., to join t with the space above m' ; but for this purpose the plate g would have to be sealed, for which operation I was not quite ready. The use of a pin- hole within and without is usually prejudicial to sensitiveness. Thus at 1,000 ohms, the deflection of 47 fringes for this condition was quite doubled when the inside pin-hole was removed. With the apparatus, figure 14, and the cock C opened at the critical point, a diapason c" blown in the vicinity of the cock was quite appreciable and the octave c'" three times as effective (15 fringes). In another adjustment, while d" gave almost 35 fringes of negative displacement (dilatation), g" gave a positive displacement of 45 fringes and d'" (shrill overtone) nearly 100 fringes, also positive. There is some misgiving in interpreting these data, as the open mouth of the pipe must usually be close to the mouth of the cock; but as d" was still appreciably effective 6 to 12 inches away, the results are probably trustworthy. The difference in sign of the compression corresponding to c" and c'" is particularly to be noticed. 21. Apparent removal of pressure decrements. — The slow growth of rel- atively enormous pressure decrements here recorded is so surprising that fur- ther experiments are needed. To begin, one may ask whether the telephone plate, held as usual by strong screw-pressure between annular plates of hard rubber, is adequately air-tight. I therefore removed the telephone and sealed all these parts with cement, rigorously. On replacing the telephone with the adjustment, as in figure 25, the be- havior had in fact changed, the negative deflection being of the small value indicated in §14, without growth in the lapse of time. In other words, the presence of the pin-hole c' within the closed region was now ineffective. We may summarize the results, so far as at present in hand, in figures 26 and 27. In an air region R, closed on one side by a vibrating telephone plate T and on the other by a quiet plate U, the pressures are distributed as if there were a maximum at T and a minimum at U. If the region R, figure 26, com- municates with the atmosphere by a pin-hole 0 of critical diameter, the pres- 22 DISPLACEMENT INTERFEROMETRY APPLIED TO sure within R is raised as a whole by the amount which the pin-hole air-valve will withstand. Again, if the closed region TU, figure 27, contains a pin-hole valve 0 within only, it does not differ essentially from the corresponding case in figure 26. But if an additional pin-hole leak 0' is supplied on the T side, figure 27, the U side gradually develops a large pressure decrement; whereas if the pin-hole is supplied on the U side, this develops the usual pressure in- crement. In the foimer case air leaks out of 0' diffusively; in the latter it leaks into 0". In figures 22, 23, 24, marked pressure decrements occur near the minima at c" and o" in case of these prolonged tests. One may therefore suspect that (as above) the decrements result from an insufficiently tight joint at the tele- phone plate. The telephone with sealed plate was therefore carried through the chromatic series of notes from /' to a", as recorded in figure 28. The curve resembles figures 22 to 24 in character, except (as was anticipated) that there are no pressure decrements at the minima. In fact, the maxima (below /' at c", g", and above a") came out more sharply than in figuies 22 to 24, and the now positive minima (near g', d", a") equally so. It seems as if the ordi- nary overtones in the key of C were in question, but further work will be needed to establish these and other relations. To obtain corroborative evidence, I finally replaced this telephone with a common bipolar, containing the usual clamped plate, and worked out the results of figure 29. These obviously again reproduce figures 22 to 24 with marked dilatations at the minima, except that the maxima at a', e", a," are now in the key of A, conformably with the changed internal volume. Thus far the argument seemed to be conclusive ; but when I re-examined the original unipolar telephone with sealed plate at a lowrer frequency c' (n = 26o), an ACOUSTICS AND GRAVITATION. 23 astonishingly large pressure decrement ( Ap = • 0.04 cm. of mercury at 100 ohms) now lurked in this region. Clearly, therefore, the pressure decrements depend on something more than the mere instrumental error suspected, and that their occurrence must be as pronounced as the pressure increments which they reciprocally precede or follow, the next paragraph will show. A final experiment may here be inserted. Using the bipolar telephone above at a frequency of 100 and with 1,000 ohms in circuit, the usual pin-hole leak c was replaced by a fine screw stop-cock (at C, fig. 1 4) . Here the degree of opening could be specified in degrees of turn of the screw. The results obtained were (after the lapse of about a minute) : Cock open, o° 45° 90° 180° quite open Fringe deflection, s = — 34 — 20 — 12 — 2 +2 fringes The pressure production at a second very small leak thus very soon counter- balances the gradual dissipation of air at the clamped telephone plate. 22. Reversal of poles of telephones changes sign of fringe deflection. — An earlier detection of this result would have saved me much mystification. Not expecting it, I did not look for it, but it seems that a reversal of the telephone current (so to speak) frequently reverses the fringe deflection symmetrically. Thus when a switch (positions 7, //) was added to the telephone the fringe deflections were r 5j sz n 2,000 ohms +27 —26 i oo per second 15 — 16 12 1,000 48 —47 100 49 -47 100 This is as close to an exact reversal as the quivering fringes can guarantee. To test the case further, I used the motor interrupter, making a survey for frequencies between g' and a", as shown in figure 30. The motor did not run very smoothly, so that the estimation of pitch was difficult; but the curve, figure 30, corresponds to the curve, figure 28, except that maxima and minima have been reversed by the switch. The maxima now dip into the positive region. Thus the apparatus, regarded as a dynamometer and with precautions as to frequency, would give both quantity and sign of the effective currents in the telephone. Moreover, in any given adjustment, pressure increments changing continuously into decrements, and vice versa, are not unusual. Since the resonating region R is vented by the pin-hole, the positions of equilibrium of the quiet and vibrating plate are ineffective. Hence it is neces- sary to assume that the vibrations of the plate are not symmetrical; or that, for instance, the impulse corresponding to the break of current at the inter- rupter is of excessive importance. Thus, if this current re-enforces the permanent magnetization, the effect is a dilatation; if it interferes with the magnetization, the effect is a pressure. If the plate is energized symmetrically (as, for instance, by a small magneto 24 DISPLACEMENT INTERFEROMETRY APPLIED TO inductor) there should be less tendency to marked fiinge deflection. Tests made between g" and c'" seemed to bear out this inference ; but the current of the small magneto was too weak to decide the case. 23. Change of volume of reservoir. — The marked effect of resonance met with in the above experiments induced me to provide resonators of different volume. The open shank K of the U-tube, figure 14, was therefore sealed with a salient cylindrical cap, closed on top by a glass plate. The volume thus added was about 135 cm.3, which is 2.8 times the original volume R' of 48 cm.3. Testing this with the mercury break of frequency 12 per second, the high cylinder (Rf) was less responsive than the low cylinder, in the ratio of about 20 to 30 fringes; but at the higher frequency, n= 100 per second, this disparity was reversed, being in the ratio of 65 to 50 fringes, both data being larger. The two cases were thus unequally near a resonance maximum. With the motor break I obtained the resonance curves, figures 31 and 32, having 1,000 and 2,000 ohms in the telephone circuit, respectively, and using the sealed telephone. The harmonics found are apparently in the key of A. Strong negative and positive fringe deflections are here successively encoun- tered; i. e., pressure increments and decrements occur together, though in different frequencies. The fringes at the maxima, or the minima, or both, left the field of the telescope ; but it does not seem that the response for the three- fold additional volume is less strong than for the original volume. If anything it is larger, which is rather an unexpected result, unless it refers to the greater mass vibrating in the former case. 24. U-tube used differentially. — I now placed the two different reservoirs R and R' of the U-tube in communication by the branch-pipe bb' (fig. 33) leading to the telephone T by way of the quill-tubes t and t'. The pin-hole is in the sec- ond branch t" at 0 and a cock C (here open) is interposed. I had anticipated a ACOUSTICS AND GRAVITATION. 25 differential fringe deflection, owing to the different resonance volumes at R and R'. The result, however, was absolutely negative. The fringes showed no motion whatever, either at any frequency (12 to 500) nor when all resist- ance was removed from the telephone circuit. Each reservoir (R, R') thus acts like an open-air communication with reference to the other, so that the effect of the pin-hole valve 0 vanishes. On using either side separately (tb'R' closed, b open, or tbR closed, b' open) the normal behavior at once appeared. The only way of securing a differential effect detected was by elongating either branch (b, for instance) by inserting a long piece of rubber tubing. Thus, for 40 cm. and 80 cm. of interposed length, displacements cf 5 and 10 fringes were obtained from the unsymmetrical adjustment. No doubt this is merely equivalent to stopping, partially, the access to either chamber, R or R'. A variety of correlative experiments were made with the simple (non- differential) apparatus, figure 14. I have already referred to the absence of fringe displacement (nearly) when the copper foil carrying the pin-hole is cemented to the mouth of a funnel- tube. Small flasks with a lateral tubulure to be joined to t", figure 14, and the device c secured with a rubber cork in the neck, similarly gave deflection of 7 to 15 fringes only ; test-tubes with lateral a V 34 C! t tubulure not above 20 fringes in contrast with a normal deflection of 40. Thus a reservoir, within and immediately preceding the pin-hole, removes its valve effectiveness in promoting pressure and virtually opens the reservoir to the air. On the other hand, the prolongation of the quill-tube t", if the diameter is not increased, is almost indefinitely permissible. Thus, by inserting 40 cm., 80 cm.t 120 cm. length of eighth-inch rubber hose between t" and the pin-hole c, no marked difference in efficiency of the valve-like action could be detected. In all experiments with closed regions, care must be taken to allow for the temperature or air-thermometer effect resulting from the entrance of light into the resonator. It is small, but gradually increases in the lapse of time. Such results are also obtained when screw-cocks are put in the branches b or b' and very slightly opened. The experiments made suggest a method of obtaining an effect which is at least apparently differential. For this purpose the cock C, figure 33, is to be closed, so that c is inactive, and the pin-hole c', figure 14, to be inserted either into the branch bf, figure 33, or b. With a normal fringe displacement of 40, the pin-hole in b' gave a displacement of 38 fringes, in b correspondingly 27 26 DISPLACEMENT INTERFEROMETRY APPLIED TO fringes. Hence the large volume Rf is more favorable to a large displacement than the smaller volume R. It will be observed at once that here these res- ervoirs act merely like the outside air in the case of the original experiment, figure 14. The acoustic pressure is produced in the volume which is in uninter- rupted communication with the telephone and is larger in proportion as the other volume (shut off nearly by the pin-hole) is larger. If the latter is as high as 200 cm.8, it nearly replaces the atmosphere, seeing that the reduced pressure here (R') acts in concert with the increased pressure on the other side (R). Thus there is no true differential effect resulting from the size of resonators, whether the region be closed, open, or partially open. 25. Conical vents reversible. Periodicity. — The conical vents cf, figure 34, may be inserted into the branch tube t" in two ways; either in the salient position a, with the convex surface around the pin-hole 0 outward , or in the re-entrant position b, with the convex surface at 0 inward. The results ob- tained are usually reversals of each other, so that a pressure excess is liable to be on the concave side of the conical vent. Thus, for the high region R' alone (R being in communication with the atmosphere), the position a gave a pressure displacement of 28 fringes, while b gave dilatation equivalent to — 70 fringes. In other experiments 37 and — 14 fringes, 25 and i fringes, were found, while the region R alone gave 45 and —1,36 and 10 fringes, etc. These results were always consistent in character; but it was soon found that the strength of the telephone current and the length / of the tube c' (fig. 34) , in the re-entrant position, greatly modified them. When there is no resistance in circuit, i. e-, when the telephone sounds harshly, reversal ceases, so that either the case a or b produces a pressure within; but even here, on closing the circuit, the fringes in case b are seen to move first toward a dilatation and then turn in the direction of pressure. The position a being the normal case investigated above, the case b was studied with 1,000 ohms in circuit, and for different lengths, I, of the quill- tube c,' beyond 0, from / = 2 to over 40 cm. These results are given in figure 35, the abscissas showing the length / of tube taken and the ordinates the fringe displacements s, both for the region R' alone (positive displacements here denoting dilatation) and for the region R (positive displacements denoting pressure). The graphs are periodic in marked degree, so that the quill-tube b is a musical instrument with a pin-hole embouchure; and in fact, while the case a is nearly silent, b audibly reproduces the sound of the telephone. The curve for R' shows two resonance maxima and one minimum; but in all cases the dilatation (positive displacement 5) within R' is sustained, merely changing in degree. The curve for R, however (dilatations for negative 5), indicates the occurence of both dilatations and pressures within this (smaller) region. Both curves are quite consistent (although R' is nearly four times more capa- cious than R) and one may infer the length of pipe ^ = 30 cm. (fig. 34), or the wave-length 2/ = 6o cm., to be an harmonic of the telephone interrupter . This was in fact close to the 4-foot c. ACOUSTICS AND GRAVITATION. 27 In a majority of cases the action of the conical vent thus recalls the behavior of the cup anemometer, as the pressure excess is on the concave side ; but the lower curve of figure 35 (for R, smaller volume) is out of keeping with this, as between lengths of 20 and 35 cm. of pipe the pressure excess is within, or on the salient side. All attempts to use larger vents, together with wider external pipes (organ- pipe form) , failed completely. It is supposed that with more powerfully reso- nating external vessels, the pin-hole valve might be dispensed with, but no corrcboration could be obtained. One may now argue that if in figure 33 a salient conical vent is attached to the branch b', opening it to the atmosphere, and similarly a re-entrant vent to the branch 6, the pressure difference within would be accentuated. Experi- ments made both with the mercury interrupter and with the motor through all available pitches distinctly negatived this surmise. The largest fringe displacements were but one or two, determinable with certainty only by mul- tiplication. Thus the pressure within the closed region is uniform in spite of the reversed valves and the impulsive condensations of the telephone. 26. Resonators of very large capacity. — The volume of the region R', fig- ure 14, was now further increased by adding an additional cylindrical tube, 6.2 cm. in diameter and 10.7 cm. high, closed on top with a glass plate. All parts were (as before) carefully cemented together, thus completing resonator III. The volume added was thus 370 cm.3, as compared with the original 48 cm.3, the ratio of volume increment being 6.7 and the ratio of total to original volume 7.7. The conical vent cf, figure 14, here acted much better than the copper-foil pin-holes, in the ratio of about 28 to 19 fringes at the frequency n = 100 per second and resistance 100 ohms. At n = 12 per second the conical vent gave about 15 fringes. Using the motor break and conical valve, the fringe displacements were observed for frequencies between the notes g' and c'" and both with 1,000 ohms and 500 ohms in the telephone circuit. The results are comprehended in the two curves in figure 36, the region R being in communication with the atmosphere and R' closed, except as to the salient conical vent specified. The curves are remarkable because of the sharpness of the maxima, which are apparently overtones in the key of B or B\>. The strong maximum near f" is fully obtained, also a small one near b', and indi- cations of a large one again below g. The advantage of the salient conical vent over one or more of the pin-holes in foil is shown in the small inset at a (mercury interrupter, n= 100). It is obvious that the fundamentals of the large closed reservoir R' will lie very low as compared with the frequencies of the diagram, and very large fringe displacements may be looked for there (§27); but the present motor would not function below g\ between a" and c'" the current seems to be uncertain. It was now thought desirable to test the conical vent in the re-entrant position, and data of this kind are given by the curve, figure 37. All the 28 DISPLACEMENT INTERFEROMETRY APPLIED TO maxima are here dilatations, laid off positively for convenience in com- parison with figure 36. The valve action (500 ohms in circuit) is much weaker in figure 37 than in figure 36, but in every other respect figure 36 is reproduced. This is a very disconcerting result, for it is not the impulsive displacement of the telephone plate alone which produces the pressure increments within, even if a reversal of the telephonic current (change of poles) changes pressure increments into pressure decrements. The coni- cal pin-hole vent in association with a given adjustment of telephone current (poles not changed) will do the same thing if reversed. Suggestions to the same effect are contained in the above data for reversible conical vents, but nothing quite so trenchant throughout all the harmonics. Further work with the resonator III consisted in modifying the motor, so that with reduced frequency of current interruption the lower overtones could be surveyed. The results are given in the graph, figure 38, for 1,000 ohms in 50 0 a' C" d" e" (f a the telephone circuit. Conformably with the large volume, the low tones are very effective and the maxima occur at B, f, b, after which there is an hiatus in which the motor refused to function, continuing with b', f", of figure 36 (lower curve) appended to figure 38. The B maximum is represented by a displace- ment of about 50 fringes, equivalent to a pressure increment of about 0.012 cm. of mercury when 100 ohms are in circuit. 27. Resonator of very small capacity. — Finally the resonator R was all but closed (resonator IV), or at least reduced to a shell-like space by a cylindrical inset, closed with a glass plate but a few millimeters above the mercury of the U-tube. At n = ioo deflections of 50 fringes were obtained with the copper- foil pin-hole, and at w = i2 but 25 fringes. The conical vent was less useful, being too large. ACOUSTICS AND GRAVITATION. 29 A survey of the fringe displacement corresponding to different harmonics is given in figure 39, when 1,000 ohms completed the telephone circuit. Their distribution is less regular than heretofore. If we place them a semitone below d' , a', a", the one between a' and a" is missing. What particularly astonishes is the occurrence of resonance at these relatively low notes, seeing that the resonator volume is here nearly negligible. The maximum displacement of 100 fringes corresponds to a pressure increment of about 0.03 cm. of mercury, which, though twice as large as obtained with the large resonator ///, is never- theless of the same order of values. A variety of other experiments made with different nozzles attached to this cup-shaped cylindrical shell were not quite consistent as to the location and number of overtones. d1 ef a1 a' c"a" er " a -/aw T * i --T-. -_i i ' ' L t f - £ 3" d & d & g, a c' d' e' gf a' e" d" e' 2' a at & cf e? a' a" o u Thus figure 40, obtained with a conical vent (somewhat too large here), shows but two maxima near g' and g", the others being obliterated or very low. A fine pin-hole 0.03 cm. in diameter in copper foil was therefore prepared and tested on the modified motor break, between A and a", with results given in figure 41. The figure shows much greater detail, there being maxima near a, a1, a", e, ef, e" Yet this is not the whole list, for the curve is probably much more closly serrated. The smaller indentations are slurred over by the imper- fections of the subjective method of observation. A special search made for maxima brought such locations as M, for instance. In one respect the present curve, figure 41, for the shell resonator IV differs conspicuously from the curve, figure 38, for the capacious resonator ///. In the latter, conformably with the large volume, the lower notes (8-foot octave) are very much more effective than the notes of the i-foot octave. The curve as a whole falls from left to right. In figure 41 the reverse is ob- served. In fact, it is rather surprising that for the case of the very small 30 DISPLACEMENT INTERFEROMETRY. volume (fig. 41) the 4-foot octave should appear at all. Finally, the highest peaks in figure 41 exceed those of figure 38. fct, The perplexing question as to what resonates in case of the small resona- tor was approached by lengthening the tubing between the telephone and U-tube from its usual length of 48cm. to 85 cm. and 120 cm. successively. The shell-like volume of the resonator IV can not much have exceeded 10 or 15 cm.3. The tubes added 10, 17, and 24 cm.3 respectively. To this the disk- like volume in the telephone (about 2 cm.3) is to be added ; nevertheless the greater part of the total volume seems to reside in the tubes. The results obtained in the survey maxima and minima are given by the curves in figure 42 . The corresponding maxima and minima are easily recog- nized in the three cases, though there has been a little shifting, part of which is necessarily mere subjective impression. What has changed actually is the amount of fringe displacement, both at the maxima and the minima. This is particularly true for the order of pitch above c" . In other words, the overtones have been differently accentuated. Thus the minimum near d" passes from positive to negative values as the tube-length increases, while the reverse is observed at the maximum near g". The case is, then, analogous to the occurrences of the vowel sounds, where different overtones are accentuated by variation of the mouth cavity. CHAPTER IV. THE PIN-HOLE PROBE FOR SOUND PRESSURES. 28. The pin-hole sonde, or probe. — It was shown above that when the tele- phone is detached but slightly at the connection-pipe t, figure 14, no pressure effect is discernible in the U-tube. This, however, may be predicted if the extremity of the pipe t is regarded as the end of an organ-pipe. Thus it seemed worth while to determine the possibilities of sounding for pressure by aid of a pin-hole valve 0 at the end of a slender tube c (as in fig. 43), communicating with the closed reservoir R' (IV) of the U-tube, the other, R, being open to the atmosphere. The telephone T was therefore provided with a tubular projection t, 12 cm. deep to the plate and about 0.75 cm. in the inner diameter. This received the probe c (0 being at the end of an aluminum tube 2 mm. in bore, 3 mm. in outer diameter, and about 20 cm. long) to different depths with a view to exploring the pressures within. The side branch with cock C and extra pin-hole 0' is useful below. Here C is to be kept closed. The probe used in the normal fashion (as in fig. 14) gave the results contained in figure 44 when 500 and 1,000 ohms were successively put in the telephone circuit. On comparing this with the usual pin-hole it was seen to be about Ocm, 4 one-third as sensitive. Moreover, the details of the resonance have been oblit- erated by the slender aluminum tube. A wider quill-tube will therefore be preferable in general, though in the present case greater width would have been permissible because of the small bore of the pipe t. Probably, however, the decreased sensitiveness here observed is of no consequence, as there is to be a minimum of vibration in the tube c when used for exploring or sounding purposes. The results obtained on inserting the probe to different depths (marked on the graphs) in the telephone pipe are given in figs. 45 and 46, compressions being laid off positively downward in these figures. The curves are unexpect- 32 DISPLACEMENT INTERFEROMETRY APPLIED TO edly complicated, but consistent in their character throughout. In place of the simple distribution of pressure in the usual closed organ-pipe, there is a maximum pressure decrement near f't another at d", a maximum nega- tive pressure increment near a', and a very pronounced positive pressure increment near f". A pipe of the same length, if wide and uniform, would have responded to e". The low pitch is thus attributable to the plate space in the telephone and possibly to the narrowness and roughness of the tube, which was of hard rubber. However, the increase or decrease of pressures respectively, from the mouth of t toward the telephone plate 12.5 cm. below, is well shown in figure 47, in which the fringe displacements s for the large minimum d" and the large maximum f" are recorded. Pressure rises (or falls) very rapidly from the mouth inward and at 6 or 8 cm. of depth already reaches its maximum value. Beyond this the curves of the figure often show a decrease, which is probably not incidental (owing, for instance, to a heated telephone or to inevitable changes in the motor-break inducing current, during so protracted a series of observations) , as it occurs frequently below. A cylindrical enlargement was now attached to the pipe t, figure 43, con- sisting of a glass tube (see fig. 48) 9 cm. long and 1.5 cm. in diameter, tapering to a narrow tube fitting t, the total length of which was now 15 cm. Explor- ations were made with the probe in the wider tube, with results also shown in figure 48, at i, 4, 6, 8 cm. from the end of the tube. These results (pressures positive upward) are again consistent. There are two definite maxima, one near a' and the other (uncertain) near d". The latter may be a residue of the case, figure 45, but the former is surely a repetition of the a'. Naturally, pres- sure increments in figure 48 are much smaller; for the main pressure changes will begin in the narrow tube / (see fig. 47). Finally, believing that the complicated distribution of maxima in figures 45 and 46 was due to roughness and narrowness or other similar incidental quality of the pipe t, figure 43, 1 replaced it by a tube of brass, i cm. in diameter and 13 cm. in depth, as far as the telephone plate. Using the pipe-blower (§ 45), this tube was found to respond weakly to the note a" and strongly to a'". The record obtained by aid of the probe, as shown in figure 49, is, in fact, an apparent simplification; there were now but two maxima determinable, both of them quite sharp, a very large one near a' and a small one near d". The one at a' coincides with the response to the blower and is accentuated when evoked by the telephone. ' e1 qi a' c1 d" e' a' o * The little maximum at d", however, is supernumerary, being the reversed d" of figure 45, so that something else resonates, possibly the plate of the tele- phone or the telephone space or the U-tube reservoir. ACOUSTICS AND GRAVITATION. 33 The rubber tube was now about halved in length, the present depth to plate being about 5 cm. and the diameter 0.75 cm. The note obtained with the blower was a faint d' and a strong g" . The survey in pitch with the pin-hole at a depth of 4 cm. is shown in figure 50, 700 ohms being in circuit. There are maxima at d' and particularly at the octave of the blown pipe d" 1 and minima at e' and a', all well marked. If we compare the present curve for the half pipe with figures 45 and 46 for the full pipe (remembering that the latter curve is inverted), there is thus a complete inversion of results, maxima taking the place of minima. Adding the discarded part of the tube, the results of figure 45 were reproduced; but on adding a part but i cm. shorter, the curve corre- sponded in character to figure 50, though with less displacement throughout. Hence a little difference in length, possibly with some incidental difference in width, may change pressure increments into decrements and greatly modify the sensitiveness. Figure 50 also shows the effect of the depth of the pin-hole below the mouth of the tube, on the fringe displacements 5, for the d" maximum. The largest displacement occurs at about 5 mm. from the plate. 