THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES fc'S/4 SCIENTIFIC PAPEBS CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANNER LONDON : FETTER LANE, E.G. 4 NEW YORK : THE MACMILLAN CO. BOMBAY \ CALCUTTA L MACMILLAN AND CO., LTD. MADRAS j TORONTO : THE MACMILLAN CO. OF CANADA, LTD.* TOKYO: MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS RESERVED SCIENTIFIC PAPEES BY JOHN WILLIAM STBUTT, BARON RAYLEIGH, O.M., D.Sc., F.R.S., CHANCELLOR OF THE UNIVERSITY OF CAMBRIDGE, HONORARY PROFESSOR OF NATURAL PHILOSOPHY IN THE ROYAL INSTITUTION. VOL. VI. 19111919 CAMBRIDGE AT THE UNIVERSITY PRESS 1920 Eiyineerinc t.i' rary v, PEEFACE rriHIS volume completes the collection of my Father's published papers. The two last papers (Nos. 445 and 446) were left ready for the press^ but were not sent to any channel of publication until after the Author's death. Mr W. F. Sedgwick, late Scholar of Trinity College, Cambridge, who had done valuable service in sending corrections of my Father's writings during his lifetime, kindly consented to examine the proofs of the later papers of this volume [No. 399 onwards] which had not been printed off at the time of the Author's death. He has done this very thoroughly, checking the numerical calculations other than those embodied in tables, and supplying footnotes to elucidate doubtful or obscure points in the text. These notes are enclosed in square brackets [ ] and signed W. F. S. It has not been thought necessary to notice minor corrections. KAYLEIGH. Sept. 1920. 803486 CONTENTS ART. PAGE 350. Note on Bessel's Functions as applied to the Vibrations of a Circular Membrane ........ 1 {Philosophical Magazme, Vol. xxi. pp. 5358, 1911.] 351. Hydrodynamical Notes . -;' 6 Potential and Kinetic Energies of Wave Motion . . 6 Waves moving into Shallower Water ..... 7 Concentrated Initial Disturbance with inclusion of Capil- larity . . ". ' 9 Periodic Waves in Deep Water advancing without change ofType 11 Tide Races .>'.'. . . 14 Rotational Fluid Motion in a Corner . ; ., , ; . - : . :..<: 15 Steady Motion in a Corner of a Viscous Fluid . . . 18 [Philosophical Magazine, Vol. xxi. pp. 177195, 1911.] 352. On a Physical Interpretation of Schlomilch's Theorem in Bessel's Functions . . . .. .- ;. V ... . . 22 [Philosophical Magazine, Vol. xxi. pp. 567571, 1911.] 353. Breath Figures 26 [Nature, Vol. LXXXVI. pp. 416, 417, 1911.] 354. On the Motion of Solid Bodies through Viscous Liquid . . 29 [Philosophical Magazine, Vol. xxr. pp. 697711, 1911.] 355. Aberration in a Dispersive Medium . .... . . 41 [Philosophical Magazine, Vol. xxn. pp. 130134, 1911.] 356. Letter to Professor Nernst 45 [Conseil scientifique sous les auspices de M. Ernest Solvay, Oct. 1911.] 357. On the Calculation of Chladni's Figures for a Square Plate . 47 [Philosophical Magazine, Vol. xxn. pp. 225229, 191 l.J 358. Problems in the Conduction of Heat 51 [Philosophical Magazine, Vol. xxu. pp. 381 396, 1911.] 359. On the General Problem of Photographic Reproduction, with suggestions for enhancing Gradation originally Invisible . 65 [Philosophical Magazine, Vol. xxii. pp. 734740, 1911.] 360. On the Propagation of Waves through a Stratified Medium, with special reference to the Question of Reflection . . . 71 [Proceedings of the Royal Society, A, Vol. LXXXVI. pp. 207 266, 1912.] viii CONTENTS ART. PAGE 361. Spectroscopic Methods 91 '[Nature, Vol. LXXXVIII. p. 377, 1912.] 362. On Departures from Fresnel's Laws of Reflexion ... 92 [Philosophical Magazine, Vol. xxin. pp. 431439, 1912.] 363. The Principle of Reflection in Spectroscopes . . . .100 [Nature, VoL LXXXIX. p. 167, 1912.] 364. On the Self-Induction of Electric Currents in a Thin Anchor-Ring 101 [Proceedings of the Royal Society, A, Vol. LXXXVI. pp. 562571, 1912.] 365. Electrical Vibrations on a Thin Anchor-Ring . . . .111 [Proceedings of the Royal Society, A, Vol. LXXXVII. pp. 193202, 1912.] 366. Coloured Photometry 121 [Philosophical Magazine, Vol. xxiv. pp. 301, 302, 1912.] 367. On some Iridescent Films 123 [Philosophical Magazine, Vol. xxiv. pp. 751755, 1912.] 368. Breath Figures 127 [Nature, Vol. xc. pp. 436, 437, 1912.] 369. Remarks concerning Fourier's Theorem as applied to Physical Problems . . 131 [Philosophical Magazine, Vol. xxiv. pp. 864 869, 1912.] 370. Sur la Resistance des Spheres dans 1'Air en Mouvement . . 136 [Comptes Rendus, t. CLVI. p. 109, 1913.] 371. The Effect of Junctions on the Propagation of Electric Waves along Conductors . . . . . . . . .137 [Proceedings of the Royal Society, A, Vol. LXXXVIII. pp. 103110, 1913.] 372. The Correction to the Length of Terminated Rods in Electrical Problems . ....... 145 [Philosophical Magazine, Vol. xxv. pp. 19, 1913.] 373. On Conformal Representation from a Mechanical Point of View . 153 [Philosophical Magazine, Vol. xxv. pp. 698702, 1913.] 374. On the Approximate Solution of Certain Problems relatrhg to the Potential. II 157 [Philosophical Magazine, VoL xxvi. pp. 195199, 1913.'] 375. On the Passage of Waves through Fine Slits in Thin Opaque Screens 161 [Proceedings of the Royal Society, A, VoL LXXXIX. pp. 194219, 1913.] 376. On the Motion of a Viscous Fluid 187 [Philosophical Magazine, VoL xxvi. pp. 776786, 1913.] 377. On the Stability of the Laminar Motion of an Inviscid Fluid . 197 [Philosophical Magazine, Vol. xxvi. pp. 10011010, 1913.] 378. Reflection of Light at the Confines of a Diffusing Medium . 205 [Nature, Vol. xcn. p. 450, 1913.] 379. The Pressure of Radiation and Carnot's Principle . . .208 [Nature, Vol. xcn. pp. 527, 528, 1914.] CONTENTS IX ART. PAGE 380. Further Applications of Bessel's Functions of High Order to the Whispering Gallery and Allied Problems . . . . 211 {Philosophical Magazine, Vol. xxvii. pp. 100 109, 1914.] 381. On the Diffraction of Light by Spheres of Small* Relative Index 220 [Proceedings of the Royal Society, A, Vol. xc. pp. 219225, 1914.] 382. Some Calculations in Illustration of Fourier's Theorem . . 227 [Proceedings of the Royal Society, A, Vol. xc. pp. 318323, 1914.] 383. Further Calculations concerning the Momentum of Progressive Waves .-,; \ -: jfe.%* , . 232 [Philosophical Magazine, Vol. xxvii. pp. 436440, 1914.] 384. Fluid Motions .... f .* ' . ,'. .';. 1 . -- . 237 [Proc. Roy. Inst. March, 1914 ; Nature, Vol. xcm. p. 364, 1914.] 385. On the Theory of Long Waves and Bores 250 Experimental . . . ' ,. . . . . ,' * . 254 [Proceedings of the Royal Society, A, Vol. xc. pp. 324 328, 1914.] 386. The Sand-Blast 255 [Nature, Vol. xcm. p. 188, 1914.] 387. The Equilibrium of Revolving Liquid under Capillary Force . 257 [Philosophical Magazine, Vol. xxvm. pp. 161170, 1914.] 388. Further Remarks on the Stability of Viscous Fluid Motion . 266 [Philosophical Magazine, Vol. xxvm. pp. 609 619, 1914.] 389. Note on the Formula for the Gradient Wind . . . ' : . 276 [Advisory Committee for Aeronautics. Reports and Memoranda. No. 147. January, 1915.] 390. Some Problems concerning the Mutual Influence of Resonators exposed to Primary Plane Waves . . . . . 279 [Philosophical Magazine, Vol. xxix. pp. 209222, 1915.] 391. On the Widening of Spectrum Lines 291 [Philosophical Magazine, Vol. xxix. pp. 274284, 1915.] 392. The Principle of Similitude . ; ;!' ;; f ' .' ^ . . 300 [Nature, Vol. xcv. pp. 6668, 644, 1915.] 393. Deep Water Waves, Progressive or Stationary, to the Third Order of Approximation "'. ' ; ' : ''? '. . . '" . ". . 306 [Proceedings of the Royal Society, A, Vol. xci. pp. 345353, 1915.] 394. jEolian Tones . . . ._/7 . .... 315 [Philosophical Magazine, Vol. xxix. pp. 433444, 1915:] 395. On the Resistance experienced by Small Plates exposed to a Stream of Fluid ! '. t "'' ; . 326 [Philosophical Magazine, Vol. xxx. pp. 179 181, 1915.] 396. Hydrodynamical Problems suggested by Pitot's Tubes . . 329 [Proceedings of the Royal Society, A, Vol. xci. pp. 503511, 1915.] * [1914. It would have been in better accordance with usage to have said " of Relative Index differing little from Unity."] CONTENTS ART. PAGE 397. On the Character of the "S" Sound 337 [Nature, VoL xcv. pp. 646, 646, 1915.] 398. On the Stability of the Simple Shearing Motion of a Viscous Incompressible Fluid . . . . . . . . 341 [Philosophical Magazine, Vol. xxx. pp. 329338, 1915.] 399. On the Theory of the Capillary Tube 350 The Narrow Tube 351 The Wide Tube . . . . . . - .356 [Proceedings of the Royal Society, A, Vol. xcn. pp. 184195, Oct. 1915.] 400. The Cone as a Collector of Sound 362 [Advisory Committee for Aeronautics, T. 618, 1915.] 401. The Theory of the Helmholtz Resonator 365 [Proceedings of the Royal Society, A, Vol. xcn. pp. 265275, 1915.] 402. On the Propagation of Sound in Narrow Tubes of Variable Section 376 [Philosophical Magazine, Vol. xxxi. pp. 8996, 1916.] 403. On the Electrical Capacity of Approximate Spheres and Cylinders 383 [Philosophical Magazine, Vol. xxxi. pp. 177186, March 1916.] 404. On Legendre's Function P n (0), when n is great and 6 has any value* 393 [Proceedings of the Royal Society, A, Vol. xcn. pp. 433 437, 1916.] 405. Memorandum on Fog Signals 398 [Report to Trinity House, May 1916.] 406. Lamb's Hydrodynamics ........ 400 [Nature, VoL xcvu. p. 318, 1916.] 407. On the Flow of Compressible Fluid past an Obstacle . . 402 [Philosophical Magazine, Vol. xxxn. pp. 16, 1916.] 408. On the Discharge of Gases under High Pressures . . . 407 [Philosophical Magazine, Vol. xxxil. pp. 177187, 1916 ] 409. On the Energy acquired by Small Resonators from Incident Waves of like Period 416 [Philosophical Magazine, Vol. xxxn. pp. 188190, 1916.] 410. On the Attenuation of Sound in the Atmosphere . . . 419 [Advisory Committee for Aeronautics. August 1916.] 411. On Vibrations and Deflexions of Membranes, Bars, and Plates . 422 [Philosophical Magazine, Vol. xxxil. pp. 353364, 1916.] 412. On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the Under Side . . . 432 Appendix ......... 444 [Philosophical Magazine, Vol. XXXII. pp. 529546, 1916.] 413. On the Dynamics of Revolving Fluids 447 [Proceedings of the Royal Society, A, Vol. xcin. pp. 148154, 1916.] * [1917. It would be more correct to say P H (cos 6), where cos lies between 1.] CONTENTS XI ART. PAGE 414. Propagation of Sound in Water ...... 454 [Not hitherto published.] 415. On Methods for Detecting Small Optical Retardations, and on the Theory of Foucault's Test 455 [Philosophical Magazine, Vol. xxxin. pp. 161 178, 1917.] 416. Talbot's Observations on Fused Nitre 471 [Nature, Vol. xcvm. p. 428, 1917.] 417. Cutting and Chipping of Glass ; 473 [Engineering, Feb. 2, 1917, p. 111.] 418. The Le Chatelier-Braun Principle . , t '. . .,;'"T""^. 475 [Transactions of the Chemical Society, Vol. cxi. pp. 250252, 1917.] 419. On Periodic Irrotational Waves at the Surface of Deep Water . 478 [Philosophical Magazine, Vol. xxxni. pp. 381389, 1917.] 420. On the Suggested Analogy between the Conduction of Heat and Momentum during the Turbulent Motion of a Fluid . 486 [Advisory Committee for Aeronautics, T. 941, 1917.] 421. The Theory of Anomalous Dispersion , , ... .. . . . 488 [Philosophical Magazine, Vol. xxxin. pp. 496499, 1917.] 422. On the Reflection of Light from a regularly Stratified Medium 492 [Proceedings of the Royal Society, A, Vol. xcm. pp. 565577, 1917.] 423. On the Pressure developed in a Liquid during the Collapse of a Spherical Cavity ........ 504 [Philosophical Magazine, Vol. xxxiv. pp. 9498, 1917.] 424. On the Colours Diffusely Reflected from some Collodion Films spread on Metal Surfaces . ..... , . , . . . 508 [Philosophical Magazine, Vol. xxxiv. pp. 423 428, 1917.] 425. Memorandum on Synchronous Signalling . . . - . . . 513 [Report to Trinity House, 1917.] 426. A Simple Problem in Forced Lubrication . . ; . . . 514 [Engineering, Dec. 14, 28, 1917.] 427. On the Scattering of Light by Spherical Shells, and by Complete Spheres of Periodic Structure, when the Refractivity is Small 518 [Proceedings of the Royal Society, A, Vol. xciv. pp. 296300, 1918.] 428. Notes on the Theory of Lubrication . ...:"'... . . 523 [Philosophical Magazine, Vol. xxxv. pp. 112, 1918.] 429. On the Lubricating and other Properties of Thin Oily Films . 534 [Philosophical Magazine, Vol. xxxv. pp. 157 162, 1918.] 430. On the Scattering of Light by a Cloud of Similar Small Par- ticles of any Shape and Oriented at Random . . . 540 [Philosophical Magazine, Vol. xxxv. pp. 373381, 1918.] 431. Propagation of Sound and Light in an Irregular Atmosphere . 547 [Nature, Vol. ci. p. 284, 1918.] Xli CONTENTS ABT. 432. Note on the Theory of the Double Resonator .... 549 [Philosophical Magazine, Vol. xxxvi. pp. 231234, 1918.] 433. A Proposed Hydraulic Experiment .' . v > . ' - 552 [Philosophical Magazine, VoL xxxvi. pp. 315, 316, 1918.] 434. On the Dispersal of Light by a Dielectric Cylinder . . .554 [Philosophical Magazine, Vol. xxxvi. pp. 365 376, 1918.] 435. The Perception of Sound 564 [Nature, VoL en. p. 225, 1918.] 436. On the Light Emitted from a Random Distribution of Luminous Sources - , . . . .565 [Philosophical Magazine, VoL xxxvi. pp. 429449, 1918.] 437. The Perception of Sound .583 [Nature, Vol. en. p. 304, 1918.] 438. On the Optical Character of some Brilliant Animal Colours . 584 [Philosophical Magazine, Vol. xxxvn. pp. 98111, 1919.] 439. On the Possible Disturbance of a Range-Finder by Atmospheric Refraction due to the Motion of the Ship which carries it . 597 [Transactions of the Optical Society, Vol. XX. pp. 125129, 1919.] 440. Remarks on Major G. I. Taylor's Papers on the Distribution of Air Pressure 602 [Advisory Committee for Aeronautics, T. 1296, 1919.] 441. On the Problem of Random Vibrations, and of Random Flights in One, Two, or Three Dimensions ..... 604 One Dimension ........ 607 Two Dimensions 610 Three Dimensions 618 [Philosophical Magazine, VoL xxxvn. pp. 321347, 1919.] 442. On the Resultant of a Number of Unit Vibrations, whose Phases are at Random over a Range not Limited to an Integral Number of Periods 627 [Philosophical Magazine, VoL xxxvn. pp. 498515, 1919.] 443. Presidential Address 642 [Proceedings of the Society for Psychical Research, Vol. xxx. pp. 275290, 1919.] 444. The Travelling Cyclone 654 [Philosophical Magazine, VoL xxxvill. pp. 420424, 1919.] 445. Periodic Precipitates 659 Hookham's Crystals 661 [Philosophical Magazine, Vol. xxxvin. pp. 738740, 1919.] 446. On Resonant Reflexion of Sound from a Perforated Wall . . 662 [Philosophical Magazine, VoL xxxix. pp. 225233, 1920.] CONTENTS PAGE CONTENTS OF VOLUMES I VI CLASSIFIED ACCORDING TO SUBJECT .... 670 I. Mathematics . . . '. 671 II. General Mechanics . V " \ " . 672 III. Elastic Solids 674 IV. Capillarity : ., 675 -V. Hydrodynamics . . . . . 677 VI. Sound . . . r - . . . 681 VII. Thermodynamics .... 688 VIII. Dynamical Theory of Gases . . 689 IX. Properties of Gases . . . . 691 X. Electricity and Magnetism . . 694 XI. Optics 700 XII. Miscellaneous 707 INDEX OF NAMES 710 ERRATA (INCLUDING THE ERRATA NOTED IN VOLUME V. PAGE XHL) VOLUME I. 86, last line. For 1882 read 1881. 89, line 10. Insert comma after maximum. 144, line 6 from bottom. For D read D, . 324, equation (8). Insert negative Bign before the single \ ^ Theofy Qf I (1894), p. 477, equation (8) and 442; line 9. After *! insert y. 443, line 9. For (7) read (8). 443, line 10. For y read . 446, line 10. For <f> read <j>'. 448, line 5. For v read c. 459, line 17. For 256, 257 read 456, 457. 492, line 7 from bottom. For r\/2n read r/\/2n. 2mr 2 2mr 2 494, lines 10 and 12. For - . . .cos 26 read +- - cos 20. n 2 - 4m 2 n 2 - 4m 2 523, line 9. For n/X read n/fc. 524, In the second term of equations (32) and following for AK' 1 read Aft.- 1 . 525, line 11. For / read ft. 526, line 13. For f : g read f\:gi. 528, line 3 from bottom. For e int read e< (<-*). 538, line 11 from bottom. This passage is incorrect (see Vol. vi. Art. 355, p. 41). 556. In line 8 after (15) add with <-$* for s<j>; in line 9 for dA t read 8A t '-, and for line 10 substitute + 8A,'as {co8$8ir + cos(^tir + r)} F. Throughout lines 12 25 for A t , A lt A 2 , ... A 6 , SA, read A t ', AI, A s ', ... A 6 ', 8A t ' ; for sin J.STT read -COS^T; and reverse the signs of the expressions for A 2 ', AJ, A$. Similarly, in Theory of Sound, Vol. i. (1894), p. 427, substitute s<j> + \ir for t<f> in (32) (see p. 424), and in lines 1126 for A,,', A t , 8A t read A t , A.', 8A,\ and for sin read + cos. Also in (43) and (47) for s z -s read s 3 - s. [In both cases the work done corre- sponding to 8A t vanishes whether s be odd or even.] VOLUME II. 197, line 19. For nature read value. 240, line 22. For dpjdx read dpjdy. 241, line 2. For du/dx read dujdy. 244, line 4. For k/n read njk. 823, lines 7 and 16 from bottom. For Thomson read C. Thompson. 345, line 8 from bottom. For as pressures read at pressures. 386, lines 12, 15, and 19. For cos CBD read cos CBB'. 389, line 6. For minor read mirror. 414, line 5. For favourable read favourably. 551, first footnote. For 1866 read 1886. ERRATA XV VOLUME III. Page 11, footnote. For has read have. 92, line 4. For Vol. I. read Vol. II. ,, 129, equation (12). For e u ( i - x )dx read e(-*>dM. , , 162, line 19, and p. 224, second footnote. For Jellet read Jellett. ,, 179, line 15. For Provostaye read De la Provostaye. 224, equation (20). For 2 X read x . ) And Theory of Sound, Vol. i. (1894), ,, ,, second footnote. For p. 179 read p. 343. ] p. 412, equation (12), and p. 423 (footnote). 231, line 5 of first footnote. For 171 read 172. 273, lines 15 and 20. For \<t>(x)}* read {<t>(x)}*dx. 314, line 1. For (38) read (39). 326. In the lower part of the Table, under Ampton for < + 4 read < + 4, and under Terling (3) for fct> + 6 read 6 + 6 (and in Theory of Sound, Vol. i. (1894), p. 393). 522, equation (31). Insert as factor of last term I/ R. 548, second footnote. For 1863 read 1868. 569, second footnote. For alcohol read water. 580, line 3. Prof. Orr remarks that a is a function of r. VOLUME IV. 14, lines 6 and 8. For 38 read 42. 267, lines 6, 10, and 20, and p. 269, line 1. For van t' Hoff read van 't Hoff. Also in. Index, p. 604 (the entry should be under Hoff). 277, equation (12). For dz read dx. 299, first footnote. For 1887 read 1877. 369, footnote. For 1890 read 1896. 400, equation (14). A formula equivalent to this was given by Lorenz in 1890. 418. In table opposite 6 for -354 read -324. 2 2 453, line 8 from bottom. For - - read -- =-. n-1 n-1 556, line 8 from bottom. For reflected read rotated. 570, line 7 (Section III). For 176 read 179. 576, liiie 7 from bottom.) V For end lies read ends he. 586, line 20. j 582, last line. For 557 read 555. 603. Transfer the entry under Provostaye to De la Provostaye. 604. Transfer the entry n 553 from W. Weber to H. F. Weber. VOLUME V. 43, line 19. For (5) read (2). 137, line 14. y. is here used in two senses, which must be distinguished. 149, line 3. For P read Pj. 209, footnote. For XLX. read xix. 241, line 10 from bottom. For position read supposition. 255, first footnote. For Matthews read Mathews. 256, line 6. For 1889 read 1899, 265, line 16 from bottom. For 351 read 251. ,, ,, 15 ,, ,, For solution read relation. 266, lines 5 and 6, and Theory of Sound, 251. An equivalent result had at an earlier date been obtained by De Morgan (see Volume vi. p. 233). 286, line 7. For a read x. Xvi ERRATA VOLUME V continued. Page 364, title, and p. ix, Art. 320. After Acoustical Notes add VH. ,, 409, first line of P.S. For anwer read answer. 444, line 2 of footnote. For p. 441, line 9 read p. 442, line 9. 496, equation (4). Substitute equation (19) on p. 253 of Volume vi. (tee pp. 251253). 549, equation (48). For <T** r read -'* r <>. 619, line 3. Omit the second expression for J, (n). > lines 11, 12, 19. For 2-1123 read 1-3447. I See the first footnote on p. 211 of line 12. For 1-1814 read 1-8558. j Volume vi. line 19. For -51342 read -8065. J VOLUME VI. 4, first footnote. After equation (8) add-. Scientific Papers, Vol. v. p. 619. See also Errata last noted above. 5, line 3. For (2n + l)*2=4n(n + l)(n + 2) read z*=2n(n + 2), so that z* is an integer. 11, last footnote. For 230 read 250 (fourth edition). 13, equation (17). For |fc 4 4 read f* 4 a 4 . 14, footnote. For 247 read 251 (fourth edition). 78, footnote. Add -.Scientific Papers, Vol. v. p. 400. 87, footnote. Add-. Thomson and Tait's Natural Philosophy, Vol. i. p. 497. 89, second footnote. For 328 read 329. 90, second footnote. Add: Math, and Phys. Papers, Vol. iv. p. 77. 138, footnote. For 1868 read 1865, and for Vol. n. p. 128, read Vol. i. p. 526. 148, footnote. Add -.Scientific Papers, Vol. iv. p. 407, and this Volume, p. 47. 155, footnote. For Vol. iv. read Vol. in. 222, second footnote. For Vol. n. read Vol. i. And in Theory of Sound, Vol. i. (1894), last line of 207, for 4-4747 read 4-4774. 223, line 5 from bottom. For 0-5772156 read 0-5772157. 225, line 1. For much greater read not much greater. ,, line 6 from bottom. For 13-094 read 3-3274. 253, equation (19). For ( - + p\ read ( - - - t J . 259, line 5. For -- % read =F- ^ . a at a dz 263, equation (24). For *^ read ^- . 282, footnote. For p. 77 read p. 71. 303, line 17. For ^(OVC/K) read v '(6wc/t). 307, line 8. For ^ read -^ . dy dy 315, line 2. Delete 195. 341, second footnote. Add : [This Volume, p. 275]. 351, line 13 from bottom. For Tgp read Tfgp. 350. NOTE ON BESSEL'S FUNCTIONS AS APPLIED TO THE VIBRATIONS OF A CIRCULAR MEMBRANE. [Philosophical Magazine, Vol. XXL pp. 5358, 1911.] IT often happens that physical considerations point to analytical con- clusions not yet formulated. The pure mathematician will admit that arguments of this kind are suggestive, while the physicist may regard them as conclusive. The first question here to be touched upon relates to the dependence of the roots of the function J n (z) upon the order n, regarded as susceptible of continuous variation. It will be shown that each root increases continually with n. Let us contemplate the transverse vibrations of a membrane fixed along the radii = and 6 ft and also along the circular arc r = 1. A typical simple vibration is expressed by* iv = J n (z ( ^r).smne.cos(z ( ^t), (I) where ^ is a finite root of J n (z) = 0, and n = IT 1/3. Of these finite roots the lowest z (l) gives the principal vibration, i.e. the one without internal circular nodes. For the vibration corresponding to z ( * ] the number of internal nodal circles is s 1. As prescribed, the vibration (1) has no internal nodal diameter. It might be generalized by taking n = vTr/fi, where v is an integer ; but for our purpose nothing would be gained, since /9 is at disposal, and a suitable reduction of /3 comes to the same as the introduction of v. In tracing the effect of a diminishing ft it may suffice to commence at /S = TT, or n=l. The frequencies of vibration are then proportional to the roots of the function /",. The reduction of /8 is supposed to be effected by * Theory of Sound, 205, 207. R. VI. 1 2 NOTE ox BESSEL'S FUNCTIONS AS APPLIED [350 increasing without limit the potential energy of the displacement (w) at every point of the small sector to be cut off. We may imagine suitable springs to be introduced whose stiffness is gradually increased, and that without limit. During this process every frequency originally finite must increase*, finally by an amount proportional to d/3', and, as we know, no zero root can become finite. Thus before and after the change the finite roots correspond each to each, and every member of the latter series exceeds the corresponding member of the former. As ft continues to diminish this process goes on until when /8 reaches ^TT, ?i again becomes integral and equal to 2. We infer that every finite root of Jj exceeds the corresponding finite root of Jj. In like manner every finite root of /, exceeds the corresponding root of J 3 , and so onf. I was led to consider this question by a remark of Gray and MathewsJ " It seems probable that between every pair of successive real roots of J n there is exactly one real root of / n+1 . It does not appear that this has been strictly proved ; there must in any case be an odd number of roots in the interval." The property just established seems to allow the proof to be completed. As regards the latter part of the statement, it may be considered to be a consequence of the well-known relation (2) When J n vanishes, J n+l has the opposite sign to J n ', botji these quantities being finite. But at consecutive roots of J n , J n ' must assume opposite signs, and so therefore must J n+l . Accordingly the number of roots of J n+1 in the interval must be odd. The theorem required then follows readily. For the first root of J n+l must lie between the first and second roots of J n . We have proved that it exceeds the first root. If it also exceeded the second root, the interval would be destitute of roots, contrary to what we have just seen. In like manner the second root of J n+l lies between the second and third roots of J H , and so on. The roots of J n+1 separate those of J n ||. Loc. cit. 83, 92 a. t [1915. Similar arguments may be applied to tesseral spherical harmonics, proportional to cos <f>, where denotes longitude, of fixed order n and continuously variable *.] * HettcVs Functions, 1895, p. 50. If /,,, J n+t could vanish together, the sequence formula, (8) below, would require that every succeeding order vanish also. This of course is impossible, if only because when n is great the lowest root of </ is of order of magnitude n. || I have since found in Whittaker's Modern Analysis, 152, another proof of this proposition, attributed to Gegenbaner (1897). 1911] TO THE VIBRATIONS OF A CIRCULAR MEMBRANE 3 The physical argument may easily be extended to show in like manner that all the finite roots of J n ' (z) increase continually with n. For this purpose it is only necessary to alter the boundary condition at r = 1 so as to make dw/dr = instead of w = 0. The only difference in (1) is that ( * } now denotes a root of /' (z) = 0. Mechanically the membrane is fixed as before along 6 = 0, 6 = /3, but all points on the circular boundary are free to slide transversely. The required conclusion follows by the same argument as was applied to J n . It is also true that there must be at least one root of J' n +\ between any two consecutive roots of J n ', but this is not so easily proved as for the original functions. If we differentiate (2) with respect to z and then eliminate J n between the equation so obtained and the general differential equation, viz. (3) * \ * / we find /' = 0. ...(4) In (4) we suppose that z is a root of J n ', so that J n ' = 0. The argument then proceeds as before if we can assume that z* n 2 and z 2 n (n + 1) are both positive. Passing over this question for the moment, we notice that Jn and J' n+1 have opposite signs, and that both functions are finite. In fact if J^' and J n ' could vanish together, so also by (3) would J n , and again by (2) J n+1 ; and this we have already seen to be impossible. At consecutive roots of /', J n " must have opposite signs, and therefore also J'n+i. Accordingly there must be at least one root of J' n+1 between consecutive roots of J n '. It follows as before that the roots of J' n +i separate those of J^. It remains to prove that z* necessarily exceeds n(n + 1). That z 2 exceeds n 2 is well known*, but this does not suffice. We can obtain what we require from a formula given in Theory of Sound, 2nd ed. 339. If the finite roots taken in order be z lt z a , ... z,..., we may write log J n (z) = const. + (n - 1) log z + 2 log (1 - 2 2 /2 t *), the summation including all finite values of z g ; or on differentiation with respect to z n z z z?-z* This holds for all values of z. If we put z = n, we get ...(5) f n" * Riemann's Partielle Di/erentialgleichungen ; Theory of Sound, 210. 12 4 NOTE ON BESSEL'S FUNCTIONS AS APPLIED [350 since by (3) J n "(n) + J n '(n) = -n-\ In (5) all the denominators are positive. We deduce *_! +*!=;+ rj* + ...>l; ............... ,6) 2n * 2 - " z* - n* and therefore z,- >n* + 2n>n(n +1). Our theorems are therefore proved. If a closer approximation to z? is desired, it may be obtained by sub- stituting on the right of (6) 2n for z? w 2 in the numerators and neglecting n 2 in the denominators. Thus Z *~ n * > 1 + 2n (z a ~* + z 3 ~* + ...) Now, as is easily proved from the ascending series for J n ', *r + *r> + *r + ... so that finally (7, When n is very great, it will follow from (7) that z? > n=+ 3n. Howevei the approximation is not close, for the ultimate form is* ^>n'+ [1*6130] ". As has been mentioned, the sequence formula (8) prohibits the simultaneous evanescence of </_, and J n , or of J n -* and J n +\- The question arises can Bessel's functions whose orders (supposed integral) differ by more than 2 vanish simultaneously ? If we change n into n + 1 in (8) and then eliminate J n , we get [**(+!) J r _r 2n : 1 f / n+i = ^n-i H /tt+2 (,") ( 2 ) ^ from which it appears that if </_! and J n+a vanish simultaneously, then either A-t-i = 0, which is impossible, or z 2 = 4n (n + 1). Any common root of /,,_! and ./ n+3 must therefore be such that its square is an integer. * Phil. Mag. Vol. M. p. 1003, 1910, equation (8). [1913. A correction is here introduced. See Nicholson, Phil. Mag. Vol. xxv. p. 200, 1913.] 1911] TO THE VIBRATIONS OF A CIRCULAR MEMBRANE 5 Pursuing the process, we find that if J n -\, Jn+3 have a common root z, then (2n + 1) z* = 4n (n + 1) (TO + 2), so that z* is rational. And however far we go, we find that the simultaneous evanescence of two Bessel's functions requires that the common root be such that 2 2 satisfies an algebraic equation whose coefficients are integers, the degree of the equation rising with the difference in order of the functions. If, as seems probable, a root of a Bessel's function cannot satisfy an integral algebraic equation, it would follow that no two Bessel's functions have a common root. The question seems worthy of the attention of mathematicians. 