29. Pressures in smooth straight pipes. — The simple results for these pipes, given in figure 49, seemed to me suspicious, and possibly referable to a dete- rioration of the pin-hole valve after prolonged use. Thus, even a slight ad- ditional sound leak, or looseness, would contribute to such simplification. The pin-hole was therefore carefully overhauled and the survey of the straight tube (13 cm. deep, i cm. diameter) between f' and g" repeated with results recorded (pressures now laid off positively) in figure 51. The suspicion was thus well founded; for the sensitiveness in the present results is enormously greater than in the case figure 49, so that even 100 ohms placed in the tele- phone circuit do not keep the fringes within the limits of the field. As I did not wish to replace the ocular micrometer by the slide-micrometer because of the danger of complications, most of the curves are open at the dilatation d". The present curves (fig. 51) therefore reproduce faithfully the pressure minima near f and d", and the pressure maxima near a' and possibly c" of figures 45 and 46, though on a much larger scale, with the exception of the latter (c"), which in both figures is uncertain; but the character of the results is the same for the rough and the smooth tubes, in spite of a difference in bore (0.75 cm. and i.oo cm.). Finally, the smooth tubes respond to a' when blown. The harmonics d" and c", which are supernumerary here, however, also occur in figure 42, for the resonator R' (IV) alone, with corresponding intensity and sign. These must therefore arise in something which has not changed, i. e., in the telephone itself one might surmise, or even in the resonator, cut off though it is by the pin-hole valve. In the inset (a' at 100 ohms) I have laid off the fringe displacement 5 in its variation with the depth of the pin-hole below the mouth of the tube. This could not be done at d", since the curves are here incomplete. As these con- trasting distributions are, however, of great interest, I made a set of inde- 34 DISPLACEMENT INTERFEROMETRY APPLIED TO pendent measurements both at a' (now with 500 ohms in the telephone circuit) and at d" with 200 ohms in circuit). At a' the pressure increments increase rapidly with the depth until within i cm. from the telephone plate, when they drop off a little. At the minimum d" the pressure decrements are continually g! a' c" larger until a point a little below the middle of the tube is reached by the pin-hole, after which they continually wane as far as the telephone plate. This tube (diameter i cm.) was now cut down to a length of about 10 cm. When placed in the telephone and blown with the pipe-blower it responded weakly to c' and strongly to d'". The survey made with the pin-hole valve is given in figure 52, the curves showing the pressure increments at depths 2, 4, 6, 8, 10 cm. below the mouth of the tube. The arrangement was remarkably sensitive, so that 500 ohms had to be put in the telephone circuit to keep the fringes in the field. Moreover, in sharp contrast to figure 51, the curves are throughout of the same kind, there being now no pressure decrements. Whereas the blown tube responded to a clear c", the chief maximum evoked by the telephone is a decidedly flatted c". There remain the two supernumerary resonances as in the preceding figure, the one at a' considerably dwarfed and soon vanishing and the one near d", now not only sharpened in pitch, but re- versed, so that it becomes a pressure increment. The accentuation of the notes near c" and d" was obvious to the ear. In the inset I have given an independent survey of the distribution of pressures in depth below the mouth of the tube. To keep the fringes well in the field, 1,000 ohms were put in the telephone circuit. Pressures increase continually, almost as far as the telephone plate, both at the c" (flat) maximum and at the d" (sharp) maximum. One may note that at the flat c" the maxi- mum fringe displacement at 1,000 ohms is 35, so that these pressure increments (about o.o i cm. of mercury) are of the same order as occur in most of the com- pletely closed regions above. The brass tube was then further cut down to a length of 7 cm. from the ACOUSTICS AND GRAVITATION. 35 telephone plate. The note evoked by the blower was now a faint e' and a strong flat e"'. The survey in pitch at a depth of 4 cm. showed a minimum at a' ,&', e", a"} and maxima (near g', a', d" , bg"; d" very pronounced) . They differed in the amount of displacement at those pitches. The additional pin-hole does not,, therefore, interfere with the action of the probe; but its efficiency is weakened nearly one-half. Moreover, since the pin-hole o' here acts like a valve, it would seem to follow that the reservoir R' vibrates. The pipe t was now closed at the outer end by inserting the perforated cork and a survey was made between g' and a" with the auxiliary pin-hole o' in action. The graph naturally differed from the preceding, having maxima near g'» b', g", the latter very pi enounced, and minima near bf, c", and a". The main pitch had thus risen from d" to g". The pin-hole o' was then shut off, but the cork in /' loosened. This left, the maxima (particularly the d" maximum) nearly intact ; but the c" minimum was shifted to e". Finally the cork at the end cf t was closed tight, so that the region was completely closed. The resulting behavior was peculiar, inasmuch as large fringe displacements were attainable, but only after the lapse of considerable time (minutes). The same conditions have been instanced above and are due to the fact that the very small leaks in the region are taken advantage of „ for on breaking the current the fringes return to zero with the same slow motion. Allowing the necessary time to elapse, however, the maxima for the corked pipe t (with pin-hole 0') were reproducible ; but the former minima respectively at c" and e" had now shifted to d". There was one other noteworthy difference, as the a" minimum, immediately following the very high g" maximum,, now appeared as a deep negative fringe displacement or dilatation. All the pipes showed a maximum, more or less developed, at g' and at g",. the unstoppered pipes being far weaker. Between these limits of pitch, the unstoppered pipe is in its regular behavior almost an inversion of the stoppered pipe. ACOUSTICS AND GRAVITATION. 37 4K. cm, 32. Closed organ-pipe. — The tubes heretofore studied were all relatively narrow, so that only the overtones strongly responded on blowing. It is therefore desirable to examine a relatively wide pipe p, figure 56, inset (di- ameter 2.6 cm., length to the telephone plate 13 cm., smooth brass) , from which a strong fundamental (#c") could be easily obtained. The brass pipe was attached with cement, axially to the mouth of the telephone T. An exami- nation both as to frequency (at 12 cm. below the mouth) and depth distri- butions of pressure, made with the pin-hole probe s, is given in the graphs of figure 56. With the exception of the small supernumerary maximum at the sharp a', the pressure relations are simple, there being only one strong, abrupt maximum at %c" coinciding with the blown note, within the range (e1 to a") surveyed ; but the suggestion of a second maximum near c'" (which could not be reached) is obvious. There were no dilatations. This distribution in depth is also a smooth curve which begins with a significant value in the plane of the mouth of the tube, and reaches a maximum a few millimeters from the telephone plate. The high pressure value (0.012 cm. of mercury) in this wide pipe is remarkable, as there were 1,000 ohms in the telephone circuit. 33. Open pipes and adjutages. — The final experiments were made with an adjutaged open pipe pp, 33 cm. long and 3 cm. in diameter, blown by tele- phone T, as shown by the inset in figure 57. The adjutage or connecting-pipe a (7.5 cm. long and over i cm. in diameter) was screwed in at both ends and cemented. The pipe pp had a hole (diameter 0.6 cm.) opposite a, through which the long pin-hole probe s (aluminum or glass quill-tube) could be in- serted to any depth; or, again, the probe could be introduced into the pipe pp longitudinally to any depth, as shown at s', the lateral hole being either left open or closed with a cork. A short flexible tube connected 5 or 5', with either (closed) reservoir of the U-tube, the other having been opened to the atmos- phere. It is interesting to note that if the two halves of the pipe pp are not equally long, measured from a, or if the notes are not harmonics, the telephone •evokes violent beating wave- trains from the two ends of pp. The survey was first made in the adjutage a within a few centimeters of the telephone plate, with results as in the two graphs given in figure 57. Not- withstanding the complication of these examples, both are in agreement and both pressure increments and decrements occur, particularly in the lower curve. As there are 5 maxima and as many minima, often close together, it seems probable that the harmonics of the open pipe are here superimposed on those of the adjutage. The pipe pp responded to the note a' when the cork opposite the adjutage was in place, and to d" (flatted) when the cork was out. Both cases were studied. The d" pipe showed but this single maximum between gr and a" and 38 DISPLACEMENT INTERFEROMETRY APPLIED TO the U-tube (when i ,000 ohms were in circuit) gave a deflection of but 5 fringes. This maximum (curiously enough) is in the middle, figure 58, in spite of the open hole there. The result appears more clearly when the resistance is re- duced about one-half (500 ohms) . It is the d" ', probably, which is impressed on the graph of the adjutage, figure 57. The a' pipe (hole opposite adjutage closed) showed two maxima, a large one of 25 fringes at a' and a smaller one of 12 fringes at e" when 1,000 ohms were in circuit. The latter is possibly impressed on pp by the e" note in the adju- tage. The distribution of compression in depth is also given in figure 58 for this pipe. It increases more gradually toward the center than in the pre- ceding case, a result which suggests a tendency to a re-entrance in the middle for the d" curve. The a' pipe is also much stronger than the d" pipe under the same conditions ; but both are usually less than half as strong in their compressions as the closed organ-pipe of the preceding section. The quill-tube glass probe functioned about equally well at d" and a', but the somewhat narrower aluminum-tube probe often refused to give results at d". As I could not detect any defect in the probe, it is probable that the phenomena of §25, relating to nodes in the pin-hole tube, are here in question. If so, pressure would be transferred to the U-tube by means of a vibrating column of air. 0 4 8 34. Reversal of poles of telephone. — The more or less complete exchange of the maxima and minima of 5, of a closed region, when the telephone current (motor break) is reversed, has already been instanced. Open pipes, however, do not quite conform to this rule. In wide tubes (2.6 cm.) of organ-pipe shape, pressure increments only are found, and these are indifferent to the direction of the telephone current. ACOUSTICS AND GRAVITATION. 39 In narrow pipes (diameter 0.7 cm., length 5 cm.) certain harmonics may be reversed and the others left unchanged when the telephone poles are changed. Thus, with the short hard-rubber tube of figure 50, the curve above g' was not subject to change on commutation, but below this there were marked differ- ences. The e' of a few negative fringe displacements in figure 50 changed to e' of about 50 positive displacements on reversing the current. Moreover > the curve with 1,000 ohms in circuit as compared with the curve for 2,000 ohms, though preserving the same character as figure 50, differed in the relative value of maxima and minima. It is, moreover, frequently possible to change negative fringe displacements into increasing positive displacements by increasing the telephone current; i. e., with the removal of extra resistance in circuit. The curves are thus de- pendent on their general location on the incidental changes of current (as one of their parameters), and this explains why on repeating a series of observations the curve, as a whole, may have risen or been depressed. In a more searching investigation the current, like the frequency n, would have to be definitely measured in its relations to s. The most interesting inversion of this kind was made with the brass pipe 13 cm. long, i cm. in diameter, from which the peculiar graphs of figure 51 were obtained. To get the whole graphs within this field, 2,000 ohms were put in the telephone circuit and there were a few other incidental modifications. Apart from this, the graph, figure 59, obtained with the pin-hole probe for the position / of the switch, is the same in character as the group in figure 5 1 , with the strong maximum at a' and strong minimum at d" '. The curve for the position // of the switch is very different. While the minimum near g' has probably been retained, the former strong maximum at a' is obliterated. Furthermore, the former negative minimum at d" has been reversed to a positive maximum and enhanced. Similarly, the same pipe cut down to 10 cm., as recorded in figure 52, with only positive fringe deflections, lost its %d" maximum on commutation of current. Finally the pipe, cut down to 7 cm. and examined at 6 cm. of depth, gave the graphs / and //, figure 60, for two positions of the commutator. The difference at g' and a' is marked; but at the pipe-note e" the position / evokes an enor- mous maximum, followed at once by an equally pronounced negative minimum, whereas in the position // the minimum remained in the positive field. Other examples will be given among many obtained, in treating the open organ-pipe on the interferometer. Referring for convenience to figure 59, let sn be the mean fringe displacement at any frequency n. Then the observed displacements at n will be sn+ A^n and 5n— Asn, respectively, for the two positions, I, II, of the switch of the telephone current. Hence the reversible effect impressed by the telephone is 2 Asn, a quantity which vanishes between a' and c" (i. e., at &') and is positive below and negative above &', so far as observed. It also vanishes at f'. Thus it seems to me probable that on operating with a sufficiently strong magneto, the 40 DISPLACEMENT INTERFEROMETRY APPLIED TO As would be nearly zero throughout in consequence of the symmetrical in- duction utilized. The magneto employed above, however, was too weak to test the case adequately Like the reversible Asn, even sn would nevertheless depend on the mode of vibration of the telephone plates. On completion of the interferometer work of the next chapter, I took up this pipe again for a more detailed survey of the pressure distribution in rela- tion to frequency, at different depths below, on the mouths of the pipe. The graphs (5,000 ohms in circuit) are given in figure 61, where the numbers on the curve show the position of the pin-hole in centimeters from the nearer end, 16 cm. being at the middle, and 8 cm. half-way to it. The a! maximum, so loud to the ear in the i6-cm. curve, is followed at once by a near a' minimum, which is easily overlooked, as it runs at once into the next c" maximum. The e" i? actually a minimum followed by the maximum near f" . At a quarter tube-length from the end (8 cm.) these features have been thoroughly modified. The a' maximum is now isolated, the former e" mini- mum has been reversed. The 8-cm. curve is somewhat low throughout, probably because the telephone gets heated in so long a series of experiments. Figure 62 shows the effect of reversing the f poles of the telephone with the pin-hole at / the middle of the tube. The position // of the switch corresponds to the preceding fig- ure; / is the alternate position. The effect produced is obvious near c" , whereas e" retains its negative character, as do also the other positive parts of the graph. - - /T n1 n» rl* 0" n" n1 f>" H P" n" n' In an adequate discussion of these results the telephone current would have to be measured as one of the parameters. 35. Open pipe on the interferometer. — It seemed to me of sufficient interest to make a direct measurement of the compression in the open pipe, by passing the component ray of the interferometer longitudinally through it, as explained in Chapter V and shown in figure 78, § 47. The reed R of that figure is here to be replaced by the telephone as in figure 57 (inset). Using the motor- break, a survey of compressions was made between a' and a" . On removing the resistance from the telephone, very sonorous and regular sinusoidal wave- forms were obtained at the harmonics a' and g" (nearly), each about one fringe in double amplitude. The occurrence of the latter note is peculiar. Referring to the a', the mean compression would be (notation as in § 42), C\ 16cm. when the height of crest above trough is one fringe. Since the mean value in question is thus Ap/p = 6.2 X io~3 per fringe and the maxi- mum compression would be IT/ 2 times larger or io~3X9-7, practically io~2. ACOUSTICS AND GRAVITATION. 41 The ratio of compression in case of the interferometer direct and the interfer- ometer and pin-hole conjointly is thus 9.7Xio-3/i.6Xio-4 = 6i,of the same order of value as estimated for the same pipe. With 1,000 ohms in the telephone circuit, however, the waves were scarcely perceptible on the interferometer, equivalent, therefore, to a double amplitude of o.i fringe, whereas the pin-hole valve showed about 25 fringes (fig. 58). The bg" overtone with smooth, sinuous waves is not easily explained, for in the normal diapason the nodes are respectively rare and dense and thus with- out effect on the interferometer. It may be plausibly argued, therefore, that in the telephone-blown pipe both nodes are together either dense or rare, an abnormal condition impressed by the telephone. The central node in the a' pipe is thus halved and the two halves symmetrically shifted outward, to be in equilibrium with the impact of air from without . Hence bg" is the fundamental of two identically sounding half -pipes and the above relations remain unchanged. The e" overtone was also strong, but the waves were no longer sinuous, consisting rather of successive arches, meeting in cusps at their abutments, and with the curves of the arches sharply serrated. The mean double ampli- tude was again about one fringe. In proportion as the vibration telescope dies down in amplitude, the fringes gradually assume the appearance of par- allel strands of beads. Notes near a' (like g', 6') usually emit strongly beating wave-trains from the two ends of the pipe, even when the halves are of equal length. The same is true of notes near bg" (g" and a") . The interferometer record usually shows waves of nearly the same amplitude as at the regular overtones ; but they are highly compound successive serrated inclines, and the like. Waves are running from end to end of the tube, without steadiness of motion. Only at c" and f" was the fringe-band approximately straight. With the d" pipe (unstop- pered) a', d", \>g" gave a sinusoidal record, while b' ', c", e" were strongly beating trains, but the maximum double amplitude did not here much exceed a half fringe, agreeing with the case of figure 58. Finally, the open pipe was tested with the small magneto actuating the telephone. Scarcely any effect was discernible, even when all extra resistance was removed from the circuit. This current is thus not strong enough to blow the pipe appreciably. If the telephone, provided with a mirror for observation on the interferometer, is examined the effect is throughout quite marked, as will be instanced (§ 56) below, Chapter VI. 36. Helmholtz spherical resonator. — It is interesting to compare with the complicated results described the graphs for a telephone-blown Konig reso- nator (c"}. These are given in figure 63 for probe depths of 2, 4, 8 cm. belowthe mouth. Here the U-tube reservoir contributes no appreciable overtones. It was necessary to withdraw the extra resistance from the telephone circuit to obtain large fringe displacements, owing to the small entrance channel (neck) for the telephone note and the relatively large diameter (7 cm.) and mouth of the resonator. For this reason, also, no appreciable effect was 42 DISPLACEMENT INTERFEROMETRY APPLIED TO obtainable from the c resonator (diameter 14 cm.), the dissipation in 3 dimensions being probably too severe. A repetition of this c" survey showed that commutation had no appreciable effect ; but indications of a small flat maximum near a' were now apparent. Similarly, a d" resonator suggested the small maximum at a', but the d" was enormously developed, over twice as high and abrupt as the preceding maximum for c". The same results were again obtained for an/7' resonator, the maximum being very high and sharp. Beyond this the narrow aluminum probe refused to function, the g" and a" resonators being characterized by very small maxima at their respective pitches, less than one-third as high as in the case for c" . The occurrence of the obscure maxima near a' induced me to replace the narrow-tubed aluminum probe by a quill-tube glass probe, the same diameter of pin-hole being used. The results so obtained with the c" resonator (given in fig. 64) are quite different from the preceding graph for c", inasmuch as the a' maximum is now highly developed. The apparatus, moreover, is so much more sensitive (auxiliary resistance here removed) that even 1,000 ohms may be put in the telephone circuit with the distinct data seen in the lower curve. The effect of commutation is again absent, the numbers I and 77 in the curve referring to the position of the switch. 66 d ' Tn. 6 74 73 from a great variety of vessels. Thus bottles, deep tumblers and beakers, flat jars (like sardine boxes), truncated cones, thistle-tubes, and even thimbles respond, often very loudly. Very disconcerting sounds are otten obtained. Thus, for a wide-mouthed cylindrical jar, 3 inches in diameter and 6 inches high, taperi ig down at the top to a mouth 1.5 inches in diameter, the fundamental appears at once (55' across the middle). If now the distance sc is decreased, the overtone will appear loudly; but it is not the fifth above but the octave itself. As the kinematics of the stationary waves are given, the overtone belongs to an original wave of 1.5 longer wave-length than the fundamental. With a flexible strip ss', like moistened paper, the response is often better as to tone quality and the clarionet note is suggested; but the instrument is less convenient.* "Reference should here be made to a more comprehensive apparatus invented by Professor Webster (Proc. Nat. Ac. Sc., July, 1919; Am. Phys. Soc., 1919; Science, Feb. 25, 1919) and adapted for blowing brass instruments. ACOUSTICS AND GRAVITATION. 51 The results with the simple tubes in question were much the same as those •obtained with organ-pipes and need no further comment. With the open pipes, where the octave vanishes, a wave-length much longer than the funda- mental is often seen, probably due to the blower in some way. 46. Interference. — This experiment succeeded beautifully with the strip ss' of the blower placed between two coaxial pipes P and P' (fig. 74), each about 10 cm. long and 2 cm. in diameter, for instance, and closed at the outer end. Either pipe alone sounds vigorously when actuated by the blower. With the two together there is a mere siffling, the wave running from end to end of the (virtually) double closed pipe PP'. Nevertheless, there is abundant room at mm for the escape of sound; indeed, one pipe, P for instance, may even be placed at right angles to the other, leaving a wide open space, and still almost the whole energy of one pipe is alternatingly captured by the other. The avidity with which one pipe absorbs the vibrations of the other is an excellent illustration of the reversal of spectrum lines. The nodes here are respectively dense and rare, i. e., always opposite in the two pipes; hence the interference. In the cross-pipe above, the nodes were necessarily identical in sign, and, therefore, gave marked response. The same will be true if the pipes PP' are each open at the further end. L 73 ^ ^ i i C}V_t ^tC 1 *=Pii y° ^ ? i1 j 76 s c/\c' s1 * ] t ^s. 9 77 * Another form of the blower is given in figures 75 and 76, where ppr is a short end of square-sectioned brass pipe, closed at one end p' and provided with a tubulure P at the other. One of the edges of the brass pipe has been filed down until a fine rift cc' can be cut into it with a thin blade. As before, gg' are the guides of an adjustable strip ss', against which the lamella from cc' plays. If tubes T of large diameter (5 cm., 10 cm., and more) are to be excited, the strip ss' should be wide as shown, so that it may completely close the end of the organ- tube, except the mouth at cc'. Brass and pasteboard tubes in this case respond strongly, provided the distance sc has been properly set. For very wide mouths an influx pipe at P' normal to ppf is often advantageous and in this case also the apparatus may be doubled, as shown in figure 77. Here there are two rifts c and c' and strips s and s' opposite each other on the common guides gg', P being the influx pipe. The apparatus is placed flat upon the pipe or object T to be energized with c or c' regulated. A tumbler of elliptic section is interesting; when gg' passes by rotation from the major to the minor axis, the pitch rises continuously about a tone. Mouths 5 to 10 cm. 52 DISPLACEMENT INTERFEROMETRY APPLIED TO and wider than the strips 5 may have to be closed laterally, wholly or partly, except at cc', by the hands, for instance. The sounding-boxes of tuning-forks respond very nicely to the apparatus 77. If two small closed pipes (say locm. long and 3 cm. diameter) be placed at c and c', figure 77, strong beats are almost always obtained, unless the embouchures at c and at c' are quite alike. This attended to, one may get notes out of drain-pipes 5 feet long as well as out of tin cups or funnels. Though this was not the case in my apparatus, it would be an advantage to have the plane scc's' quite flat, so as to repose on the tube end of any size. If obtainable, pp' would then preferably be made of triangular tubing. 47. Reed pipes, voice. — In the endeavor to obtain waves of larger amplitude the device of figure 78 was resorted to. Here the pipe PP, 32 cm. long and 3 cm. in diameter, through which the component ray L of the interferometer passes, is provided with a tubulure about i cm. in diameter at its middle and con- nected by flexible tubing t (to be made longer or shorter at pleasure) with the reed-box R. The vibrations of the reed are thus impressed on PP and conversely. The note was coarse, like a bassoon. Without PP the reed merely wheezed. With t very short, the first experiments gave a display of enormous waves, 20 or 30 fringes high. I suspected that this could be only a direct mechanical effect of the vibrating reed on the interferometer. The reed pipe was therefore mounted on a separate scaffolding, entirely independent of the interferometer, although both were ultimately attached to the large pier of the laboratory. The result was an immediate reduction of double amplitudes to 4 or 5 fringes. On replacing the reed R by a clarionet mouth-piece, the results were similar. The waves are naturally compound wave-trains, and could only be repro- duced photographically. When blowing strongly, deeply incised fundamentals appear, decorated throughout by the overtones. On blowing more gently the fundamentals receded, and the compound wave-train defied further analysis. The pipe was now moved, as a whole, out of the range of the ray L, and again sounded. To my astonishment, strong waves were again produced, though not Q Q as much so as when the pipe was in | | position for interference . In other words , U/ these reed notes act directly on certain y g _L1 parts of the interferometer and excite the parts selected by resonance. To test this further, I made use of T 79 r the voice, singing a foot or more away from the interferometer. I was again surprised to find that at certain chest notes (b1 c"}, the interference bands broke into marked waves near a fringe in double amplitude, the effect being absent from the remainder of the scale. A clarionet played about a yard from the interferometer evoked the following response : ACOUSTICS AND GRAVITATION. 53 g a b c' d' e' f g' a' f" o strong stronger max. strong weak o weak o o 1.2 fringes Resonating pipes on the interferometer made no discernible difference. The seat of activity is probably in the iron base of the apparatus acting as a sound- ing-board. Loading it depressed the maximum to b. A totally different interferometer in a new location showed the same behavior on the same base (lathe-bed slide), with a maximum at 6. This discrepancy is an exceedingly •difficult one to eliminate, as it calls for a detection of the resonant member of the interferometer, for which reason I abandoned further work with strong reed pipes. With the diapason organ-pipes used above, there is much less danger of direct influence. This is shown, for instance, in the balance obtained with nodes of opposite sign. Moreover, I made control experiments by blowing •equipitched diapason pipes strongly in the neighborhood. There is, even here, liable to be a little response. The tendency to assume wave-form may be recognized; but it is much smaller than the pipe-note proper, and quite •absent in the overtones. Finally, the elbowed pipe, figure 79, which blows -away from the interferometer, is additional guarantee. 48. General result. — The data obtained in these experiments with a di- versity of pipes (cross-pipes, closed and open organ-pipes, tubes) when sound- ing their full note, however shrill this might be in the overtones, rarely showed a compression in excess of the equivalent of one fringe. In other words, there was a remarkable constancy in the maximum amplitude throughout. Possibly in the steam-whistle pattern (which I have not yet tried) this limit may be exceeded. However, for the usual pipe, since the intensity * = a*/X2 and a is found nearly constant, the shrill notes of overtones are largely to be referred to the decrease of wave-length. Thus in the closed pipe they would be 9, 25, etc., times louder than the primitive, for equal response. To test this further, I constructed the open pipe, elbowed in the middle, E, as shown in figure 79. The pipe is embouchured by the glass plate g and there is a corresponding plate g' beyond the elbow, to allow the component ray L of the interferometer to pass through. For the funda- mental, the node is at E, so that a little more than a quarter-wave will be seen, and a little more than a semi-node. For the octave (which like the fundamental came out strongly and with the same pitch as for the straight pipe) the nodes are at P and P', and a half wave-length is completely visible. Large, handsome achromatic fringes were installed. In the course of several settings of the embouchure, the best position for loud notes did not evoke more than a fringe breadth between trough and crest; usually about 0.8 fringe was recorded,, while somewhat more than X/4 was observed. In cor- responding settings of the embouchure for the shrillest overtone, the largest -double amplitude of the zigzag waves was about 0.8 fringe. As a closed 54 DISPLACEMENT INTERFEROMETRY. pipe, this elbowed form functioned badly. No waves were obtained for the fundamental and the shrill duodecima. Nor would they be expected. The fifth, which was not loud, gave about 0.4 fringe. Returning to the loud open pipes, whose semi-length is now 12 cm., the energy content per cubic centimeter will be somewhat larger than above. In the full fundamental there would be i o3 X 2 7 ergs /cm.3 ; but more than \ /4 was seen, and the usual values were 0.8 fringe, so that 2 X io4 ergs/cm. s (allow- ing for a slight increment from the interferometer) was probably not usually exceeded. The same value holds for the shrill overtones of 0.8 fringe. Directly compared, the amplitudes of the fundamental were the larger, agreeing with the above inference. I have not been able to carry the experiment to a conclusion, but it seems as if with the amplitude in excess of the equivalent of one fringe the harmonic motion in the given pipe becomes unstable and breaks up into the next overtone in succession. In the endeavor to prove this I constructed diapason pipes of the 2-foot octave (c') about 60 cm. or more long and 5 cm. in diameter, of brass and pasteboard. They were excited by the device, figure 75, suitably secured to the pipe. The note was strong and passed continuously (to use this phrase) into the overtone c" on increasing the strength of the blast. To my regret, however, the fundamental c' again shook the interferometer, as in the case of the reed pipes above, so that measurements could not be made. One may finally ask how large a temperature effect A* should correspond to the maximum compression Ap/p = 2Xio~2 given above. If c is the specific heat at constant volume and / Joule's equivalent, JCp&t = p&P/p = 2 X io4 whence 2X10" =2.2 42 Xio6Xo.i7Xo. 00129 The same result may be obtained from the equations of adiabatic expansion : T being the absolute temperature of freezing. This is an unexpectedly large temperature increment and would seem to be easily measurable by the bolometric-telephone method, described in the last report. It is probable that the reason there given for the negative result is correct; i. e., in the case of these rapid alternations heat fails to penetrate the bolometer wire adequately to become appreciable at the telephone. CHAPTER VI. THE VIBRATION OF THE TELEPHONE PLATE. 49. Phenomena. — Intense white light (sun, electric arc) from a collimator, figure 80, passes the half -silvers hh' and the opaque mirrors mm' and is observed in the vibration telescope oe (vibrating objective o) . The slit must be very fine and the interference fringes brilliantly beaded . The very small light mirror m is at the center of the plate p of an ordinary telephone pt. This is excited by a little induction coil, with high resistance in circuit. It is also convenient to pass the current from hand to hand through the body and vary the contact at the fingers. A second telephone should be in circuit to test the loudness of response by the ear. What one observes is a very beautiful phenomenon of apparently beating wave-trains ; and this in spite of the fact that the telephone at the ear does not suggest the slightest departure from a uniform rapping; or otherwise a steady, low-pitched note. I have endeavored to analyze the phenomenon observed, though almost infinitely varied and perplexing, in figures 81 and 82. When the telephone current is sufficiently strong, it consists essentially of detached equidistant groups of fringe-waves w, w', etc. and vertical equidistant lines of white light s, which are temporary slit-images. There may be many more waves and lines than drawn and differences in amplitude; but this is immaterial here. For simplicity, one fringe only is treated. In one extreme case waves and lines may alternate, figure 81; in the other, figure 82, coincide; and there are all inter- mediate cases. If the telephone current is weak, the lines 5 vanish, but the beating-fringe wave-trains are left. In other cases the lines may be replaced by fringe-free gaps (Cf. figure 87) in the wave-field. Both the lines 5 and the grouped waves w move with the objective; and since the objective vibrates, they also vibrate, as suggested by the arrows in the two parts of each figure. The waves at any part of the field have the appearance of being on an axis alternately stretched longitudinally with loss of amplitude, and compressed *h&P i/.. /"m> ' / 80 A AAA 1 /m! 8J VW • Id / s I /\y ? 86 /JN .... f t t t 1 f 1 s\/ Supposing these occurrences might be due to accidental looseness of mirror, I took out the plates and refastened the mirror carefully. When the fringes were again found there was no appreciable change in the phenomena. As already suggested, it seems probable that the above observations as a whole are to be referred to a buckled plate of the telephone, resulting from the relatively excessive expansion of the hard-rubber case which holds it. In figures 85 and 86 there are two positions of equilibrium of the plate, and the passage from one to the other is practically instantaneous. Possibly also when the fringes are unbioken straight bands, there are strains in the plate which resist displacement under the given forces, so that straight fringe bands result. Again, two wave-trains may happen to annihilate each other. To guard against resonance vibrations in the interferometer itself, I discon- nected the telephone on the latter, and placed the auxiliary telephone strongly excited in different parts of the interferometer and even on the optical tele- phone itself. The fringe bands remained even and non-sinuous throughout, so that discrepancies of this kind are absent. Evidence with the same purport is the frequent occurrence of unbroken fringe bands in case of a sounding tele- phone, throughout the experiments. The differences between the effects of induction periods T — O.I second to r = o.oi second was chiefly this, that the former allowed each shock- wave more time to subside and there are fewer groups in the field. The occurrence of gaps ACOUSTICS AND GRAVITATION. 61 in the wave band from oscillations of the plate around a horizontal axis are often striking in the former case. Thus a succession of strong waves subsiding to fringe bands passing through a gap and then in turn to a line and a strong wave, are frequent in intense vibrations. The gaps are often in multiple with pulses between, as in figure 87. Waves seem to tear themselves apart longi- tudinally, leaving a gap between, which in turn closes by reversed motion. There is a correlative cause for the possible occurrence of gaps which must not be lost sight of. Since the fringe vibration up and down is very fast in frequency compared with the vibration of the objective, it will be chiefly at the maxima and minima, where the up-and-down motion is instantaneously zero, that they will be most easily seen in the vibration telescope, even when the mirror of the telephone is displaced translationally. Unless the (to- and-fro) amplitude of the objective is very long, the fringes will overlap in their up-and-down motion, leaving gaps in these places in the field. It is these beaded forms and gaps which which may often be transformed into waves, by making the objective travel more swiftly, i. e., to fall, from a larger amplitude. When, however, gaps occur in alternation with waves, as suggested in figure 87, it does not seem probable that they can be produced otherwise than by plate rotation, which slides the slit-images longitudinally along each other till the fringes actually vanish. 54. Synchronism. Iron-screw core. — To obtain further insight into these phenomena, it is necessary to make the period of the vibrating objective and of the interrupter of the telephone current identical and incidentally to devise means for increasing the amplitude of the objective as far as possible. For this purpose the apparatus, figure 88, was designed. Here M is the usual electromagnet held in a sleeve H, with a stem a, fitting the clutch on a vertical standard (not shown), so that M may be raised or lowered, placed nearer to or further from the steel spring s. The latter is shaped like a hacksaw blade, firmly clamped in a small vise v below. This, also, is capable of being raised or low- ered on the specified standard. The spring 5 carries the light lens L, which is the objective of the telescope. L need not be achromatic (in which case it would be too heavy), for the fringes appear almost equally well with a simple lens, if the focal length is suitable to the telescope. The standard rod to which the vise v and the electro- magnet M are adjustably clamped must be rigidly attached to the tube of the telescope, which has the usual biaxial mounting on a tripod. Hence the telescope may be pointed in any direction while the lens L is in uninterrupted vibration. This is a great convenience in viewing different parts of the fringe band, drawn out by the quivering objective. 62 DISPLACEMENT INTERFEROMETRY APPLIED TO The preliminary adjustment consists in raising or lowering the vise v on s, until the spring s vibrates naturally in step with the interrupter . When this is sufficiently nearly the case, the soft-iron core 55 of the electromagnet M is first pushed towards the springs and then gradually withdrawn from it, until the amplitude of L (using it in connection with the remainder of the telescope) is a maximum. For this purpose, the core 55 is a fine-threaded screw, snugly fitting a fixed socket within M. Thus the fine adjustment for a maximum amplitude of L or maximum width of fringe band may be made to a nicety up to the point at which the vibration becomes unstable and s drops back to its position of equilibrium. Amplitudes of i cm. at L are easily obtained, unless the spring s is too short. Care must be taken to prevent an excess of spring, 5', from interfering with the vibration of s. 55. Shattered fringes. — When the fringe- waves are observed with the syn- chronized electromagnetic apparatus just described, their appearance in general may be indicated as in figure 89. There are regions x, xf at the two ends, 55', of the fringe band, consisting of shattered waves (as I may call them) and the impression produced on the observer is that of fringe rays radiating from a center. Frequently, x' is absent and the phenomenon is confined to one side. Following x are a series of wave-lines w, of decreasing amplitude and increasing length, the form of which, however, is not sinusoidal, but rather like a succession of cycloids (either erect or inverted) , or very much like a succession of festoons. They are nearly stationary. When the telephone current is weak (20,000 ohms in circuit), the region x all but vanishes and the waves begin at the edge s of the fringe band and extend toward the middle m. When the current is increased, x rapidly increases and the waves w are driven toward the middle and ultimately (if x' is absent) across the field to the opposite edge s'. If X is the observed wave-length at the center of the field T, and a the period and amplitude of the vibration telescope, the period of the shock- waves is easily seen to be t = \T /ma. The ocular micrometer showed X = 5 and a= 100 scale-parts, while T = o.i second, so that * = io~4X8 seconds. Owing to the quiver, \ is hard to determine; but the waves affecting the interferometer in such marked degree occur with a frequency of the order of r,oooper second ACOUSTICS AND GRAVITATION. 63 , < when the induction period is but o. i second. The waves observed are imme- diately referable to the natural vibration of the plate of the telephone. They follow a magnetic shock, as the Hertzian waves follow a spark. It is now necessary to endeavor to analyze the region x of shattered waves by aid of the screw-iron core described in the preceding paragraph. For this purpose the intemtptor also should run uniformly. If it dips more or less deeply into the mercury, the effect on % is the same as removing or adding resistance to the telephone circuit. By drawing out the fringe band to a limit, one gets a great variety of results ; but their appearance on the average is as shown in figure 90. The phenomenon usually begins with an undisturbed fringe a, meaning the objective begins on its path before the inductive shock arrives at the telephone. After this a series of about 6 discontinuous wave-forms (b to c) of rapidly decreasing amplitude appear. Only the inclined and nearly straight parts (6, b', etc.) of the fringe band are visible, so that these waves drop off sheer and are akin to the faulted waves described above (figs. 85 and 86) . The slope of b is accidentally sometimes upward and at other times downward, throughout ; but the slope of b and b' is always symmetrical. At cc' these linear groups begin to coalesce and coalescence is about complete at d, though the arch is often seen to open. Beyond d the waves are a suc- cession of smooth flat cycloids with the elevation rapidly dying out toward the center of the field. If the slope of b is downward, then waves are festooned. The endeavor to obtain further information by working through a capacity failed. The whole display is seen through 0.05 microfarad about as well as if there were no interruption. With very small capacities, however, the tele- phone still sounds when the fringe bands are quite straight. 56. Large frequencies. Musical notes. — In all the above cases the impulses were distinct and the shock- waves had time to subside in the interval between the impulses. This will no longer be the case when the frequency is much increased. There must then be a coalescence into a single wave phenomenon. The attempts to trace this through with the motor contact breaker used above did not succeed. The period of the objective being 7 = 0.34 second, the note g' in the telephone showed uneven waves and suggestion of cross-hatching when the T amplitude was small. At d" there were lines and pulses; but very little was to be inferred as a whole from the compound wave-trains obtained. This interrupter was therefore replaced by a small magneto inductor, which, as a rule, gave compound but otherwise unbroken wave-trains from c" to about clv. They were about one fringe in maximum amplitude at low speeds in the absence of marked resistance in the telephone circuit; but above c'", 10,000 ohms had to be inserted to reduce the waves from broken (gaps, fig. 87) to continuous forms. At certain very definite frequencies, however, marked disturbances showed themselves. Thus at d" there was resonance and the field became filled with 64 DISPLACEMENT INTERFEROMETRY. stroboscopic .vertical lines or slit-images, very definite but faint. At a" the lines again appeared, this time very bright and present throughout the field. The same occurred at c'", the note being strident. After this the wave-trains became regular again, and were traced almost as far as clv. No doubt these are indications of resonance in the telephone plate, the vibra- tions being of the type, figure 84, and implying a rotation of mirror. In fact, at a" there were fine lines between the main lines, indicating contemporaneous presence of very high rotational overtones. Moreover, in different adjust- ments the frequency evoking resonance was not quite the same. CHAPTER VII. EXPERIMENTS MADE IN THE ENDEAVOR TO PLACE THE REVOLVING MIRROR ON THE INTERFEROMETER. 57. Apparatus. Revolving mirror normal. — The ease with which the dis- placement interferometer lends itself for the investigation of vibrating systems induced me to make an attempt to combine the revolving mirror with the interferometer, though this is obviously far more difficult, if at all feasible. Figure 91 shows the first general plan, L being a beam of white light, N, N', M, M' the mirrors of the quadratic interferometer, mm an auxiliary mirror in the normal position. The telescope is at T. With the usual adjustment, it is then easy to find the fringes, and any slight rotation of the mirror mm will be registered by their displacement. If the mirror mm is at an angle, as at m'm' in the figure, and the rays are reflected normally from a second auxiliary mirror nn, the fringes must again appear if the adjustment is accurate. Moreover, any slight rotation of m'm' will in tuin be registered by the displacement of fringes. If the mirror mm rotates, fringes will, therefore, appear for these two posi- tions of the mirror mm. They will naturally flash through the field of vision, but would be open to detection by the aid of a vibration or revolving telescope, such as I used in case of vibrating systems. In the endeavor to carry out this difficult experiment, a powerful beam of sunlight is to be introduced at L. To obtain this, I made use of a doublet of two lenses, each about 10 cm. in diameter, respectively convex and concave, and of a focal power of less than 2 diopters. By placing these at different distances apart, focal distances of 10 to 20 meters were obtainable, admitting of an easy guidance of the long beam, when the mirrors mm and nn were 7 to 10 meters apart. There was, consequently, always an overabundance of light available. The difficulty with the experiment lies in the smallness of the image obtained when the mirror nn is far off, as it must be, when measurements bearing on the velocity of light are contemplated. Unless a special lens train is put in the beam between mm and nn, this difficulty, so far as I see, is insuperable, inas- much as the mirror mm, at least, can not be very large. An additional dif- ficulty wa? encountered in the difference of the intensities of the light coming from mm when normal, and from nn. It would, for this reason, hardly be practicable to compare the former fringes with the latter as to relative dis- placement, the plan I had in view. But the normal reflection from mm may be blotted out by screens mlt m\, placed as shewn in the figure, since this part of the mirror is not used in connection with nn. Moreover, the displacement of fringes due to the time lost in passage of light from mm to nn and return may be compensated by the micrometric rotation of nn around a vertical axis, as 65 66 DISPLACEMENT INTERFEROMETRY APPLIED TO well as by the micrometric shifting of the interferometer mirrors Nr or Mr. There are thus two independent methods, apart from the ocular reading of displaced fringes. The fringes from mm and nn may be made parallel by rotating nn on a horizontal axis. If they are visible for the same micrometer position, the glass paths must be rigorously the same in the two cases, so that optic plate is needed. To make the fringes of the same size a vertical axis at nn suffices. it r xt/J' \ _. ^ ' m, /«*' , \ 6^ \ \w' r 91 t/Tv m P1" 73t -n, 58. Lens train. — This would be peculiarly difficult to apply in the present case. As a large field of view is desirable, a collimator will be needed. If F is its external focus, supposed to be on m (fig. 92 vertical plane, 93 plan), the lenses L can not be of very different focal power from the collimator, if too much light is not to be lost at the edges. If fl and fa are the focal distances, the total distances apart of the mirror m and n would not much exceed 2^-f 2 fa. It would probably not be advantageous to make this exceed 5 meters and lenses of large diameter would be needed. The main difficulty, however, is the introduction of one of these trains into each of the component beams, in particular as it implies the insertion of uncertain glass paths for the rays. In fact, though I obtained very good slit-images from each beam, the result for two beams conjointly was inadequate. 59. Estimate. — It is finally of interest to ascertain the sensitiveness of an arrangement like figure 91. If the distance 5 of mm from nn is passed in A* seconds, and v is the velocity of light, 28 = v A* and A*/ 7= A0/27r,if the angu- lar displacement A0 corresponds to A* and T is the period of the mirror. Fur- thermore, on the interferometer A0 = A/V cos t/b, where b is the breadth of the ray parallelogram, i the angle of incidence at the mirrors and AJV the microm- eter displacement (normal to mirror) which corresponds to M. Hence cos If 5 =500 cm., 6 = 10 cm., T = o.i second, AN" = 10 4cm., ioJX6.3Xio u= — — =9Xio9cm. T r>~ > y Tr\~ ' y r> T = 45°, nearly. Since AAf = io"4 cm. corresponds to about 2.5 fringes, v = 2.3 Xio10 cm. /sec. per fringe. Now both 5 and 6 may be increased and T decreased, so that the apparatus could easily be made more sensitive; nevertheless, the present experiment is itself so extremely difficult and unpromising that I did not further persevere with it. ACOUSTICS AND GRAVITATION. 67 If the velocity is expressed in miles and seconds, or if v= 1.87X10' and 2 A N cos i = X, G may be computed in miles per fringe. This gives G = 0.0045 miles or 48 feet per fringe as the sensitiveness for light- waves for the given 6= 10 cm. The experimental datum below is well within this. 60. The inclined revolving mirror and interferometer. — It is obvious that a method totally different from the preceding will have to be resorted to. Lens systems in the light-paths are treacherous complications, to be avoided. Parallel rays, preferably sunlight, should be used directly, if possible without the intervention of glass condensers of any kind, unless it be in front of the interferometer before rays separate. The solar image in a telescope of mod- erately high magnifying power (15 to 2 5) subtends a sufficiently large angle and is intense enough to be used directly. If a large field is needed the image may be enlarged by putting it out of focus, and interferences of almost equal distinctness obtained, provided the pencils of the two washed images are quite coincident. This method of using images out of focus is here of con- siderable importance; for fringe fields may be obtained in this way from images which (because of the long distance traversed) in sharp focus would be mere points. Finally, if the revolving mirror is so inclined as to move virtually f " * h^- t ^'». 95 on the surface of a cone instead of a cylinder, its advantages are retained, while rays are returned, only in the position for measurement. Figures 94 (plan) and 95 (elevation) will make these points clearer. Here L is a bundle of white parallel rays, preferably direct sunlight, from a sufficiently wide heliostat mirror, silvered in front. M,M',N,Nf are the mirrors of the quadratic interferometer, all but M' being half -silvered and M and A^ being equally thick. All mirrors are adjustable on three leveling-s crews, as usual, and M' is on a micrometer with the screw in the normal direction s. C and C' are thick glass-plate compensators, capable of rotating on a horizontal and a vertical axis. The rays from L thus fall in two component beams on the revolving mirror R, inclined at 45° to the horizontal. The prism carrying R is mounted on the table tt, about a foot in diameter, capable of rotating with .great speed around the axis A, which in practice is the shaft of an electromotor. After leaving the mirror R the two beams are reflected by the adjustable stationary mirror m, also at 45° to the horizontal, along the long path (10 meters) 5 Sf to the distant mirror m' normal to the rays. They are thus re- turned along the paths SmRN'T and S'mRNT and enter the telescope at T. When R is stationary and in the position of the figure, and when T magnifies sufficiently, two coincident images of the sun, quite intense and filling nearly lialf the field, may be obtained, capable of interference. 68 DISPLACEMENT INTERFEROMETRY APPLIED TO 61. The revolving telescope objective. — When R, in figures 94 and 9$, ro- tates, mere flashes of light would be seen at T if stationary, and the fringes would be invisible as they move broadsides on. Hence to study these phenom- ena I first availed myself of what is virtually a rotating telescope, shown in figure 96. H is the cylindrical body of a small electromotor, clamped at the foot F, and B is the horizontal shaft of the armature. To this axle the disk DD, about a foot in diameter, is attached in front and well balanced, capable, therefore, of rapid rotation. DD carries the objective P of the telescope, embedded near its outer edge, P' being a suitable counterpoise at the same distance from the axis B. The eyepiece E and tube of the telescope are carried by the ring GG, capable of rotating about the rear end of the cylindrical bcdy of the motor and to be put in any desirable position by a set-screw, as at c. E is thus stationary and the telescope is complete when E and P are temporarily coaxial. The fringes, when found in the interferometer, figures 94 and 95, without compensator, may be at any angle to the horizontal, and it is of ten troublesome to change their inclination. Hence the eyepiece E is to be set so that the 0$ R 3 4 V 6 6/ xV e A. 6 ^ V X 97 / / tangential motion of DD at E is parallel to the direction of the fringes. Since the fringes move by virtue of the rotation of the mirror R normally to their directions, they will be seen moving in the direction of the resultant. It is necessary, therefoie, that the fringes be reduced to a length not exceeding their breadth, practically to points. This may be done, for instance, by a slot in a distant screen at the heliostat, the slot being at the proper angle to the horizontal. These relations are shown in figure 97. Let a i, 2, 3 ... .a' be successive positions of the achromatic fringe, corresponding to the positions c, i, 2, 3 ... ,c' ', of the normal to the revolving mirror; let b, i, 2, 3 .... bf be the corresponding positions of the same fringe resulting from the rotation of the telescope objective. Then the fringe is seen in the line EE'. Hence the retardation due to the passage of light twice over the distances S, S' will produce a displacement of the line EE, observed in the telescope along the direction dd', normal to it. It is in this direction, therefore, that an ocular micrometer would have to be placed. ACOUSTICS AND GRAVITATION. 69 It would furthermore be necessary, to insure synchronism, or a definite ratio of frequencies in the two revolving disks tt and DD, by gearing or otherwise. Moreover, the telescope must always be in the position of observation when the revolving mirror flashes. This is a difficult thing to do with two disks. It might seem (erroneously, however) as if a single disk ti, figure 94, could be used for both purposes, P being the telescopic objective and P' a counterpoise. In such a case T is to be replaced by the auxiliary mirror n, figure 94, which reflects light to a similar mirror under the objective P, the reading being made in an eyepiece above P. The eyepiece is so placed that the telescope is com- pleted when the revolving mirror flashes. The possibilities of this plan will be considered presently. Obviously, the fringes must move in a direction differing from that of the objective. Hence they must be easily controllable as to inclination and size. Compensators C and C1 may be set symmetrically (vertical and horizontal axes) by hand for this purpose. The same result is secured by putting M and N on vertical axes (with tangent screws) and Mt and N' on horizontal axes (plane-dot-slot device) . 62. Control fringes. — As the revolving mirror R starts from rest to attain its maximum speed, the line EE' in figure 97 will be more and more deflected from the original position bb to some position as indicated (EE'). Simul- taneously, the retardation due to the time consumed by light in passing twice over the distance S will move EE' in the direction dd'. As it is this displace- ment which must be measured, the accomplishment would be difficult. One may, however, establish a set of control fringes independent of the distance 5. To do this the half-silver plate hh', figure 95, is inserted between the mirrors m and R and adjusted until the two slit-images from the distant m' and the near hh' coincide. Thus the fringes from hh' will not depend for position on the line dd' on the speed of R and may be used to define the zero of displacement. Obviously hh' must be good optic plate. If this is not the case, not only will the pairs of slit-images fail to coincide at the same time, but the fringes will differ in size and inclination and appear for different positions of the revolving mirror R, or of the micrometer s, at M1 '• 63. Sensitiveness. — At this stage of progress a large number of experiments were made, using the equipment, figures 94, 95, 96. Providing the revolving mirror R (kept stationary) with a divided arc-and-tangent screw, a number of direct tests bearing on the sensitiveness were carried out, all referring to the velocity of light to fix the ideas. Thus the distance passed in t seconds is d = ct, while in the revolving mirror of frequency n, if 6 is the corresponding angle, t = 6/2irn, whence d = c9/2irn. But if 5 is the corresponding number of ocular scale-parts passed over by the fringes and k is the constant, d*=ks; and finally, if / is the number of fringes (/) to the scale-part (5), f=ls. Hence 6 = kf/l, and if »=io, d = 70 DISPLACEMENT INTERFEROMETRY APPLIED TO The following values were obtained : / s I io4 e s io4k° d/f miles d//feet i i i.o 75 25 3.0 .016 82 i 1.4 .75 62 25 2.5 17 90 i 2.0 .5 37 25 1.4 15 77 i .7 1.5 100 25 4.0 14 74 The data are rough, as the tangent screw micrometer was not adapted for such- small angles. Thus at 10 rotations per second the distance d traversed by light per fringe is about 80 feet, or for the larger fringes 40 feet, per ocular scale- part of o.i mm. It would be possible to read within this. Since the distance 5 is passed twice, d = 25; so that 5 = 20 feet per scale-part per 10 rotations per second, may be taken as an experimental estimate. It is of the same order, 30 feet per fringe, as the theoretical value above. 