351. HYDRODYNAMICAL NOTES. [Philosophical Magazine, Vol. xxi. pp. 177195, 1911.] Potential and Kinetic Energies of Wave Motion. Waves moving into Shallower Water. Concentrated Initial Disturbance with inclusion of Capillarity. Periodic Waves in Deep Water advancing without change of Type. Tide Races. Rotational Fluid Motion in a Corner. Steady Motion in a Corner of Viscous Fluid. IN the problems here considered the fluid is regarded as incompressible, and the motion is supposed to take place in two dimensions. Potential and Kinetic Energies of Wave Motion. When there is no dispersion, the energy of a progressive wave of any form is half potential and half kinetic. Thus in the case of a long wave in shallow water, " if we suppose that initially the surface is displaced, but that the particles have no velocity, we shall evidently obtain (as in the case of sound) two equal waves travelling in opposite directions, whose total energies are equal, and together make up the potential energy of the original dis- placement. Now the elevation of the derived waves must be half of that of the original displacement, and accordingly the potential energies less in the ratio of 4 : 1. Since therefore the potential energy of each derived wave is one quarter, and the total energy one half that of the original displacement, it follows that in the derived wave the potential and kinetic energies are equal " *. The assumption that the displacement in each derived wave, when separated, is similar to the original displacement fails when the medium is dispersive. The equality of the two kinds of energy in an infinite pro- train of simple waves may, however, be established as follows. "On Waves," Phil. Mag. Vol. i. p. 257 (1876) ; Scientific Papers, Vol. i. p. 254. 1911] HYDRODYNAMFCAL NOTES 7 Consider first an infinite series of simple stationary waves, of which the energy is at one moment wholly potential and [a quarter of] a period later wholly kinetic. If t denote the time and E the total energy, we may write K.E. = E sin 2 nt, P.E. = E cos 2 nt. Upon this superpose a similar system, displaced through a quarter wave- length in space and through a quarter period in time. For this, taken by itself, we should have K.E == E cos 2 nt, P.E. = E sin 2 nt. And, the vibrations being conjugate, the potential and kinetic energies of the combined motion may be found by simple addition of the components, and are accordingly independent of the time, and each equal to E. Now the resultant motion is a simple progressive train, of which the potential and kinetic energies are thus seen to be equal. A similar argument is applicable to prove the equality of energies in the motion of a simple conical pendulum. It is to be observed that the conclusion is in general limited to vibrations which are infinitely small. Waves moving into Shallower Water. The problem proposed is the passage of an infinite train of simple infinitesimal waves from deep water into water which shallows gradually in such a manner that there is no loss of energy by reflexion or otherwise. At any stage the whole energy, being the double of the potential energy, is proportional per unit length to the square of the height ; and for motion in two dimensions the only remaining question for our purpose is what are to be regarded as corresponding lengths along the direction of propagation. In the case of long waves, where the wave-length (A.) is long in comparison with the depth (I) of the water, corresponding parts are as the velocities of propagation ( V), or since the periodic time (T) is constant, as A.. Conservation of energy then requires that (height) 2 x F = constant; (1) or since V varies as ft, height varies as / ~ ^ *. But for a dispersive medium corresponding parts are not proportional to V, and the argument requires modification. A uniform regime being established, what we are to equate at two separated places where the waves are of different character is the rate of propagation of energy through these places. It is a general proposition that in any kind of waves the ratio of the energy propagated past a fixed point in unit time to that resident in unit * Loc. cit. p. 255. 8 HYDRODYNAMICAL NOTES [351 length is U, where U is the group-velocity, equal to d<r/dk, where <r = 2-7T/T, k = 2?r/X*. Hence in our problem we must take height varies as U~^, ........................... (2) which includes the former result, since in a non-dispersive medium U = V. For waves in water ot depth I, o- 2 = #tanh kl, .............................. (3) whence 2<rU/g = tanh kl +kl(l -tanh'M) ................... (4) As the wave progresses, a remains constant, (3) determines k in terms of /, and U follows from (4). If we write *% = *', ................................ .'(5) (3) becomes kl . t&Jih kl = I' , .............................. (6) and (4) may be written 2<rU/g = kl + (l'-P)/U ......................... (7) By (6), (7) U is determined as a function of I' or by (5) of I. If kl, and therefore V, is very great, kl = /', and then by (7) if U be the corresponding value of U, 2<rU /g=l, ................................. (8) and in general U/U n = kl + (l'-r*)/kl ............................ (9) Equations (2), (5), (6), (9) may be regarded as giving the solution of the problem in terms of a known a. It is perhaps more practical to replace a in (5) by X , the corresponding wave-length in a great depth. The relation between a and \ being <r* = Zirg/Xo, we find in place of (5) l' = Zirll\ = kl ................. . ............. (10) Starting in (10) from X,, and I we may obtain I', whence (6) gives kl, and (9) gives U/U . But in calculating results by means of tables of the hyper- bolic functions it is more convenient to start from kl. We find 10 5 2 1-5 1-0 8 7 Id kl 4-999 1-928 1-358 762 531 423 UIU 1-000 1-000 1-001 1-105 1-176 1-182 1-110 1-048 322 231 152 087 039 010 964 855 722 200 Proc. Land. Math. Soc. Vol. ix. 1877 ; Scientific Papers, Vol. i. p. 326. 1911] HYDRODYNAMICAL NOTES 9 It appears that U/U does not differ much from unity between V = '23 and I' x , so that the shallowing of the water does not at first produce much effect upon the height of the waves. It must be remembered, however, that the wave-length is diminishing, so that waves, even though they do no more than maintain their height, grow steeper. Concentrated Initial Disturbance with inclusion of Capillarity. A simple approximate treatment of the general problem of initial linear disturbance is due to Kelvin*. We have for the elevation 17 at any point x and at any time t 1 f 00 77 = cos kx cos fft dk TTJO = - \ cos (kx - at) dk + - ! cos (kx + at) dk, . . .(1) 27T J o ftlf . in which o- is a function of k, determined by the character of the dispersive medium expressing that the initial elevation (t = 0) is concentrated at the origin of x. When t is great, the angles whose cosines are to be integrated will in general vary rapidly with k, and the corresponding parts of the integral contribute little to the total result. The most important part of the range of integration is the neighbourhood of places where kx at is stationary with respect to k, i.e. where In the vast majority of practical applications dar/dk is positive, so that if x and t are also positive the second integral in (1) makes no sensible contri- bution. The result then depends upon the first integral, and only upon such parts of that as lie in the neighbourhood of the value, or values, of k which satisfy (2) taken with the lower sign. If k^ be such a value, Kelvin shows that the corresponding term in vj has an expression equivalent to _ cos (aj - k& - ITT) ~ o-! being the value of a corresponding to k lt In the case of deep-water waves where a- = \/(gk), there is only one pre- dominant value of k for given values of x and t, and (2) gives k 1 = gt z {4>x n ; <r 1 = gt/2x, (4) making a-^t k^ $7r=gF/4>x - ^TT, (5) g^t (gt 2 TT) and finally rj = y . icos j'^- -^\ , the well-known formula of Cauchy and Poisson. * Proc. Roy. Soc. Vol. XLII. p. 80 (1887) ; Math, and Phys. Papers, Vol. iv. p. 303. 10 HYDRODYNAMICAL NOTES [351 In the numerator of (3) <r, and h are functions of x and t. If we inquire what change (A) in x with t constant alters the angle by 2?r, we find so that by (2) A = 27r/&j, i.e. the effective wave-length A coincides with that of the predominant component in the original integral (1), and a like result holds for the periodic time*. Again, it follows from (2) that k^x a^t in (3) may be replaced by | k^dx, as is exemplified in (4) and (6). When the waves move under the influence of a capillary tension T in addition to gravity, <r* = gk+Tl<*/p ............................... (7) p being the density, and for the wave- velocity ( F) V* = a*lk*=g/k+Tk/p, ........................... (8) as first found by Kelvin. Under these circumstances V has a minimum value when * = 9I>IT. ................................. (9) The group- velocity U is equal to darjdk, or to d (kV)/dk; so that when V has a minimum value, U and V coincide. Referring to this, Kelvin towards the close of his paper remarks " The working out of our present problem for this case, or any case in which there are either minimums or maximums, or both maximums and minimums, of wave-velocity, is particularly interesting, but time does not permit of its being included in the present communication." A glance at the simplified form (3) shows, however, that the special case arises, not when V is a minimum (or maximum), but when U is so, since then (frajdk? vanishes. As given by (3), rj would become infinite an indication that the approximation must be pursued. If k = fcj -f , we have in general in the neighbourhood of k lt In the present case where the term in f 2 disappears, as well as that in , we get in place of (3) when t is great cosa'.da, ............... (11) r<~ varying as t ~ * instead of as t ~ *. The definite integral is included in the general form (12) [+< '- -- m) 2m * Cf. Green, Proc. Roy. Soe. Ed. Vol. xxix. p. 445 (1909). 1911] HYDRODYNAMICAL NOTES 11 giving *"a=-^r(i) (13) The former is employed in the derivation of (3). The occurrence of stationary values of U is determined from (7) by means of a quadratic. There is but one such value ( U ), easily seen to be a minimum, and it occurs when '={Vf-l}f = '1547^ (14) On the other hand, the minimum of V occurs when #* = gp/T simply. When t is great, there is no important effect so long as x (positive) is less than U t. For this value of x the Kelvin formula requires the modification expressed by (11). When x is decidedly greater than U t, there arise two terms o|" the Kelvin form, indicating that there are now two systems of waves of different wave-lengths, effective at the same place. It will be seen that the introduction of capillarity greatly alters the character of the solution. The quiescent region inside the annular waves is easily recognized a few seconds after a very small stone is dropped into smooth water*, but I have not observed the duplicity of the annular waves them- selves. Probably the capillary waves of short wave-length are rapidly damped, especially when the water-surface is not quite clean. It would be interesting to experiment upon truly linear waves, such as might be generated by the sudden electrical charge or discharge of a wire stretched just above the surface. But the full development of the peculiar features to be expected on the inside of the wave-system seems to require a space larger than is- con- veniently available in a laboratory. Periodic Waves in Deep Water advancing without change of Type. The solution of this problem when the height of the waves is infinitesimal has been familiar for more than a century, and the pursuance of the approxi- mation to cover the case of moderate height is to be found in a well-known paper by Stokesf. In a supplement published in 1880J the same author treated the problem by another method in which the space coordinates x, y are regarded as functions of <f>, ty the velocity and stream functions, and carried the approximation a stage further. In an early publication! I showed that some of the results of Stokes' first memoir could be very simply derived from the expression for the * A checkered background, e.g. the sky seen through foliage, shows the waves best, t Camb. Phil. Soc. Trans. Vol. vni. p. 441 (1847) ; Math, and Phys. Papers, Vol. i. p. 197. J Loc. cit. Vol. i. p. 314. Phil Mag. Vol. i. p. 257 (1876) ; Scientific Papers, Vol. i. p. 262. See also Lamb's Hydrodynamics, % 230. 12 HYDRODYNAMICAL NOTES [351 stream-function in terms of x and y, and lately I have found that this method may be extended to give, as readily if perhaps less elegantly, all the results of Stokes' Supplement. Supposing for brevity that the wave-length is 2?r and the velocity of propagation unity, we take as the expression for the stream-function of the waves, reduced to rest, fy = y ae~ y cos x fte~ yy cos 2x ye~ 9y cos 3#, (1) in which x is measured horizontally and y vertically downwards. This expression evidently satisfies the differential equation to which ty is subject, whatever may be the values of the constants a, ft, 7. From (1) we find U* - 2gy = (d+/d*y + (d^fdyY - 2gy = 1 - 2i/r + 2 (1 - g) y + 2fte~^ cos 2# + ^e~w cos 3# + 4 2 <r 4 y + 9rf e -y + 4ctft e -*y cos x ftye-*vcosa; (2) The condition to be satisfied at a free surface is the constancy of (2). The solution to a moderate degree of approximation (as already referred to) may be obtained with omission of ft and 7 in (1), (2). Thus from (1) we get, determining i/r so that the mean value of y is zero, 7/ = a(l + fa 2 )cosa;-|a 2 cos2a; + fa 8 cos3#, (3) which is correct as far as a 8 inclusive. If we call the coefficient of cos x in (3) a, we may write with the same approximation y = a cos x $ a 2 cos 2# -H |a 8 cos 3x (4) Again from (2) with omission of ft, 7, U*-2gy = const. + 2 (1 -g - a 2 - a 4 ) y 4- a 4 cos 2x - $ of cos 3# (5) It appears from (5) that the surface condition may be satisfied with a only, provided that a 4 is neglected and that l-g-o? = (6) In (6) a may be replaced by a, and the equation determines the velocity of propagation. To exhibit this we must restore generality by introduction of &(=27r/\) and c the velocity of propagation, hitherto treated as unity. Consideration of " dimensions " shows that (6) becomes A;c 2 -#-aW = (7) or c* = g/k.(l + A?a s ) (8) Formulae (4) and (8) are those given by Stokes in his first memoir. By means of ft and 7 the surface condition ' (2) can be satisfied with inclusion of a* and of, and from (5) we see that ft is of the order a 4 and 7 of 1911] HYDRODYNAMICAL NOTES 13 the order a 5 . The terms to be retained in (2), in addition to those given in (5), are 2/3(1- 2y) cos 2# + 4 7 cos 3# + 4a cos x = 2/3 cos 2# - 2a/3 (cos x + cos 3d?) + 4 7 cos 3# + 4a/3 cos #. Expressing the terms in cos x by means of y, we get finally U 2 - -2gy = const. + 2y (1 - g - a 2 - a 4 + /3) + (a 4 + 2/3) cos 2# + (4 7 - * a 5 - 2a/3) cos 3# ....... (9) In order to satisfy the surface condition of constant pressure, we must take /3 = -^ 4 , 7 = iV s > ........................ (10) and in addition \-g-a?-*a*=Q, ........................... (11) correct to a 5 inclusive. The expression (1) for i/r thus assumes the form ^ = y - ae-y cos x + ^e'^cos 2# - ^a 8 ^ cos 3#, ......... (12) from which y may be calculated in terms of x as far as a 5 inclusive. By successive approximation, determining \/r so as to make the mean value of y equal to zero, we find as far as a 4 y = ( a + | a 3 ) cos x ( a 2 4- |a 4 ) cos 2# + fa 3 cos 3# a 4 cos 4#, ... (13) or, if we write as before a for the coefficient of cos a; y = acosx-($a? + ^a 4 ) cos 2# + fa 3 cos 3# - a 4 cos 4c, . . .(14) in agreement with equation (20) of Stokes' Supplement. Expressed in terms of a, (11) becomes g=l -a?-la* ............................... (15) or on restoration of k, c, g = kc*-}<?a*c 2 -$k s a t c?. ........................ (16) Thus the extension of (8) is c* = g/k.(l +k n -a?+%fra 4 ), ........................ (17) which also agrees with Stokes' Supplement. If we pursue the approximation one stage further, we find from (12) terms in a 5 , additional to those expressed in (13). These are 373 243 125 128 12732 It is of interest to compare the potential and kinetic energies of waves * [1916. Burnside (Proc. Land. Math. Soc. Vol. xv. p. 26, 1916) throws doubts upon the utility of Stokes' series.] 14 HYDRODYNAMICAL NOTES [351 that are not infinitely small. For the stream-function of the waves regarded as progressive, we have, as in (1), ty = ae~ y cos (x ct) 4- terms in a 4 , so that (d-^rfdxY + (d^r /dy) 3 = aVr 2 " + terms in a s . Thus the mean kinetic energy per length x measured in the direction of propagation is where y is the ordinate of the surface. And by (3) Hence correct to a 4 , K.E. = ia 2 (l+a 2 )a; ............................ (19) Again, for the potential energy P.E. =g jy*dx = %gx(^ + f a 4 ); or since g = 1 a 2 , P.E. = ia 2 (l + a 2 )tf ......................... (20) The kinetic energy thus exceeds the potential energy, when o 4 is retained. Tide Races. It is, I believe, generally recognized that seas are apt to be exceptionally heavy when the tide runs against the wind. An obvious explanation may be founded upon the fact that the relative motion of air and water is then greater than if the latter were not running, but it seems doubtful whether this explanation is adequate. It has occurred to me that the cause may be rather in the motion of the stream relatively to itself, e.g. in the more rapid movement of the upper strata. Stokes' theory of the highest possible wave shows that in non-rotating water the angle at the crest is 120 and the height only moderate. In such waves the surface strata have a mean motion forwards. On the other hand, in Gerstner and Rankine's waves the fluid particles retain a mean position, but here there is rotation of such a character that (in the absence of waves) the surface strata have a relative motion backwards, i.e. against the direction of propagation*. It seems possible that waves moving against the tide may approximate more or less to the Gerstner type and thus be capable of acquiring a greater height and a sharper angle than would otherwise be expected. Needless to say, it is the steepness of waves, rather than their * Lamb's Hydrodynamics, 247. 1911] HYDRODYNAMICAL NOTES 15 mere height, which is a source of inconvenience and even danger to small craft. The above is nothing more than a suggestion. I do not know of any detailed account of the special character of these waves, on which perhaps a better opinion might be founded. Rotational Fluid Motion in a Corner. The motion of incompressible inviscid fluid is here supposed to take place in two dimensions and to be bounded by two fixed planes meeting at an angle a. If there is no rotation, the stream-function ty, satisfying V 2 ijr = 0, may be expressed by a series of terms */ sin 7r0/a, r 27r / a sin 2ir0ja, . . . r n */ a sin mrO/a, where n is an integer, making i/r = when 6 = or 6 = a. In the immediate vicinity of the origin the first term predominates. For example, if the angle be a right angle, ^ = r 2 sin 20 = 2xy, (1) if we introduce rectangular coordinates. The possibility of irrotational motion depends upon the fixed boundary not being closed. If a < TT, the motion near the origin is finite ; but if a > TT, the velocities deduced from i|r become infinite. If there be rotation, motion may take place even though the boundary be closed. For example, the circuit may be completed by the arc of the circle r = 1. In the case which it is proposed to consider the rotation ro is uniform, and the motion may be regarded as steady. The stream- function then satisfies the general equation V-^ = d^/dx* + d^ldf = 2a>, (2) or in polar coordinates d^ 1 d^ 1 d 2 ^ . d + r^ + ;= rfiH" (3) When the angle is a right angle, it might perhaps be expected that there should be a simple expression for i/r in powers of x and y, analogous to (1) and applicable to the immediate vicinity of the origin ; but we may easily satisfy ourselves that no such expression exists*. In order to express the motion we must find solutions of (3) subject to the conditions that >/r = when 6 = and when 6 = a. For this purpose we assume, as we may do, that ^ = 2R n sin mr0/a, (4) * In strictness the satisfaction of (2) at the origin is inconsistent with the evanescence of ^ on the rectangular axes. HYDRODYNAMICAL NOTES [351 where n is integral and R n a function of r only ; and in deducing may perform the differentiations with respect to 6 (as well as with respect to r) under the sign of summation, since ^ = at the limits. Thus The right-hand member of (3) may also be expressed in a series of sines of the form 2&> = 8o>/7r . Sn- 1 sin nir0/a, ........................ (6) where n is an odd integer; and thus for all values of n we have - + r The general solution of (7) is ............. (8) the introduction of which into (4) gives ^. In (8) A n and B n are arbitrary constants to be determined by the other conditions of the problem. For example, we might make /?, and therefore >/r, vanish when r = r^ and when r = r z , so that the fixed boundary enclosing the fluid would consist of two radii vectores and two circular arcs. If the fluid extend to the origin, we must make B n = ; and if the boundary be completed by the circular arc r = 1, we have A n = when n is even, and when n is odd (9 > Thus for the fluid enclosed in a circular sector of angle a and radius unity (10) .. - 4a s ) a the summation extending to all odd integral values of n. The above formula (10) relates to the motion of uniformly rotating fluid bounded by stationary radii vectores at 6 = 0, 6 = a. We may suppose the containing vessel to have been rotating for a long time and that the fluid (under the influence of a very small viscosity) has acquired this rotation so that the whole revolves like a solid body. The motion expressed by (10) is that which would ensue if the rotation of the vessel were suddenly stopped. A related problem was solved a long time since by Stokes*, who considered the irrotational motion of fluid in a revolving sector. The solution of Stokes' problem is derivable from (10) by mere addition to the latter of i/r = - ^car 3 , for then ty + i/r satisfies V 2 (i^ + -t/r ) = ; and this is perhaps the simplest Camb. Phil, Trans. Vol. vm. p. 533 (1847) ; Math, and Phys. Papen, Vol. i. p. 305. 1911] HYDKODYNAMICAL NOTES 17 method of obtaining it. The results are in harmony; but the fact is not immediately apparent, inasmuch as Stokes expresses the motion by means of the velocity-potential, whereas here we have employed the stream -function. That the subtraction of |<or 2 makes (10) an harmonic function shows that the series multiplying ?* can be summed. In fact 2 sin (mrd/a) = cos (20 -a) I wr(>V 2 -4a 2 )~ 2 cos a 2' r 2 cos (20 - a) , ^ r nir / a sin n?r0/a so that ^/ty = |r 2 -- ^ + 8o 2 2 -r-r- -^ (11) 2 cos a UTT ( n 2 ?r 2 - 4o 2 ) In considering the character of the motion defined by (11) in the immediate vicinity of the origin we see that if a < \ir, the term in r 2 preponderates even when n= 1. When a= \tr exactly, the second term in (11) and the first term under 2 corresponding to n = 1 become infinite, and the expression demands transformation. We find in this case (6 - fr) cos ^'-Sri- (*-!) (12) the summation commencing at n = 3. On the middle line 6 = ^TT, we have The following are derived from (13) : r -W r -fer* r -W o-o ooooo 0-4 14112 0-8 13030 o-i 02267 0-5 16507 0-9 07641 0-2 06296 0-6 17306 1-0 ii -00000 0-3 10521 0-7 16210 i The maximum value occurs when r = '592. At the point r '592, 6 = ^TT, the fluid is stationary. A similar transformation is required when a = 3?r/2. When a = TT, the boundary becomes a semicircle, and the leading term (n=l) is o o (14) _3_ 8?r' which of itself represents an irrotational motion. R. VI. 18 HYDRODYNAMICAL NOTES [351 When o = 2-n; the two bounding radii vectores coincide and the containing vessel becomes a circle with a single partition wall at 6 = 0. In this case again the leading term is irrotational, being Steady Motion in a Corner of a Viscous Fluid. Here again we suppose the fluid to be incompressible and to move in two dimensions free from external forces, or at any rate from such as cannot be derived from a potential. If in the same notation as before ^ represents the stream- function, the general equation to be satisfied by ^r is V^ = 0; ................................. (1) with the conditions that when = and = a, ^ = 0, d^/d0=Q ............................ (2) It is worthy of remark that the problem is analytically the same as that of a plane elastic plate clamped at = and 6 = a, upon which (in the region considered) no external forces act. The general problem thus represented is one of great difficulty, and all that will be attempted here is the consideration of one or two particular cases. We inquire what solutions are possible such that ty, as a function of r (the radius vector), is proportional to r m . Introducing this supposition into (1), we get as the equation determining the dependence on 6. The most general value of \Jr consistent with our suppositions is thus ^ =r m {A cosm0 + Bsmm0+Ccos(m-2)0 + Dsm(m- 2)6], ...(4) where A, B, C, D are constants. Equation (4) may be adapted to our purpose by taking m = mrja, ................................. (5) where n is an integer. Conditions (2) then give A + C = Q, A + <7cos2a-Dsin2a=0, ^0820 = 0. ^ + (^ - 2) C7 sin 2a+(-- 2) a \ a / \ a / 1911] HYDRODYNAMICAL NOTES 19 When we substitute in the second and fourth of these equations the values of A and B, derived from the first and third, there results C(l-cos2a)+Z>sin2a = 0, C sin 2a - D (1 - cos 2a) = ; and these can only be harmonized when cos2a = l, or OL = STT, where s is an integer. In physical problems, a is thus limited to the values TT and 2-7T. To these cases (4) is applicable with C and D arbitrary, provided that we make + C=0, (5 bis) Thus making Cr n / jcos (^ - 20) - cos ^ J + Drl jsin (?? - 20) - ( 1 - *) sin ^l , ...(6) 1 Vf J \ nj s}' = 4 Q - l) r-*** JC cos (^ - 20) + D sin (^ - 20)} . . . .(7) When s = 1, a = TT, the corner disappears and we have simply a straight boundary (fig. 1). In this case n = l gives a nugatory result. When n = 2, we have v/r=CV 2 (l-cos20) = 2Cy, ..................... (8) Fig. 1. Fig. 2. and 8=1 When n = 3, = Or 3 (cos - cos 30) + Dr 3 (sin 0- sin 30), In rectangular coordinates (9) (10) (11) solutions which obviously satisfy the required conditions. When s = 2, a = 2-7T, the boundary consists of a straight wall extending from the origin in one direction (fig. 2). In this case (6) and (7) give -f = GY* n [cos (w0 - 20) - cos %nd\ ...... (12) = (2/i - 4) ri- 8 {C cos (%n0 - 20) + D sin - 20)}. . . .(13) 22 20 HYDRODYNAMICAL NOTES [351 Solutions of interest are afforded in the case n = 1. The C-solution is vanishing when = IT, as well as when 6 = 0, 6 = 27r, and for no other admissible value of 6. The values of i/r are reversed when we write 2?r 6 for #. As expressed, this value is negative from to TT and positive from TT to 2-TT. The minimum occurs when 6 = 109 28'. Every stream-line which enters the circle (r= 1) on the left of this radius leaves it on the right. The velocities, represented by d^jdr and r~ l dtyldd, are infinite at the origin. For the D-solution we may take ^ = rising (15) Here i/r retains its value unaltered when 2?r - is substituted for 0. When r is given, i/r increases continuously from 6 = to 6 = TT. On the line = TT the motion is entirely transverse to it. This is an interesting example of the flow of viscous fluid round a sharp corner. In the application to an elastic plate >/r represents the displacement at any point of the plate, supposed to be clamped along = 0, and otherwise free from force within the region con- sidered. The following table exhibits corresponding values of r and 6 such as to make !//= 1 in (15) : e r e r 180 1-00 60 64-0 150 1-23 20 10 4 x3-65 120 2-37 10 108x2-28 90 8-00 00 When n = 2, (12) appears to have no significance. When n= 3, the dependence on 6 is the same as when n= 1. Thus (14) and (15) may be generalized : ^r = (Ar^ +r*)cos0sin j 0, (16) ^ = (A'r* + B'r*) sin' (17) For example, we could satisfy either of the conditions ^ = 0, or difr/dr = 0, on the circle r= 1. For n = 4 the D-solution becomes nugatory ; but for the C-solution we have ^ = (7^(1 -cos 26) =2(7^81^0= 2Cy (18) The wall (or in the elastic plate problem the clamping) along 6 = is now without effect. 1911] HYDRODYNAMICAL NOTES 21 It will be seen that along these lines nothing can be done in the apparently simple problem of a horizontal plate clamped along the rectangular axes of x and y, if it be supposed free from force*. Ritzf has shown that the solution is not developable in powers of x and y, and it may be worth while to extend the proposition to the more general case when the axes, still regarded as lines of clamping, are inclined at any angle a. In terms of the now oblique coordi- nates x, y the general equation takes the form (d*/dx; 2 + d*/dy* - 2 cos a d*/dx dy)*w = 0, (19) which may be differentiated any number of times with respect to x and y, with the conditions w=Q, dw/dy = 0, wheny = 0, (20) w=0, dw/dx = Q, when # = (21) We may differentiate, as often as we please, (20) with respect to x and (21) with respect to y. From these data it may be shown that at the origin all differential coefficients of w with respect to x arid y vanish. The evanescence of those of zero and first order is expressed in (20), (21). As regards those of the second order we have from (20) d 2 w/dx* = 0, d*w(dxdy = 0, and from (21) d 2 w/dy 2 = 0. Similarly for the third order from (20) dtw/dx 3 = 0, d^wjdx^dy = 0, and from (21) d*w/dy 3 = 0, d z w \dxdf = 0. For the fourth order (20) gives d*wldx* = 0, d 4 wjda? dy = 0, and (21) gives d*w/dy* = 0, d*w/dxdy s = 0. So far d*w/dx z dy- might be finite, but (19) requires that it also vanish. This process may be continued. For the m + 1 coefficients of the rath order we obtain four equations from (20), (21) and ra 3 by differentiations of (19), so that all the differential coefficients of the rath order vanish. It follows that every differential coefficient of w with respect to x and y vanishes at the origin. I apprehend that the conclusion is valid for all angles a less than 2?r. That the displacement at a distance r from the corner should diminish rapidly with r is easily intelligible, but that it should diminish more rapidly than any power of r, however high, would, I think, not have been expected without analytical proof. * If indeed gravity act, w=x z y* is a very simple solution, t Ann. d. Phys. Bd. xxvin. p. 760, 1909. 352. ON A PHYSICAL INTERPRETATION OF SCHLOMILCH'S THEOREM IN BESSEL'S FUNCTIONS. [Philosophical Magazine, Vol. xxi. pp. 567571, 1911.] THIS theorem teaches that any function /(r) which is finite and con- tinuous for real values of r between the limits r = and r = TT, both inclusive, may be expanded in the form f(r) = a + a l J (r) + avJ (2r)+a 3 J (3r) + ... ) ......... (1) / being the Bessel's function usually so denoted ; and Schlomilch's demon- stration has been reproduced with slight variations in several text-books*. So far as I have observed, it has been treated as a purely analytical develop- ment. From this point of view it presents rather an accidental appearance ; and I have thought that a physical interpretation, which is not without interest in itself, may help to elucidate its origin and meaning. The application that I have in mind is to the theory of aerial vibrations. Let us consider the most general vibrations in one dimension which are periodic in time 2?r and are also symmetrical with respect to the origins of and t. The condensation s, for example, may be expressed s = & + &,cos|: cos + & 2 cos2cos2 + .................. (2) where the coefficients b , b lt &c. are arbitrary. (For simplicity it is supposed that the velocity of propagation is unity.) When t 0, (2) becomes a function of only, and we write J f () = & + 6 1 cos + 6 8 cos2+..., .................. (3) in which F(^) may be considered to be an arbitrary function of from to TT. Outside these limits F is determined by the equations (4) * See, for example, Gray and Mathews' Sestets Functions, p. 30; Whittaker's Modern Analysis, 165. 1911] SCHLOMILCH'S THEOREM IN BESSEL'S FUNCTIONS 23 We now superpose an infinite number of components, analogous to (2) with the same origins of space and time, and differing from one another only in the direction of , these directions being limited to the plane xy, and in this plane distributed uniformly. The resultant is a function of t and r only, where r = J(a? + /), independent of the third coordinate z, and therefore (as is known) takes the form s = a + i J (r) cos t + a*, / (2r) cos It + a 3 J (3r) cos 3 + . . ., . . .(5) reducing to (1) when t = 0*. The expansion of a function in the series (1) is thus definitely suggested as probable in all cases and certainly possible in an immense variety. And it will be observed that no value of greater than TT contributes anything to the resultant, so long as r < TT. The relation here implied between F and / is of course identical with that used in the purely analytical investigation. If < be the angle between f and any radius vector r to a point where the value of / is required, = r cos <ft, and the mean of all the components F (%) is expressed by (6) The solution of the problem of expressing F by means of / is obtained analytically with the aid of Abel's theorem. And here again a physical, or rather geometrical, interpretation throws light upon the process. Equation (6) is the result of averaging F(%) over all directions indifferently in the xy plane. Let us abandon this restriction and take the average when f is indifferently distributed in all directions whatever. The result now becomes a function only of R, the radius vector in space. If 9 be the angle between R and one direction of , = R cos 0, and we obtain as the .'o R where FJ = F. This result is obtained by a direct integration of F (f ) over all directions in space. It may also be arrived at indirectly from (6). In the latter f(r) represents the averaging of F (g) for all directions in a certain plane, the result being independent of the coordinate perpendicular to the plane. If we take the average again for all possible positions of this plane, we must recover (7). Now if be the angle between the normal to this plane and the radius vector R, r = R sin 0, and the mean is l*"f(Rsm0)sm0d0 (8) * It will appear later that the a'a and fc's are equal. 24 ON A PHYSICAL INTERPRETATION OF [352 We conclude that which may be considered as expressing F in terms off. If in (6), (9) we take F(R) = cos R, we find* fi | J (R sin 0) sin d0 = R~* sin R. . o Differentiating (9), we get F/ m_ (**/(# sin 0) sin d0 + # I **/'( sin 0)(i -cos 2 0) d6. ...(10) .'o ><> Now U f * cos 1 0f (R sin 0) d0 = [ cos . df(R sin 0) = ~/(0) + [ /(-R sin 0) sin d0. Accordingly F()-/(0) + .B| f(Rsm0)d0 (11) That /(r) in (1) may be arbitrary from to TT is now evident. By (3) and (6) 2 ft* f(r)= - d<j> [b + &! cos (r cos <f>) + 6 2 cos (2r cos <) + . . . } 771 . o where 6 = (|)^, 6^ - cos nf F(|) df. ......... (13) Further, with use of (11) b =/() + ^ J " rf M f V (f sin v de > ............ ( u ) 6,-- r<2|.fooen|. f^/'tfim^clft ............ (15) *T J - f by which 'the coefficients in (12) are completely expressed when / is given between and TT. The physical interpretation of Schlomilch's theorem in respect of two- dimensional aerial vibrations is as follows : Within the cylinder r = TT it is possible by suitable movements at the boundary to maintain a symmetrical motion which shall be strictly periodic in period 2-7T, and which at times t = 0, t = 2-rr, &c. (when there is no velocity), shall give a condensation which * Enc. Brit. Art. "Wave Theory," 1888; Scientific Papers, Vol. in. p. 98. 1911] SCHLOMILCH'S THEOREM IN BESSEL'S FUNCTIONS 25 is arbitrary over the whole of the radius. And this motion will maintain itself without external aid if outside r TT the initial condition is chosen in accordance with (6), F '() for values of greater than TT being determined by (4). A similar statement applies of course to the vibrations of a stretched membrane, the transverse displacement w replacing s in (5). Reference may be made to a simple example quoted by Whittaker. Initially let/(r) = r, so that from to TT the form of the membrane is conical. Then from (12), (14), (15) b = -j- , b n = (n even), b n = -- - (n odd) ; and thus ..., ...... (16) the right-hand member being equal to r from r = to r = TT. The corresponding vibration is of course expressed by (16) if we multiply each function J (nr) by the time-factor cos nt. If this periodic vibration is to be maintained without external force, the initial condition must be such that it is represented by (16) for all values of r, and not merely for those less than TT. By (11) from to TT, F(g) fa, from which again by (4) the value of F for higher values of follows. Thus from TT to 2-7T, *()* i* (2* -.); from 2-rr to STT, F(& = fa(j;- 2-ir); and so on. From these / is to be found by means of (6). For example, from 7T tO 2-7T, r sin 9 = ir/r r ain 8 = \ f(r) = r\ m*6d0+\ (27r-rsin0)<20 /O .' sin0 = ir/r = r - 2 V(r 2 - 7T 2 ) + 27r cos- 1 (7r/r), ........................ (17) where cos" 1 (TT/T) is to be taken in the first quadrant. It is hardly necessary to add that a theorem similar to that proved above holds for aerial vibrations which are symmetrical in all directions about a centre. Thus within the sphere of radius TT it is possible to have a motion which shall be strictly periodic and is such that the condensation is initially arbitrary at all points along the radius. 353. BREATH FIGURES. [Nature, Vol. LXXXVI. pp. 416, 417, 1911.] THE manner in which aqueous vapour condenses upon ordinarily clean surfaces of glass or metal is familiar to all. Examination with a magnifier shows that the condensed water is in the form of small lenses, often in pretty close juxtaposition. The number and thickness of these lenses depend upon the cleanness of the glass and the amount of water deposited. In the days of wet collodion every photographer judged of the success of the cleaning process by the. uniformity of the dew deposited from the breath. Information as to the character of the deposit is obtained by looking through it at a candle or small gas flame. The diameter of the halo measures the angle at which the drops meet the glass, an angle which diminishes as the dew evaporates. That the flarne is seen at all in good definition is a proof that some of the glass is uncovered. Even when both sides of a plate are dewed the flame is still seen distinctly though with much diminished intensity. The process of formation may be followed to some extent under the microscope, the breath being led through a tube. The first deposit occurs very suddenly. As the condensation progresses, the drops grow, and many of the smaller ones coalesce. During evaporation there are two sorts of behaviour. Sometimes the boundaries of the drops contract, leaving the glass bare. In other cases the boundary of a drop remains fixed, while the thickness of the lens diminishes until all that remains is a thin lamina. Several successive formations of dew will often take place in what seems to be precisely the same pattern, showing that the local conditions which determine the situation of the drops have a certain degree of permanence. An interesting and easy experiment has been described by Aitken (Proc. Ed. Soc. p. 94, 1893). Clean a glass plate in the usual way until the breath deposits equally. 1911] BREATH FIGURES 27 " If we now pass over this clean surface the point of a blow-pipe flame, using a very small jet, and passing it over the glass with sufficient quickness to prevent the sudden heating breaking it ; and if we now breathe on the glass after it is cold, we shall find the track of the flame clearly marked. While most of the surface looks white by the light reflected from the de- posited moisture, the track of the flame is quite black ; not a ray of light is scattered by it. It looks as if there were no moisture condensed on that part of the plate, as it seems unchanged ; but if it be closely examined by a lens, it will be seen to be quite wet. But the water is so evenly distributed, that it forms a thin film, in which, with proper lighting and the aid of a lens, a display of interference colours may be seen as the film dries and thins away." "Another way of studying the change produced on the surface of the glass by the action of the flame is to take the [plate], as above described, after a line has been drawn over it with the blow-pipe jet, and when cold let a drop of water fall on any part of it where it showed white when breathed on. Now tilt the plate to make the drop flow, and note the resistance to its flow, and how it draws itself up in the rear, leaving the plate dry. When, however, the moving drop comes to the part acted on by the flame, all resistance to flow ceases, and the drop rapidly spreads itself over the whole track, and shows a decided disinclination to leave it." The impression thus produced lasts for some days or weeks, with diminish- ing distinctness. A permanent record may be obtained by the deposit of a very thin coat of silver by the usual chemical method. The silver attaches itself by preference to the track of the flame, and especially to the edges of the track, where presumably the combustion is most intense. It may be protected with celluloid, or other, varnish. The view, expressed by Mr Aitken, which would attribute the effect to very fine dust deposited on the glass from the flame, does not commend itself to me. And yet mere heat is not very effective. I was unable to obtain a good result by strongly heating the back of a thin glass in a Bunsen flame. For this purpose a long flame on Ramsay's plan is suitable, especially if it be long enough to include the entire width of the plate. It seems to me that we must appeal to varying degrees of cleanliness for the explanation, cleanliness meaning mainly freedom from grease. And one of the first things is to disabuse our minds of the idea that anything wiped with an ordinary cloth can possibly be clean. This subject was ably treated many years ago by Quincke (Wied. Ann. n. p. 145, 1877), who, however, seems to have remained in doubt whether a film of air might not give rise to the same effects as a film of grease. Quincke investigated the maximum edge-angle possible when a drop of liquid stands upon the surface of a solid. In general, the cleaner the surface, the smaller the 28 BREATH FIGURES [353 maximum edge-angle. With alcohol and petroleum there was no difficulty in reducing the maximum angle to zero. With water on glass the angle could be made small, but increased as time elapsed after cleaning. As a detergent Quincke employed hot sulphuric acid. A few drops may be poured upon a thin glass plate, which is then strongly heated over a Bunsen burner. When somewhat cooled, the plate may be washed under the tap, rinsed with distilled water, and dried over the Bunsen without any kind of wiping. The parts wetted by the acid then behave much as the track of the blow-pipe flame in Aitken's experiment. An even better treatment is with hydrofluoric acid, which actually renews the surface of the glass. A few drops of the commercial acid, diluted, say, ten times, may be employed, much as the sulphuric acid, only without heat. The parts so treated condense the breath in large laminae, contrasting strongly with the ordinary deposit. It must be admitted that some difficulties remain in attributing the behaviour of an ordinary plate to a superficial film of grease. One of these is the comparative permanence of breath figures, which often survive wiping with a cloth. The thought has sometimes occurred to me that the film of grease is not entirely superficial, but penetrates in some degree into the substance of the glass. In that case its removal and renewal would not be so easy. We know but little of the properties of matter in thin films, which may differ entirely from those of the same substance in mass. It may be recalled that a film of oil, one or two millionths of a millimetre thick, suffices to stop the movements of camphor on the surface of water, and that much smaller quantities may be rendered evident by optical and other methods. 354. ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID. [Philosophical Magazine, Vol. XXI. pp. 697711, 1911.] 1. THE problem of the uniform and infinitely slow motion of a sphere, or cylinder, through an unlimited mass of incompressible viscous liquid otherwise at rest was fully treated by Stokes in his celebrated memoir on Pendulums*. The two cases mentioned stand in sharp contrast. In the first a relative steady motion of the fluid is easily determined, satisfying all the conditions both at the surface of the sphere and at infinity ; and the force required to propel the sphere is found to be finite, being given by the formula (126) -F=Qir t MV i (1) where p, is the viscosity, a the radius, and V the velocity of the sphere. On the other hand in the case of the cylinder, moving transversely, no such steady motion is possible. If we suppose the cylinder originally at rest to be started and afterwards maintained in uniform motion, finite effects are propagated to ever greater and greater distances, and the motion of the fluid approaches no limit. Stokes shows that more and more of the fluid tends to accompany the travelling cylinder, which thus experiences a con- tinually decreasing resistance. 2. In attempting to go further, one of the first questions to suggest itself is whether similar conclusions are applicable to bodies of other forms. The consideration of this subject is often facilitated by use of the well- known analogy between the motion of a viscous fluid, when the square of the motion is neglected, and the displacements of an elastic solid. Suppose that in the latter case the solid is bounded by two closed surfaces, one of which completely envelopes the other. Whatever displacements (a, #, 7) be imposed at these two surfaces, there must be a corresponding configuration * Camb. Phil. Trans. Vol. ix. 1850; Math, and Phys. Papers, Vol. in. p. 1 30 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354 of equilibrium, satisfying certain differential equations. If the solid be incompressible, the otherwise arbitrary boundary displacements must be chosen subject to this condition. The same conclusion applies in two dimensions, where the bounding surfaces reduce to cylinders with parallel generating lines. For our present purpose we may suppose that at the outer surface the displacements are zero. The contrast between the three-dimensional and two-dimensional cases arises when the outer surface is made to pass off to infinity. In the former case, where the inner surface is supposed to be limited in all directions, the displacements there imposed diminish, on receding from it, in such a manner that when the outer surface is removed to a sufficient distance no further sensible change occurs. In the two-dimensional case the inner surface extends to infinity, and the displacement affects sensibly points however distant, provided the outer surface be still further and sufficiently removed. The nature of the distinction may be illustrated by a simple example relating to the conduction of heat through a uniform medium. If the temperature v be unity on the surface of the sphere r = a, and vanish when r = b, the steady state is expressed by When 6 is made infinite, v assumes the limiting form a/r. In the corre- sponding problem for coaxal cylinders of radii a and 6 we have v = ^gb-\ogr \ogb-\oga' But here there is no limiting form when 6 is made infinite. However great / may be, v is small when 6 exceeds r by only a little ; but when b is great enough v may acquire any value up to unity. And since the distinction depends upon what occurs at infinity, it may evidently be extended on the one side to oval surfaces of any shape, and on the other to cylinders with any form of cross-section. In the analogy already referred to there is correspondence between the displacements (a, yQ, 7) in the first case and the velocities (u, v, w) which express the motion of the viscous liquid in the second. There is also another analogy which is sometimes useful when the motion of the viscous liquid takes place in two dimensions. The stream-function (i/r) for this motion satisfies the same differential equation as does the transverse displacement (w') of a plane elastic plate. And a surface on which the fluid remains at rest (-^ = 0, d-^r/dn = 0) corresponds to a curve along which the elastic plate is clamped. In the light of these analogies we may conclude that, provided the square of the motion is neglected absolutely, there exists always a unique steady 1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 31 motion of liquid past a solid obstacle of any form limited in all directions, which satisfies the necessary conditions both at the surface of the obstacle and at infinity, and further that the force required to hold the solid is finite. But if the obstacle be an infinite cylinder of any cross-section, no such steady motion is possible, and the force required to hold the cylinder in position continually diminishes as the motion continues. 3. For further developments the simplest case is that of a material plane, coinciding with the coordinate plane x = and moving parallel to y in a fluid originally at rest. The component velocities u, w are then zero ; and the third velocity v satisfies (even though its square be not neglected) the general equation dv d*v in which v, equal to p,jp, represents the kinematic viscosity. In 7 of his memoir Stokes considers periodic oscillations of the plane. Thus in (4) if v be proportional to e int , we have on the positive side v = Ae int e~ x ^ < l ' w /"> ............................... (5) When x = 0, (5) must coincide with the velocity ( V) of the plane. If this be V n e int , we have A = V n \ so that in real quantities s{nt-xJ(n/2v)} .................. (6) corresponds with V = V n cos nt .............................. (7) for the plane itself. In order to find the tangential force ( T 3 ) exercised upon the plane ; we have from (5) when x = - Fn^vW"), ........................ (8) and T a =-p (dv/dx\ = p V n e int </(inv) = p^^nv).(l+i)V n e int = p^n V ).(v + - ?), ......... (9) \ n Qii / giving the force per unit area due to the reaction of the fluid upon one side. " The force expressed by the first of these terms tends to diminish the amplitude of the oscillations of the plane. The force expressed by the second has the same' effect as increasing the inertia of the plane." It will be observed that if V n be given, the force diminishes without limit with n. In note B Stokes resumes the problem of 7 : instead of the motion of the plane being periodic, he supposes that the plane and fluid are initially at rest, and that the plane is then (i = 0) moved with a constant velocity V. 32 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354 This problem depends upon one of Fourier's solutions which is easily verified*. We have v=V -- e~*dz ......................... (11) V7T Jo For the reaction on the plane we require only the value of dv/dx when x = 0. And Stokes continues f " now suppose the plane to be moved in any manner, so that its velocity at the end of the time t is V (t). We may evidently obtain the result in this case by writing V (T) dr for V, and t T for t in [12], and integrating with respect to T. We thus get dv\ 1 [< V'(r)dr 1 r ft, , )o = ~V(-)J_ 00 7(^r) = ~V(^)Jo ' -^ - (1< and since T s = fidv/dx , these formulae solve the problem of finding the reaction in the general case. There is another method by which the present problem may be treated, and a comparison leads to a transformation which we shall find useful further on. Starting from the periodic solution (8), we may generalize it by Fourier's theorem. Thus corresponds to* Jo where V n is an arbitrary function of n. Comparing (13) and (14), we see that It is easy to verify (16). If we substitute on the right for V (T) from (15), we get and taking first the integration with respect to T, when (16) follows at once. * Compare Kelvin, Ed. Tram. 1862 ; Thomson and Tait, Appendix D. t I have made some small changes of notation. 1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 33 As a particular case of (13), let us suppose that the fluid is at rest and that the plane starts at t = with a velocity which is uniformly accelerated for a time TJ and afterwards remains constant. Thus from oo to 0, F(r) = 0; from to T I} F(T) = /*T; from T, to t, where t > r lt V(r) = hr l . Thus (0 < t < T,) and Expressions (17), (18), taken negatively and multiplied by /i, give the force per unit area required to propel the plane against the fluid forces acting upon one side. The force increases until t = r l , that is so long as the acceleration continues. Afterwards it gradually diminishes to zero. For the differential coefficient of *Jt \/(t rO is negative when t > TJ ; and when t is great, V* - V(* - TO = T, ~ * ultimately. 4. In like manner we may treat any problem in which the motion of the material plane is prescribed. A more difficult question arises when it is the forces propelling the plane that are given. Suppose, for example, that an infinitely thin vertical lamina of superficial density a begins to fall from rest under the action of gravity when t = 0, the fluid being also initially at rest. By (13) the equation of motion may be written dV 2p^f'V'(r)dr_ dF + ^oT^)-"' ' the fluid being now supposed to act on both sides of the lamina. By an ingenious application of Abel's theorem Boggio has succeeded in integrating equations which include (19)*. The theorem is as follows: If ^ (t) be defined by M,.. ......................... (20 ) then ^CO -</>(<>)} ...................... (21) Jo ($-T>* For by (20), if (t - r) 4 = y, * Boggio, Rend. d. Accad. d. Lincei, Vol. xvi. pp. 613, 730 (1907) ; also Basset, Quart. Journ. of Mathematics, No. 164, 1910, from which I first became acquainted with Boggio's work. R. VI. 3 34 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354 so that rt,h.( T \(i r / = 2 o (* - T)* - <#> (0)}, o where r* = a? + y s . Now, if 2' be any time between and t, we hav, as in (19), Multiplying this by (< t') * eft' and integrating between and t, we get (' V'(f)dt' >> t> dt' fV'(r)dr_f' df '. 7^0*" W Jo (-!)*> "^7 ~'-''(t-f?' In (22) the first integral is the same as the integral in (19). By Abel's theorem the double integral in (22) is equal to 7rV(t), since F(0)=0. Thus <> If we now eliminate the integral between (19) and (23), we obtain simply %-?*-.-.+ ..................... (-> as the differential equation governing the motion of the lamina. This is a linear equation of the first order. Since V vanishes with t, the integral may be written (25) VTT vf in which t' = t . 4p*v/o*. When t, or ', is great, .C/""^ = ^r( 1 -5? + -) ; .................. (26) -r= 2 r'- Ultimately, when t is very great, .K I(L\ P V V 7rv / 1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 35 5. The problem of the sphere moving with arbitrary velocity through a viscous fluid is of course more difficult than the corresponding problem of the plane lamina, but it has been satisfactorily solved by Boussinesq* and by Basset f . The easiest road to the result is by the application of Fourier's theorem to the periodic solution investigated by Stokes. If the velocity of the sphere at time t be V= V n e int , a the radius, M' the mass of the liquid displaced by the sphere, and s = */(n/2v), v being as before the kinematic viscosity, Stokes finds as the total force at time t F = -M'V n n (fi + . ) t + . (l + -}\ *" ..... (29) (\2 40a/ 4sa V saj) Thus, if V=\ V n <P*dn, ...................... (30) J Of the four integrals in (31), the first = [ in V n e int dn = V ; the fourth = ^ [" V n 0* dn = ^ V. Also the second and third together give t )r J and this is the only part which could present any difficulty. We have, however, already considered this integral in connexion with the motion of a plane and its value is expressed by (16). Thus lldV 9v v **[> V'(T)dr\ - M+r+ " The first term depends upon the inertia of the fluid, and is the same as would be obtained by ordinary hydrodynamics when v = 0. If there is no acceleration at the moment, this term vanishes. If, further, there has been no acceleration for a long time, the third term also vanishes, and we obtain the result appropriate to a uniform motion SvM'V T7 jr F = -- = QirapvV = Q-n-fiaV, as in (1). The general result (32) is that of Boussinesq and Basset. * C. R. t. c. p. 935 (1885) ; Theorie Analytique de la Chaleur, t. n. Paris, 1903. t Phil. Trans. 1888 ; Hydrodynamics, Vol.- n. chap. xxn. 1888. 32 36 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354 As an example of (32), we may suppose (as formerly for the plane) that 7(0 = from - oo to 0; V(t) = ht from to T, ; V(t)**hr lt when t > T,. Then if *<T,, and when t>r l , (34) When i is very great (34) reduces to its first term. The more difficult problem of a sphere falling under the influence of gravity has been solved by Boggio (loc. rit.). In the case where the liquid and sphere are initially at rest, the solution is comparatively simple ; but the analytical form of the functions is found to depend upon the ratio of densities of the sphere and liquid. This may be rather unexpected ; but I am unable to follow Mr Basset in regarding it as an objection to the usual approximate equations of viscous motion. 6. We will now endeavour to apply a similar method to Stokes' solution for a cylinder oscillating transversely in a viscous fluid. If the radius be a and the velocity Fbe expressed by V= V n e int , Stokes finds for the force F=-M'inV n e int (k-ik f ) ...................... (35) In (35) M' is the mass of the fluid displaced ; k and k' are certain functions of r, where m = ^a J(njv), which are tabulated in his 37. The cylinder is much less amenable to mathematical treatment than the sphere, and we shall limit ourselves to the case where, all being initially at rest, the cylinder is started with unit velocity which is afterwards steadily maintained. The velocity V of the cylinder, which is to be zero when t is negative and unity when t is positive, may be expressed by in which the second term may be regarded as the real part of dn (37) n We shall see further below, and may anticipate from Stokes' result relating to uniform motion of the cylinder, that the first term of (36) contributes /. nothing to F; so that we may take ~ 1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 37 corresponding to (37). Discarding the imaginary part, we get, corresponding to (36), F=- ( (kcosnt + k' sin nt) dn. . . . .(38) 7T JO Since k, k' are known functions of m, or (a and v being given) of n, (38) may be calculated by quadratures for any prescribed value of t. It appears from the tables that k, k' are positive throughout. When m = 0, k and k' are infinite and continually diminish as m increases, until when m = oc , k = 1, k' = 0. For small values of m the limiting forms for k, k' are 1+ m 2 (logm) 2> k = ~m a logm' ^ from which it appears that if we make n vanish in (35), while V n is given, F comes to zero. We now seek the limiting form when t is very great. The integrand in (38) is then rapidly oscillatory, and ultimately the integral comes to depend sensibly upon that part of the range where n is very small. And for this part we may use the approximate forms (39). Consider, for example, the first integral in (38), from which we may omit the constant part of k. We have ^ , TT [ x cos nt dn 4nrv ("* cos (4iva~* t.x)dx I K cos nt dn = T I -T-T, = I T . . ...