64. Experiments with the rotating telescope. Fringes on washed images. — Having found the fringes for a short distance S, the revolving mirror was left at rest and the revolving telescope, figure 96, tested for adequacy of light. With the collimator and a long- focus condensing lens to illuminate the slit, the fringes were seen distinctly and manipulated; but they appeared too faint for practical purposes. Removing the collimator and lens completely and using sunlight directly, the fringes (after adjustment) were displayed strongly on the intense images of the heliostat mirror. In the revolving telescope they came out beautifully, easily recording small micrometer displacements, or slight rotations of the revolving mirror. At long distance 5 another difficulty arose* as the heliostat image (except for mirrors exceptionally large) was apt to be point-like. The light, however, was still abundant. Hence the device was adopted of putting the ocular out of focus. A large white field or glare is thus obtained, quite satisfactory for the display of fringes. A variety of important experiments was now made, relating to fringes obtained on images out of focus ; for this glare may easily be enlarged to fill the whole field by drawing out the ocular, or the reverse. It is necessary, of course, that the two component washed images be in coincidence; and this without a guide as to the identification of the points of their areas is a difficult adjustment to make. But if precautions are taken that the two pencils entering the objective of the telescope are quite coincident, the fringes will be large and they will not vanish in any position of the ocular. The reason for this is clear, since parallel pencils, partially coincident on entering the telescope, will coincide at the focus of the objective only. Thus the fringes will soon vanish for other positions of the ocular than the one corresponding to this focus. One difficulty occurs here which must be guarded against. An ocular drawn in or out from the principal focus implies an object at a finite distance. Such ACOUSTICS AND GRAVITATION. 71 an object is virtually identified with the mirror M' and is displaced by its micrometer-screw. The rays passing through the optical center of the objective of the telescope thus become inclined to each other in proportion as they are nearer, and the inclination angle vanishes only for objects at infinity. Hence one component image will pass across the other slowly when the micrometer moves, and this is attended with a change of fringes. Beautiful and intense fringes were obtained by the present method, dis- played in the glare of an ocular out of focus and observed with a stationary tel- escope. They remained equally serviceable when viewed with the revolving telescope, where advantages in sharpness are often obtained by slightly reset- ting the ocular. Displacements produced by moving the micrometer or the revolving mirror were also examined with success. The effect was strengthened by using a long-focus condenser in front cf the interferometei . If the device of obtaining fringes on images out of focus is rejected, the full solar disk may also be used. This, however, requires large mirrors of strictly optic plate in the interferometer and its accessories. Taking the solar diameter at 0.53° and remembering that the distance 5 is doubled, mirrors 20 cm. square would be needed for each 10 meters of 5, and the interferometer should be placed close to the heliostat. 65. The combined revolving mirror and telescope. — This apparatus, which has already been partially described in connection with figures 94 and 95, is shown in figures 98 and 99, plan and elevation. Sunlight enters through the long-focus doublet L, just in front of M, the focus being at the telescope, 10 or more meters distant. The component beams leaving the mirror Nf are received by the mirrors n and n' in succession. The latter, being below the disk ti' of the revolving mirror R, reflects them up vertically thi ough the ob- jective P embedded in it! They are seen when P is in position, thiough the eye-piece E. P' is a counterpoise, relative to the axis A of the electromotor. The mirror n is at the intersection of a diameter of it' and the rays NN'. When the telescope is completed, the eye will see a composition of the displacements due to the revolving mirror R and that of the objective P. The latter moves tangentially to ti' and always in the same sense. The dis- placement resulting from a turn of mirror may be taken as parallel to the face of the mirror underneath it' and hence will be normal to the line nPPr (diameter prolonged), or in other cases normal to the tangents ns' and ns" , supposing the objective embedded at s' or s". It follows, therefore, that at P the displacements are opposite in direction, and if they are equal there will be no displacement. With the objective at P' the displacements are in the same sense, and the resultant equal to their sum. At r's' and r"s", finally, the displacements are more or less at right angles and the resultant a diagonal. Quantitatively, if /is the focal length of the telescope, a the distance of the objective from the axis A, since the angle of reflection only is in question, 72 DISPLACEMENT INTERFEROMETRY APPLIED TO the displacement will be/ojAJ due to rotation of mirror and waA* due to shift of the objective. Hence, if f = a, the image at P will be stationary. At Pf it would move with the speed w(a+/). These conditions are easily tested by putting a fixed, vertical meshwork of wire gauze (sieve) between the doublet D and the mirror M of figure 98 and focussing the eye-piece E upon it. If a =/ the meshwork will remain stationary and clear, however fast the revolving mirror rotates, for any directions of the wires of the meshwork. In the same way a fine slit will remain in position dur- ing rotation. Moving the wire gauze between M and D, the position of maxi- mum definition is easily found. With higher magnification, finer diagonal lines often appear, superimposed on the image. In case of the objective positions S'r' and S"r", the wires of the grating are to be placed at 45° to the horizontal. In such a case one or the other of the parallel sets comes out strongly on rotation, the resultant displacement being along their direction. In these experiments one must distinguish between the mere displacement of the spot of light which depends largely on the effective area of the mirrors, and the displacement of an actual image like that of the grating, the latter being here alone in question. The position S'"r'" is also interesting, inasmuch as here the speed of the revolving mirror is virtually doubled; but no experiments were made with it. Obviously the mirrors N and N' must be screened off to admit of the passage of the parallel pencils at the same time only. Otherwise there will be three reflec- tions in succession, one from N, one from N and N', and one from N'- The first and last are single pencils and are not desirable. Removing the objective from it' and using the revolving mirror in connection with the apparatus, figure 96, not synchronized, the result was like a display of occasional shooting stars, moving in parallel in all parts of the field, as would be supposed; but the light conditions were not unsatisfactory. The combined plan just sketched is interesting for experimental purposes; but it does not meet the conditions of the problem, as may be indicated as follows: In the first place, it is desirable that the fringes be vertical and to move with the rotating mirror, therefore, in the horizontal direction. This in figure 98 implies fringes moving in the direction r. Hence the positions sr and s'"r'" in the figure are not available; for the fringes would overlap in their motion and vanish. In the position s'r' and s"r", however, the fringes moving ACOUSTICS AND GRAVITATION. 73 in the direction r are drawn out in the direction s, nearly normal to r. Thus if the fringes are very short, they will be displayed as fringe bands, as heretofore. So far, therefore, the conditions are met. If, however, we return to figure 97, it appears that if the fringe ai has been carried by the rotation of mirror to the position 02, the objective will have simultaneously carried it to 62 etc. In other words, the fringe i will always remain on the line EE', regardless of the retardation of light; i. e., the line EE' will not move in the directions dd' as in the case of the independent revolving mirror and telescope. The effect of the revolving mirror has simply been compensated. Nevertheless, this method of approaching the problem is well worth while, since it affords opportunities for correcting the details of the experiment. Strong fringes were produced without much difficulty and viewed through the telescope EP in the s'r' position. But in spite of many trials, it has thus far been impossible to secure a sufficiently rigid and independent mounting of the disk tt' and revolving mirror to hold the fringes in the field during rotations of 5 to 10 turns per second. CHAPTER VIII. ON THE TORSIONAL MEASUREMENT OF VARIATIONS OF THE ACCEL- ERATION OF GRAVITY, BY INTERFERENCES. 66. Apparatus. — These experiments were undertaken at the suggestion of Dr. R. S. Woodward, in order to find how far interferometer methods might contribute to the measurement of g, under circumstances in which the pendu- lum is inapplicable. The apparatus, as a whole, is a horizontal torsion balance with the deflection readable in terms of the displacement of the achromatic interference fringes. The method of registry developed would be applicable in case of an ordinary chemical balance. In figure 100, M, M, N, N' are the mirrors of the interferometer (Nf being on a micrometer P, normal to its face), the white light coming from a collimator at L and observed by the telescope at T. CMM'C' is the balance beam, sup- ported by the tense horizontal torsion wire w, normal to the beam, mm' being the auxiliary mirrors of the interferometer. This beam is of steel, U-shaped in section, figure 101, and the wire ww' passing through two notches at the top edges of the beam, normally to its length, is looped through a hole in the bottom. One end of ww' is wound around the threads of a screw, so that ww' may be kept tense at pleasure; the other end is fastened to a torsion-head, the index of which is registered as to position by the graduated circular scale 55, coaxial with the wire. The ends of the beam mm' are filed flat and bent L-shaped and at their ends carry two vertical needle-points, as shown in figure 100. On these rest the light caps c, c', from which depend the light disks v, v', like the pans of an ordinary balance. To give the system the necessary damping, the disks v, v' are loosely inclosed by the cylindrical capsules or dash pots d, dr, free from contact. The whole balancing system is inclosed by the case CCf, the top face of which, g, is plate glass. Vertical slots 5 in the long sides of the case serve for the admis- sion of the torsion-wire w. These are afterwards nearly covered by glass plates. The ends of CC are perforated with wide holes, also to be closed by glass plates eef. It is through these windows that the weight to be measured is introduced, by placing it either on the vane v or v'. To do this smoothly the weight is of wire bent along the edges of a tetrahedron. It may thus always be placed or picked up and removed with the aid of a small rod or hook. As this weight is constant, its equivalent torsion may be registered (fig. 102) by two micrometer tangent screws /, t', on the graduated circle 5, which serve as stops to the index a of the torsion-head. Smaller changes of weight may, therefore, be registered either by the micrometers t and t', which modify the torsion, or at N', which displaces the fringes. The whole apparatus was erected on a heavy cast-iron base about a foot in diameter. The collimator must be attached to this if a line across the slit is to 74 ACOUSTICS AND GRAVITATION. 75 be the index in measuring the fringe displacement. If the telescope with cross-hairs is so to be used, this must also be rigidly attached to it. I used the former method in preference. After the stops t and t' (fig. 102) to the index a have been fixed, the measure- ment consists in moving the micrometer P at N', until the fringes coincide with the guide-wire across the slit -image. This wire should be very fine. If the micrometer-screws at t or t' are moved, the images of the cross-wire and fringes both move, preferably in opposite directions ; but the fringes move several- hundred times faster, so that the slit-image is relatively stationary, or can just be seen to move. In the first experiments a phosphor-bronze wire 0.025 cm- in diameter and about 35 cm. long was inserted. This was adapted for small weights of the order of a few centigrams. The double deflections here amounted to about 1.7° per milligram and measured about 5 mm. on the scale 5. As a rule, how- ever, thicker wire and larger weights will be preferable, contributing to a more robust apparatus. As to dimensions of the apparatus constructed, I may add that the beam cmm'c' was 22 cm. long, the disks v about 5 cm. in diameter, and the distance cv about the same. The breadth b of the ray parallelogram (fig. 100) was 10 cm. The apparatus stood about 50 cm. high and should have been covered with a case to obviate air-currents, in addition to the case CC. The wire is to be kept free from twist, except when used. The degree to which the twist, and therefore the sensitiveness, may be increased without an error from viscosity must be left to trial. In the earlier apparatus the balance was rigidly attached to the base A. In the later forms with thicker wires of steel, the balance and appurtenances were mounted on an independent framework of iron, which could then be slid underneath the interferometer, as shown in figure 100. This makes it much easier to insert new wires w, and adjust the mirrors mm' to approximate paral- lelism, by aid of a beam of sunlight or the like. In the above apparatus caps on pivots were chosen at the scale-pans, merely for convenience of construction. It is obvious that knife-edges and plates would, in general be preferable, by reason of greater stability. With the air damping above the apparatus is practically aperiodic for thin wires (0.02 cm.) and the beam takes its place at once. It is still sufficiently 76 DISPLACEMENT INTERFEROMETRY APPLIED TO so for thicker wires (0.05 cm.). The work may, therefore, be done rapidly. Oil damping, even if not otherwise objectionable, was found to be too slow, as a rule. Thermal discrepancies (air convection) decrease when the wires are thicker; but the disturbances from viscosity increase. The frame carrying the wire and the torsion-heads must be quite rigid. If the torsion-head works stiffly, the twist of the framework is always percep- tible. This must be scrupulously guarded against. 67. Measurements. — With the original phosphor-bronze wire, a trial showed a double deflection at 5 of 1.7° per milligram. Half of this is available at the mirrors m, mr. The equation of the interferometer may be written b A0 = AN cos i = X/2, b being the breadth of the ray parallelogram, Ad the rotation at the mirrors m, m', corresponding to the displacement of micrometer AN when * is the angle of incidence (45°) and X the wave-length. Thus for the double deflection per milligram AN = bA9/cos t = ioX(i. 7/2) Xo.oi75/o.7i = 0.21 cm. As AN = io~4/2 cm. can be registered on the micrometer, io~4X2.4 mg. may be estimated. Thus, if a weight of 25 mg. is used, the sensitiveness would be within io~6. Per fringe the conditions are somewhat better. For here A0 = X/2&=io~5X6/2o = 3Xio-6 radians and this corresponds to 3 X io~6/o.85 Xo.oi75 = 2 X io~4 mg., while a fraction of a fringe is still determinable. To make this fringe sensitiveness available, the tangent screw-micrometer must be used, as the achromatic fringes serve merely as an index. Here A0 = X/2& at the mirrors corresponds to 2 A0 at the end of the wire. If AS be the displacement of the micrometer at the tangent screw and R= 18.2 cm. the radius of the graduated circle 5, A5 = ^X/&=i8.2X6Xio-5/io=i.iXio-4cm. per fringe. Fractions of a fringe can therefore here be recorded. If t is the modulus of torsion and m the mass weighed at the end of the lever- arm, b'= 1 1 cm., mg b' = tAd, and therefore t cos i A , , tn\ 2mbb' for a passage of « fringes. In case of the phosphor-bronze wire of diameter 0.025 crn-> the values of / in terms of milligrams would thus be *= 10X981 Xn/8.5°Xo.oi75 = 7.2X10* Hence if Ag corresponds to one fringe, 7.2Xio5X6Xio-5 Ag= — = 2X10- 2X10X10X11 or if a mass of 20 mg. is used ACOUSTICS AND GRAVITATION. 77 Much larger weights than this are available without exceptional risk from the viscosity of the wire. Moreover, if the mass weighed increases as the torsion coefficient of the wire, the sensitiveness is not changed, again with a reservation as to viscosity. The experiments with the phosphor-bronze wire were carried out both in terms of AN at the mirror micrometer and of AS at the tangential micrometer. There was no difficulty in finding or controlling the fringes, so far as the apparatus was concerned, either at the zero position or at the two elonga- tions. They appear whenever the guide-wire across the slit is sharply in coincidence with the intersections of cross-wires in the telescope, so that telescope and collimator are to be rigidly fastened, although only the collimator cross-wire (fringes coinciding with its image) determines the measurement. To this extent the damping seemed to be sufficient. Fringes large and small were tested; but in a steam-heated room the fringes were always in slow and deliberate motion over half the length of the slit-image, irregularly to and fro. Possibly this might have been avoided by placing a second case over the apparatus as a whole, to obviate air-currents and unequal distributions of temperature. This however, would have been inconvenient, particularly so as a modification apparatus with a more rigid wire and larger weights conduces to further practical advantages. The next experiments were made with a steel wire (music wire) 0.02 2 4 cm. in diameter and very hard drawn. The modification already referred to, of an independent scaffolding for the wire, its end supports (torsion- head, etc.), the balance beam and dash-pots, was now introduced. Again the operations for measurement proceeded very satisfactorily ; but the fringes were still in continual motion, though perhaps not so much so as in the preceding case. The weight used was just short of 0.05 gram, the double deflection about 68.7°. In some 12 alternations made between the stops on 5, the fringes, though moving, always reappeared in place in the field. If at the scale 5, 89 = \/b and A0 = 68.7°Xo.oi75, dg_8d_ 6Xio~5 _6 g " A0~ 10X68.7X0.0175" per fringe passing the cross-wire of the slit-image. It seems a pity, therefore, that this sensitiveness could not be utilized. 68. Thicker wire. — As the observations with the above wire were useless, because of convective temperature discrepancies, I inserted a thicker drawn- steel wire 0.05 cm. in diameter. The forces being thus 20 times larger, the apparatus shewed much greater freedom from the annoyances specified. The behavior was far steadier, though the damping coefficient had proportionately decreased. It was not, however, even now possible to rely on a single fringe in a heated room; but in the summer time this would be the case. Tempera- ture also modifies the elastic constants seriously. On the other hand, the effect of viscosity was now much more obvious, there being a proportionately greater twist of the individual fibers of the wire. 78 DISPLACEMENT INTERFEROMETRY APPLIED TO A weight of the order of i gram was selected and the double twist produced by this was A0= 138.2° — 88. 2° = 50.0°. The weight was passed from panto pan without difficulty and the fringes set to coincide with the shadow of the cross-wire on the slit. Great care had to be taken to avoid twisting the frame of the carrying wire and torsion heads. If the beam is of steel, fluctuations of the magnetic field are also a menace. The data of the interferometer being identical with the above, we may write dg_ dd_ AN cosi _ _io~4Xo.7i g A0 = i.6Xio-6 \e &A0/2 10X25X0.0175 and as AAT may be read within io~4, the sensitiveness is about io~6, the gram counterpoise in question presupposed. The error to be apprehended from viscosity may be obtained from the following observations (table 2 , series 0) of the yield of the wire with a gram excess on the scale pan : TABLE 2. — Viscous deformation of hard-drawn steel wire. 0.05 cm. in diameter. Total Same wire tempered blue by Same wire (blue). Second length 35 cm., double electrical twist. deflection 50°, counterpoise current. First twist. i gram. Series 0. Series i. Series 2. Time, ANXIO4 (dg/g) Time, A-ZVXio4 (dg/g) Time, A./VXI04 (dg/g) min. cm. Xio5 min. cm. Xio6 min. cm. Xio5 o o 0 o o o 3 21 33 i 5 8 4 21 33 10 43 69 7 16 26 ii 36 58 17 61 98 21 26 42 66 87 139 30 78 125 74 49 78 JI3 115 73 113 in 59 154 133 163 156 176 79 219 152 196 173 259 87 284 165 276 197 1,200 93 340 1 68 359 217 *i,400 3H *2J88o 493 * Room colder. The observations marked [*] were made in a very cold room and give evidence of increased rigidity. The table shows that if measurement is made within 3 minutes after twisting, the error from viscosity, if ignored, would not exceed dg/g = 33Xicr6; within 10 minutes it would not exceed 7oXio~5. If, however, the coefficient of viscosity is known, from a preliminary examination of the wire, an error of dg/g = ^X icr5 is improbable. Hand-drawn steel, though admirably resilient, is a metal of somewhat low viscosity, particularly when subjected to additional tensile stress between the torsion-heads. To remedy this, the wire must be tempered. Samples were taken and tempered in molten lead. It was found that they had lost but little of their resilience, contrary to expectations; but their viscosity is necessarily increased, owing to the elimination of molecular instabilities. ACOUSTICS AND GRAVITATION. 79 The frame carrying the wire in the above experiment was therefore removed and an electric current (10 amperes) passed through the wire from end to end, till it showed the blue oxide coat virtually equivalent to tempering in lead. The adjustments of the wire in the frame were not otherwise disturbed. Hence on putting the frame back in the interferometer, the fringes were found at once. Observed in the zero position (no stress), the wire showed no displace- ment of fringes in the lapse of time, due to viscosity. The observations (table 2, series i) were now made for the viscous yield- ing of the wire with a gram weight on one scale-pan. The wire is thus about 2.5 times more rigid than in the hard-drawn state. In fact, if r= AN/ AN', the ratio of yielding caet. par., in the two cases, the results obtained by graphic interpolation show Time 10 20 40 70 120 minutes r= 0.42 0.41 0.39 0.41 0.46 Variations at the beginning are attributable to fluctuations of temper- ature. It must be noticed, however, that the change of stress is here from zero and thus but half the change (reversal) occurring in the preceding experiment. The effective increase of viscosity is thus not so large as was looked for. Twisting the wire in the opposed direction by shifting the grams to the opposed scale-pan, the viscosity, therefore, diminished con- siderably, as was expected. The second twist, moreover, is always character- ized by maximum yield. The readings (beginning about 4 minutes after twisting) are given in table 2, series 2. The rigidity gained by the tempered wire as compared with the original wire is about 30 per cent, not as much as was hoped. TABLE 3. — Viscous deformation of hard-drawn steel wire, tempered blue. Diam. 0.05 cm., length 35 cm .(one-half effective). Double twist 50°, due to counterpoise of i gram. Series. Time. lo'XAJV Series. Time. lO'XAN Series. Time. io«XAAr min. cm. min. cm. min. cm. o o o 0 o o 6 28 2 12 3 17 13 42 8 34 9 • 9 30 3 33 61 6 16 47 54 60 76 83 59 76 83 71 125 103 in 89 . 1,090 128 • 1,074 174 , o o • 0 0 3 15 o o 3 22 6 26 6 18 10 8 34 15 44 7 • ii 28 32 63 4 ' 24 62 82 76 122 98 72 103 157 102 123 H3 183 126 o O O o f 0 o 3 22 3 22 3 22 8 • 8 36 *n 7 37 , 5 8 37 36 64 12 49 1 06 90 92 87 68 96 * After twisting^back and forth about 100 times. 80 DISPLACEMENT INTERFEROMETRY APPLIED TO Successive alternations were now made by twisting the wire between fixed stops, using the gram weight specified as a counterpoise. Table 3 is a summary of these results. The whole of the 10 series for the blue state are further shown in figure 103, which also contains the graph for the hard-drawn state. As a rule, the curves for tempered wire, so obtained, differ but little. When they do, some irregularity in twisting the wire, or an accidental delay in finding the first fringes or different periods of application of the preceding stress, 40 60 so ICQ #0 160 iso 200 differences of temperature, etc., are in question. In the last series, No. n, the wire had been previously twisted to and fro upwards of 100 times to test the stops. The effect is some decrease of viscosity. In general, the odd series show less yield than the even series, probably because the stress is not quite symmetric on the two sides. 69. Equations. — Equations for the description of the yield in the lapse of time have been proposed by different authors, in particular by Kohlrausch, the original investigator. They are usually rather complicated higher expo- nentials. It is interesting to test a tentative form. Let it be supposed that the rate of yield dy/dt is proportional to the number dn/dt of instabilities van- ishing per second, so that (1) — dy/dt = Cdn/dt or A— y = Cn Furthermore, let the decay of instabilities vary as the square of the effective number n present; *'. e., (2) — dn/dt = a'n- or = a't n n0 If equation (i) is inserted, (2) takes the form (3) A-y i 'A ACOUSTICS AND GRAVITATION. 81 since for t = 0, y = 0. Equation (3) may be written (4) i = ( Blue (i). Blue (2). Blue (3). Blue (4). 33 62 109 152 87 103 no 160 1 80 50 60 28 75 54 53 25 45 54 60 49 55 107 93 no where /3 and A are constants. Such an equation will necessarily fail in the lapse of time, since A is the limit of y in (3) ; whereas experimentally y probably increases indefinitely. On the other hand, however, thermal instabilities are always present, apart from stress, so that equation (2) can only refer to the stress part of the phenomenon. Furthermore, the exact time of the beginning of viscous deformation can not be adequately specified, since there are a few minutes of irregularity in exchanging the weight and placing the fringes. The initial observation (t = 0) is thus inevitably a few minutes late. Hence an equation which fits the data of the first 30 minutes nearly enough for practical purposes is all that could be at issue. Yet even this modest expectation is not realized. Thus if we combine the first and third, second and fourth observations, the data are, for example: Wire : Drawn. io*A = 100 125 lo1/? = 90 54 io8C = 79 10* AN at ioom= 125 A rapidly increases and /3 diminishes in the lapse of time (here about an hour or two) . If Aj3 = C, the equation takes the form indicating that at least another power of time must also occur in the denomi- nator if the equation is to reproduce the data. The reduction is thus not simple. There would not, however, be any real hardship in adapting a function by which the amount of twist at any definite time after twisting could be accu- rately computed. The real difficulty lies in the time-loss in the shifting of the weight, and the subsequent adjustment, for fringes, etc. Twist is thus not imparted suddenly or all at once. The initial time (* = 0) is too indefinite. Again, since the balance-beam remains horizontal, but one-half of the wire is twisted, the other remaining without stress and idle, unless two symmetrical torsion-heads, one at each end of the wire, are used. For these reasons I discontinued the present method temporarily in favor of the next, § 71, where t is very large, the yield proportionally small, and where certain advantages of stability would accrue to the method. It indicates the importance to be attached to the thermal coefficient of rigidity. 