(40) Jo 4 J o m 2 (log ra) 2 a * J x (log x) 2 Writing 4>vt/a? = t', we have to consider f cost'x.dae l^^f (41) In this integral the integrand is positive from x = to x = 7r/2t', negative from 7r/2' to 37r/2', and so on. For the first part of the range, if we omit the cosine, /W da_ fdlog* ^_. log#) 2 J (logar) 2 log(27ir)' ' o tfog# og and since the cosine is less than unity, this is an over estimate. When t' is very great, \og (2t' /TT) may be identified with log', and to this order of approximation it appears that (41) may be represented by (42). Thus if quadratures be applied to (41), dividing the first quadrant into three parts, we have COS 7T/12 37T[" 1 1 1 57r[ 1 1 1 log Qt'lir + >S 12 [log 3#/ir ~ log 6*771- J + S 12" Llog2'/7r l^pF/^J ' of which the second and third terms may ultimately be neglected in com- parison with the first. For example, the coefficient of cos(37r/12) is equal to log 2 H- log . log . 38 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354 Proceeding in this way we see that the cosine factor may properly be identified with unity, and that the value of the integral for the first quadrant may be equated to I/log t'. And for a similar reason the quadrants after the first contribute nothing of this order of magnitude. Accordingly we may take f * k cos id dn = -*- . . . . .(43) Jo a 8 log* For the other part of (38), we get in like manner 8i/ f" sin t'x .dx Sv [* sin x'dx k swntdn = = _ ___. (44) ' ft a 8 J # log x a 2 J # log (*'/<) log a; In the denominator of (44) it appears that ultimately we may replace log (t'/x'} by log t' simply. Thus f 00 . 47Ti/ Jo = a 2 log tf ' ' ' so that the two integrals (43), (45) are equal. We conclude that when t is great enough, F~**~ ..frff' (46) a 2 log t a 2 log (4>vt/ a 2 ) But a better discussion of these integrals is certainly a desideratum. 7. Whatever interest the solution of the approximate equations may possess, we must never forget that the conditions under which they are applicable are very restricted, and as far as possible from being observed in many practical problems. Dynamical similarity in viscous motion requires that Vajv be unchanged, a being the linear dimension. Thus the general form for the resistance to the uniform motion of a sphere will be F=p V Va.f(Va/), (47) where / is an unknown function. In Stokes' solution (I)/ is constant, and its validity requires that Vajv be small*. When F is rather large, experi- ment shows that F is nearly proportional to F 2 . In this case v disappears. " The second power of the velocity and independence of viscosity are thus inseparably connected''^. The general investigation for the sphere moving in any manner (in a straight line) shows that the departure from Stokes' law when the velocity is not very small must be due to the operation of the neglected terms involving the squares of the velocities ; but the manner in which these act has not yet been traced. Observation shows that an essential feature in rapid fluid motion past an obstacle is the formation of a wake in the rear of the obstacle ; but of this the solutions of the approximate equations give no hint. * Phil. Mag. Vol. xxxvi. p. 854 (1893) ; Scientific Papers, Vol. iv. p. 87. t Phil. Mag. Vol. xxxiv. p. 59 (1892); Scientific Papers, Vol. HI. p. 576. 1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 39 Hydrodynamical solutions involving surfaces of discontinuity of the kind investigated by Helmholtz and Kirchhoff provide indeed for a wake, but here again there are difficulties. Behind a blade immersed transversely in a stream a region of " dead water " is indicated. The conditions of steady motion are thus satisfied ; but, as Helmholtz himself pointed out, the motion thus defined is unstable. Practically the dead and live water are continually mixing ; and if there be viscosity, the layer of transition rapidly assumes a finite width independently of the instability. One important consequence is the development of a suction on the hind surface of the lamina which contributes in no insignificant degree to the total resistance. The amount of the suction does not appear to depend much on the degree of viscosity. When the latter is small, the dragging action of the live upon the dead water extends to a greater distance behind. 8. If the blade, supposed infinitely thin, be moved edgeways through the fluid, the case becomes one of " skin-friction." Towards determining the law of resistance Mr Lanchester has put forward an argument * which, even if not rigorous, at any rate throws an interesting light upon the question. Applied to the 'case of two dimensions in order to find the resistance F per unit length of blade, it is somewhat as follows. Considering two systems for which the velocity V of the blade is different, let n be the proportional width of corresponding strata of velocity. The momentum communicated to the wake per unit length of travel is as nV, and therefore on the whole as nV- per unit of time. Thus F varies as nV 2 . Again, having regard to the law of viscosity and considering the strata contiguous to the blade, we see that F varies as V/n. Hence, nV 2 varies as V/n, or V varies as n~*, from which it follows that F varies as F 3 /' 2 . If this be admitted, the general law of dynamical similarity requires that for the whole resistance , .............................. (48) where I is the length, b the width of the blade, and c a constant. Mr Lanchester gives this in the form Flp = cv*-A*V\ ............................. (49) where A is the area of the lamina, agreeing with (48) if I and b maintain a constant ratio. The difficulty in the way of accepting the above argument as rigorous is that complete similarity cannot be secured so long as b is constant as has been supposed. If, as is necessary to this end, we take b proportional to n, it is bV/n, or V (and not V/n), which varies as nV 2 , or bV 2 . The conclusion is then simply that bV must be constant (v being given). This is merely the usual condition of dynamical similarity, and no conclusion as to the law of velocity follows. * Aerodynamics, London, 1907, 35. 40 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354 But a closer consideration will show, I think, that there is a substantial foundation for the idea at the basis of Lanchester's argument. If we suppose that the viscosity is so small that the layer of fluid affected by the passage of the blade is very small compared with the width (6) of the latter, it will appear that the communication of motion at any stage takes place much as if the blade formed part of an infinite plane moving as a whole. We know that if such a plane starts from rest with a velocity V afterwards uniformly maintained, the force acting upon it at time t is per unit of area, see (12), (50) The supposition now to be made is that we may apply this formula to the element of width dy, taking t equal to y/V, where y is the distance of the element from the leading edge. Thus ............ (51) which agrees with (48) if we take in the latter c = 2/^ir. The formula (51) would seem to be justified when v is small enough, as representing a possible state of things ; and, as will be seen, it affords an absolutely definite value for the resistance. There is no difficulty in extending it under similar restrictions to a lamina of any shape. If 6, no longer constant, is the width of the lamina in the direction of motion at level z, we have F-*p(9l*pV*]b*d* (52) It will be seen that the result is not expressible in terms of the area of the lamina. In (49) c is not constant, unless the lamina remains always similar in shape. The fundamental condition as to the smallness of v would seem to be realized in numerous practical cases ; but any one who has looked over the side of a steamer will know that the motion is not usually of the kind supposed in the theory. It would appear that the theoretical motion is subject to instabilities which prevent the motion from maintaining its simply stratified character. The resistance is then doubtless more nearly as the square of the velocity and independent of the value of v. When in the case of bodies moving through air or water we express V, a, and v in a consistent system of units, we find that in all ordinary cases v/Va is so very small a quantity that it is reasonable to identify f( v f Va) with/(0). The influence of linear scale upon the character of the motion then disappears. This seems to be the explanation of a difficulty raised by Mr Lanchester (Joe. cit. 56). 355. ABERRATION IN A DISPERSIVE MEDIUM. [Philosophical Magazine, Vol. xxii. pp. 130134, 1911.] THE application of the theory of group- velocity to the case of light was discussed in an early paper* in connexion with some experimental results announced by Young and Forbes f. It is now, I believe, generally agreed that, whether the method be that of the toothed wheel or of the revolving mirror, what is determined by the experiment is not V, the wave-velocity, but U, the group-velocity, where U=d(kV)jdk, k being inversely as the wave-length. In a dispersive medium V and U are different. I proceeded: "The evidence of the terrestrial methods relating exclu- sively to U, we turn to consider the astronomical methods. Of these there are two, depending respectively upon aberration and upon the eclipses of Jupiter's satellites. The latter evidently gives U. The former does not depend upon observing the propagation of a peculiarity impressed upon a train of waves, and therefore has no relation to U. If we accept the usual theory of aberration as satisfactory, the result of a comparison between the coefficient found by observation and the solar parallax is V the wave- velocity." The above assertion that stellar aberration gives V rather than U has recently been called in question by EhrenfestJ, and with good reason. He shows that the circumstances do not differ materially from those of the toothed wheel in Fizeau's method. The argument that he employs bears, indeed, close affinity with the method used by me in a later paper . "The * Nature, Vols. xxiv., xxv. 1881 ; Scientific Papers, Vol. i. p. 537. t These observers concluded that blue light travels in vacuo 1-8 per cent, faster than red light. J Ann. d. Physik, Bd. xxxm. p. 1571 (1910). Nature, Vol. XLV. p. 499 (1892); Scientific Papers, Vol. in. p. 542. 42 ABERRATION IN A DISPERSIVE MEDIUM [355 explanation of stellar aberration, as usually given, proceeds rather upon the basis of the corpuscular than of the wave-theory. In order to adapt it to the principles of the latter theory, Fresnel found it necessary to follow Young in assuming that the aether in any vacuous space connected with the earth (and therefore practically in the atmosphere) is undisturbed by the earth's motion of 19 miles per second. Consider, for simplicity, the case in which the direction of the star is at right angles to that of the earth's motion, and replace the telescope, which would be used in practice, by a pair of perforated screens, on which the light falls perpendicularly. We may further imagine the luminous disturbance to consist of a single plane pulse. When this reaches the anterior screen, so much of it as coincides with the momentary position of the aperture is transmitted, and the remainder is stopped. The part transmitted proceeds upon its course through the aether independently of the motion of the screens. In order, therefore, that the pulse may be transmitted by the aperture in the posterior screen, it is evident that the line joining the centres of the apertures must not be perpendicular to the screens and to the wave-front, as would be necessary in the case of rest. For, in consequence of the motion of the posterior screen in its own plane, the aperture will be carried forward during the time of passage of the light. By the amount of this motion the second aperture must be drawn backwards, in order that it may be in the place required when the light reaches it. If the velocity of light be V, and that of the earth be v, the line of apertures giving the apparent direction of the star must be directed forwards through an angle equal to v/V." If the medium between the screens is dispersive, the question arises in what sense the velocity of light is to be taken. Evidently in the sense of the group-velocity ; so that, in the previous notation, the aberration angle is v/U. But to make the argument completely satisfactory, it is necessary in this case to abandon the extreme supposition of a single pulse, replacing it by a group of waves of approximately given wave-length. While there can remain no doubt but that Ehrenfest is justified in his criticism, it does not quite appear from the above how my original argument is met. There is indeed a peculiarity imposed upon the regular wave-motion constituting homogeneous light, but it would seem to be one imposed for the purposes of the argument rather than inherent in the nature of the case. The following analytical solution, though it does not relate directly to the case of a simply perforated screen, throws some light upon this question. Let us suppose that homogeneous plane waves are incident upon a "screen " at z = 0, and that the effect of the screen is to introduce a reduction of the amplitude of vibration in a ratio which is slowly periodic both with respect to the time and to a coordinate x measured in the plane of the screen, represented by the factor cos m (vt - x). Thus, when t = 0, there is no effect 1911] ABERRATION IN A DISPERSIVE MEDIUM 43 when x = 0, or a multiple of 2?r ; but when x is an odd multiple of IT, there is a reversal of sign, equivalent to a change of phase of half a period. And the places where these particular effects occur travel along the screen with a velocity v which is supposed to be small relatively to that of light. In the absence of the screen the luminous vibration is represented by (f> = cos(nt-kz), .............................. (1) or at the place of the screen, where z = 0, by </> = cos nt simply. In accordance with the suppositions already made, the vibration just behind the screen will be <f> = cos m (vt x) . cos nt = cos {(n + mv) t mx} + $ cos {(n - mv) t + mx] ; ...... (2) and the question is to find what form $ will take at a finite distance z behind the screen. It is not difficult to see that for this purpose we have only to introduce terms proportional to z into the arguments of the cosines. Thus, if we write <}> = ^ cos {(n + mv) t mx ^ z} + $ cos \(n mv) t -t- mx fJL 2 z], . . .(3) we may determine fr, ^ so as to satisfy in each case the general differential equation of propagation, viz. In (4) V is constant when the medium is non-dispersive ; but in the contrary case V must be given different values, say V 1 and F 2 , when the coefficient of t is n + mv or n mv. Thus (n 4- mvf = Fj" (m 2 + mf), (n - mv) 2 = F 2 2 (m 2 + ra a 2 ) ....... (5) The coefficients /^, yu, 2 being determined in accordance with (5), the value of <f> in (3) satisfies all the requirements of the problem. It may also be written = cos {mvt -mx -%([*>!- ^ z} . cos {nt - | Oi + fa) z}, ...... ( 6 ) of which the first factor, varying slowly with t, may be regarded as the amplitude of the luminous vibration. The condition of constant amplitude at a given time is that mx+ ^(fa fa) z shall remain unchanged. Thus the amplitude which is to be found at x on the screen prevails also behind the screen along the line -x/z = ^(^-fa)/m, ........................... (7) so that (7) may be regarded as the angle of aberration due to v. It remains to express this angle by means of (5) in terms of the fundamental data. 44 ABERRATION IN A DISPERSIVE MEDIUM [355 When m is zero, the value of n is n/F; and this is true approximately when m is small. Thus, from (5), t, 8 -/*. 9 2mv nVl with sufficient approximation. Now in (8) the difference F 2 - F, corresponds to a change in the coefficient of t from n + mv to n mv. Hence, denoting the general coefficient of t by <r, of which F is a function, we have and (8) may be written Again, F=er/&, U=da/dk, <r dV , dV <r dk and thus -^-i- j- F do- rfo- A; do- ' o- rfF <r <2fc F -F^ 25 ^^^^' where f7 is the group-velocity. Accordingly, -x/t-v/U .............................. (10) expresses the aberration angle, as was to be expected. In the present problem the peculiarity impressed is not uniform over the wave-front, as may be supposed in discussing the effect of the toothed wheel ; but it exists never- theless, and it involves for its expression the introduction of more than one frequency, from which circumstance the group-velocity takes its origin. A development of the present method would probably permit the solution of the problem of a series of equidistant moving apertures, or a single moving aperture. Doubtless in all cases the aberration angle would assume the value v/U. 356. LETTER TO PROFESSOR NERNST. [Conseil scientifique sous les auspices de M. Ernest Solvay, Oct. 1911.] DEAR PROF. NERNST, Having been honoured with an invitation to attend the Conference at Brussels, I feel that the least that I can do is to communicate my views, though I am afraid I can add but little to what has been already said upon the subject. I wish to emphasize the difficulty mentioned in my paper of 1900* with respect to the use of generalized coordinates. The possibility of representing the state of a body by a finite number of such (short at any rate of the whole number of molecules) depends upon the assumption that a body may be treated as rigid, or incompressible, or in some other way simplified. The justification, and in many cases the sufficient justification, is that a departure from the simplified condition would involve such large amounts of potential energy as could not occur under the operation of the forces concerned. But the law of equi-partition lays it down that every mode is to have its share of kinetic energy. If we begin by supposing an elastic body to be rather stiff, the vibrations have their full share and this share cannot be diminished by increasing the stiffness. For this purpose the simplification fails, which is as much as to say that the method of generalized coordinates cannot be applied. The argument becomes, in fact, self-contradictory. Perhaps this failure might be invoked in support of the views of Planck and his school that the laws of dynamics (as hitherto understood) cannot be applied to the smallest parts of bodies. But I must confess that I do not like this solution of the puzzle. Of course I have nothing to say against following out the consequences of the [quantum] theory of energy a pro- cedure which has already in the hands of able men led to some interesting * Phil. Mag. Vol. XLIX. p. 118 ; Scientific Papers, Vol. iv. p. 451. 46 LETTER TO PROFESSOR NERNST [356 conclusions. But I have a difficulty in accepting it as a picture of what actually takes place. We do well, I think, to concentrate attention upon the diatomic gaseous molecule. Under the influence of collisions the molecule freely and rapidly acquires rotation. Why does it not also acquire vibration along the line joining the two atoms ? If I rightly understand, the answer of Planck is that in consideration of the stiffness of the union the amount of energy that should be acquired at each collision falls below the minimum possible and that therefore none at all is acquired an argument which certainly sounds paradoxical. On the other hand Boltzmann and Jeans contend that it is all a question of time and that the vibrations necessary for full statistical equi- librium may be obtained only after thousands of years. The calculations of Jeans appear to show that there is nothing forced in such a view. I should like to inquire is there any definite experimental evidence against it ? So far as I know, ordinary laboratory experience affords nothing decisive. I am yours truly, RAYLEIGH. 357. ON THE CALCULATION OF CHLADNI'S FIGURES FOR A SQUARE PLATE. [Philosophical Magazine, Vol. xxn. pp. 225229, 1911.] IN my book on the Theory of Sound, ch. x. (1st ed. 1877, 2nd ed. 1894) I had to speak of the problem of the vibrations of a rectangular plate, whose edges are free, as being one of great difficulty, which had for the most part resisted attack. An exception could be made of the case in which //, (the ratio of lateral contraction to longitudinal elongation) might be regarded as evanescent. It was shown that a rectangular plate could then vibrate after the same law as obtains for a simple bar, and by superposition some of the simpler Chladni's figures for a square plate were deduced. For glass and metal the value of p is about \, so that for such plates as are usually experi- mented on the results could be considered only as rather rough approxi- mations. I wish to call attention to a remarkable memoir by W. Ritz* in which, somewhat on the above lines, is developed with great skill what may be regarded as a practically complete solution of the problem of Chladni's figures on square plates. It is shown that to within a few per cent, all the proper tones of the plate may be expressed by the formulae w mn = u m (x) u n (y) + u m (y) u, n (x), w' mn = u m (x) u n (y) - u m (y) u n (#), the functions u being those proper to a free bar vibrating transversely. The coordinate axes are drawn through the centre parallel to the sides of the square. The first function of the series u (x) is constant ; the second t*i (x}=x . const. ; u 2 (x) is thus the fundamental vibration in the usual sense, with two nodes, and so on. Ritz rather implies that I had overlooked the * "Theorie der Transversalschwingimgen einer quadratischen Platte mit freien Randern,' 1 Annalen df.r Physik, Bd. xxvni. S. 737 (1909). The early death of the talented author must be accounted a severe loss to Mathematical Physics. 48 ON THE CALCULATION OF [357 necessity of the first two. terms in the expression of an arbitrary function. It would have been better to have mentioned them explicitly ; but I do not think any reader of my book could have been misled. In 168 the inclusion of all* particular solutions is postulated, and in 175 a reference is made to zero values of the frequency. For the gravest tone of a square plate the coordinate axes are nodal, and Ritz finds as the result of successive approximations = u l v l + '0394 (! - -0040^3 - -0034 (U,W B + ,,) + -0011 in which u stands for u(x) and v for u (y). The leading term M,^, or xy, is the same as that which I had used ( 228) as a rough approximation on which to found a calculation of pitch. As has been said, the general method of approximation is very skilfully applied, but I am surprised that Ritz should have regarded the method itself as new. An integral involving an unknown arbitrary function is to be made a minimum. The unknown function can be represented by a series of known functions with arbitrary coefficients accurately if the series be continued to infinity, and approximately by a few terms. When the number of coefficients, also called generalized coordinates, is finite, they are of course to be deter- mined by ordinary methods so as to make the integral a minimum. It was in this way that I found the correction for the open end of an organ-pipe f, using a series with two terms to express the velocity at the mouth. The calculation was further elaborated in Theory of Sound, Vol. II. Appendix A. I had supposed that this treatise abounded in applications of the method in question, see 88, 89, 90, 91, 182, 209, 210, 265 ; but perhaps the most explicit formulation of it is in a more recent paper J, where it takes almost exactly the shape employed by Ritz. From the title it will be seen that I hardly expected the method to be so successful as Ritz made it in the case of higher modes of vibration. Being upon the subject I will take the opportunity of showing how the gravest mode of a square plate may be treated precisely upon the lines of the paper referred to. The potential energy of bending per unit area has the expression * Italics in original t Phil. Tram. Vol. CLXI. (1870) ; Scientific Papers, Vol. i. p. 57. * "On the Calculation of the Frequency of Vibration of a System in its Gravest Mode, with an Example from Hydrodynamics," Phil. Mag. Vol. XLVII. p. 556 (1899); Scientific Papers, Vol. iv. p. 407. 1911] CHLADNI'S FIGURES FOR A SQUARE PLATE 49 in which q is Young's modulus, and 2h the thickness of the plate ( 214). Also for the kinetic energy per unit area we have T = phi&, (2) p being the volume-density. From the symmetries of the case w must be an odd function of x and an odd function of y, and it must also be symmetrical between # and y. Thus we may take w = q^asy + q z xy (a? + f) + q 3 xy (x* + y*) + q 4 o?y 3 + (3) In the actual calculation only the two first terms will be employed. Expressions (1) and (2) are to be integrated over the square; but it will suffice to include only the first quadrant, so that if we take the side of the square as equal to 2, the limits for x and y are and 1. We find 167 2 2 , (4) &w d *+*k*tW (5) Thus, if we set _4jr^_ 3 (! we have V' = ^q l 2 + 2^^ + |<? 2 2 + , _ (7) In like manner, if = ~9~ ' ' ' When we neglect q z and suppose that ^ varies as cospt, these expressions give 2 _ 6qh? 9Qqh 2 /im P _ /i _i ,,\ _ /i _i_ ,,\ r ,4 ' v^W if we introduce a as the length of the side of the square. This is the value found in Theory of Sound, 228, equivalent to Ritz's first approximation. In proceeding to a second approximation we may omit the factors already accounted for in (10). Expressions (7), (9) are of the standard form if we take 2, (7 = o If-f, R. VI. 50 ON THE CALCULATION OF CHLADNl'S FIGURES FOR A SQUARE PLATE [357 and Lagrange's equations are (A-p*L) qi +(B-p*M)q, = QA (B-^M) qi + (C-^N)q, = 0,}" while the equation for jp* is the quadratic p*(LlT-M*)+F(2MB-LC-NA) + AC-&=0 ........ (12) For the numerical calculations we will suppose, following Ritz, that /* = '225, making C =1 1*9226. Thus LN-M* = -13714, ^(7-^ = 7-9226, 2MB -LC-NA = -2x 4-3498. The smaller root of the quadratic as calculated by the usual formula is 9239, in place of the 1 of the first approximation ; but the process is not arithmetically advantageous. If we substitute this value in the first term of the quadratic, and determine jp 2 from the resulting simple equation, we get the confirmed and corrected value p* = '9241. Restoring the omitted factors, we have finally as the result of the second approximation 96g* x -9241 p(l+f*)a* ' in which /z = '225. The value thus obtained is not so low, and therefore not so good, as that derived by Ritz from the series of w-functions. One of the advantages of the latter is that, being normal functions for the simple bar, they allow T to be expressed as a sum of squares of the generalized coordinates q lt &c. AH a consequence, p* appears only in the diagonal terms of the system of equations analogous to (11). From (11) we find further q 2 /qi = - '0852, so that for the approximate form of w corresponding to the gravest pitch we may take (14) in which the side of the square is supposed equal to 2. 358. PROBLEMS IN THE CONDUCTION OF HEAT. [Philosophical Magazine, Vol. xxn. pp. 381396, 1911.] THE general equation for the conduction of heat in a uniform medium may be written dv d?v d?v d 2 v dt-St + ty + d*-**- ........................ (1) v representing temperature. The coefficient (v) denoting diffusibility is omitted for brevity on the right-hand of (1). It can always be restored by consideration of " dimensions." Kelvin* has shown how to build up a variety of special solutions, applicable to an infinite medium, on the basis of Fourier's solution for a point-source. A few examples are quoted almost in Kelvin's words : I. Instantaneous simple point-source ; a quantity Q of heat suddenly generated at the point (0, 0, 0) at time t = 0, and left to diffuse .through an infinite homogeneous solid: where r 2 = ad 2 + y 2 + z 2 . [The thermal capacity is supposed to be unity.] Verify that and that v = when t = ; unless also x = 0, y 0, z = 0. Every other solution is obtainable from this by summation, II. Constant simple point-source, rate q : K The formula within the brackets shows how this obvious solution is derivable from (2). * " Compendium of Fourier Mathematics, &c.," Ene. Brit. 1880; Collected Papers, Vol. 11. p. 44. 42 52 PROBLEMS IN THE CONDUCTION OF HEAT [358 III. Continued point-source ; rate per unit of time at time t, an arbitrary function, f(t): (4) IV. Time-periodic simple point-source, rate per unit of time at time t, q sin 2nt : t, = ^i-e -* sin Ot-Ar] ...................... (5) Verify that v satisfies (1) ; also that 4nrr 2 dv/dr = q sin 2nt, where r = 0. V. Instantaneous spherical surface-source ; a quantity Q suddenly gener- ated over a spherical surface of radius a, and left to diffuse outwards and inwards : To prove this most easily, verify that it satisfies (1) ; and further verify that r Jo and that v = when t = 0, unless also r = a. Remark that (6) becomes identical with (2) when a = ; remark further that (6) is obtainable from (2) by integration over the spherical surface. VI. Constant spherical surface-source; rate per unit of time for the whole surface, q : [f * e~ (r ~ a} s/4t e~ (r+a) 1/4t 1 -J.* **art* \ = 9/47rr (r > a) = qj^tra (r < a). The formula within the brackets shows how this obvious solution is de- rivable from (6). VII. Fourier's "Linear Motion of Heat"; instantaneous plane-source; quantity per unit surface, a- : (7) Verify that this satisfies (1) for the case of v independent of y and z, and that r+ao vdx tr. Remark that (7) is obtainable from (6) by putting Q/^ira 3 = <r, and a = oo ; or directly from (2) by integration over the plane. 1911] PROBLEMS IN THE CONDUCTION OF HEAT 53 In Kelvin's summary linear sources are passed over. If an instantaneous source be uniformly distributed along the axis of z, so that the rate per unit length is q, we obtain at once by integration from (2) From this we may deduce the effect of an instantaneous source uniformly distributed over a circular cylinder whose axis is parallel to z, the superficial density being <r. Considering the cross-section through Q the point where v is to be estimated, let be the centre and a the radius of the circle. Then if P be a point on the circle, OP = a,OQ = r, PQ = p, z POQ = 0; and p* = a?+r*- 2ar cos 0, (9) / (#), equal to J (iac), being the function usually so denoted. From (9) we fall back on (8) if we put a = 0, Z-rraor = q. It holds good whether r be greater or less than a. When x is very great and positive, so that for very small values of t (9) assumes the form vanishing when t = 0, unless r = a. Again, suppose that the instantaneous source is uniformly distributed over the circle % = 0, = a cos 0, 77 = a sin <, the rate per unit of arc being q, and that v is required at the point x, 0, *. There is evidently no loss of generality in supposing y 0. We obtain at once from (2) '0 where r 2 = ( - ocf + i) 2 + z* = a? + x 2 + z 2 - 2ax cos <f>. from which if we write q = <rdz, and integrate with respect to z from oo to + oo , we may recover (9). 54 PROBLEMS IN 'THE CONDUCTION OF HEAT [o5S If in (12) we put q = <rda and integrate with respect to a from to oo , we obtain a solution which must coincide with (7) when in the latter we substitute z for x. Thus ..................... (13) a particular case of one of Weber's integrals*. It may be worth while to consider briefly the problem of initial in- stantaneous sources distributed over the plane (=0) in a more general manner. In rectangular coordinates the typical distribution is such that the rate per unit of area is er cos lj~ . cos mrj ............................... (14) If we assume that at x, y, z and time t, v is proportional to cos Ix . cos my, the general differential equation (1) gives so that, as for conduction in one dimension, a Z/4t / , .................. (15) yt /+ and v dz = 2 yV . A cos Ix cos my er ^^ . J -oo Putting t = 0, and comparing with (14), we see that By means of (2) the solution at time t may be built up from (14). In this way, by aid of the well-known integral e-^ cos 2cx dx =. e"" 2 / ' , (17) a we may obtain (15) independently. The process is of more interest in its application to polar coordinates. If we suppose that v is proportional to cos nd . J n (kr), d*v I dv 1 <Pv * Gray and Mathews' BeueVt Functions, p. 78, equation (160). Put n=0, X=0. See also (31) below. 1911] PROBLEMS IN THE CONDUCTION OF HEAT 55 so that (1) gives and v = Acosnej n (kr)e~ ktt r - ...................... (20) Vc From (20) j +0 vdz = 2^7r.Acosn8J n (kr)e-* t ................ (21) .' 00 If the initial distribution on the plane z = be per unit area o-cosn0J n (kr), ........................... (22) it follows from (21) that as before "' .......................... < 23 > We next proceed to investigate the effect of an instantaneous source distributed over the circle for which = 0, = a cos <f>, rj = a sin <, the rate per unit length of arc being q cos n<j>. From (2) at the point x, y, z j" 27r q cos nd> e^ 1 * 1 ad6 *-j, - - in which = a? if x = pcos0, y = psm&. The integral that we have to consider may be written f W cos 116 ep' cos <*-*> d$ = I cos n (0 + ^) e?' 9 * d*<lr .'o .' f - f where TJr = (f>0, and p' = ap/2t. In view of the periodic character of the integrand, the limits may be taken as TT and + TT. Accordingly /+JT fir I cos w^r e^' cos * d-fy = 2 / cos n^- ^ cos (-+JT I sin??,i/reo'cos*^ =0; and f "" cos n<#> &'**<+-*> d<f> = 2 cos ?i^ / * cos nty e"' 008 * d-^r ....... (26) Jo Jo The integral on the right of (26) is equivalent to irl n (p), where (27) 56 PROBLEMS IN THE CONDUCTION OF HEAT [358 J n being, as usual, the symbol of Bessel's function of order n. For, if n be even, f cos 11+ ef '* * d& = t' cos n-dr (ei*' 00 ** + e-"' *) cty Jo Jo = I cos ni/r cos (ip' cos +) d+ = 7ri~ n J n (ip') = trl n (p') ; and, if n be odd, J COS 71-^- go' 008 * Cty = - r COS Wl/r (e~P'>8* _ eP'cos*) ^ = i I cos n-^r sin (ip' cos -/r) d-fy = 7r/ n (p'). In either case TcOS?^^' 008 *^ = 7T/ n (/3 / ) (28) Jo Thus f * cos n<f> ep' 00 ^*-*) d<^> = 2?r cos nB I n (p'\ (29) and (24) becomes This gives the temperature at time and place (p, z) due to an initial instantaneous source distributed over the circle a. The solution (30) may now be used to find the effect of the initial source expressed by (22). For this purpose we replace q by <rda, and introduce the additional factor J n (ka), subsequently integrating with respect to a between the limits and oo . Comparing the result with that expressed in (20), (23), we see that is a common factor which divides out, and that there remains the identity ^ J" adar+H* J n (ka) I n (|) = J n (kp) e~ ......... (31) This agrees with the formula given by Weber, which thus receives an interesting interpretation. Reverting to (30), we recognize that it must satisfy the fundamental equation (1), now taking the form ffiv ffiv Idv Id* dv. ~dz* + d? + pdt + ?dP = di" and that when t = v must vanish, unless also z = 0, p = a. 1911] PROBLEMS IX THE CONDUCTION OF HEAT 57 If we integrate (30) with respect to z between + oo , setting q = adz, so that <r cos 116 represents the superficial density of the instantaneous source distributed over the cylinder of radius a, we obtain T I " Ma which may be regarded as a generalization of (9). And it appears that (33) satisfies (32), in which the term d 2 v/dz* may now be omitted. In V. Kelvin gives the temperature at a distance r from the centre and at time t due to an instantaneous source uniformly distributed over a spherical surface. In deriving the result by integration from (2) it is of course simplest to divide the spherical surface into elementary circles which are symmetrically situated with respect to the line OQ joining the centre of the sphere to the point Q where the effect is required. But if the circles be drawn round another axis OA, a comparison of results will give a definite integral. Adapting (12), we write a = csin#, c being the radius of the sphere, a = OQ sin & = r sin 6', z-=r cos 6' c cos 0, so that C r sin sin 0\ rcc08 * C080 ' (34) This has now to be integrated with respect to 6 from to TT. Since the result must be independent of 6', we see by putting 6' = that t * 7 (p sin 6 sin 0'} tf cose cos6 ' sin d0 Jo = ^(tf-e-o\ . ...(35) Using the simplified form and putting q = <rcd0, where a- is the superficial density, we obtain for the complete sphere (e-ry (c+r)\ .(36) agreeing with (6) when we remember that Q = 47rcV. We will now consider the problem of an instantaneous source arbitrarily distributed over the surface of the sphere whose radius is c. It suffices, of course, to treat the case of a spherical harmonic distribution; and we suppose that per unit of area of the spherical surface the rate is S n . Assuming that v is everywhere proportional to S n , we know that v satisfies (37 > 58 PROBLEMS IN THE CONDUCTION OF HEAT [358 0, to being the usual spherical polar coordinates. Hence from (1) v as a function of r and t satisfies dv _ d?v 2 dv n (n + .)v _ When n = 0, this reduces to the same form as applies in one dimension. For general values of n the required solution appears to be most easily found indirectly. Let us suppose that S n reduces to Legendre's function P n (/*), where /4 = cos0, and let us calculate directly from (2) the value of v at time t and at a point Q distant r from the centre of the sphere along the axis of p. The exponential term is r+e2 rcn r+c 2 W- e ^ = e*r<r, ......................... (39) if p = rc/2t. Now (Theory of Sound, 334) (40) whence P n (,*) " dp = 2i*H ^ ^+i (- V>. ............. ( 41 ) or, as it may also be written by (27), -V) 7 "*^ ........................... (42) Substituting in (2) (43) we now get for the value of v at time t, and at the point for which p = r, n+iJ-^+c'lAit , v (44) It may be verified by trial that (44) is a solution of (38). When /a is not restricted to the value unity, the only change required in (44) is the introduction of the factor P n (fi). When n=0, P n (/*)=!, and we fall back upon the case of uniform distribution. We have < 45 > Using this in (44), we obtain a result in accordance with (6), in which Q, representing the integrated magnitude of the source, is equal to 4nrc* in our present reckoning. 1911] PROBLEMS IN THE CONDUCTION OF HEAT 59 When n = l,P 1 ( A t) = ^, and ................... (47) and whatever integral value n may assume J n +i is expressible in finite terms. We have supposed that the rate of distribution is represented by a Legendre's function P n (/i). In the more general case it is evident that we have merely to multiply the right-hand member of (44) by S n , instead of P n . So far we have been considering instantaneous sources. As in II., the effect of constant sources may be deduced by integration, although the result is often more readily obtained otherwise. A comparison will, however, give the value of a definite integral. Let us apply this process to (33) repre- senting the effect of a cylindrical source. The required solution, being independent of t, is obtained at once from (1). We have inside the cylinder v = Ap n cos nd, and outside v = Bp~ n cos n6, with Aa n = Ba~ n . The intensity of the source is represented by the differ- ence in the values of dv/dp just inside and just outside the cylindrical surface. Thus a-' cos nd = n cos n9 (Ba~ n ~ l + Aa n ~*\ whence Aa n = Bar = <r'a/'2n, a' cos nd being the constant time rate. Accordingly, within the cylinder -" ........................... and without the cylinder '" (49) These values are applicable when n is any positive integer. When n is zero, there is no permanent distribution of temperature possible. These solutions should coincide with the value obtained from (33) by putting o- = <?' dt and integrating with respect to t from to x . Or (5o) the + sign in the ambiguity being taken when p < a, and the - sign when p > a. I have not confirmed (50) independently. 60 PROBLEMS IN THE CONDUCTION OF HEAT [358 In like manner we may treat a constant source distributed over a sphere. If the rate per unit time and per unit of area of surface be S n , we find, as above, for inside the sphere (c) and outside the sphere and these forms' are applicable to any integral n, zero included. Comparing with (44), we see that which does not differ from (50), if in the latter we suppose n = integer + . The solution for a time-periodic simple point-source has already been quoted from Kelvin (IV.). Though derivable as a particular case from (4), it is more readily obtained from the differential equation (1) taking here the form see (38) with n = d* (rv) _ d* (rv) ' or if v is assumed proportional to e ipt , d*(rv)ldr*-ip(rv) = 0, ......................... (54) giving rv = Ae*** e- { *P* r , .............................. (55) as the symbolical solution applicable to a source situated at r = 0. Denoting by q the magnitude of the source, as in (5), we get to determine A, so that v = -2- &* -****' ........................... (56) WTT If from (56) we discard the imaginary part, we have (57) corresponding to the source q cos pt. From (56) it is possible to build up by integration solutions relating to various distributions of periodic sources over lines or surfaces, but an inde- pendent treatment is usually simpler. We will, however, write down the integral corresponding to a uniform linear source coincident with the axis of z. If p* = a? + y 2 , r 2 = z* + p 8 , and (p being constant) rdr = z dz. Thus putting in (56) q = q l dz, we get -' R . (58) 1911] PROBLEMS IN THE CONDUCTION OF HEAT 61 In considering the effect of periodic sources distributed over a plane xy, we may suppose v x cos lac. cos my, ........................... (59) or again v oc J n (kr) . cos nff, ........................... (60) where r 2 = a? + y 2 . In either case if we write I 3 + m> = It?, and assume v proportional to e ipt , (1) gives (61) Thus, if 2 + ip _ j ( C os a + i sin a), ....................... (62) where A includes the factors (59) or (60). If the value of v be given on the plane z = 0, that of A follows at once. If the magnitude of the source be given, A is to be found from the value of dv/dz when z = 0. The simplest case is of course that where k = 0. If Ve ipt be the value of v when z = 0, we find v = V&& tr 2 * n \ ............................ (64) or when realized v= Ve- z ^^cos{pt-z^(p/'2)}, ................... (65) corresponding to v = V cos pt when z = 0. From (64) - (^ = ^(ip) . Ve ipt = ^6^, .................. (66) if <r be the source per unit of area of the plane regarded as operative in a medium indefinitely extended in both directions. Thus in terms of <r, (67) ^p or in real form v = 5^- e-W<pM cos {pt - ITT - z \f(p/'2)}, ............... (68) L \Jp corresponding to the uniform source <r cos pt. In the above formulae z is supposed to be positive. On the other side of the source, where z itself is negative, the signs must be changed so that the terms containing z may remain negative in character. When periodic sources are distributed over the surface of a sphere (radius = c), we may suppose that v is proportional to the spherical surface harmonic S n . As a function of r and t, v is then subject to (38) ; and when we introduce the further supposition that as dependent on t, v is proportional to e ipt , we have (69) 62 PROBLEMS IN THE CONDUCTION OF HEAT [358 When n = -0, that is in the case of symmetry round the pole, this equation takes the same form as for one dimension; but we have to distinguish between the inside and the outside of the sphere. On the inside the constants must be so chosen that v remains finite at the pole (r = 0). Hence rv^AJr t (r'JW-er r *'to>), (70) or in real form rv = Ae r < '^ cos {pt + r V(p/2)j - Ae^ W*> cos {pt - r V(^/2)|. . . .(71) Outside the sphere the condition is that rv must vanish at infinity. In this ............................. (72) or in real form rv = Be-^-JW cos{pt-r^/(p/2)} ................... (73) When n is not zero, the solution of (69) may be obtained as in Stokes' treatment of the corresponding acoustical problem (Theory of Sound, ch. XVII). Writing r \/(ip) = z, and assuming rv = Ae z + Be-*, ............................. (74) where A and B are functions of z, we find for B The solution is B = B f n (z), ............................... (76) where B is independent of z and (77) . g as may be verified by substitution. Since n is supposed integral, the series (77) terminates. For example, if n = 1, it reduces to the first two terms. The solution appropriate to the exterior is thus rv = B S n e i v t e- r 'JWf n (i*p ii r). ............... ...... (78) For the interior we have rv = A.W [r"J * / (tVr) - e^ */ (- i*jpr)}, ...... (79) which may also be expressed by a Bessel's function of order n + . In like manner we may treat the problem in two dimensions, where everything may be expressed by the polar coordinates r, 6. It suffices to consider the terms in cos nd, where n is an integer. The differential equation analogous to (69) is now d*v 1 dv n* + -- V = ^ ........................... < SO > 1911] PROBLEMS IN THE CONDUCTION OF HEAT which, if we take r J(ip) = z, as before, may be written and is of the same form as (69) when in the latter n is written for n. As appears at once from (80), the solution for the interior of the cylinder may be expressed v = A cosnde^Jntfltp^r), ..................... (82) J n being as usual the Bessel's function of the nth order. For the exterior we have from (81) A = B cos 116 ew* e~ r ^ (l / n _ ^ (i*p* r ), ............... (83) where 1.2. '-5*) -1 i ' 1.2. 3. The series (84), unlike (77), does not terminate. It is ultimately divergent, but may be employed for computation when z is moderately great. In these periodic solutions the sources distributed over the plane, sphere, or cylinder are supposed to have been in operation for so long a time that any antecedent distribution of temperature throughout the medium is with- out influence. By Fourier's theorem this procedure may be generalized. Whatever be the character of the sources with respect to time, it may be resolved into simple periodic terms ; and if the character be known through the whole of past time, the solution so obtained is unambiguous. The same conclusion follows if, instead of the magnitude of the sources, the temperature at the surfaces in question be known through past time. An important particular case is when the character of the function is such that the superficial value, having been constant (zero) for an infinite time, is suddenly raised to another value, say unity, and so maintained. The Fourier expression for such a function is the definite integral being independent of the arithmetical value of t, but changing sign when t passes through ; or, on the understanding that only the real part is to be retained, (Rft\ 2 ~*~ _ / "jr vW 64 PROBLEMS IN THE CONDUCTION OF HEAT [358 We may apply this at once to the case of the plane z = which has been at temperature from t = oo to t = 0, and at temperature 1 from t = to t=oo. By (64) If* 6**-*^ {i & = + -^ -dp (87) ITTJo P P By the methods of complex integration this solution may be transformed into Fourier's, viz. --- .. ...(88) dz V (TO 2 f*/v v = l--=- e-^da, ........................ (89) V7T./0 which are, however, more readily obtained otherwise. In the case of a cylinder (r = c) whose surface has been at up to t = and after wards at v = 1, we have from (83) with n = } ............ /- 1 (**!>* <0 J> of which only the real part is to be retained. This applies to the region out- side the cylinder. It may be observed that when t is negative (87) must vanish for positive z and (90) for r > c. 359. ON THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRO- DUCTION, WITH SUGGESTIONS FOR ENHANCING GRADATION ORIGINALLY INVISIBLE. [Philosophical Magazine, Vol. xxn. pp. 734 740, 1911.] IN copying a subject by photography the procedure usually involves two distinct steps. The first yields a so-called negative, from which, by the same or another process, a second operation gives the desired positive. Since ordinary photography affords pictures in monochrome, the reproduction can be complete only when the original is of the same colour. We may suppose, for simplicity of statement, that the original is itself a transparency, e.g. a lantern-slide. The character of the original is regarded as given by specifying the transparency (t) at every point, i.e. the ratio of light transmitted to light incident. But here an ambiguity should be noticed. It may be a question of the place at which the transmitted light is observed. When light penetrates a stained glass, or a layer of coloured liquid contained in a tank, the direction of propagation is unaltered. If the incident rays are normal, so also are the rays transmitted. The action of the photographic image, con- stituted by an imperfectly aggregated deposit, differs somewhat. Rays incident normally are more or less diffused after transmission. The effective transparency in the half-tones of a negative used for contact printing may thus be sensibly greater than when a camera and lens is employed. In the first, case all the transmitted light is effective ; in the second most of that diffused through a finite angle fails to reach the lens*. In defining t the transparency at any place account must in strictness be taken of the manner in which the picture is to be viewed. There is also another point to be considered. The transparency may not be the same for different kinds * In the extreme case a negative seen against a dark background and lighted obliquely from behind may even appear as a positive. K. VI. 5 66 ON THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION [359 of light. We must suppose either that one kind of light only is employed, or else that t is the same for all the kinds that need to be regarded. The actual values of t may be supposed to range from 0, representing complete opacity, to 1, representing complete transparency. As the first step is the production of a negative, the question naturally suggests itself whether we can define the ideal character of such a negative. Attempts have not been wanting ; but when we reflect that the negative is only a means to an end, we recognize that no answer can be given without reference to the process in which the negative is to be employed to produce the positive. In practice this process (of printing) is usually different from that by which the negative was itself made; but for simplicity we shall suppose that the same process is employed in both operations. This require- ment of identity of procedure in the two cases is to be construed strictly, extending, for example, to duration of development and degree of intensifica- tion, if any. Also we shall suppose for the present that the exposure is the same. In strictness this should be understood to require that both the intensity of the incident light and the time of its operation be maintained ; but since between wide limits the effect is known to depend only upon the product of these quantities, we may be content to regard exposure as defined by a single quantity, viz. intensity of light x time. Under these restrictions the transparency 1f at any point of the negative is a definite function of the transparency t at the corresponding point of the original, so that we may write t'=f(t\ .................................... (1) / depending upon the photographic procedure and being usually such that as t increases from to 1, t' decreases continually. When the operation is repeated upon the negative, the transparency t" at the corresponding part of the positive is given by (2) Complete reproduction may be considered to demand that at every point t" = t. Equation (2) then expresses that t must be the same function of t' that If is of t. Or, if the relation between t and t' be written in the form F(t, O = 0, ................................. (3) F must be a symmetrical function of the two variables. If we regard t, t' as the rectangular coordinates of a point, (3) expresses the relationship by a curve which is to be symmetrical with respect to the bisecting line t' = t. So far no particular form of /, or F, is demanded ; no particular kind of negative is indicated as ideal. But certain simple cases call for notice. Among these is t + t'=l, ................................. (4) 1911] ON THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION 67 which obviously satisfies the condition of symmetry. The representative curve is a straight line, equally inclined to the axes. According to (4), when t = 0, t' = I. This requirement is usually satisfied in photography, being known as freedom from fog no photographic action where no light has fallen. But the complementary relation t' = when t = 1 is only satisfied approximately. The relation between negative and positive expressed in (4) admits of simple illustration. If both be projected upon a screen from independent lanterns of equal luminous intensity, so that the images fit, the pictures obliterate one another, and there results a field of uniform intensity. Another simple form, giving the same limiting values as (4), is + '' = !; (5) and of course any number of others may be suggested. According to Fechner's law, which represents the facts fairly well, the visibility of the difference between t and t + dt is proportional to dt/t. The gradation in the negative, constituted in agreement with (4), is thus quite different from that of the positive. When t is small, large differences in the positive may be invisible in the negative, and vice versa when t approaches unity. And the want of correspondence in gradation is aggravated if we substitute (5) for (4). All this is of course consistent with complete final reproduction, the differences which are magnified in the first operation being correspondingly attenuated in the second. If we impose the condition that the gradation in the negative shall agree with that in the positive, we have dt/t = -dtf/t', (6) whence t.t' = C, (7) where C is a constant. This relation does not fully meet the other require- ments of the case. Since t' cannot exceed unity, t cannot be less than C. However, by taking C small enough, a sufficient approximation may be attained. It will be remarked that according to (7) the negative and positive obliterate one another when superposed in such a manner that light passes through them in succession a combination of course entirely different from that considered in connexion with (4). This equality of gradation (within certain limits) may perhaps be considered a claim for (7) to represent the ideal negative ; on the other hand, the word accords better with defini- tion (4). It will be remembered that hitherto we have assumed the exposure to be the same in the two operations, viz. in producing the negative and in copying from it. The restriction is somewhat arbitrary, and it is natural to inquire whether it can be removed. One might suppose that the removal would allow a greater latitude in the relationship between t and t' ; but a closer scrutiny seems to show that this is not the case. 5-2 68 ON THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION [359 The effect of varying the exposure (e) is the same as of an inverse alteration in the transparency; it is the product et with which we really have to do. This refers to the first operation ; in the second, t" is dependent in like manner upon e't'. For simplicity and without loss of generality we may suppose that e = 1 ; also that e'/e = m, where m is a numerical quantity greater or less than unity. The equations which replace (1) and (2) are now t'=f(t), t = t"=f(mt'y, ........................ (8) and we assume that / is such that it decreases continually as its argument increases. This excludes what is called in photography solarization. We observe that if t, lying between and 1, anywhere makes t' = t, then m must be taken to be unity. For in the case supposed and this in accordance with the assumed character of /cannot be true, unless m = 1. Indeed without analytical formulation it is evident that since the transparency is not altered in the negative, it will require the same exposure to obtain it in the second operation as that by which it was produced in the first. Hence, if anywhere t' = t, the exposures must be the same. It remains to show that there is no escape from a local equality of t and t'. When t = 0, t' = 1, or (if there be fog) some smaller positive quantity. As t increases from to 1, t' continually decreases, and must therefore pass t at some point of the range. We conclude that complete reproduction requires m = 1, i.e. that the two exposures be equal ; but we must not forget that we have assumed the photographic procedure to be exactly the same, except as regards exposure. Another reservation requires a moment's consideration. We have inter- preted complete reproduction to demand equality of f and t. This seems to be in accord with usage ; but it might be argued that proportionality of t" and t' is all that is really required. For although the pictures considered in themselves differ, the effect upon the eye, or upon a photographic plate, may be made identical, all that is needed being a suitable variation in the intensity of the luminous background. But at this rate we should have to regard a white and a grey paper as equivalent. If we abandon the restriction that the photographic process is to be the same in the two operations, simple conclusions of generality can hardly be looked for. But the problem is easily formulated. We may write *'=/,(<*), t = t"=/ 3 (e't'\ ..................... (9) where e, e are the exposures, not generally equal, and f lt / 2 represent two functions, whose forms may vary further with details of development and intensification. But for some printing processes / 2 might be treated as a fixed function. It would seem that this is the end at which discussion 1911] OX THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION 69 should begin. When the printing process is laid down and the character of the results yielded thereby is determined, it becomes possible to say what is required in the negative ; but it is not possible before. In many photographs it would appear that gradation tends to be lost at the ends of the scale, that is in the high lights and deep shadows, and (as a necessary consequence, if the full range is preserved) to be exaggerated in the half-tones. For some purposes, where precise reproduction is not desired, this feature may be of advantage. Consider, for example, the experimental problem, discussed by Huggins, of photographing the solar corona without an eclipse. The corona is always present, but is overpowered by atmospheric glare. The problem is to render evident a very small relative difference of luminous intensity. If the difference is exaggerated in a suitably exposed and developed photograph, so much the better. A repetition of successive copyings might render conspicuous a difference originally invisible. At each operation we may suppose a factor a to be introduced, a being greater than unity. After n copyings dtft becomes a n dt/t. Unless the gain each time were very decided, this would be a slow process, and it would be liable to fail in practice owing to multiplication of slight irregular photographic markings. But a method proposed by Mach* and the present writer f should be of service here. By the aid of reflexion light at each stage is transmitted twice through the picture. By this means alone a is raised to equality with 2, and upon it any purely photographic exaggeration of gradation is superposed. Three successive copyings on this plan should ensure at least a ten-fold exaltation of contrast. Another method, simpler in execution, consists in superposing a consider- able number (n) of similar pictures. In this way the contrast is multiplied n times. Rays from a small, but powerful, source of light fall first upon a collimating lens, so as to traverse the pile of pictures as a parallel beam. Another condensing lens brings the rays to a focus, at which point the eye is placed. Some trials on this plan made a year ago gave promising results. Ten lantern-slides were prepared from a portrait negative. The exposure (to gas-light) was for about 3 seconds through the negative and for 30 seconds bare, i.e. with negative removed, and the development was rather 'light. On single plates the picture was but just visible. Some rough photometry indicated that each plate transmitted about one-third of the incident light. In carrying out the exposures suitable stops, cemented to the negative, must be provided to guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means. When only a few plates are combined, the light of a Welsbach mantle suffices ; but, as was to be expected, the utilization of the whole number (ten) * Eder's Jahrbuchf. Photographic. t Phil. Mag. Vol. XLIV. p. 282 (1897) ; Scientific Papers, Vol. iv. p. 333. 