70. Absolute viscosity of the wire. — The curves given may be used to obtain the absolute viscosity 17 of the wire, for the given rate of twist and diameter, and at any time after twisting. As but half the wire is used, the full torque FL (where L is the half length of the beam (n cm.), F the weight of the 82 DISPLACEMENT INTERFEROMETRY APPLIED TO counterpoise, io6 dynes) acts on the length of 17.5 cm. of wire. Hence, if T is the twist per linear centimeter, n the rigidity of the steel wire of radius r = 0.02 5 cm., and dd/dt the angular yield per second, 2PL n-c 71 ~Trr*de/dt~ dd/dt This equation (as I showed elsewhere *) is found by an integration of the vis- cosity equation, from the axis to the circumference of the wire. But dd/dt = A(dN /di) cos i/b in the above notation for the interferometer, so that zPLb i 7i~irr4cosi A(dN/dt) A(dN jdi) being the micrometer displacement per second, given in the curves, figure 103. If r4=io~7X3.9, 2XiogXiiXio i =TrLuv 2-53 3:MX3-gXio~7Xo.7i If equation (3) were true, ^(dN/dt)=/3(A—y)i/A, where /3 must be reduced to seconds. Taking &(dN/dt) from the graphs, figure 103, in case of series 0 (drawn hard) and 2 (tempered) at successive times t, the following data hold per second respectively t— 5 io 20 50 100 min. A(dN/dt')Xioi = 6.7 6.0 4.7 4.0 3.0 2.0 1.2 i.o 0.8 0.8 7jXio~20 = o.38 0.42 0.54 0.63 0.84 1.26 2.1 2.5 3.2 3.2 Thus the absolute viscosity for the given twist and diameter in these cases increases in 100 minutes from about 4Xio19 to 3Xio20 and thereafter is nearly constant. The single rate of twist is about (M) (50 717.5) = 1.4° per linear centimeter; but in alternating the twist the effect of this is doubled. It is clear the 77 must depend on the rate of twist in relation to the thickness of wire, inasmuch as the outer fibers probably yield most. This is among the reasons for the inadequacy of equation (2). If this is used in the form (3), the mean viscosity between 3™ and i7m comes out 7j = 5Xio19 and between iom and 30™ 8Xio18 for the drawn wire. 71. Twist in one direction only. — The actual torsi onal weighing by passing the counterpoise from pan to pan is in itself a delicate operation when inter- ference fringes are used. Moreover, the variation of viscosity is particularly large at the beginning. If the weight remains on one side, the viscous de- formation not only grows indefinitely smaller and subject to simpler equa- tional conditions, but the apparatus itself may be made less cumbersome and steadier. Thus in figures 104, 105, if w is the wire stretched between torsion-heads (the wire running normal to the balance-beam) and mm' are the auxiliary minors of the interferometer rigidly attached to the beam, the latter "Phil. Mag., XXIX, p. 337-55, 1890; Clark Univ. Lectures in Physics, 1909, p. 149. ACOUSTICS AND GRAVITATION. 83 may be prolonged and terminate in the thin metallic plates p, p'. These are to be surrounded by the narrow cases c, c', to secure air-damping. The device is now much more efficient than in the former apparatus, for there are no loose connections or scale-pans and the cases c, c' may fit rather closely to the parts p, p' of the beam, with but a small opening in front, for ^dbration. The counter- poise W, by which the torsion of the beam is to be maintained, is adjustable but rigidly attached at some convenient point. The cases c and c' were each placed on three leveling-screws below, and held in place by a stout Vertical spring above, pressing downward. In this way it was easy to free the plates p, p', particularly as the cases were adapted to slide (with friction) on the leveling-screws. The new method thus depends on the degree to which the viscosity of the wire vanishing at a retarded rate through infinite time may be adequately treated as a correction. The experiments showed, however, that it also depends to an alarm- ing extent on the thermal coefficient of rigidity, if ordinary metals are used, though it remains to be seen in how far the Guillaume alloys will meet the conditions. 72. Observations on the permanently twisted wire. — The readings were made two or three times daily, as a rule. They were adequately recorded in the graphs, figures 106 and 107, for a period of about a month. The ordinates show the displacement of the micrometer A N, the divisions of the graph being in steps of A7V=o.ooi cm. As the twist produced by the excess weight of about i gram was here (as above) about 50°, we have dg/g= i.6X io~4 for each of the divisions (o.ooi cm.) of AjV. The laboratory temperature was very variable from day to day, and the effect of this in the graphs is astonishing, but contributes essentially to the interpretation. In figure 106, the average rate of yield may be estimated as AW = 0.00091 cm. per day, as the result of viscosity. This makes dg/g vary 0.00014 per day, a quantity in itself too large to be used as a correction. Even a more serious consideration is the ther- mal increase of rigidity. It is this feature which makes the graph so exceed- ingly jagged. If we take the large drop between the eighth and tenth days to estimate this effect, the data would be Temp. t = 22.0° 14.8° 19.8° io3A]V= S.i 2.6 8.7 which is equivalent to A7V/A* = 0.00095 centimeter per degree, or dg/g = i.6Xio~4 for each degree of fall of temperature as the equivalent of in- creased rigidity. * The graph, figure 107, is actually worse; for here, between the twentieth and * Excellent observations on the torsional rigidity of iron and steel, due to Pisati, are given in Landolt and Boernstein's tables, 1905, p. 44. The coefficients are 2.1 xio~4 for iron and 2.3 X io~4 for steel, therefore even larger than the above estimate. Moreover, the above method could easily be modified to measure this constant accurately. 84 DISPLACEMENT INTERFEROMETRY APPLIED TO twenty-fifth days, in which the temperature falls from 23° to 14°, there is no adequate recovery afterwards. It would seem if this were a dislocation within the interferometer, though I noticed nothing differing from the usual be- havior. A slight change of parallelism in the mirrors, due to drop of temper- ature, is always first corrected. 0 24 Z6 28 SO, It follows, therefore, that even if the temperature variations have here been purposely exaggerated to exhibit the evidence more strikingly, there is very little hope, in the case of the common metals, of using the permanent torsional deflection produced by a weight for the measurement of variations of gravitational acceleration. For the incidental and spurious variations of dg/g amounting to over lo"4 per day for the case of viscous yielding and nearly 2 X icr4 per degree of temperature for the case of rigidity, are too large to be treated adequately as corrections. The above method of direct weighing, even apart from its inconvenience, would not be more trustworthy, as it must contain the same inherent errors. 73. Further experiments with the preceding apparatus. — The damping in case of the apparatus, figures 104 and 105, when the plates p, p' are about 5X5 cm. square and within i or 2 mm. of the top and bottom of the shallow cases c, c' is exceedingly good. The beam mm', even with the relatively thick steel wire w used, is practically aperiodic. Hence, on putting a wide plate of brass underneath the system pmm'p' and close to it for an additional protection from air-currents, no further casing is needed. If the mirrors m, m' are wide, ACOUSTICS AND GRAVITATION. 85 notched tubes may be placed around them, standing on the plate in question ; but these are hardly needed. The beam mm' is thus easily accessible without any opening of the doors of a case. The fringes occasionally move, but they at once return to their equilibrium position, which is thus easily recognized. It seemed worth while to test this by removing the counterpoise W and pro- ceeding with the method of weighing at the two ends of the beam mm', as in the first part of this paper. For this reason stiff hooks, h, figure 108, were fastened rigidly at each end of the beam m. The plates p, p' in the cases c, c' were soldered to the bottom of h, as shown. The weight b was bell-shaped and reposed on the sharp point of the hook h. A small projection on top allowed of its easy removal with forceps. Adjusting the stops of the torsion-head of the wire w, the bell-weight 6 could be passed (with the requisite torsion) U-.-Ji— Oni — • -O 23 June 10 WJufytO ZOJIva J 7^ ty&ptS from one end of the beam to the other, without any inconvenience, the fringes in either case of adjustment appearing at once in their proper position rela- tive to the cross-wire at the slit of the collimator. Of course, h must be stout enough to be practically free from flexure relative to b. With adequate damping a similar arrangement should also suffice for a knife-edge balance, though here the plan of figure 102 would probably be more available. Many experiments were made with the apparatus, figure 108. Naturally, however smooth the results, they are ultimately all subject to the temperature coefficient of torsional rigidity. An example of continuous results extending over 3 months is given in figure 109. The ordinates show the displacements of micrometer in centimeters. The kinks in the curve are the results chiefly of change of temperature in its effect on the rigidity of the wire. Not impossibly rapid temperature changes, such as occur in the summer, may produce some yielding within the interfer- ometer itself. 86 DISPLACEMENT INTERFEROMETRY. The interesting feature of the curve is this : When observations are made daily, implying, therefore, daily torsional interference with the wire, the vis- cous yield is relatively great. When the wire is left quiet without interference, as during July and a part of August, the yield is much less pronounced. It is not improbable that the molecular disturbances due to the slight inadver- tent twisting (vibration) during observation may account for this result. The motion of the undisturbed wire may even be retrograde, as in August. If this inference is correct, it points seriously to the instability of structure of a permanently twisted wire, even when the twist is not excessive. CHAPTER IX. A PNEUMATIC METHOD OF MEASURING VARIATIONS OF ACCELERATION OF GRAVITY. 74. Introductory. — Some years ago I made an extended series of experi- ments* on the diffusion of gases through water, the gas having been im- prisoned in a Cartesian diver ; the very sensitive conditions under which the diver floats at a given level were made the criterion of measurement. Inasmuch as such experiments consist virtually of a comparison of weights with the forces derived from air-pressures, it must therefore be possible to obtain the acceleration of gravity in terms of these pressures, just as in the preceding chapter of this report the endeavor is made to evaluate the changes of g in terms of torsion. It will not, of course, be possible to determine g absolutely in this way, because so little air is used; but the question as to what degree the changes of g should, in a proper environment, be determinable with some precision is worth investigating. 75. Apparatus. — The chief difficulty with the experiment at the outset seems to be the elimination of loss of gas by diffusion. This, in the case of a cylin- drical diver, open below, amounted in the experiments cited to 0.41 per cent per day. It would not, therefore, be long before the whole of the gas would be lost. To avoid this, two resources apparently suggest themselves : (i) to place the diver about midway between two layers of air, one below and the other above it; and (2) to provide the diver with a long slender neck below, precisely what was to be avoided in cases of diffusion. I shall describe two forms of apparatus in which these precautions are taken, the first intended for laboratory experiments or information, merely; the other suggesting a possibly definite form of apparatus. Figure 1 10 is the experimen- tal type. Here AA is a stand-glass containing water ww, the diver ab, and the lower air-chamber cd. The vessel is closed above by a rigid cork, carrying the thermometer T in a central perforation, and a lateral tube / for exhaustion. The water-level at e is at a distance h above the water-level a in the diver when the latter is floating. The air at c below is contained in a porous cup, wedged in place, and the difference of heights of water-levels at a and cd (the top of the porous cup) is also to be about h. The bottom of the diver ends in a narrow tube, only just wide enough to admit of easy egress or influx of water, subject to changes of pressure. Under these circumstances the loss of air by diffusion from a to e will be practically balanced by the opposed diffusion from do a. The narrow neck also obviates danger from ingress of any minute air-bubbles. When not floating, the diver should repose with its mouth at some distance above c. The bottom of the thermometer bulb determines the * Carnegie Init. Wash. Pub. No. 186, 1913. 87 88 DISPLACEMENT INTERFEROMETRY APPLIED TO floating level, the diver being essentially unstable, and either rising or falling with acceleration. To operate the apparatus the air is exhausted at t until the rider just strike? the thermometer bulb. The top should, therefore, be flat. A very slight leak is then made at t, and the air-pressure and temperature (at T) read off at the moment when the rider and thermometer separate. With a light diver this is determinable with remarkable exactness, particularly if the influx of air is very slow. To determine the air-pressure, a mercury manometer and barome- ter are here necessary. The former should be adjusted to 10 or 20 cm. Pressures within o.i mm. of mercury and temperatures to a few hundreths degree are required. As the exhaustion proceeds, the water-level at c rises, owing to the expansion of air both at a and at c. The head h refers, therefore, to the instant of flota- tion. Such a change of level is naturally undesirable, as is also the double mercury apparatus, manometer and barometer, for air-pressure. To obviate this, the type of apparatus, figure in, was designed. The vessel A A, swimmer ab, air-supply cylinder cd, thermometer T, etc., are practically the same as before; but an auxiliary tubulure with a stop-cock g has been added. Two lateral tubulures t and tr, however, are here available; one of these /' communicates with the attached barometer D; the other, t, with the mercury receivers B and C, connected by a flexible pipe, /. Between the terminal mercury columns at B and D, the whole system, with the exception of the air-spaces at a and c, is filled with water. If the mercury heads HI and H2 and water heads h' and h" be determined as shown in the diagram, the air-pressure at the level a will be when B is the barometric height and TT the vapor-pressure of water-vapor. ACOUSTICS AND GRAVITATION. 89 Hence, if the vessel C is placed on a vertical micrometer-screw, so that C may be raised and lowered with precision, the pressure needed to just float the diver may be applied at C, and its value accurately read off at D. This obviates the need of a special barometer reading. The vessels B and C must be large enough to admit of the expansion of the air a.t a and at c. The en- deavor should also be made to eliminate the water-head h" '. With divers about an inch or more in diameter and two or more inches long the apparatus need not be cumbersome. The height of the stand-glass need not exceed a foot. Pressures of about 60 cm. at D and 16 cm. at C, B are convenient. There i? the outstanding objection, however, that the air-bubble is stored under pres- sure. This will be treated below. 76. Equations. — If M is the mass of the diver, m that of the air contained at the absolute temperature T and pressure h+(H — ir)pm/pv,, in centimeters of water (h being the incidental water-head, figure no, H the residual mercury- head, and TT the vapor-pressure in centimeters of mercury, pm,p*,ps, the densi- ties of mercury, water, and glass, all read off at T), we may write for the gravi- tational acceleration, g: Rmr g The first term in the denominator is to be multiplied by ' whence 2/Y But in § 83, x = 0.01455 y, where y is the displacement reading on the centimeter scale in the telescope. If for short alternations of the period t, Ay is the double amplitude observed in the telescope, _~ o.o 1 45 5 Ay = ~~~ Hence, if we put R = 4.2^ cm., M = Q4p grams, then 7 = icr4Xii.o Ay//2 or, if we express t in minutes, as observed, 7 = io-830.6(Ay/*2) Thus it is merely necessary to divide the double telescopic amplitude by the square of the time in minutes, apart from the constant. This would be very easy if the motion of the needle were not at times cobwebby, so much so that I have frequently passed a wire around it, inside the case, several times to insure myself of its freedom. In figures 122 and 123 I have given the data (scale readings successive minutes apart) obtained on July 5 for one-minute and two-minute periods in the switch-over of M. The turning-times of M are indicated by little circles. Since the needle is invariably drifting, it is necessary to compute the 108 DISPLACEMENT INTERFEROMETRY APPLIED TO double amplitudes from triplets. This has been done, and the value of Ay is inscribed in each of the consecutive zigzags of reasonable shape. The mean values are Series i t=im Ay = o.i62 io3 ,m O.6OO 8-73 4-6 The apparent value of 7 in each case is but little more than two-thirds of the standard value and the deficiency must be ascribed to the coefficient of friction neglected. It is interesting, however, to note that relatively equation (i) is roughly corroborated, the displacement A% being (here) 4 times as large in two minutes as in one minute, though the motion itself is far from uniformly accelerated. In figure 122 the needle moves, as it were, without inertia and the phase difference (Ap) relative to the forces is therefore 90°, but the period is too short for much discrimination, particularly in view of the drift. In figure 123, however, the same anomalous condition reappears and, what is more striking, is the retarded motion between the turning-points. For this is of the nature of a lag, or indicates an apparently waning gravitational effect, or rather the growth of some repulsive force. 8 2 £L _5 i (Vf 1 W\, • • 0w, y0 $0 40 On July 7 , I made another series of observations given in figures 1 2 4 to 127 for one, two, and three minute periods. Figures 124 and 126 made before and after figure 125 and for one-minute periods show no gravitational effect, but a mere drift with several induplications. At least, if triplets were constructed, they would have a value not exceeding A# = 0.0007 centimeter about one- twelfth the value in figure 122. During the first minute gravity shows no appreciable preponderance. During the ensuing one and two minutes, *. e., for ACOUSTICS AND GRAVITATION. 109 two and three minute periods, the results in figures 125 and 127 become very definite, as shown in the zigzags of the curves. The mean results would be Series 4 * = 3m Aj=i.2Q io3 Ax= 18.77 io87 = 4-4 6 2m 0.30 4.36 2.4 The former series (4) is still consistent with the preceding, showing that Ax increased as P. The graph, moreover, is such as we should expect, the needle being carried by its inertia after the turning-time of the force. In figure 127, however, the inertia retardations are just as large and the effect of this is to depress the double amplitude and therewith the 7 value to one-half that in series 2. It is probable, therefore, that the inaction during the first minute, present to the same degree in series 3 to 6, or figures 124 to 127, needs some other explanation, being referable to some peculiar repulsion, the nature of which is to be ascertained. It is interesting to note in passing that if in figure 125 the displacement between maximum elongations had been taken, the mean value would beA^ = 2.o, so that 7 = io~8X6.S would be the result. This, however, is merely accidental. On July 8 five series of experiments were made. The one-minute periods failed, but the motion during the two-minute periods, one of which is shown in the chart, figure 128, proceeded without apparent inertia. In series n. (two- minute periods) the drift was excessive. The results were Series 8 t = 2m Ay = 0.58 io3A* = 8.4 io87 = 4-4 9 2 .53 7.7 4-0 10 2 .55 7.9 4-2 11 2 .53 7.7 4-0 The experiments made on July 9 and io finally exhibit a new state of affairs, for here the needle is frequently in the same phase with the alternating force, so that the motion of the needle changes sign midway between the turning - times of the weight. I give two examples of these curious results in figures 129 and 130. In the former the general drift is very large and downward; in the latter about as large, but upward. The value of the triplets is shown in the indentations as before. The results were Series 15 * = 3m Ay = 0.96 io3A# = i4 io87 = 3 17 , 3 1.81 26 6 19 3 °-57 8 2 20 3 0.72 ii 2 In figures 129 and 130 the oscillating motion of the needle is quite without apparent inertia and the turning-points of the needle are such as to indicate some periodic repelling force counteracting the gravitational attraction. In order to get any data from these curves, the double amplitude of the needle must be taken between the elongations, though their meaning then becomes obscure. As compared with these figures, the phase conditions in figures 131 and 132 are again totally changed; for here the unknown repulsions operate appreciably just before the attracting weight is turned. It is astonishing that results such as these should be obtainable with a simple, carefully constructed apparatus, in a dark, damp, half subterranean 110 DISPLACEMENT INTERFEROMETRY APPLIED TO room, of practically constant temperature. In any given series, for instance, the behavior is consistent. The repulsive force retains its relations relative to the gravitational attraction, changes sign with it in a consistently different phase, in each of the series of results. If there were any approach to reso- nance between the period of the alternating force and the needle, an explana- tion of the results might be attempted ; but the semi-period of the needle may be estimated at 10 or 20 minutes. It is aperiodic, practically creeping. 87. Further observations. — The needle was now dismounted and the obser- vations of § 84 on the elastic coefficient of the needle were concluded. After it had been put together again with improvements, and readjusted, the observations of figures 133, 134, 135, one and two minute periods, were then taken. The observer throughout kept at a distance from the apparatus except at the time of reading the telescope. It will be seen that the observations for one-minute periods again fail in the usual way. For the three-minute periods the results are normal, as the needle swings with apparent inertia after turning o™ itr ar U i I** y/l«" 9/T*- 1(T SO" 0* 1 the attracting weight M. The observations (fig. 135) for two-minute periods are also normal in this respect; but it is necessary to reckon from observations midway between the turning-points if triplets are to be available. The data are : Series 21 * = 3m Ay =1.30 io3A% = i8.p 10*7 = 4.4 23 2m .61 8.9 4.6 about of the same order as above. The displacements A* are again roughly in the ratio of 9 14. But the new results are no improvement on the old. As a final condition of safety the hooks of the quartz fiber were additionally secured with sealing-wax to the torsion-head above and the needle below, so that the system throughout was free from any loose parts or movable joints. Results so obtained for three-minute and one-minute periods are given in fig- ures 136 and 137. The latter again completely fail. The former are not essen- tially different or better than the above. The fiber was a little shortened by accident. The value of the successive triplets is given on the curve. Their mean value is Series 24 * = 3m A>>=i.i7cm. io3Ax = i.7 cm. What is particularly noticeable in figure 136 is the uniform motion within the branches of the zigzag. From the data as a whole we may conclude that, in addition to gravitation, some other force is present, attractive or repulsive, and in any given experiment ACOUSTICS AND GRAVITATION. Ill always in a consistent phase-difference relative to the alternating gravitational force. In other words, the discrepant force develops in an orderly manner after the weight M is turned. Furthermore, there is in most of the graphs a marked tendency toward uniform motion of the needle between the turning- points or their equivalents. We may thus explain the ratio of total excursions i : 4 : 9, for one, two, and three minute periods, more consistently with the obser- vations as follows : In the first minute after turning there is a mere accommo- dation of force conditions. Hence the one-minute periods nearly always fail to show interpretable results. In the two-minute and three-minute periods the excursion times have thus been only 2-1 minutes and 3-1 minutes, the latter twice the former. The approximate ratios 4: 9 are thus more probably to be taken f or i : 2 or the mere total excursions to be expected from uniform motion. Furthermore, if there is uniform motion, the gravitational attraction is counterbalanced by frictional resistance and the latter may be taken as pro- portional to the speed of the needle. These rates of motion must therefore be looked upon as of much greater interest than the value of the triplets as com- puted above. Moreover, it there is marked drift the value of the triplets will be correspondingly changed, for the force in action is large as compared with gravitation. 88. New apparatus. — Another needle, of the same kind as above, was now installed in a smaller case, capable of exhaustion and useful for other experi- mental work. This consisted (fig. 138 elevation, fig. 139 plan) of a hollow rectangle ww of waxed wood, 1.8 cm. thick, to which glass plates gg' were attached, soft sealing-wax having been melted around their edges, reinforced by steel clips. The needle with the two shots at m, m', 0.56 gram each, and a light mir- ror at n, was supported by the quartz fiber q, from a torsion-head above, carried by the glass tube t. The whole was rendered air- tight by melting soft wax into all crevices and joints. The hooks of the quartz fiber were also cemented together, so that there was nothing loose. The needle was put in place with one of the glass plates off. This was then fastened as stated. It could again be removed by melting the wax at the edges. The case was fastened nearly vertically, by aid of the brass rods r, r', held in clamps anchored in the pier. The final leveling and freeing of the needle was accomplished by a screw-pusher p abutting in the pier. The tube t was also held in a clamp from the pier, to prevent excessive tremor. The attracting weight M had a separate mounting on the same pier, and by a crank-like mechanism could be smoothly and easily passed between adjust- able stops, from M to M'. The observations were made in the same dark, 112 DISPLACEMENT INTERFEROMETRY APPLIED TO damp, semi-subterranean room, as in the former case, and the two instruments were not very far apart. It was necessary to put the scale and telescope somewhat nearer (265 meters) to the mirror n, than before. The new instrument is thus apparently, though not really, less sensitive than the other. As the scale-parts are given in centimeters, the displacement of shot will be ii. 357(2X265) = ( cm. the needle being 22.7 cm. (between centers) long. 89. Trial observations. Radiation effect. — On testing the apparatus roughly with a brass kilo, it was noticed that the first excursions were always abnor- mally large. In the course of the day these died off to values scarcely one-tenth of the original. An attracting ball of lead M, weighing M = 1,602 grams and capable of approaching the shot to R = 5 cm., was then installed. Here again the first triplet (14) obtained was enormous (19.2, 1.4, 10.6 scale-parts), but it gradually diminished and on the next day the excursions were steadily of normal value (about 3). It seemed probable, therefore, that an extremely subtle radiation effect was here in action. To test this, the lead ball M was removed and just exposed to a few passes of the Bunsen burner, so as to be warm to the touch. On being replaced, the alternations (in approximately 8 141 B ^ 7 cL \ z\ 6 I Ii. I 9 Q \ »\_yr.. r_v \'_'.i \U /t>\ I\ A" A ;\ . k/8r\ /co\ /k_\ //vA A..\ /..\ 600* half -hour periods) obtained (fig. 140 a, scale one-tenth of the preceding) were enormous, the first excursion being up to the wall of the case. They gradually decreased, as the figure shows, but the effect was still very great, the initial triplets (within the first hour) being 22 and 1 8, or 6 or 7 times as large as the normal gravitational effect and of the same sign throughout. The remarkable feature of this is the long time needed for the effect to pass off; for, after more than an hour, the ball M was again quite cold on the touch, but its radiation activity was nevertheless still pronounced. The next day the same ball, whose natural excursion was but 2.5 (fig. 1406), was merely warmed by the hand (fig. 140 c). The result is an excursion of ACOUSTICS AND GRAVITATION. 113 15, or 6 times the normal value. The ball was next cooled slightly by sub- merging it in tap-water, with the results of figures 140 d and 141 d (enlarged). There is a drop of y = 3 s. p. It was now again submerged, dried, and replaced, with the result of a further drop in deflection of 3.1. On alternating the posi- tion of M from the front (F) to the rear (R), the effect of gravity was found to be just exceeded by the repulsive radiant force from the cooled ball ; for the F deflections are normally the larger. Finally, the case itself was covered (except at the mirror) with a thin sheet of metal closely fitting it, and the effects of alternating the position of M were tested through this metallic sheath, with the results of figure 141 e- The excursions of the needle are not only large and irregular, but on the whole a reversal of the gravitational effect. Hence the metallic sheath was again discarded as introducing unnecessary complications and affording no advantages. But after long repose (days), the new apparatus functioned quite as well as the old. This may be seen both in figure 142 and others below, or in the three-minute alternation in figure 1 43 . There is little drift and the mean value of the triplets as indicated on the curve is Ay = 0.866, somewhat smaller than in the old apparatus, where M is nearer m. It is natural to refer the radiation effect to the reduction of pressure into the energy of motion. The convection between the shot m and heated ball M may be regarded as in excess of that on the other side of the shot, and hence a larger amount of the former pressure is changed to kinetic energy. This would similarly account for the observed reversal in case of a cold ball. There is thus a peculiar radiometer effect in a plenum of air, tending to vanish on exhaustion, to which further attention is given in the next paragraph. 90. Radiation effect observed on exhaustion. — The effect of exhausting the case in small steps of 10 cm., slowly, is to cool the air contained. Hence the result should be like that produced by a hot body on the outside, provided the needle is not quite symmetrically placed with respect to the glass windows of the case and provided the needle end on the side of the hot body is also the nearer. The needle in the present experiments was, accidentally, so hung. Apart from this, the present experiments have nothing to do with the outside environment, if it does not change. They are then merely concerned with the interaction of the needle and the case when the air within is successively and slowly exhausted. In the experiments summarized by figure 144 a, the needle was at rest at 10.7 scale-parts. Exhaustions were then made from o to 30 cm. (lost), 30 to 40 cm., 40 to 50 cm., etc., finally 70 to 74 cm. ; the curve shows the effect produced. After every exhaustion the needle was allowed to return to its zero-point (10.7, see arrows) nearly. The result is very interesting, for the effect of exhaustion changes sign after about 60 or 70 cm. of pressure have been removed. If the effect was like an attraction before, it is like a repulsion after. The curve b, with a somewhat lower zero-point (9.2), shows the same effect for somewhat irregular steps of slow exhaustion, o to 10 cm., 10 to 20, etc., as 114 DISPLACEMENT INTERFEROMETRY APPLIED TO indicated by the little circles. The evidence is of the same nature, remember- ing that, quantitatively, the effect must depend on the rapidity of exhaustion, etc., which is kept low to avoid influencing the needle by a current of air directly and also to avoid large temperature decrements. There are thus two radiation effects, respectively positive and negative, which pass continuously into each other. One of these may be ascribed to the change of pressure into kinetic energy on the hotter side or side of convection excess; i. e., there is pressure deficiency on the hotter side. The other, at higher vacua, is the radiometer effect, which is of opposite sign; i. e., there is pressure excess on the hotter side. Hence, between the two, somewhere near 65 cm. here, the two phenomena are in equilibrium and the effect of exhaustion is nil. The investigation resulting in figure 144 was merely made for practical purposes and the deflections depend on many subsidiary details. It will be carried out under standard conditions presently. There is a correlative test for the effect of radiation, from without, on the needle in an exhausted case. To apply this, the case was exhausted to above 70 cm. and a ball heated to 60° or 70° placed outside. No discernible effect was produced, or, in other words, the effect of external radiation is now small. Consequently the drift of the needle must vanish and the triplets become static repetitions of each other. That this is the case the experiments of the next paragraph will show. It follows, also, that the resistance encountered by the needle is now referable to the viscosity of air and therefore possibly open to computation. At a later opportunity I returned to these interesting phenomena again, with the object of obtaining more definite results. The case of the needle was, as above, exhausted in steps of 10 cm. of mercury, successively and quite slowly, so as not to disturb the motion of the needle by air-currents. For this reason, pressure increments of 10 cm., *. e., the return passage, did not suc- ceed very well, as the needle was apt to be influenced by any accidental puff of air. During exhaustion there is no such danger. Experiments were first made with the ball M removed, and they are given in figure 145 b. After each exhaustion the excursion of the needle was allowed to return to the equilibrium position, marked with little circles on the diagram. The period observed in these cases was relatively short and did not exceed six minutes, three minutes for the semi-period. Moreover, the equilibrium position varies somewhat, possibly because more than one swing should have been waited for; but this would have enormously extended the experiments and introduced other drift-like errors. ACOUSTICS AND GRAVITATION. 115 The needle lay somewhat obliquely across the case, the front end being nearer the front window and the rear end nearer the rear, to the effect that an increased potency (high temperature) of these windows would deflect the needle toward larger figures. This is the case in figure 145, where the ordinates are the scale-readings in centimeters and the abscissas change in steps of three minutes, roughly. The amount of the exhaustion o to 10 cm., 10 to 20 cm., etc., is inscribed on the curve, as to the larger number, remembering that the steps except the last (70 to 72, 72 to 74) are each 10 cm. At the end are two cases of influx with a reversal of result. On account of the slow and cautious exhaustions made, the correspond- ing temperature decrement of the air within the case must have been very small ; and yet the excursions of the needle due to the attraction of the nearer and relatively hotter glass window is at first very marked, for the deflection produced by M=i.s kg., at 5 cm. should be but 3 cm., roughly, in the same scale. The amount diminishes, however, until between 60 and 70 there is no effect, or an equilibrium of radiant forces. Thereafter, 60 to 70, 70 to 72, 72 to 74, the effect changes sign and these repulsions increase at a relatively very large rate. The effect is reversed on compression. The experiment was now varied, merely by the changing of the environment ; i. e., the ball M = 1,600 grams was placed within 5 cm. of the needle, in such a way that its gravitation attraction or its radiant force for a plenum and hot ball M would deflect the needle toward smaller numbers; i. e., to act against the effect of the window or case. The results are given in figure 146 and are totally different from the preceding, as there is a double inversion for each exhaustion. At first the case (c) acts as in figure 145; thereafter the effect of the more massive ball M (6) supervenes and finally the needle returns to its equilibrium position. The null effect here occurs between 50 and 60 cm. The inversion, 60 to 70, 70 to 74, is as marked as before. The circles in figure 145 are higher than in figure 146, owing to the gravitational attraction of M. _ UC. JV* '„— 60 80 Finally, figures 147 and 148, the torsion-head of the needle was slightly rotated so as to place it in a more symmetrical position to the walls. The re- sults are again noteworthy. In the first place, the phenomenon, figure 147, as a whole, has been reversed through partaking of the same general character. Furthermore the interval of absence of radiant force has been lowered to be- tween 40 and 50 cm. At the same time the radiometer effect at the end has 116 DISPLACEMENT INTERFEROMETRY APPLIED TO dwindled in importance. The plenum effect, however, refuses to be so easily dissipated. The circles in figure 148 are higher than in figure 147, owing to the attraction of M. Figure 148 has the same characteristics as figure 147, except that the ball M was left in front, thus tending, if warmer, to produce deflections toward larger numbers. Consequently figure 148 should be an inversion of the other, as it is . All these results were very definite and the figures might have been drawn to a larger scale with advantage. 91. Tendency of needle to stick to glass window. — This very annoying phenomenon, frequently mentioned in these experiments, now finds a rational explanation. If the glass plates of the case are different in temperature from the needle, there is a radiant force on one side of it (the side depending on the exhaustion), which increases as the needle approaches the glass window, suf- ficiently to hold the needle there against the torque of the quartz fiber. In the damp laboratory room electrical forces are, I think, out of the question. Radium close at hand does not change the phenomenon. I may cite a marked example here. In order to locate a leak in apparatus II, the whole case was submerged in a water-bath. After replacing the apparatus, well-dried, etc., at about i p. m. , the east end of the needle first stuck tothe front window. If shaken off by tapping the case, it vibrated, but finally returned to the same position. Later, from 2 to 5 p. m. it similarly adhered to the rear. Still later it stuck to the front again, after tapping. It was found so the next morning; i. e. after more than twelve hours. Now, however, after tapping, it soon took its position in the middle of the case and was free. The Ly was a normal value, but both readings y were still high, showing that the window attraction was not quite spent. In all this work the case was kept exhausted to a pressure less than 5 cm. It is such results as these that lead one to question, even granting the law of cooling, which associates long times of cooling with small temperature dif- ferences, whether something more than mere temperature is not in question. It is probable, also, that the continuous drift of the zero-point, in case of the daily readings of apparatus /, is attributable to a slight quasi-temperature excess of the front plate (east) as compared with the rear plate. 92. Needle excursions under increasing pressure. — The excursions of the needle of apparatus II in a plenum of air were similar to those in apparatus J, but less variable, as has been already stated. Thus in figure 142 the mean data for July 30 to August 2 are July 30. July 31. Aug. i. Aug. 2. A. M., Ay = 4.oo 4.14 2.83 2.86 P. M., 4.40 3.90 which is a drop in Ay corresponding to the case of apparatus /, notwithstanding the east- west needle in apparatus II. The details in figure 142 are also similar. As this apparatus may be evacuated, it was thought well to observe the ACOUSTICS AND GRAVITATION. 117 effect of a slow influx of air into the apparatus, when exhausted to different pressures. The conditions of influx were throughout about the same, being about 7 cm. of mercury per 30 minutes at the highest vacua. The phenomenon is thus very nearly isothermal, and yet the results of the heating produced are phenomenally marked. The individual readings are given in figure 142, where the numbers show the exhaustion. Figure 149 contains the mean results Ay at the different vacuum-pressures p. The irregularity of results is to be ascribed to the change of Ay, even for the plenum, and to the difficulty of controlling the influx; but apart from this the excursions Ay rapidly diminish, nearly proportionally to p, throughout the large part of the curve, while at very low pressures the effect is possibly even accelerated. It often appears as if for p = 0 there would be no excursion. These results are so striking that one may well ask whether temperature is here only in question. The effect of influx (10 to 15 cm. per hour) is to heat the interior relatively to the ball M, without. Hence the region between the shot m and ball M is relatively cool as compared with the other side of the needle, and the result would be a repulsion on the cold side counteracting grav- itational attraction. So far the reduced results are consistent with the experi- ments of figures 144 to 148, though the graph figure 149 is incorrect as to slope. Again, the earlier data demand an inversion of the effect at about (say) 50 cm. of exhaustion; *'. e., an attraction. The present data here show no inver- sion, but rather an accentuated repulsion. True, the inversion in figures 147 and 148 is not marked. It is present, however, while in figure 149 Ay in a vacuum of 50 cm. is but one-half of the value for a plenum. The work done by this influx, so nearly isothermal, would be Rmr log p/pf, where m is the mass of gas, p and p' are the initial and final pressures, and m varies as the mean pressure, nearly. Thus Pressures 43 -36 cm. 33-25 23-16 11-3 7-2 Work done oc 31 35 31 40 28 Log P'/P = °-°8 0.12 o. 16 0.56 0.54 A0 = 14° 23° 30° 108° 104° The work done is therefore not so different in the various cases, but as the mass is inversely as the pressure, the rise A0 of temperature would vary as log p'/p and would thus be greatly in excess at the higher vacua. In this respect the results of figure 149 may in a general way be interpreted, if the radiometer effect is ignored. It would have to be nil in a perfectly symmetrical needle, but what is curious is the persistence of the plenum effect. There is a persistent pressure on the colder side of the needle at all exhaustions. The total amount of heat produced and its temperature effect may be com- puted. If T is absolute temperature, m the mass of gas, the work done is in the usual notation for a pressure increment of dp, dW = RmT(dp/p) If dd is the corresponding rise of temperature without loss by radiation, = Jkmdd 118 DISPLACEMENT INTERFEROMETRY APPLIED TO Thus, after integration of the equated values of dW, Rr p' A0 = —rr- log — JK p The constant factor is about 83 X2.3, since Naperian logarithms are needed. In this way the values above given were obtained. They represent the total accumulation of heat in over half an hour ; but it is at once radiated by the gas and finds lodgment on the walls of the apparatus. Even if these radiated nothing, their temperature increment would be infinitesimal in the ratio principally of their mass to the mass of the air-content (probably for the plenum 0.3 gram to several kilograms), which is then further diminished by radiation. The available heat does not differ as much at all the pressures as does the temperature. I question whether such small thermal effects, if they are such and so circumstanced, could be detected in any other way. 93. Experiments with the exhausted case. — The case, figure 139, having been carefully sealed, measurements were made in partial vacua, as shown in figure 150, for an exhaustion between 20 and 30 cm. and figure 151, for an I«I;\K: 0.85 0.86 exhaustion between 37 to 45 cm. In the former an error was made in the second triplet, which is therefore discarded. It is seen at once that, as the ex- haustion increases, the drift grows less and the triplets are nearly repetitions of each other. The double amplitude changes but little; thus, for Plenum, £ = 76 cm. Vacuum, 51 cm. Vacuum, 35 cm. a result to be anticipated, if the viscosity of air is independent of pressure, and in view of the very slow motion of the needle, the resistance is not due to tur- bulent motion of the air. The case leaked slightly, but this does not seem to be of moment here. An important consequence follows : Since the frictional resistance is inde- pendent of pressure, it may be computed as a case of viscosity, the problem being that of a cylindrical shaft rotating on a normal axis, at its middle point, in a viscous medium. This is further borne out by the fact that in the above three-minute triplet curves, the branches, after being released from the turning-points, are very nearly straight lines. Moreover, midway between the turning-points the elastic force of the fiber is necessarily zero. In the pre- ceding report, many instances were given in which, even for larger periods, the curve soon consisted practically of zigzagged straight lines. ACOUSTICS AND GRAVITATION. 119 Notwithstanding the steadiness of the new results, one-minute periods, as shown in figure 152, are again impossible, while the three -minute periods follow- ing succeed at once. In figure 153, the highest mean exhaustion available dur- ing thirty minutes was applied. The value of the triplets has not practically changed size: Vacuum, £ = 30 cm. A;y = o.79 £ = 8 0.83 the smaller values being referable to slightly greater temperatures within the case. Figure 153, moreover, is appreciably sinusoidal, all the branches being first accelerated and finally retarded. There is no apparent inertia effect. The double inflection might be associated with the elastic resilience of the fiber. It is much more probably, however, the result of a repulsive radiant force which requires a minute or so for development. Thus, when the weight is turned the counter radiant force acts for a time with the new pull of the weight. Hence the acceleration. The radiant force gradually vanishes in turn, passing through zero, then to become negative and constituting an increasing resistance to the new pull. Thus the retardation ensues. In other words, the radiant force needing time to develop, is in phase behind the gravitational pull, which acts instantaneously. The two observations between the turning-points should therefore be without appreciable radiant force and the data of § 95 for the rates of uniform motion bear this out very fully. 94. Tentative estimate. — The resistance experienced by a sphere of radius r, moving in a viscous fluid (77) with a velocity v = lu, is well known to be 6irrjrv. I have not found the corresponding expression for a cylinder (figure 139, m m') of radius r, semi-length /, and with hemispherical ends, moving with constant speed, broad-sides on. To get at an order of values, however, we may postulate that for equal frontal areas, 7rr2 = 2rA/, the resistances are alike. Thus the element of resistance is dF = 6irrjrv = 6V VTJ/WV/ Vr2 = 6v/7n?a>/v/2?'.A/ — and we may meet the further conditions by integrating for the length 2 1 of the needle. To carry out the integration* put l = nX2r, where n is a serial number. The equation becomes and the problem reduces itself to the summation of a series of cubes, 2\/2ns = n(n+i), the length being 2/, Hence, finally, for two masses, M, m, at a distance R apart, disregarding corrections, The constants of the second apparatus were : M = 1 6o2g, 1*1 = 0.563% R = 5.i cm., 2r = o.4 cm., 2^ = 22.8 cm., 17 = 0.00019, 11 = 28.5 *A more acceptable upper limit would be y=*(B?/Mm)yrtQr*<*n(n+'l). 120 DISPLACEMENT INTERFEROMETRY APPLIED TO In figure 151 the last three scale-rates of the lines prolonged have the mean value 2.17 cm. per five minutes, for the scale-distance of 265 cm. Hence co = 2.17/300X530 = 0.00001364 radian per second. Inserting these data, 7 = io~8X6.2, which is much closer to the standard value than from the inade- quate theory and improvised apparatus (straw shaft) I had expected to get. It sufficiently substantiates, I think, the assumed viscous character of the resistance, and moreover shows that the Newtonian constant may be found with precision, in terms of the resistance to the uniform motion, broad-side on, of a cylinder with hemispherical ends, in air. TABLE 5. — -Apparatus II. Values of co. p cm. wXio7 P cm. wXio7 P cm. «Xio7 P cm. «Xio7 76 Mean 76 cm. 131 1 08 ISO i«5 121 126 IOI 98 1 08 92 126 121 IOI 137 5i Mean 51 cm. 150 131 143 143 157 150 137 35 Mean 35 cm. 126 131 131 105 1 08 137 98 137 116 143 30 Mean 30 cm. 8 Mean 8 cm. 112 112 121 IOO "1.5 121 157 126 143 131 137 116 116 144.3 123.3 II6.2 I3I-0 Mean total: co =0.000,012,53 • • • 8 cm. to 76 cm. 95. Angular velocity at different pressures. — The favorable result obtained in the preceding tentative computation of 7 induced me to make a summary of the values of co occurring in the different experiments of figures 143, 150, 151, 152, 153, for the needle of apparatus II. They were obtained graphically, by prolonging the lines of the triplets as shown in the figures and dividing the scale-rate (y) per minute by 60X2X265, the scale being at 265 cm. from the mirror. These results with their mean values are given in table 5. Except in case of the last series (p = 8 cm), the rate-lines pass through three consecutive points after turning. In the case of p = 8 cm., figure 1 53, they pass through but two, the curve being sinuous. The variation of co can not be ascribed to insufficiently rapid turning of M, though this was done by hand. The individual values of co are rather more variable than I hoped to find them, and this is also true to the mean values. The data do not, however, show any systematic relation to pressure, the differences being clearly referable to other incidental causes. 96. Water-bath. — The next step in the pursuance of this subject consisted in putting the apparatus II in a capacious water-bath of copper, sufficiently large to contain the movable weight M. An additional object to be gained was ACOUSTICS AND GRAVITATION. 121 IS i u 1 [ the location of the small leak in the apparatus, which after several complete overhaulings of the apparatus had escaped detection. The chief purpose, how- ever, was that of surrounding the needle with a medium of large heat-content and slow variation of temperature. The arrangement is shown in figures 154 (plan) and 155 (elevation), where mm is the needle with its mirror n, quartz fiber q suspended from a sealed torsion-head at the top of the tube t. The needle is surrounded by the case ww of waxed (impregnated) wood, closed on both vertical sides by plate-glass windows. The case is supported by the posts rr held in clutches anchored in the pier. The tube for exhaustion is at s. The water-bath BB surrounds the case completely, and the parts rrs pass through gasketed devices to prevent leak- 154 age. The rod v, also gasketed and clutched <73 by an arm fastened in the pier, carries the c sleeve u within the water-bath. It is to this that the crank-like arm carrying the weight M is attached and free to revolve from M to M,' the arc being limited by appropriate stops. The circular trough of the water-bath was about 30 cm. in diameter and 20 cm. high; the rectangular part 25 cm. long and 14 cm. broad. It contained over 30 pounds of water. The lead weight M, whose effective density is now 1 1 .3-1 , was approachable to i or 2 mm. of the glass walls of the case, the distance between centers M, m being left unchanged. The case was leveled and the tube t supported by separate clutches, as above. Moreover, to keep the glass window z parallel to the case, it also was secured by a special clamp from the pier. The results obtained with this complicated apparatus were not at first as orderly as expected from so large a mass of water, nor was the leak of about 5 cm. per hour on complete exhaustion removed. The conformity to the tem- perature of the water-bath was extremely slow. At first, at the high vacua, p = 4 to 8 cm., the results seemed to give promise; the readings were ^ = 6.95, 7.50, 7.00, 7.60, 6.95, 7.60, etc., in five-minute periods. Later, however, at a slight exhaustion, p = 68 cm., the mean excursions in successive hours were Ay = 0.69, 0.93, 1.02, etc., the normal excursion in the absence of the water- bath being over twice as large. It therefore seemed preferable to begin the work with long-period observation (per hour) for some time, in a manner simi- lar to the case of figures 119, 120 of apparatus I. Next day, at £ = 67 cm., the results as shown in figure 156 a conformed to Ay=i.i6; but they are otherwise no improvement on the prior results. Thereafter the apparatus developed a leak and had to be taken down to be thoroughly overhauled. It was then replaced and observations were made under full air-pressure. The apparatus having had a day for cooling, the excursion in the absence of water in the bath was again normal in behavior, 122 DISPLACEMENT INTERFEROMETRY APPLIED TO Ay = 2. 7. Water which had stood in the same room for days, alongside of the apparatus, was now poured in and observations of y made in half- hour periods throughout the afternoon, August 18. The readings as given in figure 156 b indicate a drifting needle, modified according as the drift to larger numbers was with the gravitational pull (5) or against it (N). This force is wholly secondary and almost masked. Its mean effectiveness is but Ay = 0.73 cm. The same experiment (half-hour periods) was continued throughout the whole of the ensuing day, August 19, the greatest attention being paid to carefully sitrring the water. The results are given in figure is6cand show only slight improvement. There is still an excess of drift, but alternations of gravitational pull are now usually apparent. The mean excursion is but Ay = 0.65 in the morning and Ay =0.42 in the afternoon. The change of temperature of the water-bath could not have exceeded a few tenths degree; but to give greater definiteness to this statement, the experiments were con- tinued in the same way during a third day, August 20, and parallel observa- tions made on the temperature of the water-bath by a tenth-degree thermom- eter. The results also given in figure 156 d are quite as erratic as the pre- ceding. In the morning the gravitational and radiant (repulsive) forces are all but equal. In the afternoon, with a different adjustment, the effect of grav- itation gradually emerges, frequently only as an acceleration or an impedi- ment on the drift. In fact, the needle is in continual motion and the results would have been more rational if the periods instead of being 30 minutes, had been shorter. As it is, the mean (p. m.) gravitational excess is but Ay= 0.42 cm. The gradual increase of the temperature of the water-bath scarcely amounted to one-fifth of a degree for the day. Now the actual temperature ACOUSTICS AND GRAVITATION. 123 differences of m and M must be less than this, since there is some accommo- dation, all of which shows the minute temperature discrepancies which are quite adequate to annul the effect of gravitation. Next day, August 21, with the water-bath removed and a free case in air, the alternations throughout the day given in figure 156 e were obtained, show- ing that the full deflection should have been (deducting of course the 7/11.3 part due to the presence of water) : &y = 4.45 a. m., and 4.30 p. m. The be- havior of apparatus /, however, announced this to have been a day of large excursions; but at least one-half of the Ay should have been exceeded. The drift observed must be ascribed to the very slightly rising temperature of the water-bath, of which the case partakes. It is an increasing pull toward the front of the case or, better, a decreasing push toward the rear, and agrees in sense with the pull of the weight M when in front. There is also marked drift in the air values in figure 156 e. I made these experiments with scrupulous care, as they furnish the only quantitative estimate of the thermal relations involved. It is astonishing that this standard and effective method of obtaining temperature constancy here utterly breaks down, notwithstanding the 30 pounds of water used. While the mere exposure in a medium of air (fig. 156 e) gives results which are fluctu- ating indeed, but not abnormal. In speculating as to a cause, it seems prob- able that the surface temperatures, on radiation through air, tend to become equal to a greater extent than happens in the other cases. 97. Attraction in vacuo. — The remarkable result embodied in figure 149, in which the gravitational attraction decreases in a high vacuum to about one- third of its value in a plenum, induced me to give the subject further attention and ultimately to construct a metal apparatus for the further study of this strange behavior. Unfortunately it was not possible to get apparatus II (case of wood, impregnated when hot with melted soft sealing-wax) quite tight. The leak was originally about 7 cm. per hour at the highest exhaustions. After many improvements and much labor I reduced this to 2 cm. per hour. I then found, however, that a supply of air was absorbed in the wood. This came out very gradually on reducing the internal pressure, but was reabsorbed on increasing it. Thus one meets the curious result of a gradually increasing or a decreasing vacuum, according to the direction of the change of the internal air-pressure. An increase by as much as 10 cm. or more was recorded at mean pressures. These annoyances, as they must be accompanied by slight temper- ature variations in the wood complicate the results. They can not possibly be smooth. I hoped, however, to get a mean result by operating both with increasing and with decreasing internal pressure. There is the additional difficulty of the change of atmospheric temperature during these essentially long-period observations, which operates with a lag. This induces a change of Ay, as explained above (figs. 119, 120, and 121). My first endeavor to eliminate it consisted in comparing the Ay2 of apparatus II with A^t of apparatus/, placed on the ad joining face of the pier; but although 124 DISPLACEMENT INTERFEROMETRY APPLIED TO there is here a general similarity observable, it is not sufficient for such a reduction, as tabulation showed. The plan finally adopted for want of a better was somewhat cumbersome and as follows: Those observations in which complete series of results were available between, say, p = 2 cm. and 76 cm. were first constructed and the mean result taken; i. e., the mean graph drawn through them, assuming Aj2 = 3 for a plenum. By aid of this, the data taken at all other pressures (incomplete series) were then reduced, assuming Aj2 = 3 for the plenum, throughout. The table of corrected data is too bulky for insertion here and the summary in figure 157 must suffice. The rates are sometimes too low, probably from escape of air from the wood, sometimes too high for the reversed reason of absorption ; but as to their testimony as a whole and for this particular apparatus II there can be no question, I think. The effective gravitational attraction when one of the bodies is in a partial vacuum decreases at a mean rate of about A^2 = 0.03 or about i per cent per centi- meter of mercury-pressure. This makes a drop of over Ay =2 for the com- plete range from plenum to vacuum, about as originally obtained. It has been stated that this result is inconsistent with the efflux results of figures 145 to 148, the interpretation of which is straightforward. Moreover, these effects are relatively transient, whereas in the case of figure 157 the results persist. If the graphs, figure 1 49 or 1 5 7 , are considered due to the influx resulting from the leak with consequent rise of temperature, then these graphs should slope downward from left to right and not upward. Under any cir- cumstances, moreover, the thermal effect should vanish in a partial vacuum of 50 to 60 cm. ; i. e., the initial or plenum A;y2 should be regained. There is no suggestion of such an inversion in figures 149 or 157. The effect continues in the same manner indefinitely, so far as observed. One can not but conclude, therefore, that so far as this evidence goes, there is here a different phenomenon superimposed on gravitation; in other words, that the effective gravitational attraction on a body in vacuum decreases i per cent per centimeter of pressure, for reasons not understood. 