70 ON THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION [359 requires a more powerful source. Good results were obtained with a lime- light ; the portrait, barely visible at all on the single plates, came out fairly well under this illumination. If it were proposed to push the experiment much further by the combination of a larger number of plates, it would probably be advantageous to immerse them in benzole contained in a tank, so as to obviate the numerous reflexions at the surfaces. It has been mentioned that in the above experiment the development of the plates was rather light. The question may be raised whether further development, or intensification, might not make one plate as good as two or three superposed. I think that to a certain extent this is so. When in a recent experiment one of the plates above described was intensified with mercuric chloride followed by ferrous oxalate, the picture was certainly more apparent than before, when backed by a sufficiently strong light. And the process of intensification may be repeated. But there is another point to be considered. In the illustrative experiment it was convenient to copy all the plates from the same negative. But this procedure would not be the proper one in an attempt to render visible the solar corona. For this purpose a good many independent pictures should be combined, so as to eliminate slight photographic defects. As in many physical measurements, when it is desired to enhance the delicacy, the aim must be to separate feeble constant effects from chance disturbances. It may be that, besides that of the corona, there are other astronomical problems to which one or other of the methods above described, or a com- bination of both, might be applied with a prospect of attaining a further advance. 360. ON THE PROPAGATION OF WAVES THROUGH A STRATIFIED MEDIUM, WITH SPECIAL REFERENCE TO THE QUESTION OF REFLECTION. [Proceedings of the Royal Society, A, Vol. LXXXVI. pp. 207266, 1912.] THE medium is supposed to be such that its properties are everywhere a function of but one coordinate x, being of one uniform quality where x is less than a certain value x lt and of another uniform quality (in general, different from the first) where x exceeds a greater value x m _ l \ and the principal problem is the investigation of the reflection which in general ensues when plane waves in the first medium are incident upon the strati- fications. For the present we suppose the quality to be uniform through strata of finite thickness, the first transition occurring when x = x lt the second at x=x z , and the last at x=x m _ 1 . The expressions for the waves in the various media in order may be taken to be and so on, the A's and B's denoting arbitrary constants. The first terms represent the waves travelling in the positive direction, the second those travelling in the negative direction ; and our principal aim is the determina- tion of the ratio BJA^ imposed by the conditions of the problem, including the requirement that in the final medium there shall be no negative wave. As in the simple transition from one uniform medium to another (Theory of Sound, 270), the symbols c and b are common to all the media, the first depending merely upon the periodicity, while the constancy of the second is required in order that the traces of the various waves on the surfaces of 72 ON THE PROPAGATION OF WAVES transition should move together equivalent to the ordinary law of refrac- tion. In the usual optical notation, if V be the velocity of propagation and 6 the angle of incidence, c = 2irV/\, b = (27T/X) sin 0, a = (27r/\)cos 6, (2) where V/\, X" 1 sin 6 are the same in all the strata. On the other hand a is variable and is connected with the direction of propagation within the stratum by the relation a = 6cot0. (3) The a's are thus known in terms of the original angle of incidence and of the various refractive indices. Since the factor e { (et+b > runs through all our expressions, we may regard it as understood and write simply (4) (5) (6) <}> m = A m e-*>- <*-*->' + B m e^ *-*- (7) In the problem of reflection we are to make B m = 0, and (if we please) A m = l. We have now to consider the boundary conditions which hold at the surfaces of transition. In the case of sound travelling through gas, where < is taken to represent the velocity-potential, these conditions are the continuity of d<f>/dx and of cr<$>, where <r is the density. Whether the multiplier attaches to the dependent variable itself or to its derivative is of no particular significance. For example, if we take a new dependent variable ty, equal to <r<f>, the above conditions are equivalent to the con- tinuity of -fy and of o-'ctyr/c&r. Nor should we really gain generality by introducing a multiplier in both places. We may therefore for the present confine ourselves to the acoustical form, knowing that the results will admit of interpretation in numerous other cases. At the first transition x = x l the boundary conditions give a, (B, - A,) = a 2 ( 2 - A 9 ), a, (B, + A,) = o- 8 (5 2 + A,) (8) If we stop here, we have the simple case of the juxtaposition of two media both of infinite depth. Supposing 5 2 = 0, we get #1 __ q-2/o-i QS/C^ _ flra/o-! cot tfg/cot 0j A! ~~ <r t /<r l + Otfa^ ~ a-y/ffi + cot tf a /cot t) l ' For a further discussion of (9) reference may be made to Theory of Sound (loc. tit.). In the case of the simple gases the compressibilities are 1912] THROUGH A STRATIFIED MEDIUM, ETC. 73 the same, and a l sin 2 ft = ov, sin 2 ft. The general formula (9) then identifies itself with Fresnel's expression tan (ft -0.) tan (ft + ft)' On the other hand, if 0%, = <r l , the change being one of compressibility only, we find ,a\ sin (ft - ft) (9) = sin(ft + ft)' ^ U > Fresnel's other expression. In the above it is supposed that a 2 (and 6. 2 ) are real. If the wave be incident in the more refractive medium and the angle of incidence be too great, 2 becomes imaginary, say to, 7 . In this case, of course, the reflection is total, the modulus of (9) becoming unity. The change of phase incurred is given by (9). In accordance with what has been said these results are at once available for the corresponding optical problems. If there are more than two media, the boundary conditions at x = x 3 are a 2 [Bttfr**-** - A 2 e-^**-*J} = a 3 (B 3 - A 3 ), (12) a- 2 {B 2 e ia ^~^+A 2 e- ia ^-^} = <T 3 (B 3 + A 3 ), (13) and so on. For extended calculations it is desirable to write these equations in an abbreviated shape. We set B 2 -A 2 = H 2 , B 2 + A 2 = K 2 , etc., (14) i sin a^ (x 2 x^) = s : , etc., (15) 0-3/0-2 = /9 2) etc.; (16) and the series of equations then takes the form (17) (18) (19) and so on. In the reflection problem the special condition is the numerical equality of H and K of highest suffix. We may make H=-l, K = + I (20) As we have to work backwards from the terms of highest suffix, it is convenient to solve algebraically each pair of simple equations. In this way, remembering that c 2 s 2 =l, we get (21) (22) (23) 74 ON THE PROPAGATION OF WAVES [360 and so on. In these equations the c's and the f?s are real, and also the a's, unless there is " total reflection " ; the s's are pure imaginaries, with the same reservation. When there are three media, we are to suppose in the problem of reflection that H> = -l,Kt= 1. Thus from (21), (22), B l _K,^-H l _ Cl (&& - a 1 a a ) + g. (oy8, - a, ~ If there be no " total reflection," the relative intensity of the reflected waves is o, a (A A - o^) 2 - fr ( 2 & - , &) , 2 , ^(AA+W-ViteA+^A) 1 ' ' where d 2 = cos 2 a 2 (x z x^, s^ = sin 2 Og (x 2 Xj). ......... (26) The reflection will vanish independently of the values of Cj and s 1} i.e., whatever may be the thickness of the middle layer, provided AA -!. = (>, 8 A-iA = 0; or & = ,, & = a 8 , since these quantities are all positive. Reference to (9) shows that these are the conditions of vanishing reflection at the two surfaces of transition considered separately. If these conditions be not satisfied, the evanescence of (25) requires that either C, or Sj be zero. The latter case is realized if the intermediate layer be abolished, and the remaining condition is equivalent to 0-3/0-^ = o 3 /a, , as was to be expected from (9). We learn now that, if there would be no reflection in the absence of an intermediate layer, its introduction will have no effect provided a^x^-x-^ be a multiple of TT. An obvious example is when the first and third media are similar, as in the usual theory of " thin plates." On the other hand, if c l , or cos a 2 (# 2 #i)> vanish, the remaining require- ment for the evanescence of (25) is that yS 2 /a 2 = y9i/ai. In this case &ZJ?! AZ! . ft + l & + / so that by (9) the reflections at the two faces are equal in all respects. In general, if the third and first media are similar, (25) reduces to {,/, - ,/&} 2 sin' 02 fa -x,} 4 cos 1 a, (x, - x,) + {ft/a, + *,/,} sin 8 a* (x, - xj ' which may readily be identified with the expression usually given in terms of (9). It remains to consider the cases of so-called total reflection. If this occurs only at the second surface of transition, a,, a 2 are real, while o s is a 1912] THROUGH A STRATIFIED MEDIUM, ETC. 75 pure imaginary. Thus j is real, and a 2 is imaginary; d is real always, and s l is imaginary as before; the yS's are always real. Thus, if we separate the real and imaginary parts of the numerator and denominator of (24), we get ~ - , of which the modulus is unity. In this case, accordingly, the reflection back in the first medium is literally total, whatever may be the thickness of the intermediate layer, as was to be expected. The separation of real and imaginary parts follows the same rule when a. 2 is imaginary, as well as a s . For then a l is imaginary, while a 2 , Sj are real. Thus iCr 2 A remains real, and c^a^, s^fa remain imaginary. The reflection back in the first medium is total in this case also. The only other case requiring consideration occurs when a z is imaginary and 3 real. The reflection is then total if the middle layer be thick enough, but if this thickness be reduced, the reflection cannot remain total, as is evident if we suppose the thickness to vanish. The ratios a lt 2 are now both imaginary, while s x is real. The separation of real and imaginary parts stands as in (24), and the intensity of reflection is still expressed by (25). If we take a 2 = iaj, we may write in place of (25), (& A - i a 2 ) 2 cosh 2 a,' (# 2 - a?!) - (,& - !&)* sinh 2 q 2 ' (# 2 - Xl ) (&& + a, cr 2 ) 2 cosh 2 a/ (# 2 - ofi - ( 2 & + a!&) 2 sinh 2 a/ (# 2 - #,) ' ' When x z x^ is extremely small, this reduces to (/8 X A + a, a 2 ) 2 ' (0-3 in accordance with (9). When on the other hand # 2 #1 exceeds a few wave-lengths, (29) approaches unity, as we see from a form, equivalent to (29), viz., (& 2 - i 2 ) (& 2 - a* 2 ) cosh 2 o 2 / (cc z - i 2 ) (& 2 - 2 2 ) cosh 2 a 2 ' (# 2 - a?,) + (O.A + ^ 2 ) 2 It is to be remembered that in (30), a^, a 2 2 , a^ have negative values. The form assumed when the third medium is similar to the first may be noted. In this case ttjOg = 1, /3]/3 2 = 1, and we get from (29) (ft/gj - a^) 2 sinh 2 a g ' fa - ap ,^ inh 2 a,' (i - a*) - 4 ' ' In this case, of course, the reflection vanishes when # 2 ^ is sufficiently reduced. Equations (21), etc., may be regarded as constituting the solution of the general problem. If there are m media, we suppose H m = 1, K m =l, 76 ON THE PROPAGATION OF WAVES [360 thence calculate in order from the pairs of simple equations H m -\, ^m-i5 .Hw-s, K m .t, etc., until J5T, and K l are reached ; and then determine the ratio BijA^ The procedure would entail no difficulty in any special case numerically given ; but the algebraic expression of H 1 and K^ in terms of H m and K m soon becomes complicated, unless further simplifying conditions are introduced. Such simplification may be of two kinds. In the first it is supposed that the total thickness between the initial and final media is small relatively to the wave-lengths, so that the phase-changes occurring within the layer are of subordinate importance. In the second kind of simplification the thicknesses are left arbitrary, but the changes in the character of the medium, which occur at each transition, are supposed small. The problem of a thin transitional layer has been treated by several authors, L. Lorenz*, Van Rynf, DrudeJ, >chott, an( l Maclaurin||. A full account will be found in Theory of Light by the last named. It will therefore not be necessary to treat the subject in detail here ; but it may be worth while to indicate how the results may be derived from our equations. For this purpose it is convenient to revert to the original notation so far as to retain a and <r. Thus in place of (17), etc., we write (32) etc. ...(33) In virtue of the supposition that all the layers are thin, the c's are nearly equal to unity and the s's are small. Thus, for a first approximation, we identify c with 1 and neglect * altogether, so obtaining a 1 H l = a 2 H 2 =... = a m H m , ^K^ <r 2 K 2 = ... = <r m K m . ...(34) The relation of H lt K^ to H m , K m is the same as if the transition between the extreme values took place without intermediate layers, and the law of reflection is not disturbed by the presence of these layers, as was to be expected. For the second approximation we may still identify the c's with unity, while the s's are retained as quantities of the first order. Adding together the column of equations constituting the first members of (32), (33), etc., we find a, H l + a,,*, K 9 +a,8tK, + ...+ a m _, s m _ 8 K m ^ = a m H m ; (35) and in like manner, with substitution of <r for a and interchange of K and H, 1 = <r m K m (36) * Pogg. Ann. 1860, Vol. cxi. p. 460. t Wied. Ann. 1883, Vol. xx. p. 22. J Wied. Ann. 1891, Vol. xi.ni. p. 126. Phil. Tram. 1894, VoL CLXXXV. p. 823. II Roy. Soc. Proc. A, 1905, Vol. LXXVI. p. 49. 1912] THROUGH A STRATIFIED MEDIUM, ETC. 77 In the small terms containing s's we may substitute the approximate values of H and K from (34). For the problem of reflection we suppose H m + K m = Q. Hence o- w In (37), s t = ia z (# 2 a^), and so on, so that 7 7 a 2 the integration extending over the layer of transition. One conclusion may be drawn at once. To this degree of approximation the reflection is independent of the order of the strata. It will be noted that the sums in (37) are pure imaginaries. In what follows we shall suppose that a m is real. As the final result for the reflection, we find A-^-H-^'"' < 39 > where R = ^"V * ~ a "V * , ...(40) tan a = 2^ m (41) - - To this order of approximation the intensity of the reflection is unchanged by the presence of the intermediate layers, unless, indeed, the circumstances are such that (40) is itself small. If <r m l<r\ = ^m/di absolutely, we have -^f 1 a m j a- } (42) and S = ^TT. This case is important in Optics, as representing the reflection at the polarising angle from a contaminated surface. The two important optical cases : (i) where <r is constant, leading (when there is no transitional layer) to Fresnel's formula (11), and (ii) where <r sin 2 6 is constant, leading to (10), are now easily treated as special examples. Introducing the refractive index //,, we find after reduction for case (i) ->, o = where X,, /^ relate to the first medium, /* m is the index for the last medium, and the integration is over the layer of transition. The application of (43) 78 ON THE PROPAGATION OF WAVES [360 should be noticed when the layer is in effect abolished, either by supposing /* = /*> or, on the other hand, /t* = fa. In the second case (42), corresponding to the polarising angle, becomes 7T (44) In general for this case Q J Xl (/C -/*!) (co* 0, - -^ sin' 0. /*i ...... (45) The second fraction in (45) is equal to the thickness of the layer of transition simply, when we suppose /* = /Ltj. /(/*' -/iW-fr 8 )^ Further, 8"- 8 ' = -^4 - fi_ - , ...... (46) Xl ^ -* cos^-^lsin^ A*nt the difference of phase vanishing, as it ought to do, when /* = /*!, or ^ Hl , or again, when # x = 0. It should not escape notice that the expressions (10) and (11) have different signs when 1 and 2 are small. This anomaly, as it must appear from an optical point of view, should be corrected when we consider the significance of B" &'. The origin of it lies in the circumstance that, in our application of the boundary conditions, we have, in effect, used different vectors as dependent variables to express light of the two polarisations. For further explanation reference may be made to former writings, e.g. " On the Dynamical Theory of Gratings*." If throughout the range of integration, /*, is intermediate between the terminal values fr, p. m , the reflection is of the kind called positive by Jamin. The transition may well be of this character when there is no contamination. On the other hand, the reflection is negative, if /JL has throughout a value outside the range between /^ and /i m . It is probable that something of this kind occurs when water has a greasy surface. The formulae required in Optics, viz. (43), (44), (45), (46), are due, in substance, to Lorenz and Van Ryn. The more general expressions (41), (42) do not seem to have been given. There is no particular difficulty in pursuing the approximation from (32), etc. At the next stage the second term in the expansion of the c's * Roy. Soc. Proc. A, 1907, Vol. LXMX. p. 413. 1912] THROUGH A STRATIFIED MEDIUM, ETC. 79 must be retained, while the s's are still sufficiently represented by the first terms. The result, analogous to (37), (38), is [ d { x - a. Jo Jo , , . m I - a. .dx + i dx a! ( d d i f* . a m C d . 1-1 -. o-dx.dx + t <rdx Jo 0" Jo <r m J .(47) in which the terminal abscissae of the variable layer are taken to be and d, instead of ^ and x m _^. I do not follow out the application to particular cases such as cr = constant, or <r sin 2 6 = constant. For this reference may be made to Maclaurin, who, however, uses a different method. The second case which allows of a simple approximate expression for the reflection arises when all the partial reflections are small. It is then hardly necessary to appeal to the general equations : the method usually employed in Optics suffices. The assumptions are that at each surface of transition the incident waves may be taken to be the same as in the first medium, merely retarded by the appropriate amount, and that each partial reflection reaches the first medium no otherwise modified than by such retardation. This amounts to the neglect of waves three times reflected. Thus A &-i , &T^[ An interesting question suggests itself as to the manner in which the transition from one uniform medium to another must be effected in order to obviate reflection, and especially as to the least thickness of the layer of transition consistent with this result. If there be two transitions only, the least thickness of the layer is obtained by supposing in (48) and 2a 2 (# 2 - a^) = TT ; .............................. (50) and this conclusion, as we have seen already, is not limited to the case of small differences of quality. In its application to perpendicular incidence, (50) expresses that the thickness of the layer is one-quarter of the wave- length proper to the layer. The two partial reflections are equal in magnitude and sign. It is evident that nothing better than this can be done so long as the reflections are of one sign, however numerous the surfaces of transition may be. If we allow the partial reflections to be of different signs, some reduction of the necessary thickness is possible. For example, suppose that there are two intermediate layers of equal thickness, of which the first is similar to the final uniform medium, and the second similar to the initial uniform medium. Of the three partial reflections the first and third are similar, but the second 80 ON THE PROPAGATION OF WAVES [360 is of the opposite sign. If three vectors of equal numerical value compensate one another, they must be at angles of 120. The necessary conditions are satisfied (in the case of perpendicular transmission) if the total thickness (11) is X, in accordance with The total thickness of the layer of transition is thus somewhat reduced, but only by a very artificial arrangement, such as would not usually be contemplated when a layer of transition is spoken of. If the progress from the first to the second uniform quality be always in one direction, reflection cannot be obviated unless the layer be at least \ thick. The general formula (48) may be adapted to express the result appropriate to continuous variation of the medium. Suppose, for example, that cr is constant, making ft = 1, and corresponding to the continuity of both <f> and d<f>/dx*. It is convenient to suppose that the variation commences at x 0. Then (48) may be written a at any point x being connected with the angle of propagation by the usual relation (3). In the special case of perpendicular propagation, a = 27r/A/\i/Lti, H being refractive index and \ lt /^ relating to the first medium. A curious example, theoretically possible even if unrealizable in experi- ment, arises when the variable medium is constituted in such a manner that the velocity of propagation is everywhere constant, so that there is no refraction. Then a is constant, = 1, and (48) gives irJi 6 " 2 ^ < 52 > Some of the questions relating to the propagation of waves in a variable medium are more readily treated on the basis of the appropriate differential equation. As in (1), we suppose that the waves are plane, and that the medium is stratified in plane strata perpendicular to x, and we usually omit the exponential factors involving t and y, which may be supposed to run through. In the case of perpendicular propagation, y would not appear at all. Consider the differential equation Aty = 0, (53) in which (unless # can be infinite) it is necessary to suppose that <f> and d<j>{dx are continuous ; # is a function of x, which must be everywhere * These wonld be the conditions appropriate to a stretched string of variable longitudinal density vibrating transversely. 1912] THROUGH A STRATIFIED MEDIUM, ETC. 81 positive when the transmission is perpendicular, as, for example, in the case of a stretched string. When the transmission is oblique to the strata, k* may become negative, corresponding to " total reflection," but in most of what follows we shall assume that this does not happen. The continuity of and d(f>/dx, even though k 2 be discontinuous, appears to limit the applica- tion of (53) to certain kinds of waves, although, as a matter of analysis, the general differential equation of the second order may always be reduced to this form*. In the theory of a uniform medium, we may consider stationary waves or progressive waves. The former may be either (f> A cos k x cospt, or <f> = B sin k x sin pt ; and, if B= A, the two may be combined, so as to constitute progressive waves $ = A cos (pt k Q x). Conversely, progressive waves, travelling in opposite directions, may be combined so as to constitute stationary waves. When we pass to variable media, no ambiguity arises respecting stationary waves ; they are such that the phase is the same at all points. But is there such a thing as a pro- gressive wave ? In the full sense of the phrase there is not. In general, if we contemplate the wave forms at two different times, the difference between them cannot be represented by a mere shift of position proportional to the interval of time which has elapsed. The solution of (53) may be taken to be where ty(x), %(#) are real oscillatory functions of x; A', B, arbitrary constants as regards x. If we introduce the time-factor, writing p in place of the less familiar c of (1), we may take $ = A cospt . i]r(x) + B sinpt . %(#); ................ (55) and this may be put into the form 4>=Hcos(pt-0), ........................... (56) where Hcos d = Aty (x), Hsin0=Bx(x), ................ (57) or H* = A*[+(x)]* + B*[ x (x)y, ..................... (58) - (59) But the expression for <f> in (56) cannot be said to represent in general a progressive wave. We may illustrate this even from the case of the uniform medium where i/r (x) = cos Tex, % (x) = sin kx. In this case (56) becomes - tan" 1 (-^ tan fac . . . . * Forsyth's Differential Equations, % 59. <j> = {A 2 cos 2 kx + B* sin 2 kx}* cos \pt - tan" 1 -^ tan facj . . . .(60) 82 ON THE PROPAGATION OF WAVES [360 If BA, reduction ensues to the familiar positive or negative pro- gressive wave. But if B be not equal to A, (65), taking the form <i> = (A + B) cos (pt -kx) + \(A-B) cos (pt + kx), clearly does not represent a progressive wave. The mere possibility of reduction to the form (57) proves little, without an examination of the character of H and 0. It may be of interest to consider for a moment the character of 6 in (60). If B/A, or, say, m, is positive, 6 may be identified with kx at the quadrants but elsewhere they differ, unless m = l. Introducing the imaginary ex- pressions for tangents, we find 6 = kx + M sin 2kx + pf 2 sin 4>kx + $M S sin Qkx + . . . , ...... (61) where ^ = ^ZT ................................. < 62 > m + 1 When k is constant, one of the solutions of (53) makes </> proportional to e -ite Acting on this suggestion, and following out optical ideas, let us assume in general <t> = < n e- i l adx , ............................... (63) where the amplitude 77 and a are real functions of x, which, for the purpose of approximations, may be supposed to vary slowly. Substituting in (53), we find a 2 )7 7 -2ta(a) = ................... (64) For a first approximation, we neglect d*r)/dx*. Hence k = a, $r) = C, ................................. (65) so that <f> = Ck-*e ipt e- i $ kdx ............................ (66) or in real form, <f> = Ck~^cos(pt -fkdx) ......................... (67) If we hold rigorously to the suppositions expressed in (65), the satis- faction of (64) requires that d'rj/dx* = 0, or d'k ~ ^/dx 2 = 0. With omission of arbitrary constants affecting merely the origin and the scale of x, this makes k 2 = x~ l , corresponding to the differential equation * 4 | + * = ' ............................... (68 > whose accurate solution is accordingly (69) In (69) the imaginary part may be rejected. The solution (69) is, of course, easily verified. In all other cases (67) is only an approximation. 1912] THROUGH A STRATIFIED MEDIUM, ETC. 83 As an example, the case where k* = n*/x* may be referred to. Here fkdx = ft log # - e, and (67) gives <f> = Cx* cos (pt n log x + e) (70) as an approximate solution. We shall see presently that a slight change makes it accurate. Reverting to (64), we recognize that the first and second terms are real, while the third is imaginary. The satisfaction of the equation requires therefore that <**n = C, (71) and that & 2 = C^- 4 - - ~^- ; (72) while (63) becomes ( f> = r,e~ i ^ r '~ 2dx (73) Let us examine in what cases 77 may take the form Dx r . From (72), If r = 0, k z is constant. If r 1, k 2 = G 4 D~ 4 x~*, already considered in (68). The only other case in which & is a simple power of x occurs when r = \, making k 2 = (C*D~* + J) x~* = n 2 #~ 2 (say) (75) Here 77 = Dx*, C' 2 I 77-" dx = <7 2 /D 2 . log x - e, and the realized form of (73) is which is the exact form of the solution obtained by approximate methods in (70). For a discussion of (76) reference may be made to Theory of Sound, second edition, 148 b. The relation between a and 77 in (71) is the expression of the energy condition, as appears readily if we consider the application to waves along a stretched string. From (53), with restoration of e ipt , If the common phase factors be omitted, the parts of d<f>/dt and dfyjdx which are in the same phase are as prj and 0^77, and thus the mean work transmitted at any place is as arf. Since there is no accumulation of energy between two places, a77 2 must be constant. When the changes are gradual enough, a may be identified with k, and then 77 oc k~ , as represented in (67). If we regard 77 as a given function of x, a follows when C has been chosen, and also k 3 from (72). In the case of perpendicular propagation k 3 cannot be negative, but this is the only restriction. When 77 is constant, k 3 is constant ; 62 84 ON THE PROPAGATION OF WAVES [360 and thus if we suppose 77 to piss from one constant value to another through a finite transitional layer, the transition is also from one uniform A? to another; and (73) shows that there is no reflection back into the first medium. If the terminal values of rj and therefore of fc 2 be given, and the transitional layer be thick enough, it will always be possible, and that in an infinite number of ways, to avoid a negative A?, and thus to secure complete transmission without reflection back ; but if with given terminal values the layer be too much reduced, A? must become negative. In this case reflection cannot be obviated. It may appear at first sight as if this argument proved too much, and that there should be no reflection in any case so long as fc 2 is positive throughout. But although a constant rj requires a constant k-, it does not follow con- versely that a constant A? requires a constant 17, and, in fact, this is not true. One solution of (72), when Ar* is constant, certainly is if = C*lk; but the complete solution necessarily includes two arbitrary constants, of which C is not one. From (60) it may be anticipated that a solution of (72) may be rf = A 2 cos 2 kx + & sin' kx = ( A 2 + &) + $ (A 2 - B 2 ) cos 2kx. . . .