98. Apparatus No. Ill, brass and glass. — It was necessary, therefore, to endeavor to construct a new apparatus, which could both be freed from leak- age and would be without the porous annoyances of wood. The new apparatus /// was constructed much like No. II, except that a rectangular frame of square brass tubing, the frame being 23 cm. long and 5 cm. high in the clear, replaced the wood. Thick glass plates were cemented on in front and rear, the distance apart of their outer faces being 3 cm. and 1.3 cm. between their inner faces. The joints at the torsion-head, etc., were inclosed in cups, into which melted soft sealing-wax was poured, obviating leakage. The quartz fiber was somewhat finer than the above and, the center of the weight M = i .6 kg. being able to approach the center of m to # = 4-5 cm., the deflections, other things being equal, were larger. The needle moved more slowly. The greater deflection here, however, is a doubtful advantage in view of the longer period, for the readings are always very sharp. The needle when complete ACOUSTICS AND GRAVITATION. 125 with mirror weighed 1.813 grams; of this, the two shots contributed 0.595 gram each, the straw shaft 20.9 cm. long (between centers), the remainder, 0.303 gram, being in the mirror and hangers. The shaft thus weighed but about half as much as either shot. The space within which the needle was to vibrate was only a little over a centimeter; but it usually sufficed here, after the apparatus had been thor- oughly and uniformly cooled. Tested in a plenum of air, the apparatus showed itself extraordinarily sen- sitive to temperature, as instanced in the half -hour periods in figure 158 a, where the gravitational attraction is at first ineffective but eventually devel- ops. A part of this is due to the day, as apparatus I also shows it, though not to the same degree. A>'3 as high as 6 was finally obtained. The effect of exhaustion is shown at figure 1586 and there is here an apparent increase of Aj3 with the pressure p marked on the curves. But while A^3 increased from 1.6 to 3.1, the corresponding reading of apparatus 7 also increased (for a plenum) from A^x = 1.2 to 2.1. Thus in these complica- tions there seems to be no margin left for the above effect of exhaustion. The marked irregularities which a variable atmospheric temperature may produce is instanced in figure 158 c, observations obtained with apparatus II on the preceding day (N north and 5 south position of the attracting mass M) . It is not improbable, therefore, that the metallic frame III will in a different way respond to the anomalies which the wood frame showed at reduced pressures. Apparatus 777, as stated, was installed on the east-west wall of the pier, in front or facing south, replacing apparatus 77. The latter was now placed in a niche in the rear of the pier, facing a brick wall about i meter to the south. In this position it was completely screened from any direct radiation from the walls of the room having outside exposure. Apparatus 7 on the north-south face of the pier was left in its old position fronting (east) the 30-inch wall exposed to the morning sunlight outside, when present. The wall east-west faced by apparatus 777 was always in shadow. It was there- 126 DISPLACEMENT INTERFEROMETRY APPLIED TO fore thought interesting to pursue the records of these three instruments for a reasonable time, to determine their differences, and in particular to compare I and II at atmospheric pressure with the contemporaneous behavior of III under exhaustion. Figure 159 shows the mean results of experiments of this kind, made in half-hour periods at intervals of over 2 hours apart. Apparatus I and II were at full atmospheric pressure and in the location stated. The day (August 31) was overcast, with rain. Apparatus /// was kept exhausted to different degrees at the pressures marked on the curve. Although the three instruments differ in sensitiveness, they at first tell the same story, even as to No. II in its dark niche. Moreover, the high or moderate exhaustions in No. 777 have seemingly produced no marked difference, such as appears for instance in figures 149 and 157. One would thus be obliged to conclude, in the latter cases, that the air absorbed by the wood, in its issue or reentrance, was responsible for the effect in by obtained. During the course of the experiments (fig. 159) the apparatus developed a slight leak. Though this did not apparently influence the results, as the figure shows, it was thought well to remove it. In the long search which was necessary to find it, the fiber of the needle was unfortunately broken and in the trouble- some repairs one-half of it had to be sacrificed. This reduces the sensitiveness of 777, from its high value above, to an intermediate grade between apparatus 7 and 77, the latter being least sensitive. Though 777 was at first quite tight, it again after usage developed a leak of about 3.5 cm. per hour at complete exhaustion. As this on trial made no difference with the behavior of the metallic case 777, it was disregarded. In the work with the wooden case, however, even the smaller leak of 2 cm. per hour at the highest exhaustion was held accountable for the startling effect of figures 149 and 157. Such an infer- ence is therefore no longer tenable. One is left to surmise that the non- conducting wood retains the heat almost indefinitely, while the thin metal frame (half -inch tube of square section) loses it relatively soon. A comparison of the records of apparatus 7, 77, and 777 in their several locations will now be given (fig. 160), in order to discriminate between a possible absolute temperature effect, if it exists, and the relative effect which would naturally follow from changes of atmospheric temperature. The comparisons in figures 160 and 161 begin with the cloudy days on Sep- tember 2 (observations taken at the same time are in the same vertical), when the three instruments still show some similarity of behavior, though it is very meager. With the appearance of sunshine in the atmosphere without, the three instruments part company. Initially No. 777 was observed at different pressure (numbers, cm. of mercury on the curve) . At first the excursions under p = j6 and £ = 3 cm. were about the same, so that the final high values at p = ?6 cm. or at lower pressures must obviously be referred to other causes in addition to pressure. For this reason I temporarily abandoned the pressure work and made some of the subsequent observations for No. 777 in a plenum. They soon became extremely erratic. On September 4 the needle adhered to ACOUSTICS AND GRAVITATION. 127 the window and the torsion of the fiber had to be changed. Subsequently, repulsion in the morning and attraction in the afternoon were the rule, as shown in figures 162 a, 6, and c, N and S denoting a north and south posi- tion of M. The mean excursions, Ay3, were, for instance, as follows: Sept. 5. Sept. 6. Sept. 7. nh Ayj=— 2.00 nh Ay, = — 1.67 nh Ayt=— 2.92 I* —1-77 ih —0.48 ib —0.45 5h +2.73 4h +1.14 5h +2.14 etc. This apparatus, with a narrow (1.3 cm., inside) metallic case, and an east- west position of needle, is thus singularly and immediately responsive to tem- perature in a way opposite to No. /. No. Ill was therefore again abandoned as useless for plenum work. /4 15 tf 1T ' 18 ' 19 Apparatus II in a brick niche, narrowly and completely surrounded by brick-work, without outside exposure, and kept in the dark, nevertheless soon began to show a remarkable drift; yet it rarely behaved abnormally. On September 5, when this drift was exceptionally great, apparatus 77 and 7 are out of accord; but on a number of the following days the march of results is similar to apparatus 7, in spite of the difference of environment. The zigzags in 77 are much subdued, as one would anticipate from the secondary reflections received. In fact, No. 7 confronts an exteriorly illuminated wall in the morning and 777 in the afternoon. Thus the former is high in the morning and 777 in the afternoon, indicating the rotation of the sun. It is none the less difficult to understand why 777 registers repulsions in the morning (fig. 162). Heat seems to pass more rapidly through the thin metal case than, so far as temper- ature effect is concerned, it can enter the massive ball M. Finally, after observing that the needle, if stuck to the sides, could always be freed by exhausting the case, experiments with 777 were resumed at low pres- 128 DISPLACEMENT INTERFEROMETRY APPLIED TO sures throughout. These are marked on the curves, figures 160 and 161. It was inferred that as the air is removed, the effectiveness of convection currents ceases more and more, and therefore the earlier or low-pressure radiant forces should disappear. The resumption of the work with No. Ill under partial vacua (marked on the curve) after September 8 soon showed that pressures even as low as 35 centimeters were quite inadequate to remove the radiant repulsion specified. With these No. Ill behaved in a way which was exactly the opposite of No. 7. Consequently, after September 14, No. Ill was observed at high vacua only, the pressures ranging from p=i to 2 cm. of mercury. Under these circum- stances the records of No. 7 (observed in plenum) and No. Ill (observed in vacuum) are substantially alike, a result, at first, quite perplexing. It follows, therefore, that whereas in case of No. II (wood) the vacuum re- moves a radiant attraction, in case of No. Ill (metal) the vacuum removes a radiant repulsion, the radiant forces being throughout large as compared with gravitation. In fact, even at p < i . 5, the effect of exhaustion has not subsided, for direct tests showed that dy/dp = o.s; i. e., an addition of 5 mm. to y for each mercury centimeter of pressure reduction was still outstanding; twice as much, therefore, for the double amplitude Aj. This is a relatively enormous discrepancy. Thus, there must be an inversion of radiant forces with decreasing pressure p, from the repulsions at high pres- sure through zero to the attractions at low pressures or vacua. This complete opposition in the behavior of two apparatus (II, III) in the same position and alike, except as to the wood case of one and the metal case of the other, is difficult to explain. The active needle-end is relatively hot in the metal case and cold in the wood case, or vice versa, under like circumstances, if the simple theories adhered to above are applicable. With regard to the metal case 771, it is easy to show, by using a hot body in place of the mass M, that the radiant forces are increasingly repulsive for p < 4 cm. and increasingly attractive as pressures are larger. The exact pressure corresponding to radiant equilibrium is a little more difficult to determine. In relation to apparatus 7 and 777, therefore, this implies that in the former case M is warm in the morning (eastern exposure, radiant force attractive) and for apparatus 777, M is relatively warm in the afternoon (southern exposure). Consequently the afternoon repulsive force, operative in apparatus 777 under high exhaustion and warm M, balances the afternoon repulsive force in the plenum apparatus 7, with a relatively cold M. Hence the curves for the two apparatus are ti:e same in kind. In other words, one can thus, in a dark room, demonstrate the rotation of the sun. 99. Metallic case cooled by efflux. — The marked difference in behavior of apparatus 777 (glass and brass) as compared with apparatus 7 and 77 (glass and wood) makes it seem probable that the results of §90 will also be modified in the present experiments; and this is markedly the fact. Unfortunately, in No. 777 the needle hung very near the metallic bottom of the case, so that ACOUSTICS AND GRAVITATION. 129 great care in exhaustion (slow change of pressure) was necessary; but I do not think this had any influence on the following experiments. These are given in figure 163 and made (as above) by exhausting the case slowly in steps (v) of the pressure-gage, v=o to 15, 15 to 25, etc., cm., as marked on the curves, so that the pressures fall as £ = 76 to 61, 61 to 51 cm., etc. In the series 2, 3, 4, the weight M was removed and the case and needle reciprocate in the absence of M. In series i, the weight M was in front, attract- ing toward larger numbers y, of the scale. Series i and 2 were made after sun- down, at night ; series 3 and 4 in the day-time. The first exhaustion in series i (0.15 cm.) was obviously too fast; but all these series consistently show a new result, different from § 90 ; there is no inversion of the immediate effects of ex- haustion, according as they are made at high vacua or at low vacua. The first effect of exhaustion is always an increase of pressure on the hotter side; i. e., the radiometer effect prevails throughout. It seems, however, to pass through a minimum somewhere between 40 and 50 cm., though, as the rate of efflux can not be controlled, this is difficult to assert. In high vacua (about 70 cm.) the effect rapidly increases as it did above. Here there is no difference. If we compare series i and 2 another inference may be drawn. The equilib- rium positions from which exhaustions were made are marked by little circles. In series i these rise rapidly, which means that the weight M was at a lower temperature than the needle (and case) and that the radiant forces are largely removed by the exhaustion. Above 70 cm., however, they are again partially restored, and this is also the case in series 3 and 4, where the circles descend after v = 70 cm. Series 2 in the absence of the weight M shows that needle and case were about at the same temperature and that the exhaustion, therefore, has less influence, there being only relatively small radiant forces. In series 3 and 4, made in the afternoon, the case was colder than the needle and the 130 DISPLACEMENT INTERFEROMETRY. strong radiant forces resulting, rapidly dwindle as the exhaustion proceeds. They are never, however, quite removed, as was possible in the case of inversion of radiant forces, above. The effect of the successive exhaustions on the motion of the needle washere also very different from observations in the above work. The needle reaches its low positions within two minutes after the exhaustion. It then merely creeps to larger numbers y. I usually waited four or five minutes. By waiting longer, larger numbers would have been reached and no doubt the minima immediately after exhaustion would have been larger. Creeping re- sponds to a developing change of radiant force. The effect of exhaustion in removing radiant forces may be used in freeing the needle when it adheres to the windows of the case. As this in No. Ill was very narrow (1.3 cm. inside), the annoyance of needle pushing with its ends the two opposite windows was quite frequent on sunny days, as stated above. It is merely necessary to exhaust the case, without changing the torsion-heads to free the needle, and the apparatus then behaves normally; but if p is variable the attractions of M are not constant, nevertheless showing that a small part of the radiant force persists residually to modify the results. To use III with advantage, it must therefore be kept exhausted to above at least v = 50 cm. or £ = 25 cm., as is ultimately done in figure 161. In the experiments with the wood case, I was under the impression that the radiant forces for low and for high vacua would always be opposite in sign. The experiments of figure 163 show that this is not generally true. It is dif- ficult to ascertain the reasons which govern the sign of the former, and it is probably associated with distributions of temperature in the very narrow metallic case, in a complicated way. Thus by changing the position of the needle, results (of which series 6, fig. 163, is an example) were obtained in which the radiant forces for low vacua are at first positive but become negative at about 30 cm. of exhaustion. Inversion is thus met with even with the metallic case, though the pressure at which radiant equilibrium occurs is not a definite quantity. The fact that so much of the earlier radiant force may be removed by exhaustion may be regarded as a proof of its dependence on convection. The other proof is its gradual development; for it is not present immediately after turning the weight M. CHAPTER XL GRAVITATIONAL EXPERIMENTS. 100. Slender needles. — The object in making the above experiments was at the outset a mere endeavor to read the deflections of the gravitational needle by interferometry. For this reason a straw shaft was used, as it facilitated the adjustments. The plan succeeded without serious hardships. It was soon found, however, that what was being measured was not the gravitational de- flection, but a much larger value, resulting from the radiation forces simulta- neously present. Unfortunately, the fiber broke in the vibration experiments made to find its torsion coefficient. Using the microscope to measure the diameter of the fiber and using the known rigidity, the Newtonian constant came out 10*7 = 21, or about three times too large, nor were the individual deflections even approximately constant. Thereafter, I retained the needle, as I became specially interested in this interaction of radiation and gravitational forces, and a series of experiments showing a very striking response to solar radiation (even when screened off from the apparatus by the semi-submerged dark basement room) was carried out. The results given in the preceding paragraphs point out, incidentally, that if measurements of 7 are aimed at, the straw shaft is inadmissible and that needles having the thinnest possible framework should be used, as offering the least surface for the application of radiation pressures. One is tempted to interpret the discrepancies in question (at least with the needle in vacuo), directly in terms of light pressure or heat-wave pressures. Now, if the full solar constant be taken as 0.05 dyne per cm.2 per second, this pressure would be /=7oXio~6 dyne per cm.2 and in the case of the straw shaft considerably more than a square centimeter would be screened off by the large lead ball outside. The corresponding gravitational forces in the above experiments were about F— 2.4Xio~6 dyne. Hence the ratio f/F is over 30; so that if considerably less than one-thirtieth of the solar radiation penetrates the walls of the room its effect is in conflict with the gravita- tional forces. Figure 164 will make this tentative explanation clearer. Let mm' be the gravitation needle with the shot at its ends, B the large external lead ball. The radiation RR, acting symmetrically at both ends of the needle, remains ineffective. The radiation R'R' on the other side is screened at B, supposing B to radiate less, which is almost always the apparent case in the following work, though the reason for this effective screening is not quite clear to me. Hence B acts radiationally as if it were an attracting body. It is not necessary that R and R' be of equal intensity, but the drift of the needle (which is some- times excessive) would depend on this difference. Curiously enough, these forces are present in marked degree, even when the needle is a framework of 131 132 DISPLACEMENT INTERFEROMETRY APPLIED TO filamentary wire; i. e., the balls at the end of the needle offer an appreciable surface in any case. What militates seriously against such a straightforward explanation, how- ever, is the enormous effect of the presence of air. In a partial vacuum i to 10 cm. the error will not exceed 20 per cent in the summer (constant temperature), or 30 per cent in the winter (steam-heated room) . In a plenum of air, the results obtained may be 5 or 10 times too large, depending on the lateral area of the needle. It is difficult to escape the inference, therefore, that the presence of the ball B pro- 1/ e duces air-currents within the apparatus between ball and needle, and that it does this by modify- ing the radiation on its own side as compared with the radiation on the other side of the needle. The effect of a ball B, colder or hotter than the effective temperature on the other three sides, might thus be the same and its presence would be accompanied by a spurious attraction in relation to gravitation. If the ball B were to tranquilize the medium between m and B, there would be repulsion, and this practically never supervenes, so far as the present experiments go. 101. Experiments with slender needles.— To carry out the suggestions of the preceding paragraph, three types of apparatus were used. In the first of these (IV), the quartz fiber of the above apparatus (No. 7) which had shown deflections from A^ = 2.6 to 7.2 cm. in its old location, was inserted. The case, the needle, and the position, however, were new. The case, specially made for experiments in vacua, was of the form shown in cross-section (normal to the needle) in figure 165. The rectangular body BB was of cast brass with an internal clear space 1.2 cm. wide, 8 cm. high, and 25 cm. long. Long, thick, rectangular rubber gaskets r, r were placed on the inner rim, on which the glass plates P, P' reposed. The channel between the plate and the body was filled with resinous cement cc, poured in when molten and the brass body warm. The joint so made proved to be perfectly tight, though it was a little difficult to remove the plates for the change of needles, etc. The needle N was supported by the quartz fiber q, surrounded by a glass tube, and the joint made by an annular cup C, into which melted cement was poured for sealing, as indicated in the figure. A similar cup-like contrivance sealed the torsion-head at the top of the tube. To turn the head, the wax was temporarily melted. To mount the two shot (0.581 gram each and about 0.45 mm. in diameter) a phosphor-bronze wire, 22 cm. long, 0.64 mm. in diameter, and weighing 0.649 gram, was found to be adequate. The finished needle, with mirror and hanger, weighed 1.931 grams. The stem weighing but 0.0296 gram per centimeter, an allowance could be made for the gravitational attractions actuating it. ACOUSTICS AND GRAVITATION. 133 As the external weight, M = g^g grams, was the same as before, and the dis- tances between centers 4.2 cm., as well as length of needle also, the double deflections Ay to be anticipated in comparison with the former apparatus would depend merely on the scale-distances from the mirror. This ratio was, with all allowances, about 0.7, so that the new deflections should not have been much smaller than the old. They were, however, 5 to 10 times smaller, showing that with the new needle and new location, radiation forces of enor- mous amount (relatively) had been eliminated. The first experiments were made in a plenum of air. They are given in figure 1 66, the scale-readings of the chart (y) being taken at regular intervals about half an hour apart, both in the morning and afternoon. Even if we discard the readings on October 17, when the sun accidentally entered the room, the deflections are far from regular and there is considerable drift. Table 6 shows the mean double deflections (scale-readings) Ay in centimeters (scale- distance 261 cm.) for the mornings and afternoons of successive days. The values are least on dark days and large on clear, sunny days, as in the work above. In spite of the filamentary shaft of the needle, therefore, the data are still quite unsatisfactory, ranging from 0.70 cm. to 1.5 cm., and in the morning they are usually larger. The apparatus was now exhausted to about i cm. of mercury, and the experi- ment repeated in the same way. The new data (scale-readings y) are given in fig. 167, p. 138. The results are not only smaller in amplitude , but much more regular, showing that much of the radiation effect has been eliminated. Never- theless, there is some drift in the lapse of time, to be attributed to radiation- pressure. The double amplitudes Ay in table 6 , constructed graphically in fig . 1 68 (p . 1 3 7 , on a tenfold larger scale) , bear out the same inference. Ay varies 134 DISPLACEMENT INTERFEROMETRY APPLIED T O TABLE 6. — Values of double amplitudes Ay, mean per day. Apparatus IV, M =949 grams; m = 0.581 grams ;R= 4.2 cm.; ^=261 cm.; 2/=2i-9 cm.; Bronze needle. Plenum. Apparatus IV. Needle in vacuo. Date. A.M. Ay P.M. Ay Remarks. Date. A. M. Ay P.M. Ay Remarks. Sept. 14 15 16 17 18 19 20 21 0.84 .70 .72 1.28 •99 I.I? i .04 .78 Dark day. Do. , Do. Sun. Do. Do. Do. Sept. 24 25 26 27 28 29 30 Oct. i 2 3 4 5 0.46 .41 •45 •45 •44 .46 .42 .40 •40 •44 •43 Sun. Do. Do. Dark. Sun. Rain. Steam heat off. Do. 0.90 .68 1.90 1.18 1.48 i-47 94 0-43 •44 •44 •45 •48 4i •39 •45 •45 •44 .48 Apparatus II. M = i )590grams;m= 0.600 gram 1^=4.8 cm.; L=3io cm.; 21—22 cm.; old needle. Apparatus II. Needle with glass stem. Date. A.M. Ay P.M. Ay Remarks. Sept. 15 16 17 18 19 20 21 1.07 1.18 1.14 I.OI 1 .24 0.50 .61 •71 •03 •30 •57 •58 Dark. Do. Sun. Do. Do. Do. Date. A.M. Ay P.M. Ay Remarks. Sept. 28 29 30 Oct. i 2 3 4 1.46 1.48 1.69 2.40 —4.06 -2-57 -1.65 1.97 2.38 3-40 1.48 1. 06 1.05 Rain. Steam heat. Apparatus V. Af = 1,490 grams; old needle. Date. A. M. Ay P.M. Ay Remarks. Sept. 17 18 19 20 21 0.51 .90 •63 .70 0.80 •55 •87 .69 •33 Apparatus V. (continued) Apparatus V. Needle with glass stem. Date. A.M. Ay P.M. Ay Remarks. Date. A.M. Ay P.M. Ay Remarks. Sept. 25 26 27 28 29 30 0.79 1.19 .92 •45 .64 •42 0.80 i .01 .81 •54 •56 .40 Rain. Oct. i 2 3 4 0.44 • 53 •40 .27 0-55 •39 •35 ACOUSTICS AND GRAVITATION. 135 between 0.39 and 0.45 (if the exceptional results are excluded), depending on changes of temperature outside of the laboratory. The two graphs, morning and afternoon results, as a whole are similar. After removing the needle, the torsion coefficient of the quartz fiber was found by the vibration method, two small brass cylinders of moments of inertia AT=o.o4g8 and 0.0486 being used in succession, the periods being 7=5.51 and ^=5-39 seconds. Unfortunately, both these data are rough, for it was found difficult to make the small cylinders rotate axially quite at will. Accepting the mean temporarily, the value of the constant 7 may be then computed as y from the double deflection Ay, if R = 4.2 cm. is the distance between the center of the masses M=Q49 grams and m= 0.581 gram, L = 26i cm. the scale-dis- tance, and /=n cm. the semi-length of the needle. The mean value thus obtained was 7=io~8Xi8.2 Ay so that for y = io~8,X6.66 the deflection should have been 0.37 cm. Even the vacuum values are thus excessive, except on the dark days, September 30 and October i ; but it is probable that with allowance for the stem attraction, the data for Ay found will not be far from the correct values. At all events, it is clear that observations made with the needle in vacua show deflections which are an enormously closer approximation to the truth than the plenum values, and that further pursuit along these lines will probably lead to trustworthy data. The mean a. m. and p. m. displacements were Ay =0.44 cm. and Ay = 0.43 cm. This is an excess of less than 16 per cent, most of which is referable to the attraction of the stem of the needle, as appears in the next paragraph. 102. Torque exerted on the stem. — If the center of the mass M is at right angles to the needle of semi-length /, and at a distance R from it; if * be measured from the end of the needle to its center (* = /), and if dx be at a distance r from the center of M, it is easily seen that the torque will be lMpdxR ,, . *> where p is the mass of stem per unit of length. The integral^of this expression is or t=(yMp/R) where the coefficient yMp/R is the total attraction in the direction R for a long needle. IfT=(vMm/R*)l, then, 136 DISPLACEMENT INTERFEROMETRY APPLIED TO For the given needle p = o. 65/22 =0.03 gram per linear cm., /=n cm., 4.2 cm., m = 0.58 gram. Thus Hence the discrepancy due to the attraction of stem of needle actually amounts to nearly 15 per cent of excess in 7. If now we write T=yTf, t = yt', and if T is the modulus of torsion and y' the above approximate constant, T== whence where y' = KAy. For K the more accurate mean value of §104 should be in- serted, viz, K = io~7 X i .73 ; while i/T = o. 145 and Ay = 0.437 (means in preceding paragraph) . Hence 1+0.145 This is within i per cent of a normal result, and in comparison with the orig- inal errors supplies the strongest evidence in favor of the exhaustion method pursued. But to rate the last result as to its trustworthiness, it is nevertheless necessary to consult figure 168 or the corresponding table on the individual values of Ay, which are quite variable. It seems, however, that in a series of results extending over a long time interval the radiation error is eliminated in the vacuum experiments. 103. Contemporaneous experiments. — To accentuate the importance of the vacuum method, a series of parallel observations were made at the same time with apparatus V and the old apparatus II (closed chamber of waxed wood) . In the work with No. V, the glass case of No. I was again utilized in its old position, but a new quartz fiber was inserted. This was, unfortunately, thicker than was desirable. Tested with a small brass cylinder (mass 1.774 grams, diameter 0.474 cm.) of moment of inertia N = 0.0498, the period was found to be 7 = 2.51 seconds. Using the preceding equation, where now L = 378 cm., is the scale-distance from needle and /= n cm. the semi-length of the latter, M= 1,490 grams, ^ = 4.24 cm., 9.87X0.0498 (4.24)' 378XiiX(2.5i)2 1490X0.59 Hence a deflection of only 2 mm. was to be expected. The results in figure 169 are very much larger, the old straw-shafted needle of apparatus No. / being used. In fact, in the experiments of the first three days there was so much drift that they were thrown out. There is even an ACOUSTICS AND GRAVITATION. 