(77) From this we find on differentiation and thus (72) is satisfied, provided that &A*B* = C* ................................. (78) It appears then that (77) subject to (78) is a solution of (72). The second arbitrary constant evidently takes the form of an arbitrary addition to x, and 77 will not be constant unless J. 2 = B 2 . On the supposition that 77 and a are slowly varying functions, the approximations of (65) may be pursued. We find (79) (80) The retardation, as usually reckoned in optics, is fkdx. The additional retardation according to (80) is i/r As applied to the transition from one uniform medium to another, the retardation is less than according to the first approximation by dx (81) 1912] THROUGH A STRATIFIED MEDIUM, ETC. 85 The supposition that 77 varies slowly excludes more than a very small reflection. Equations (79), (80) may be tested on the particular case already referred to where k = njx. We get 1 / 1 \ a = ( n -8-n )' so that \adx=(n ^- V on When n~* is neglected in comparison with unity, n ^n~ l may be identified with V(w 2 - I)- Let us now consider what are the possibilities of avoiding reflection when the transition layer (# 2 a?,) between two uniform media is reduced. If i7i> &i 3 ^2, &2 are the terminal values, (79) requires that k* = (frir*. & 2 2 = CV*. We will suppose that ^ 2 >^i- If the transition from ^ to ij 2 be made too quickly, viz., in too short a space, d 2 i}/dx* will become somewhere so large as to render Tc 1 negative. The same consideration shows that at the beginning of the layer of transition (a^), drj/dx must vanish. The quickest admissible rise of 17 will ensue when the curve of rise is such as to make jfc 2 vanish. When 17 attains the second prescribed value 17,, it must suddenly become constant, notwithstanding that this makes k 2 positively infinite. From (72) it appears that the curve of rise thus defined satisfies (82) The solution of (82), subject to the conditions that 17 = 171, dr)/dx when x = x l , is Again, when 77 = 172, x = ao 2 , so that giving the minimum thickness of the layer of transition. It will be observed that the minimum thickness of the layer of transition necessary to avoid reflection diminishes without limit with ^ k 2 , that is, as the difference between the two media diminishes. However, the arrange- ment under discussion is very artificial. In the case of the string, for example, it is supposed that the density drops suddenly from the first uniform value to zero, at which it remains constant for a time. At the end of this it becomes momentarily infinite, before assuming the second uniform value. The infinite longitudinal density at x. z is equivalent to a finite load 86 ON THE PROPAGATION OF WAVES [360 there attached. In the layer of transition, if so it may be called, the string remains straight during the passage of the waves. If, as in the more ordinary use of the term, we require the transition to be such that k? moves always in one direction from the first terminal value to the second, the problem is one already considered. The minimum thickness is such that k? has throughout it a constant intermediate value, so chosen as to make the reflections equal at the two faces. It would be of interest to consider a particular case in which k 3 varies continuously and always in the one direction. As appears at once from (72), d*iilda?, as well as drj/dx, must vanish at both ends of the layer, and there must also be a third point of inflection between. If the layer be from x = to x = ft, we may take jJ2-4*(*-)(*-) ......................... (85) We find that ft = 2a, and that From these k 2 would have to be calculated by means of (72), and one question would be to find how far a might be reduced without interfering with the prescribed character of fc 2 . But to discuss this in detail would lead us too far. If the differences of quality in the variable medium are small, (72) simplifies. If T/ O , k be corresponding values, subject to k * = C 4 ^^, we may take r) = Vo + r) ' ) & = &<? + $&, ....................... (88) where 77' and 8k 2 are small, and (72) becomes approximately % ........................... (89) Replacing x by t, representing time, we see that the problem is the same as that of a pendulum upon which displacing forces act; see Theory of Sound, 66. The analogue of the transition from one uniform medium to another is that of the pendulum initially at rest in the position of equilibrium, upon which at a certain time a displacing force acts. The force may be variable at first, but ultimately assumes a constant value. If there is to be no reflection in the original problem, the force must be of such a character that when it becomes constant the pendulum is left at rest in the new position. If the object be to effect the transition between the two states in the shortest possible time, but with forces which are restricted never to exceed the final value, it is pretty evident that the force must 1912] THROUGH A STRATIFIED MEDIUM, ETC. 87 immediately assume the maximum admissible value, and retain it for such a time that the pendulum, then left free, will just reach the new position of equilibrium, after which the force is reimposed. The present solution is excluded, if it be required that the force never decrease in value. Under this restriction the best we can do is to make the force assume at once half its final value, and remain constant for a time equal to one-half of the free period. Under this force the pendulum will just swing out to the new position of equilibrium, where it is held on arrival by doubling the force. These cases have already been considered, but the analogue of the pendulum is instructive. Kelvin* has shown that the equation of the second order * ............................ o> can be solved by a machine. It is worth noting that an equation of the form (53) is solved at the same time. In fact, if we make ~Tx> we get on elimination either (90) for y lf or for y z . Equations (91) are those which express directly the action of the machine. It now remains to consider more in detail some cases where total reflection occurs. When there is merely a simple transition from one medium (1) to another (2), the transmitted wave is ( f) 2 = A 2 e~ ia ^ x - x ^e i(ct+b y } ......................... (93) If there is total reflection, a 2 becomes imaginary, say ia^ ; the trans- mitted wave is then no longer a wave in the ordinary sense, but there remains some disturbance, not conveying energy, and rapidly diminishing as we recede from the surface of transition according to the factor $-' <*-*.>. From (2) or (94) It appears that soon after the critical angle is passed, the disturbance in the second medium extends sensibly to a distance of only a few wave-lengths. The circumstances of total reflection at a sudden transition are thus very simple ; but total reflection itself does not require a sudden transition, and * Roy. Soc. Proc. 1876, Vol. xxiv. p. 269. 88 ON THE PROPAGATION OF WAVES [360 takes place however gradual the passage may be from the first medium to the second, the only condition being that when the second is reached the angle of refraction becomes imaginary. From this point of view total reflection is more naturally regarded as a sort of refraction, reflection proper depending on some degree of abruptness of transition. Phenomena of this kind are familiar in Optics under the name of mirage. In the province of acoustics the vagaries of fog-signals are naturally referred to irregular refraction and reflection in the atmosphere, due to temperature or wind differences ; but the difficulty of verifying a suggested explanation on these lines is usually serious, owing to our ignorance of the state of affairs overhead *. The penetration of vibrations into a medium where no regular waves can be propagated is a matter of considerable interest ; but, so far as I am aware, there is no discussion of such a case, beyond that already sketched, relating to a sudden transition between two uniform media. It might have been supposed that oblique propagation through a variable medium would involve too many difficulties, but we have already had opportunity to see that, in reality, obliquity need not add appreciably to the complication of the problem. To fix ideas, let us suppose that we are dealing with waves in a membrane uniformly stretched with tension T, and of superficial density p, which is a function of a; only. The equation of vibration is (Theory of Sound, 194) or, if be proportional to e^+W, as in (1), > ........................ (95) agreeing with (53) if k* = (?p/T-b* ............................... (96) The waves originally move towards the less dense parts, and total reflection will ensue when a place is reached, at and after which Jc 2 is negative. The case which best lends itself to analytical treatment is when p is a linear function of x. k 1 is then also a linear function ; and, by suitable choice of the origin and scale of x, (95) takes the form * An observation daring the exceptionally hot weather of last summer recalled my attention to this subject. A train passing at high speed at a distance of not more than 150 yards was almost inaudible. The wheels were in full view, but the situation was such that the line of vision passed for most of its length pretty close to the highly heated ground. It seemed clear that the sound rays which should have reached the observers were deflected upwards over their heads, which were left in a kind of shadow. 1912] THROUGH A STRATIFIED MEDIUM, ETC. 89 The waves are now supposed to come from the positive side and are totally reflected at x = 0. The coefficient and sign of x are chosen so as to suit the formulae about to be quoted. The solution of (97), appropriate to the present problem, is exactly the integral investigated by Airy to express the intensity of light in the neighbourhood of a caustic*. The line # = is, in fact, a caustic in the optical sense, being touched by all the rays. Airy's integral is W=[ Jo (98) It was shown by Stokes -f* to satisfy (97), if x (in his notation n) = (%ir) zl3 m ................... (99) Calculating by quadratures and from series proceeding by ascending powers of m, Airy tabulated W for values of m lying between m = 5'6. For larger numerical values of m another method is necessary, for which Stokes gave the necessary formulas. Writing <^|=2(^) 3 / 2 =7r(^w) 3 / 2 ; ..................... (100) where the numerical values of m and x are supposed to be taken when these quantities are negative, he found when in is positive W = 2* (3m)~i {R cos (<f> - TT) + S sin (< - )}, ......... (101) 1.5.7.11 1.5.7.11.13.17.19.23 Where 1.2.8.4(720* 1.5 1.5.7.11.13.17 5=8 17720 - 1.2. 3 (720)3 When m is negative, so that W is the integral expressed by writing - m for m in (98), -... ....... (104) The first form (101) is evidently fluctuating. The roots of W=0 are given by 0-028145 0-026510 nft .v ---. + ..., ......... (II * being a positive integer, so that for i = 2, 3, 4, etc., we get TO = 4-3631, 5-8922, 7'2436, 8'4788, etc. For i=l, Airy's calculation gave m = 2'4955. * Camb. Phil. Trans. 1838, Vol. vi. p. 379 ; 1849, Vol. vin. p. 595. t Camb. Phil. Trans. 1850, Vol. ix. ; Math, and Phys. Papers, Vol. n. p. 328. J Here used in another sense. 90 ON THE PROPAGATION OF WAVES, ETC. [360 The complete solution of (97) in series of ascending powers of # is to be obtained in the usual way, and the arbitrary constants are readily determined by comparison with (98). Lommel* showed that these series are expressible by means of the Bessel's functions Jj, /-$. The connection between the complete solutions of (97), as expressed by ascending or by descending semi- convergent series, is investigated in a second memoir by Stokesf. A repro- duction of the most important part of Airy's table will be found in Mascart's Optics (Vol. I. p. 397). As total reflection requires, the waves in our problem are stationary as regards x. The realized solution of (95) may be written (f>= Wcos(ct + by) (106) W being the function of a; already discussed. On the negative side, when x numerically exceeds a moderate value, the disturbance becomes insensible. * Studien fiber die BesseVschen Functionen, Leipzig, 1868. t Camb. Phil. Trans. 1857, Vol. x. p. 106. 361. SPECTROSCOPIC METHODS. [Nature, Vol. LXXXVIII. p. 377, 1912.] IN his interesting address on spectroscopic methods, Prof. Michelson falls into a not uncommon error when he says that, in order to obtain a pure spectrum, " two important modifications must be made in Newton's arrange- ment. First, the light must be allowed to pass through a veiy narrow aperture, and, secondly, a sharp image of this aperture must be formed by a lens or mirror." Both these modifications were made by Newton himself, and with a clear understanding of their advantages. In Opticks, Exper. 11, we read: "In the Sun's Light let into my darkened Chamber through a small round hole in my Window shut, at about 10 or 12 feet from the Window, I placed a Lens, by which the Image of the hole might be distinctly cast upon a sheet of white Paper, placed at the distance of six, eight, ten, or twelve Feet from the Lens.... For in this case the circular Images of the hole which comprise that Image... were terminated most distinctly without any Penumbra, and therefore extended into one another the least that they could, and by conse- quence the mixture of the Heterogeneous Rays was now the least of all." And further on : " Yet instead of the circular hole F, 'tis better to substitute an oblong hole shaped like a long Parallelogram with its length Parallel to the Prism ABC. For if this hole be an Inch or two long, and but a tenth or twentieth part of an Inch broad or narrower : the Light of the Image pt will be as Simple as before or simpler [i.e. as compared with a correspondingly narrow circular hole], and the Image will become much broader, and therefore more fit to have Experiments tried in its Light than before." Again, it was not Bunsen and Kirchhoff who first introduced the collimator into the spectroscope. Swan employed it in 1847, and fully described its use in Edin. Trans. Vol. xvi. p. 375, 1849. See also Edin. Trans. Vol. xxi. p. 411, 1857 ; Pogg. Ann. C, p. 306, 1857. These are very minor matters as compared with what Prof. Michelson has to tell of his own achievements and experiences, but it seems desirable that they should be set right. 362. ON DEPARTURES FROM FRESNEL'S LAWS OF REFLEXION. [Philosophical Magazine, Vol. xxin. pp. 431 439, 1912.] IN the summer of 1907, in connexion with my experiments upon re- flexion from glass at the polarizing angle*, I made observations also upon the diamond, a subject in which Kelvin had expressed an interest. It was known from the work of Jamin and others that the polarization of light reflected from this substance is very far from complete at any angle of incidence, and my first experiments were directed to ascertain whether this irregularity could be plausibly attributed to superficial films of foreign matter, such as so greatly influence the corresponding phenomena in the case of waterf. The arrangements were of the simplest. The light from a paraffin flame seen edgeways was reflected from the diamond and examined with a nicol, the angle being varied until the reflexion was a minimum. In one important respect the diamond offers advantages, in comparison, for instance, with glass, where the surface is the field of rapid chemical changes due presumably to atmospheric influences. On the other hand, the smallness of the available surfaces is an inconvenience which, however, is less felt than it would be, were high precision necessary in the measure- ments. Two diamonds were employed one, kindly lent me by Sir W. Crookes, mounted at the end of a bar of lead, the other belonging to a lady's ring. No particular difference in behaviour revealed itself. The results of repeated observations seemed to leave it improbable that any process of cleaning would do more than reduce the reflexion at the polarizing angle. Potent chemicals, such as hot chromic acid, may be employed, but there is usually a little difficulty in the subsequent prepa- ration. After copious rinsing, at first under the tap and then with distilled water from a wash-bottle, the question arises how to dry the surface. Any ordinary wiping may be expected to nullify the chemical treatment; but if Phil. Mag. Vol. xvi. p. 444 (1908) ; Scientific Papers, Vol. v. p. 489. t Phil. Mag. Vol. xxxui. p. 1 (1892) ; Scientific Papers, Vol. in. p. 496. 1912] ON DEPARTURES FROM FRESNEL'S LAWS OF REFLEXION 93 drops are allowed to dry on, the effect is usually bad. Sometimes it is possible to shake the drops away sufficiently. After a successful operation of this sort wiping with an ordinarily clean cloth usually increases the minimum reflexion, and of course a touch with the finger, however prepared, is much worse. As the result of numerous trials I got the impression that the reflexion could not be reduced below a certain standard which left the flame still easily visible. Rotation of the diamond surface in its own plane seemed to be without effect. During the last few months I have resumed these observations, using the same diamonds, but with such additions to the apparatus as are necessary for obtaining measures of the residual reflexion. Besides the polarizing nicol, there is required a quarter-wave mica plate and an analysing nicol, to be traversed successively by the light after reflexion, as described in my former papers. The analysing nicol is set alternately at angles /3 = 45. At each of these angles extinction may be obtained by a suitable rotation of the polarizing nicol ; and the observation consists in determining the angle of a between the two positions. Jamin's k, representing the ratio of reflected amplitudes for the two principal planes when light incident at the angle tan" 1 yu, is polarized at 45 to these planes, is equal to tan (a' a). The sign of a! - a is reversed when the mica is rotated through a right angle, and the absolute sign of k must be found independently. Wiped with an ordinarily clean cloth, the diamond gave at first a' a = 2 0> 3. By various treatments this angle could be much reduced. There was no difficulty in getting down to 1. On the whole the best results were obtained when the surface was finally wiped, or rather pressed repeatedly, upon sheet asbestos which had been ignited a few minutes earlier in the blowpipe flame ; but they were not very consistent. The lowest reading was 0'4; and we may, I think, conclude that with a clean surface a a would not exceed 0- 5. No more than in the case of glass, did the effect seem sensitive to moisture, no appreciable difference being observable when chemically dried air played upon the surface. It is impossible to attain absolute certainty, but my impression is that the angle cannot, be reduced much further. So long as it exceeds a few tenths of a degree, the paraffin flame is quite adequate as a source of light. If we take for diamond a' a = 30', we get k = tan & (' - a) = '0044. Jamin's value for k is '019, corresponding more nearly with what I found for a merely wiped surface. Similar observations have been made upon the face of a small dispersing prism which has been in my possession some 45 years. When first examined, it gave a - a. = 9, or thereabouts. Treatment with rouge on a piece of 94 ON DEPARTURES FROM FRESNEL'S LAWS OF REFLEXION [362 calico, stretched over a glass plate, soon reduced the angle to 4 or 3, but further progress seemed more difficult. Comparisons were rendered some- what uncertain by the fact that different parts of the surface gave varying numbers. After a good deal of rubbing, a' a. was reduced to such figures as 2, on one occasion apparently to 1. Sometimes the readings were taken without touching the surface after removal from the rouge, at others the face was breathed upon and wiped. In general, the latter treatment seemed to increase the angle. Strong sulphuric acid was also tried, but without advantage, as also putty-powder in place of or in addition to rouge. The behaviour did not appear to be sensitive to moisture, or to alter appreciably when the surface stood for a few days after treatment. Thinking that possibly changes due to atmospheric influences might in nearly half a century have penetrated somewhat deeply into the glass, I re-ground and polished (sufficiently for the purpose) one of the originally unpolished faces of the prism, but failed even with this surface to reduce a a below 2. As in the case of the diamond, it is impossible to prove absolutely that a' a cannot be reduced to zero, but after repeated trials I had to despair of doing so. It may be well to record that the refractive index of the glass for yellow rays is T680. These results, in which k (presumably positive) remained large in spite of all treatment, contrast remarkably with those formerly obtained on less refractive glasses, one of which, however, appears to contain lead. It was then found that by re-polishing it was possible to carry k down to zero and to the negative side, somewhat as in the observations upon water it was possible to convert the negative k of ordinary (greasy) water into one with a small positive value, when the surface was purified to the utmost. There is another departure from Fresnel's laws which is observed when a piece of plate glass is immersed in a liquid of equal index*. Under such circumstances the reflexion ought to vanish. The liquid may consist of benzole and bisulphide of carbon, of which the first is less and the second more refractive than the glass. If the adjust- ment is for the yellow, more benzole or a higher temperature will take the ray of equal index towards the blue and vice versd. " For a closer exami- nation the plate was roughened behind (to destroy the second reflexion), and was mounted in a bottle prism in such a manner that the incidence could be rendered grazing. When the adjustment of indices was for the yellow the appearances observed were as follows : if the incidence is pretty oblique, the reflexion is total for the violet and blue ; scanty, but not evanescent, for the yellow ; more copious again in the red. As the incidence becomes more and more nearly grazing, the region of total reflexion advances from the blue * "On the Existence of Reflexion when the relative Refractive Index is Unity," Brit. Astoc. Report, p. 585 (1887) ; Scientific Papers, Vol. HI. p. 15. 1912] ON DEPARTURES FROM FRESNEI/S LAWS OF REFLEXION 95 end closer and closer upon the ray of equal index, and ultimately there is a very sharp transition between this region and the band which now looks very dark. On the other side the reflexion revives, but more gradually, and becomes very copious in the orange and red. On this side the reflexion is not technically total. If the prism be now turned so that the angle of incidence is moderate, it is found that, in spite of the equality of index for the most luminous part of the spectrum, there is a pretty strong reflexion of a candle-flame, and apparently without colour. With the aid of sunlight it was proved that in the reflexion at moderate incidences there was no marked chromatic selection, and in all probability the blackness of the band in the yellow at grazing incidences is a matter of contrast only. Indeed, calculation shows that according to Fresnel's formulas, the reflexion would be nearly insensible for all parts of the spectrum when the index is adjusted for the yellow." It was further shown that the reflexion could be reduced, but not destroyed, by re-polishing or treatment of the surface with hydrofluoric acid. I have lately thought it desirable to return to these experiments under the impression that formerly I may not have been sufficiently alive to the irregular behaviour of glass surfaces which are in contact with the atmosphere. 1 wished also to be able to observe the transmitted as well as the reflected light. A cell was prepared from a tin-plate cylinder 3 inches long and 2 inches in diameter by closing the ends with glass plates cemented on with glue and treacle. Within was the glass plate to be experimented on, of similar dimensions, so as to be nearly a fit. A hole in the cylindrical wall allowed the liquid to be poured in and out. Although the plate looked good and had been well wiped, I was unable to reproduce the old effects ; or, for a time, even to satisfy myself that I could attain the right com- position of the liquid. Afterwards a clue was found in the spectra formed by the edges of the plate (acting as prisms) when the cell was slewed round. The subject of observation was a candle placed at a moderate distance. When the adjustment of indices is correct for any ray, the corresponding part of the spectrum is seen in the same direction as is the undispersed candle-flame by rays which have passed outside the plate. Either spectrum may be used, but the best for the purpose is that formed by the edge nearer the eye. There was now no difficulty in adjusting the index for the yellow ray, and the old effects ought to have manifested themselves ; but they did not. The reflected image showed little deficiency in the yellow, although the incidence was nearly grazing, while at moderate angles it was fairly bright and without colour. This considerable departure from Fresnel's laws could only be attributed to a not very thin superficial modification of the glass rendering it optically different from the interior. In order to allow of the more easy removal and replacement of the plate under examination, an altered arrangement was introduced, in which the 96 ON DEPARTURES FROM FRESNELS LAWS OF REFLEXION [362 aperture at the top of the cell extended over the whole length. The general dimensions being the same as before, the body of the cell was formed by bending round a rectangular piece of tin-plate A (fig. 1) and securing the ends, to which the glass faces B were to be cemented, by enveloping copper wire. The plate C could then be removed for cleaning or polishing without breaking a joint. In emptying the cell it is necessary to employ a large funnel, as the liquid pours badly. The plate tried behaved much as the one just spoken of. In the reflected light, whether at moderate angles or nearly grazing, the yellow-green ray of equal index did not appear to be missing. A line or rather band of polish, by putty-powder applied with the finger, showed a great alteration. Near grazing there was now a dark band in the spectrum of the reflected light as formerly described, and the effect was intensified when the polish affected both faces. In the transmitted light the spectrum was shorn of blue and green, the limit coming down as grazing is approached a consequence of the total reflexion of certain rays which then sets in. But at incidences far removed from grazing the place of equal index in the spectrum of the reflected light showed little weakening. A few days' standing (after polishing) in the air did not appear to alter the behaviour materially. On the same plate other bands were treated with hydrofluoric acid commercial acid diluted to one- third. This seemed more effective than the putty-powder. At about 15 off grazing, the spectrum of the reflected light still showed some weakening in the ray of equal index. In the cell with parallel faces it is not possible to reduce the angle of incidence (reckoned from the normal) sufficiently, a circumstance which led me to revert to the 60 bottle-prism. A strip of glass half an inch wide could be inserted through the neck, and this width suffices for the obser- vation of the reflected light. But I experienced some trouble in finding the light until I had made a calculation of the angles concerned. Supposing the plane of the reflecting surface to be parallel to the base of the prism, let us call the angle of incidence upon it , and let 6, <f> be the angles which the ray makes with the normal to the faces, externally and internally, measured in each case towards the refracting angle of the prism. Then X = 60 - <t>, $ = sin- 1 ( sin 6). The smallest % occurs when = 90, in which case ^ = 18 10'. This value cannot be actually attained, since the emergence would be grazing. If X = 90, giving grazing reflexion, = -48 36'. Again, if = 0, ^ = 60; 1912] ON DEPARTURES FROM FRESNEL'S LAWS OF REFLEXION 97 and if ^ = 45, = 22 51'. We can thus deal with all kinds of reflexion from x = 90 down to nearly 18, and this suffices for the purpose. The strip employed was of plate glass and was ground upon the back surface. The front reflecting face was treated for about 30" with hydro- fluoric acid. It was now easy to trace the effects all the way from grazing incidence down to an incidence of 45 or less. The ray of equal index was in the yellow-green, as was apparent at once from the spectrum of the reflected light near grazing. There was a very dark band in this region, and total reflexion reaching nearly down to it from the blue end. The light was from a paraffin flame, at a distance of about two feet, seen edgeways. As grazing incidence is departed from, the flame continues at first to show' a purple colour, and the spectrum shows a weakened, but not totally absent, green. As the angle of incidence % still further decreases, the reflected light weakens both in intensity and colour. When ^ = 45, or thereabouts, the light was weak and the colour imperceptible. After two further treatments with hydrofluoric acid and immediate examination, the light seemed further diminished, but it remained bright enough to allow the absence of colour to be ascertained, especially when the lamp was temporarily brought nearer. An ordinary candle-flame at the same (2 feet) distance was easily visible. In order to allow the use of the stopper, the strip was removed from the bottle-prism when the observations were concluded, and it stood for four days exposed to the atmosphere. On re-examination it seemed that the reflexion at % = 45 had sensibly increased, a conclusion confirmed by a fresh treatment with hydrofluoric acid. It remains to consider the theoretical bearing of the two anomalies which manifest themselves (i) at the polarizing angle, and (ii) at other angles when both media have the same index, at any rate for a particular ray. Evidently the cause may lie in a skin due either to contamination or to the inevitable differences which must occur in the neighbourhood of the surface of a solid or fluid body. Such a skin would explain both anomalies and is certainly a part of the true explanation, but it remains doubtful whether it accounts for everything. Under these circumstances it seems worth while to inquire what would be the effect of less simple boundary conditions than those which lead to Fresnel's formula;. On the electromagnetic theory, if 6, 6 l are respectively the angles of incidence and refraction, the ratio of the reflected to the incident vibration is, for the two principal polarizations, tan fl/tan - p/ft tan #!