137 abundance of drift in the last three days. Figure 170 gives the successive mean double deflections Ay, for a. m. and p. m. These deflections are three to four times too large on the average. The straw shaft in apparatus No. V was now withdrawn and a needle with a slender glass shaft substituted. Its dimensions were: stem-length 22 cm.; diameter o.i cm. ; weight 0.326 ; each shot 0.620 gram; total weight with mirror and hanger 1.573 grams. The readings made in a plenum of air are given in figure 171, p. 138. They are much more regular and there is less drift than in the straw-shaft experiments, so that some advantage has been gained, but not nearly enough. The double deflections given in table 6 and figure 170 show a curiously gradual approach to the correct value of Ay, but this is probably accidental. Even the final deflections are about twice too large on the average. The experiments with the old apparatus II, also in its former location behind the pier, shut in on all sides but one by heavy brick walls within less than a meter, are shown for the straw-shaft needle in figure 172. They are very irregular and the drift is excessive. On September 21 (R and F refer to rear and front positions of M) there is an inversion. The data for Ay in figure 173 are equally erratic. This needle was then replaced by one with a glass shaft, 22 cm. long, o.i cm. in diameter, and stem-weight 0.380 gram; the weight of each shot was 0.600 gram and the total weight of the needle 1.585 grams. The new readings given in figure 174 are even worse than the preceding. The values of Ay in table 6 are equally lawless with enormous inversions (F and R exchanged), mean- ing that repulsions are in excess. It is extremely difficult to surmise why this apparatus, in what would be regarded an ideal location, in the dark and virtually constant temperature, should behave in this way; moreover, with the filamentary stem even at a disadvantage as compared with the straw shaft. Electrical charges in a damp basement room are hardly possible. It is more probable that the heavy walls are interchanging heat reservoirs and that the apparatus is in the midst of the radiation. Tested with the brass cylinder, JV= 0.0498, the period of the fiber was 7 = 6.13 seconds. The other quantities in the equation were M= 1590 grams, w = o.6oo grams, 7^ = 4.8 cm., L = 3io cm., I— n cm., so that 7 = io~89-26Ay Thus Ay should have been about 7 mm. Apart from drift, these low values were never reached, even with the straw shaft. Ay, moreover, was always markedly larger in the afternoon. In No. IV they were usually larger in the morning. 104. Filamentary needle. — I thought it desirable to carry the slenderness of needle one step further by making the stem out of the lightest wire possible. 138 DISPLACEMENT INTERFEROMETRY APPLIED TO A resilient, straight bronze wire, 21=22 cm. long, was selected for the purpose, 0.24 mm. in diameter and weighing 0.0044 gram per centimeter. This was trussed (fig. 175), at about one-fifth of the length from the end of the needle by a thinner brass wire, stretched from the top of the hanger and mirror sup- port. As the truss was not sufficiently stiff, laterally, to support the two shots, (m = 0.6295 gram each), it had to be strengthened by a thin glass stem 0.47 mm. in diameter and 14 cm. long. This was tied on at its ends with fine silk thread and a little wax. Hence the stem would offer no appreciable purchase to the radiant forces except (unavoidably) at the shot, which were about 4.5 mm. in diameter. The results showed that the limit had already been reached in the preceding needle, so far as the stem is concerned. The constants* of the apparatus were thus M = 949 grams. m = 0.6295 gram. ^ = 4.3 cm. /=n.ocm. L = 26icm. Tested by a small sphere (ball-bearing, diameter 0.633 c"1-. mass 1-0415 grams) of moment of inertia TV = 0.041 73, the slightly modified quartz torsion- fiber showed a period of T = 5.06 seconds. This, in the equation, § 101, makes 7 = io~7Xi-735 Ay Ay 0 = 0. 3 84 cm. The same result was obtained with the needle itself vibrating in vacua. The moment of inertia was here TV = 152.4, to which 5.2 per cent were to be added to allow for the accessories (wire truss, hanger, and mirror) . The period found was T = 3i3 seconds. This in the equation gives 7 = io~7Xi.726 Ay, or Ay 0 = 0.385 cm. In fact, it seems to me that with care as to the increments of N contributed by the parts of needle, this method is preferable, in spite of the tediousness of finding the long period. *R and / were found by measurement with a distant telescope and a scale near Mm. ACOUSTICS AND GRAVITATION. 139 105. Observations. — The beginning and end of a series are indicated by circles. These were made on the same plan as above, readings, y, being taken at intervals about 30 minutes apart, and they are given in figure 176. The double amplitudes, Ay, are shown in figure 177. The circles distinguish a. m. and p.m. means. The observations in a steam-heated room with its variable temperature (20° to 30°) would be inadmissible; but here, where a test is aimed at, this change of temperature is even desirable. At the outset a few observa- tions were taken for a plenum of air. These have the usual excessive value, being on the average Ay = 2 . i cm. for the morning and Ay = i .6 for the after- noon periods. It should be Ay = 0.38 cm. The smaller values of Ay occur when the steam heat is shut off. 09 2{V 177 =• ou-o' 1 Itacuwn-l Oct. The observations for the needle in vacua (fig. 176) lie indifferent regions, for the reason that a number of adjustments had to be made at the distant telescope and scale. These moved from day to day as the temperature of the room varied, without, however, affecting the differences or double amplitudes Ay. The latter (fig. 177) have a definitely larger value as a whole than in the preceding paragraph, ranging here from Ay = 0.3 7 cm. (rare) to Ay = 0.55, if we discard the exceptional value (0.66) obtained with a open window. The mean values in the morning were Ay = 0.49 cm. and in the afternoon Ay= 0.47 cm. Both are much too large (nearly 30 per cent) and the curve of suc- cessive values of Ay is inadmissibly meandering. The mean summer data of the preceding paragraph for the same apparatus (IV) and location were Ay= 0.44 cm. in the morning and Ay = 0.43 cm. in the afternoon. The effect of a heated room is thus an increment from Ay = 0.43 to Ay = 0.48. Moreover, in the latter case there is no equally appreciable correction for stem attraction as in the former. Vacua from 0.6 cm. to 6 cm. were employed, but no difference was detected attributable to this cause. Obviously, the radiant forces now impinge on the shot, the thin stem of the needle having no advantage over the preceding cases. 106. The residual radiant forces. It is interesting to inquire as to the amount of pressure variation and velocity of air-currents implied in the esti- mate for radiant forces. The energy equation may be written, if (p— Ap) = p nearly, 140 DISPLACEMENT INTERFEROMETRY- where A£ is the decrement of pressure and Afl the corresponding increment of the velocity of air (from rest) at constant energy. If a is the sectional area of the shot and/ the radiant force, K its value in terms of the contemporaneous gravitation force, it follows further that Ap =//a, and, therefore, apR? For a plenum of air (p = o.ooi3 and a = o.i6 cm8.) this comes out as (Aa)2 = KXo.o2i. Thus, if K = $, Aa is about 3 mm. per second and A^ = 6Xio~5 dyne per cm.2 In the experiments in vacua, however, K has decreased to 0.3 and p to about 0.001293/76 = 0.000017. Thus (Afl^-KX 0.02 1X76 or Az> = o.7 cm. /sec., while Ap is but io~6 X4-i dynes /cm2. That currents of 7 mm. per second should be possible is difficult to conceive ; but the momentum (vp) carried for the 7Xi present case and the plenum is in the ratio of - - =0.03, which accounts 3X76 for the advantage of the vacuum. CHAPTER XII. MISCELLANEOUS EXPERIMENTS. I. HEAVY GRAVITATIONIAL SYSTEMS. 107. Attractions in case of a heavy needle. — The above gravitational ex- periment had to be discontinued when the steam heat was turned into the building. It seemed worth while, however, to make a few tests with a needle so heavy, that the radiant forces might be small in comparison with the grav- itational attractions. Here, moreover, the air resistances to the motion of the needle could possibly be disregarded, and the excursions, therefore, treated as a case of nearly uniformly varied motion, resulting from gravi- tational attraction. 108. Apparatus. — The apparatus for this purpose had the usual form (fig. 178), MM' being two heavy balls of lead, each weighing about 1,500 grams originally. They were counterpoised in suspension from the three-eighths-inch aluminum tube KB', effectively R = 29.3 cm. long. This was supported by the thin brass tube t and brass sleeves, with an appropriate brace of bronze wire b. The torsion- wire w, of thinnest hand-drawn steel (music wire) , was adjustably attached above to a torsion-head, anchored in the pier and below to the tube t . By passing w around the threads of a fine screw above on the torsion-head, the needle could be raised and lowered at pleasure. The length of the wire was 154 cm. and its diameter about 0.022 cm. Tested with a variety of brass cylin- ders of known moments of inertia, the torsion coefficient was about r= 154 in the original installment. The apparatus was surrounded (except as to the wire w) by a closely fitting glass case. Care was taken to avoid steel or iron in moving parts. The attract- ing mass corresponding to M was on the outside of the case and moved easily on a circular track, as in the earlier work. To read off the deflections, a small mirror m, with telescope and scale at a distance of Rf = 2go cm. was first adopted. To use the interferometer it was merely necessary to replace m by a silvered strip of thin glass plate; for small additions of weight are here of little consequence. Hence the force by which M is drawn from without will be r T s 154 where s is the deflection in centimeters. If we estimate the moment of inertia of the needle as 2X1, 500X302 = 2. 7 X io6, the period should be about 14 minutes. We may therefore assume that the period will not exceed 15 minutes in interpreting the excursions. The gravitational force for an attracting weight of the same value (M = 1,500 grams and 6 cm. between centers) would be roughly 141 142 DISPLACEMENT INTERFEROMETRY APPLIED TO Thus the deflection would be 4-2X10-* s = — — - = 0.47 cm. 9X10- which, on being doubled by commutation, is a little short of a centimeter. A tenth millimeter of the scale is thus i per cent. The wire, however, would be capable of sustaining much larger weights and the external mass could be increased at pleasure. On the quadratic interferometer with achromatic fringes, if b is the breadth of the ray parallelogram, i the inclination of mirrors to rays, AJV the play of the micrometer, the angle s/2R' of the needle corresponds to ') = AN cos i If b= 10 cm. and 1 = 45°, therefore (2(5/2^') =0.94) AAT = 10X0.94 = 0.023 cin- °-7°7Xs8o which may be read to io~4 cm. directly or to 4Xio"5 per fringe. With good fringes the accuracy, so far as mere reading is concerned, should be within o.i per cent. -/23456T89 179 When the telescope and scale are used, the divisions at a long distance are still sharp, but it is often difficult to clearly distinguish the small numbers. For this reason I used the notation shown in figure 179. These dots may easily be put on a glass or other scale in any size, with a sharp pen and india ink. They are very clear at all distances. They have the further advantage that they are equally serviceable, whether the scale be upside down or reversed right to left. 109. Observations. — The chief purpose of the observations in the cold sea- son was to determine the effect of the steam-heated room on the degree or quiescence of so heavy a needle, weighing over 3 kilograms. In the earlier work the absence of iron in the framework was not considered, and the data, though of the same character, were discarded; but the lead-aluminum needle behaved not much better. The hand-drawn steel wire, however, disclosed a second discrepancy of considerable interest, incidentally, as it showed marked ACOUSTICS AND GRAVITATION. 143 viscous deformation, due to residual torsional strain. Figure 180 is an example, observations being taken several times a day, for as many days as the essential narrowness of the case permitted. Apart from the daily kinks, the curve shows steady progress at a rate of about 0.08° of torsion per day. The wire, in other words, developes torque at the rate of about 0.21 dyne — cm. per day, wholly as the result of the breakdown of unstable molecular groups which promote viscous deformation. How the wire acquired this concealed torsional strain is hard to understand. It could not have developed out of the intense tractional strain which it carries, for there would be no reason to account for the particular direction of rotation. The wire had been reeled on a drum with a diameter of about 8 inches ; it is probable, therefore, that it was reeled helically, or in a way to impart a perma- nent twist, allowing the intense tensile strain to develop a torsional strain in the lapse of time. The free wire resting on a plane assumed a screw shape 44 with a pitch of about 2 feet. Under the heavy weight M, the steel wire is at relatively low viscosity; for this implies accentuated continuous breakdown of molecular groups, and therefore greater facility for the exit of the set tor- sional strain. The reason to be given for the kinks in the curve is more uncertain. The telescope and scale were attached to one of the ground walls in the basement of the building; but this does not imply complete freedom from thermal and other discrepancies. Again, interferences at the suspension, induced by tremors in the building, might put the needle in slow vibration. Finally, the possibility of radiant forces, from the closely fitting case surrounding the needle and from the lead weight, can not at once be dismissed, without measurement. Prob- ably all these factors enter, and the average discrepancy is about as large as the gravitational attraction (on the side) to be measured. Unless immense masses of lead are used on the outside, the experiment has little to recom- mend it, even though it is here performed under the unfavorable conditions of a steam-heated room. With regard to the graph, figure 180, it appears that the increase of irregularities (which become more marked as the room is more vigorously heated) is accompanied by correspondingly increased 144 DISPLACEMENT INTERFEROMETRY APPLIED TO slow vibration of the needle. It follows, therefore, that in spite of the large masses used, the radiant forces are by no means negligible, even in the pres- ent cumbersome experiment.* Finally, it is interesting to note, since the torque is T = 0.265, the angle 6 = 0.00175, and the daily detorsion 5 = 0.8 cm., that energy is being dissipated at the rate of 70/2, or 0.00014 erg per day, by this slight torsional strain alone. Much more is doubtless released by the decay of the intense traction (wire- drawn) which the wire carries. If the metal were not exceedingly opaque it would probably be phosphorescent. II. THE TORSIONAL MAGNETIC ENERGY ABSORPTION OF AN IRON CONDUCTOR. 110. Apparatus. — The relations of torsion and magnetization have been studied by Wiedemann, Auerbach, and many others since, chiefly in longitudi- nal fields. The torsion effect produced by a circular field is very small and difficult to ascertain ; therefore, I thought it of interest to make some measure- ments of this kind, using the displacement interferometer and achromatic fringes. The results were very definite and would easily have admitted of precision. The apparatus is shown in figure 181, where i-m, H TJTI AB is a thin low-carbon steel tube, effectively 55 cm. long, j having an average diameter of 0.875 cm- and walls 0.076 cm. thick. The tube is firmly clutched below by a clamp, but free 181 above. It carries the mirror mm', which is a strip of thin plate glass, silvered, and slightly adjustable about vertical and horizontal axes. The ends receive the component rays of the interferometer, so that any slight rotation of mm' about the vertical axis is at once registered by the displacement of fringes. Finally, a strong electric current may be passed through the length of the tube, entering at A and leaving by the mercury cups C. The current must be reversible at pleasure. ill. Observations. — The fringes are displaced (i. e., the tube receives mag- netic set) immediately after closing the circuit. Closing it any number of times thereafter is ineffective, to the fraction of a fringe. There is practically no temporary effect. On reversing the current, the fringes are markedly displaced in the opposite direction, again, to hold the new position, however often, the current is made and broken thereafter. The effect of reversal is static and may be repeated indefinitely. To obtain a temporary effect, I surrounded AB with a massive iron tube, about 6 inches long and 2 inches in diameter, clamped at the top B, but other- wise free from it. Even now, with currents up to 20 amperes, I observed no temporary effect in excess of the quiver of the fringes. *The relatively large temperature coefficient of torsional viscosity is also effective. Similarly, the thermal changes of rigidity reciprocate with the concealed torsional stress. ACOUSTICS AND GRAVITATION. 145 The following data were obtained for the fringe displacement AJV, showing the permanent effect of reversing current. 3 amperes, AN = ? 10 amperes, AN = 0.00031 20 amperes AN = 0.00065 Below 3 amperes I was unable to observe a displacement. For larger currents the twist increases proportionally to the current passing the tube. It is prob- able that here, as in similar cases in magnetization, fields below a certain small value are ineffective, as though there were static friction. A further observation is to be made. If a certain magnetic set is produced by a stronger current (say 20 amperes), then a weaker current (say 10 amperes) is unable to modify it, provided both currents are in the same direction. If this weaker current is reversed, the change of twist corresponding to the cur- rent will appear. The absolute value of the set thus depends upon the past history of the iron, however varied this may have been, with the understanding that the set produced by the maximum current in either direction is character- istic of the strength of that current. Differential values, in other words, re- main the same. 112. Data. — Finally, the numerical equivalents of the observations made may be stated. The elastic torsional coefficient for a tube of the dimensions given may be computed with the usual equation, using the differential form dT/dr = 2Trr3d/l with the mean radius r and length / and taking the rigidity n as 8.2Xion. If T is the torque corresponding to the twist of 6 radians, the relation was found to be T=io9X4-70. On the interferometer, if the breadth of ray parallelogram is b = 10 cm. and A9 is the rotation of the mirror mm' around a vertical axis corresponding to the displacement of mirror AN (the mirror being at 2 = 45° to the rays) the relation will be A0 = 0.071 AN, since A6 = AN cos i/b. Hence, if we assume that the resistance to magnetic set is the same as the elastic resistance for the same twist Ad, the above values of AN will correspond to the following data : Current. AJVXio5 Torque Xio~6 Energy. E/vo\. 10 amperes 31 cm 1.03 1.13 ergs o.io 20 amperes 65 2.16 4.97 0.43 The energy E potentialized by the magnetic set is computed as TAB/ 2 . From the volume of iron in the tube, 11.5 cm3., the data of the last column follow. My conception of this phenomenon is that of two concentric circular fields in opposite directions, one within, the other on the outside of the tube, intro- ducing a rather intense vortex sheet in the thin walls of the tube, in which the circular fields terminate. 113. Longitudinal field. — The longitudinal strain produced by a longi- tudinal field is well known. The question may be asked whether, in this case, in a free iron bar there is any corresponding torsion. This was easily answered by slipping a helix over the steel tube. Feeding it with I amperes, the micro- meter displacement AN was successively 146 DISPLACEMENT INTERFEROMETRY APPLIED TO /= 1.5 4.5 9.4 1.5 am. io6 AAf = 15 10 10 5 cm. This means, no doubt, that the residual torsion left by the last experiments is being gradually eliminated by the longitudinal field. For the relation to / is merely that of a decay. The field of the helix was H = 2-ji within, but it covered less than one-half the tube. At io amperes the mean field was prob- ably less than 100 gausses. Nevertheless, I can not assert, in the lapse of time and many applications, to have quite eliminated the torsion strain. A per- sistent displacement of one or two fringes (set) may have remained to the end, with each reversal of current ; for violent commotion due to the longitudi- nal strain here interferes with the measurement. III. LIQUID REFRACTION NEAR A SOLID SURFACE. 1 14. Methods and first experiment. — An interesting discovery has recently been announced by Mr. F. Twyman (Nature, Nov. 20, 1919, p. 3X5- v°l- I04). indicating a marked increase of the index of refraction of certain liquids at the surface of contact with a polished glass surface. As the method of experiment is not given, I was induced to make a few trials with the same end in view, with the aid of the self-adjusting interferometer which happened to be available. The advantage here is this, that separate installments are immediately possible and require no long searching for fringes and that the latter may be had in any size. At the same time, there is sufficient ray separa- tion possible for all purposes of the present kind. £ \ AM, 182 The interferometer is shown in figure 182, receiving light at L, reflected by the four mirrors M, Mr, N, N', and observed in a telescope at T. CC is a long (15 cm.) metallic cell about 1.5X1.5 cm.2 in square section, with plate-glass faces at the ends. Through it pass the two component rays ab, cd, of the inter- ferometer. The liquid is introduced through a small hole h in the top of the cell, closed by a glass plate. The cell CC is on a horizontal micrometer m, so that it may be moved trans- versely to the rays ab, cd. By this means either one or the other of these rays is brought as near the metallic edge of the cell as desirable, until the fringes disappear with the extinction of the ray, the other being alone in the field. I examined benzol and ether but in neither case was there the slightest slipping of fringes with the motion of the micrometer-screw m, up to the point at which they vanished. The experiments were varied in many ways which need not be detailed here. I did not, however, expect any positive result from ACOUSTICS AND GRAVITATION. 147 this experiment, seeing that the range within which any such capillary effect would be active would probably be too narrow to admit of visible fringes, even for a thin blade of light at L. 1 15. Second experiment. — The next experiment was more promising. The cell CC in figure 182 was removed and replaced by a plate of glass A A, figure 1 83 , normal to the rays ab, cd. This is without effect on the fringes. Two strips of glass B and C of identical thickness were then mounted on opposite sides of AA and held in place by wooden clips. Through them the rays ab and cd, respectively, passed. The strips BC are without effect on the fringes if AA is firmly clamped in a vertical plane. If, however, a drop of water or other liquid is placed at the plane of contact of one strip only (at CA, for instance), the fringes are immediately displaced in view of the entrance of a capillary film of liquid between A and C. Even with- out any spacing inserted between C and A, the displacement is quite large, four fringes for instance. These experiments are necessarily made with the achromatic fringes, since the displacement is practically instantaneous. They may be obtained in any size by rotating the mirrors N, Nr, or both, around a horizontal axis. A vertical axis is necessary to secure coincidence of slit-images. The rays ab, cd may be separated at pleasure by sliding the mirrors M, M' (together) in the direction of the rays. The fringes are necessarily horizontal and the index obtained holds for mean wave-length. True, with the use of the spectro- telescope the corre- sponding spectrum fringes are immediately at hand ; but it would not be easy to determine their displacement, even with the clue given by the achromatics. For guidance as to the quantitative relations, the equations of the last report* may be used. If one of the films is air, the index M= i — zB/\- of the liquid, at the wave-length X, for a thickness of liquid film e, is (1) ft = i - 2£/X2-fC &x/e* where C is the constant of micrometer when its displacement reading is A#; so that for n fringes (2) n\ = C&x Ax may be in arbitrary units, like those of the Billet wedge micrometer, for instance. The usual value of the factor C is of the order of C = 3 X io~4. Hence in equation (i), M — i-f 2J3/X2 being constant for a given liquid and X, A# is proportional to e. If the parenthesis is taken as 0.6, for instance, A* is 500 times larger than e. Again, on differentiating (i) apart from details and if 5x is one scale-part, edn = 3 X io~4; or e= 0.3 0.03 0.003 cm. 'Carnegie Inst. Wash. Pub. No. 249, Part IV, p. 81. 1919. 148 DISPLACEMENT INTERFEROMETRY APPLIED TO Thus as o.i scale-part is given by the vernier, a film 0.03 mm. thick should still insure an accuracy of one unit in the second place of M. Moreover, if Sx— i, the number of fringes is A fringe thus corresponds to 0.2 scale-part and is again capable of being subdivided. Tests would, therefore, be narrowed down to finding in what degree dx/x — 8e/e is persistently zero, or dx/de constant, as e decreases indefinitely. It is again improbable that the interferometer will register a sufficiently small e. An experiment made on the difference of an air-and-water film, figure 183, devoid of spacing, showed Ax = 0.6 to 0.7 scale-part. On standardizing the fringes, it was found that 5.8 fringes came to the scale-part. Hence the increase 8p. of refraction due to the film is equivalent to o.65X5-8 = 3-8 fringes. If for water 6^ = 0.33, equations (2) and (3) give us BA and CA, figure 183, not being optic plate, do not approach closer than this. The strips were now left in place and a film of air and ether and water were compared in succession. In this case it was possible to see the ether evaporate while the fringes changed from the air to the ether position. The water does not so easily evaporate between the plates of glass, hours being necessary be- fore it is gone. The mean results of four experiments, each corresponding to fixed plates, were Ether, AX=I.IO Water, Ax = o.99 We may with the above equations (approximately) put 0-333 /*' — So that for ether M = i .37 . This is quite as near the actual index of ether (M = 1.36) as the work with such thin films will admit. And yet the films are not thin enough, probably, to suggest an increase of /*, as a result of the Twyman surface effect. The interferometer, so far as I see, is thus not adapted for work of this kind. IV. COMPARISON OF TWO INDEPENDENT SETS OF FRINGES. 116. Apparatus. — This experiment was also made with the self-adjusting interferometer, for the purpose of obtaining two juxtaposed sets of fringes, of nearly the same size, one pair being used as a sort of vernier on the other. If converted into overlapping spectrum fringes, these should appear and vanish periodically. Another method is given in §62. The apparatus is shown in figure 184, where MM, NN' are vertical mirrors ' half -silvered), L, L' two sources of white light, C a Billet wedge compensa- ACOUSTICS AND GRAVITATION. 149 tor, T the telescope. The rays from L follow the paths a'bab' and b'aba' and again emerge toward L, but are caught in part and reflected by the plate of glass m and sent to T. The same is true of the rays from L'. Hence four rays pass the side of the ray rectangle and interfere in pairs at T. By moving the set of mirrors NNf bodily right or left, these rays may be separated into two pairs of two rays each, passing the corresponding plates of the compensator C. The observer at the telescope T sees two vertical slit-images, which may be adjusted to lie exactly side by side with their near edges just touching. Each carries its horizontal achromatic interferences, nearly v but not quite of the same size, since the glass-paths of /\L> the two rays will not, in general, be rigorously the same. »/y / .-^ Hence, in moving the screw of the compensator C, intro- ^/ " a/ '• / ducing a difference of glass-paths, the two sets of fringes <** ""/JT 184 move in the same direction, but are only periodically horizontal prolongations of each other. With the achromatics i to 3 complete phase reversals were thus producible within the vertical limits of the slit- image. The experiment succeeded best with sunlight and short collimators close at hand, giving high slit-images. To make the experiment more useful, the four rays should be separated, so so that if path-difference is introduced into one pair and not into the other, the result may be accentuated. To do this requires a broader Billet compen- sator than I possessed or a special device, and I did not therefore carry it out. The horizontal spectrum fringes, even if from a wide slit, require much light, particularly in view of the plate-glass m. Sunlight with a condenser lens of long focus is necessary for clearness. The two sets of fringes are then merged in the same banded spectrum and alternately strengthen or destroy each other. DISPLACEMENT INTERFEROMETRY APPLIED TO ACOUSTICS AND TO GRAVITATION By CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON WASHINGTON, 1921 MBL/WHOI LIBRARY LIH IfiLY R