/tan + p/^ ' ' and tan fl/tan 0- tan 0,/tantf + #/#,' " 98 ON DEPARTURES FROM FRESNEL'S LAWS OF REFLEXION [362 in which K, /* are the electric and magnetic constants for the first medium, K lt fr for the second*. The relation between B and 0, is IT,**, : A> = sin*0 : sin'fl, .......................... (C) It is evident that mere absence of refraction will not secure the evanescence of reflexion for both polarizations, unless we assume both ^ = /u, and K^ = A". In the usual theory ^ is supposed equal to /* in all cases. (A) then identifies itself with Fresnel's sine-formula, and (B) with the tangent-formula, and both vanish when K^^K corresponding to no refraction. Further, (B) vanishes at the Brewsterian angle, even though there be refraction. A slight departure from these laws would easily be accounted for by a difference between /A, and y., such as in fact occurs in some degree (diamagnetism). But the effect of such a departure is not to interfere with the complete evanescence of (B), but merely to displace the angle at which it occurs from the Brewsterian value. If yu,j //* = 1 + k, where h is small, calculation shows that the angle of complete polarization is changed by the amount n being the refractive index. The failure of the diamond and dense glass to polarize completely at some angle of incidence is not to be explained in this way. As I formerly suggested, the anomalies may perhaps be connected with the fact that one at least of the media is dispersive. A good deal depends upon the cause of the dispersion. In the case of a stretched string, vibrating transversely and endowed with a moderate amount of stiffness, the boundary conditions would certainly be such as would entail a reflexion in spite of equal velocity of wave-propagation. All optical dispersion is now supposed to be of the same nature as what used to be called anomalous dispersion, i.e. to be due to resonances lying beyond the visible range. In the simplest form of this theory, as given by Maxwell f and Sellmeier, the resonating bodies take their motion from those parts of the aether with which they are directly connected, but they do not influence one another. In such a case the boundary conditions involve merely the continuity of the displacement and its first derivative, and no complication ensues. When there is no refraction, there is also no reflexion. By introducing a mutual reaction between the resonators, and probably in other ways, it would be possible to modify the situation in such a manner that the boundary conditions would involve higher derivatives, as in the case of the stiff string, and thus to allow reflexion in spite of equality of wave- velocities for a given ray. On the Electromagnetic Theory of Light," Phil. Mag. Vol. xn. p. 81 (1881) ; Scitntific Paper*, Vol. i. p. 521. t Cambridge Calendar for 1869. See Phil. Mag. Vol. XLVIII. p. 151 (1899); Scientific Papcrt, Vol. nr. p. 418. 1912] ON DEPARTURES FROM FRESNEL'S LAWS OF REFLEXION 99 P.S. Jan. 15. Some later observations upon a surface of fused quartz are of interest. The plate, prepared by Messrs Hilger, was inch square, and the surfaces were inclined at a few degrees so as to separate the reflexions. From these surfaces the reflexion at the polarizing angle sensibly disappears. The image of the paraffin flame could be quenched by the operation of the polarizing nicol alone. When the quarter wave-plate and analysing nicol were introduced, o' and a could not be distinguished, the difference probably not exceeding -05, i.e. 3 minutes of angle. In order to examine the reflexion when the quartz was in contact with a liquid of equal index, I had to mix alcohol with the benzole. The behaviour was then much the same as with glass of which the surface had been renewed by hydrofluoric acid. Xo precise measures could be taken, but the reflexion at 45 incidence seemed less than from the glass, though still easily visible. In spite of repeated trials with intermediate cleanings, it was difficult to feel sure that the residual effect might not be due to foreign matter, the more as differences could sometimes be detected between various parts of the surface*. Even if the surface could be regarded as clean on immersion, there is no certainty that a capillary film of some sort might not be deposited upon it from the liquid. The cause of the small residual reflexion must remain for the present an open question. * At the top of the plate, where it was attached to a handle, a slight invasion of gelatine (used as a cement) gave rise to a copious reflexion ; but this film was easily visible in the air. 72 363. THE PRINCIPLE OF REFLECTION IN SPECTROSCOPES. [Nature, Vol. LXXXIX. p. 167, 1912.] THE application of a reflector to pass light back through a prism, or prisms, is usually ascribed to Littrow. Thus Kayser writes (Handbuch der Spectroscopie, Bd. I. p. 513), "Der Erste, der Rtickkehr der Strahlen zur Steigerung der Dispersion verwandte, war Littrow " (O. v. Littrow, Wien. Ber. XLVII. ii. pp. 26-32, 1863). But this was certainly not the first use of the method. I learned it myself from Maxwell (Phil. Trans. Vol. CL. p. 78, 1860), who says, " The principle of reflecting light, so as to pass twice through the same prism, was employed by me in an instrument for combining colours made in 1856, and a reflecting instrument for observing the spectrum has been constructed by M. Porro." I have not been able to find the reference to Porro ; but it would seem that both Maxwell and Porro antedated Littrow. As to the advantages of the method there can be no doubt. 364. ON THE SELF-INDUCTION OF ELECTRIC CURRENTS IN A THIN ANCHOR-RING. [Proceedings of the Royal Society, A, Vol. LXXXVI. pp. 562 571, 1912.] IN their useful compendium of " Formulae and Tables for the Calculation of Mutual and Self-Inductance*," Rosa and Cohen remark upon a small discrepancy in the formulae given by myself f and by M. WienJ for the self- induction of a coil of circular cross-section over which the current is uniformly distributed. With omission of n, representative of the number of windings, my formula was 8a 7 p 2 /. Sa l --j+l? (log 7 + g where p is the radius of the section and a that of the circular axis. The first two terms were given long before by Kirchhoff. In place of the fourth term within the bracket, viz., + -fap* /a?, Wien found - -0083,o 2 /a 2 . In either case a correction would be necessary in practice to take account of the space occupied by the insulation. Without, so far as I see, giving a reason, Rosa and Cohen express a preference for Wien's number. The difference is of no great importance, but I have thought it worth while to repeat the calculation and I obtain the same result as in 1881. A con- firmation after 30 years, and without reference to notes, is perhaps almost as good as if it were independent. I propose to exhibit the main steps of the calculation and to make extension to some related problems. The starting point is the expression given by Maxwell || for the mutual induction M between two neighbouring co-axial circuits. For the present * Bulletin of the Bureau of Standards, Washington, 1908, Vol. in. No. 1. t Roy. Soc. Proc. 1881, Vol. xxxn. p. 104 ; Scientific Papers, Vol. n. p. 15. Ann. d. Physik, 1894, Vol. LIII. p. 934 ; it would appear that Wien did not know of my earlier calculation. Pogg. Ann. 1864, Vol. cxxi. p. 551. || Electricity and Magnetism, 705. 102 ON THE SELF-INDUCTION OF [364 purpose this requires transformation, so as to express the inductance in terms of the situation of the elementary circuits relatively to the circular axis. In the figure, is the centre of the circular axis, A the centre of a section B through the axis of symmetry, and the position of any point P of the section is given by polar coordinates relatively to A, viz., by PA (p) and by the angle PAC(<f>). If p l , fa\ p 2 , fa be the coordinates of two points of the section P,, P 2 , the mutual induction between the two circular circuits represented by P,, P 2 is approximately t cos fa /?,' + pS + 2/j t a sin 3 fa + 2p a 2 sin 8 fa 16a a 2p,ptcos(fa-fa) + 4> Pl p a smfasmfa\ 8a 16a 10g T _ 9 _ Pi COS fa + pi COS fa 2o 3 (pi 8 -f p g a ) - 4 (pi 2 sin 2 fa + p 2 2 sin 2 fa) + 2^ p 2 cos (fa - fa} 16a 2 in which r, the distance between PI and P 2 , is given by Further details will be found in Wien's memoir ; I do not repeat them because I am in complete agreement so far. For the problem of a current uniformly distributed we are to integrate (2) twice over the area of the section. Taking first the integrations with respect to fa, fa, let us express <*> of which we can also make another application. The integration of the terms which do not involve logr is elementary. For those which do involve log r we may conveniently replace fa by fa + <, where <(> = fa-fa, and take first the integration with respect to fa fa being constant. Subsequently we integrate with respect to fa. It is evident that the terms in (2) which involve the first power of p vanish in the integration. For a change of fa, fa into TT fa, IT fa 1912] ELECTRIC CURRENTS IN A THIN ANCHOR-RING 103 respectively reverses cos fa and cos fa, while it leaves r unaltered. The definite integrals required for the other terms are* I log (p! 2 + p 2 2 2pn p. z cos <) d<f> = greater of 4-rr log p 2 and 4nr log p 1} (5) I cos nt(t> log (pi* + p 2 2 2pip 2 cos <) d$ = - - x smaller of f P -*Y and f &Y" , ... .(6) m W \pJ = - m being an integer. Thus g reater of lo g ^ and log pj. (7) So far as the more important terms in (4) those which do not involve p as a factor we have at once log (80.) 2 greater of log p 2 and log p l ................ (8) If p 2 and p l are equal, this becomes log(8a/p)-2 .................................. (9) We have now to consider the terms of the second order in (2). The contribution which these make to (4) may be divided into two parts. The first, not arising from the terms in log r, is easily found to be (10) The difference between Wien's number and mine arises from the inte- gration of the terms in log r, so that it is advisable to set out these somewhat in detail. Taking the terms in order, we have as in (7) I r+Tf r+ir I I log r dfa d(f> 2 = greater of log p 2 and log Oj ........ (11) 47T 2 J_ n .J_ n . In like manner 1 1 sin 2 (/>! log r dfa dfa = % [greater of log p 2 and log p,], . . ..(12) an( l I S i n 2 2 log r dfa dfa has the same value. Also by (6), with m = l, -7- a 1 1 cos (fa - fa) log r d fad fa = - [smaller of p^p^ and pjpz]. . . . (13) Finally j 2 sin fa sin fa log r dfa dfa 1 r+7T ,~+ir = dfa sin fa (sin fa cos < + cos fa sin 47T J _. j = -| [smaller of p. 2 /p 1 and pj/pj ......................... (14) * Todhunter's Int. Calc. 287, 289. 104 ON THE SELF-INDUCTION OF [364 Thus altogether the terms in (2) of the second order involving log r yield in (4) _ PL+J& [greater of log p. and log Pl ] - | a [smaller of and ] . ...(15) The complete value of (4) to this order of approximation is found by addition of (8), (10), and (15). By making p 2 and p l equal we obtain at once for the self-induction of a current limited to the circumference of an anchor-ring, and uniformly dis- tributed over that circumference, (16) p being the radius of the circular section. The value of L for this case, when /> a is neglected, was virtually given by Maxwell*. When the current is uniformly distributed over the area of the section, we have to integrate again with respect to p l and p 2 between the limits and p in each case. For the more important terms we have from (8) jj dpS dpf [log 8a - 2 - greater of log & and log p,] = log- ................................ (17) A similar operation performed upon (10) gives In like manner, the first part of (15) yields For the second part we have " 8^y I I ****** [ smaller of P*> Pfl = ~ 24^ ; thus altogether from (15) ...(19) The terms of the second order are accordingly, by addition of (18) and (19), Electricity and Magneti$m, 692, 706. 1912] ELECTRIC CURRENTS IN A THIN ANCHOR-RING 105 To this are to be added the leading terms (17) ; whence, introducing 4-Tra, we get finally the expression for L already stated in (1). It must be clearly understood that the above result, and the corresponding one for a hollow anchor-ring, depend upon the assumption of a uniform distribution of current, such as is approximated to when the coil consists of a great number of windings of wire insulated from one another. If the conductor be solid and the currents due to induction, the distribution will, in general, not be uniform. Under this head Wien considers the case where the currents are due to the variation of a homogeneous magnetic field, parallel to the axis of symmetry, and where the distribution of currents is governed by resistance, as will happen in practice when the variations are slow enough. In an elementary circuit the electromotive force varies as the square of the radius and the resistance as the first power. Assuming' as before that the whole current is unity, we have merely to introduce into (4) the factors (a + p t cos fa) (a + p z cos <fr 2 ) a MM retaining the value given in (2). The leading term in (21) is unity, and this, when carried into (14), will reproduce the former result. The term of the first order in p in (21) is (p! cos </>! + p 2 cos <f>z)/a, and this must be combined with the terms of order p and p 1 in (2). The former, however, contributes nothing to the integral. The latter yield in (4) Pi + Pz M j i i smaller of p^ and o 2 2 . '- L ^~- (log 8a-l -greater of log Pl and log p 2 } + - ^ - (22) The term of the second order in (21), viz., /3jp 2 / 2 - cos </h cos $ 2 > needs to be combined only with the leading term in (2). It yields in (4) smaller of pf and /j 2 2 .__. 4a 2 If PJ and p 2 are equal (p), the additional terms expressed by (22), (23) become If (24), multiplied by 4nra, be added to (16), we shall obtain the self- induction for a shell (of uniform infinitesimal thickness) in the form of an anchor-ring, the currents being excited in the manner supposed. The result is (25) 106 ON THE SELF-INDUCTION OF [364 We now proceed to consider the solid ring. By (22), (23) the terms, additional to those previously obtained on the supposition that the current was uniformly distributed, are smaller of pS&ndpJ + ?L+ ?* a 1 log 8a - 1 - greater of log p l and log p 2 | . ... (26) The first part of this is p s /6a 2 , and the second is ^ (log 8a - 1 - log p 4- The additional terms are accordingly These multiplied by 4nra are to be added to (1). We thus obtain 7 (28) for the self-induction of the solid ring when currents are slowly generated in it by uniform magnetic forces parallel to the axis of symmetry. In Wien's result for this case there appears an additional term within the bracket equal to - O092 p a /a j . A more interesting problem is that which arises when the alternations in the magnetic field are rapid instead of slow. Ultimately the distribution of current becomes independent of resistance, and is determined by induction alone. A leading feature is that the currents are superficial, although the ring itself may be solid. They remain, of course, symmetrical with respect to the straight axis, and to the plane which contains the circular axis. The magnetic field may be supposed to be due to a current x l in a circuit at a distance, and the whole energy of the field may be represented by T = \M u x* + P/rf + M n xf + ... + M lz x lXz + M^x.x, +... + M & x y x 3 + .......... (29) x z , x 3 , etc., being currents in other circuits where no independent electro- motive force acts. If a?, be regarded as given, the corresponding values of x it a-,, ... are to be found by making T a minimum. Thus M 12 ar, + 3/22*2 + M x x 3 + . . . = 0, 3Q M *, + 3/230:2 + M a x 3 + . . . = 0, and so on, are the equations by which x*, etc., are to be found in terms of x^ What we require is the corresponding value of T', formed from T by omission of the terms containing a^. The method here sketched is general. It is not necessary that x z , etc., be currents in particular circuits. They may be regarded as generalized 1912] ELECTRIC CURRENTS IX A THIN ANCHOR-RING 107 coordinates, or rather velocities, by which the kinetic energy of the system is defined. For the present application we suppose that the distribution of current round the circumference of the section is represented by ( + ! cos <j + 2 cos 2<j + ...} ^ , ................. (31) so that the total current is cr . The doubled energy, so far as it depends upon the interaction of the ring currents, is I J(a + a 1 cos</> 1 + a 2 cos2< 1 + ...)(a + a 1 cos< 2 + ...) M^dfadfa, (32) where M lz has the value given in (2), simplified by making p l and p 2 both equal to p. To this has to be added the double energy arising from the interaction of the ring currents with the primary current. For each element of the ring currents (31) we have to introduce a factor proportional to the area of the circuit, viz., TT (a + p cos c^) 2 . This part of the double energy may thus be taken to be H I dfa (a + p cos fa) 2 (o + i cos fa + a 2 cos 2 fa +...), that is 27r#{(a 2 + / 3 2 )a + a / 3a 1 + p 2 a 2 }, .................. (33) 3 , etc., not appearing. The sum of (33) and (32) is to be made a minimum by variation of the o's. We have now to evaluate (32). The coefficient of 2 is the quantity already expressed in (16). For the other terms it is not necessary to go further than the first power of p in (2). We get 47m a ' log l + - 2 - 2 *|^(^l)^ ......... (34) Differentiating the sum of (33), (34), with respect to er , a,, etc., in turn, find H (a- + tf) + 4a* jlog ?? (l + ;) - 2} + p., (log ^ - i) = 0, (35) ^ /0 \ 0, .................................. (36) (37) 108 ON THE SELF-INDUCTION OF [364 The leading term is, of course, a,,. Relatively to this, a a is of order p, o s of order p*, and so on. Accordingly, cr a , a,, etc., may be omitted entirely from (34), which is only expected to be accurate up to />* inclusive. Also, in a t only the leading term need be retained. The ratio of or, to o is to be found by elimination of H between (35), (36). We get (38 > Substituting this in (34), we find as the coefficient of self-induction The approximate value of er in terms of H is A closer approximation can be found by elimination of a a between (35), (36). In (39) the currents are supposed to be induced by the variation (in time) of an unlimited uniform magnetic field. A problem, simpler from the theoretical point of view, arises if we suppose the uniform field to be limited to a cylindrical space co-axial with the ring, and of diameter less than the smallest diameter of the ring (2a 2/o). Such a field may be supposed to be due to a cylindrical current sheet, the length of the cylinder being infinite. The ring currents to be investigated are those arising from the instantaneous abolition of the current sheet and its conductor. If 7r& 2 be the area of the cylinder, (33) is replaced simply by < > ................ (41) The expression (34) remains unaltered and the equations replacing (35), (36) are thus + 4o log l + . - 2 + pa, log - = 0, ....(42) The introduction of (43) into (34) gives for the coefficient of self-induction in this 7*+-*-*7-i (44) It will be observed that the sign of a, /a, is different in (38) and (43). 1912] ELECTRIC CURRENTS IN A THIN ANCHOR-RING 109 The peculiarity of the problem last considered is that the primary current occasions no magnetic force at the surface of the ring. The consequences were set out 40 years ago by Maxwell in a passage* whose significance was very slowly appreciated. " In the case of a current sheet of no resistance, the surface integral of magnetic induction remains constant at every point of the current sheet. " If, therefore, by the motion of magnets or variations of currents in the neighbourhood, the magnetic field is in any way altered, electric currents will be set up in the current sheet, such that their magnetic effect, combined with that of the magnets or currents in the field, will maintain the normal component of magnetic induction at every point of the sheet unchanged. If at, first there is no magnetic action, and no currents in the sheet, then the normal component of magnetic induction will always be zero at every point of the sheet. "The sheet may therefore be regarded as impervious to magnetic in- duction, and the lines of magnetic induction will be deflected by the sheet exactly in the same way as the lines of flow of an electric current in an infinite and uniform conducting mass would be deflected by the introduction of a sheet of the same form made of a substance of infinite resistance. " If the sheet forms a closed or an infinite surface, no magnetic actions which may take place on one side of the sheet will produce any magnetic effect on the other side." All that Maxwell says of a current sheet is, of course, applicable to the surface of a perfectly conducting solid, such as our anchor-ring may be supposed to be. The currents left in the ring after the abolition of the primary current must be such that the magnetic force due to them is wholly f+n tangential to the surface of the ring. Under this condition I M lz d(j>. 2 must J -it be independent of </>!, and we might have investigated the problem upon this basis. In Maxwell's notation a, @, 7 denote the components of magnetic force, and the whole energy of the field T is given by (45) Moreover a ,the total current, multiplied by 4-n- is equal to the "circulation" of magnetic force round the ring. In this form our result admits of imme- diate application to the hydrodynamical problem of the circulation of * Electricity and Magnetism, 654, 655. Compare my "Acoustical Observations," Phil. Mag. 1882, Vol. xm. p. 340 ; Scientific Papers, Vol. n. p. 99. 110 SELF-INDUCTION OF ELECTRIC CURRENTS IN A THIN ANCHOR-RING [364 incompressible frictionless fluid round a solid having the form of the ring ; for the components of velocity u, v t w are subject to precisely the same conditions as are a, fi, 7. If the density be unity, the kinetic energy T of the motion has the expression T=_ x (circulation) 5 , (46) O7T L having the value given in (44). P.S. March 4. Sir W. D. Niven, who in 1881 verified some other results for self-induction those numbered (11), (12) in the paper referred to has been good enough to confirm the formulae (1), (28) of the present communi- cation, in which I differ from M. Wien. 365. ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING. [Proceedings of the Royal Society, A, Vol. LXXXVII. pp. 193 202, 1912.] ALTHOUGH much attention has been bestowed upon the interesting subject of electric oscillations, there are comparatively few examples in which definite mathematical solutions have been gained. These problems are much simplified when conductors are supposed to be perfect, but even then the difficulties usually remain formidable. Apart from cases where the propagation may be regarded as being in one dimension*, we have Sir J. Thomson's solutions for electrical vibrations upon a conducting sphere or cylinder^. But these vibrations have so little persistence as hardly to deserve their name. A more instructive example is afforded by a conductor in the form of a circular ring, whose circular section is supposed small. There is then in the neighbourhood of the conductor a considerable store of energy which is more or less entrapped, and so allows of vibrations of reasonable persistence. This problem was very ably treated by PocklingtonJ in 1897, but with deficient explanations . Moreover, Pocklington limits his detailed conclusions to one particular mode of free vibration. I think I shall be doing a service in calling attention to this investigation, and in exhibiting the result for the radiation of vibrations in the higher modes. But I do not attempt a complete re-statement of the argument. Pocklington starts from Hertz's formulae for an elementary vibrator at the origin of coordinates , y, f> where H = e^ e^/p, .................................. (2) * Phil. Mag. 1897, Vol. XLIII. p. 125 ; 1897, Vol. XLIV. p. 199 ; Scientific Papers, Vol. nr. pp. 276, 327. t Recent Researcties, 1893, 301, 312. [1913. There is also Abraham's solution for the ellipsoid.] t Camb. Proceedings, 1897, Vol. ix. p. 324. Compare W. M C F. Orr, Phil. Mag. 1903, Vol. vi. p. 667. 112 ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING [365 in which P, Q, R denote the components of electromotive intensity, 2-Tr/jp is the period of the disturbance, and 2ir/a the wave-length corresponding in free fether to this period. At a great distance p from the source, we have from (1) The resultant is perpendicular to p, and in the plane containing p and . Its magnitude is where x * s tne angle between p and f. The required solution is obtained by a distribution of elementary vibrators of this kind along the circular axis of the ring, the axis of the vibrator being everywhere tangential to the axis of the ring and the coefficient of intensity proportional to cos m<j>, where m is an integer and <j>' defines a point upon the axis. The calculation proceeds in terms of semi-polar coordinates *, -or, <f>, the axis of symmetry being that of z, and the origin being at the centre of the circular axis. The radius of the circular axis is a, and the radius of the circular section is e, e being very small relatively to a. The condition to be satisfied is that at every point of the surface of the ring, where (vr a) 8 -I- z* = e 2 , the tangential component of (P, Q, R) shall vanish. It is not satisfied absolutely by the above specification; but Pocklington shows that to the order of approximation required the speci- fication suffices, provided a be suitably chosen. The equation determining a expresses the evanescence of that tangential component which is parallel to the circular axis, and it takes the form w 2 -a 2 a 2 cos<) = 0, .................. (5) g*a[*+4nra8tn > "" In (5) we are to retain the large term, arising in the integral when < is small, and the finite term, but we may reject small quantities. Thus Pocklington finds | * (aa cos - ra j ) cos m<j> d<j> w s ) cos m<f> d<f> 0, (7) the condition being to this order of approximation the same at all points of a cross-section. 1912] ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING 113 The first integral in (7) may be evaluated for any (integral) value of w. Writing </> = i|r, we have a v / }e a /4a 2 + sin 2 i/r} The large part of the integral arises from small values of ty. We divide the range of integration into two parts, the first from to ^ where ty, though small, is large compared with e/2a, and the second from i/r to ^TT. For the first part we may replace cos 2-\Jr, cos 2mty by unity, and sin 2 ty by 2 . We thus obtain Thus to a first approximation aa = + m. In the second part of the range of integration we may neglect 2 /4a 2 in comparison with sin 2 -\Jr, thus obtaining m 2 ) cos 2m-^r cZ^ a sn The numerator may be expressed as a sum of terms such as cos 2n ty, and for each of these the integral may be evaluated by taking cos ty = z, in virtue of Accordingly fi" COS 2n x ' '**"** when small quantities are neglected. For example, The sum of the coefficients in the series of terms (analogous to cos 2n >/r) which represents the numerator of (10) is necessarily a 2 o 2 - w 2 , since this is the value of the numerator itself when <fr = 0. The particular value of vjr chosen for the division of the range of integration thus disappears from the sum of (9) and (10), as of course it ought to do. When m = l, corresponding to the gravest mode of vibration specially considered by Pocklington, the numerator in (10) is 4a 2 o 2 cos 4 ^ - (4a s a 2 + 2) cos 2 ^ + a*a 2 + 1, R. VI. 114 ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING [365 and the value of the integral is accordingly To this is to be added from (9) a 1 * 1 - 1 making altogether for the value of (8) ' (12) The second integral in (7) contributes only finite terms, but it is important as determining the imaginary part of o and thus the rate of dissipation. We may write it e * where a? = 4a 2 et 2 = 4m 2 approximately. Pocklington shows that the imaginary part of (13) can be expressed by means of Bessel's functions. We may take (14) fjir oixsin^ _ 1 ,'_ rx whence J^ (tycos2^ - ^ - = - J ^ {/, (*) + i K w (x)} dx ...... (15) Accordingly, (13) may be replaced by so that \*dx{J 9m + s -2J am + J"^- 2 } = 4 J' m = 2/ 2m _ 1 - 2/ 2m+1 ...... (17) The imaginary part of (13) is thus simply ^{SMty-SmC*)] ......................... (18) A corresponding theory for the K functions does not appear to have been developed. When m = 1, our equation becomes (19) Compare Theory of Sound, 302. f Gray and Mathews, Bestel's Functions, p. 13. 1912] ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING 115 and on the right we may replace x by its first approximate value. Referring to (2) we see that the negative sign must be chosen for o and x, so that x = 2. The imaginary term on the right is thus For the real term Pocklington calculates 0*485, so that, L being written for log (8a/e), (0-243 + 0-352 (20) " Hence the period of the oscillation is equal to the time required for a free wave to traverse a distance equal to the circumference of the circle multiplied by 1 0'243/i, and the ratio of the amplitudes of consecutive vibrations is 1 e~-' 2l ' L or 1 - 2'21/L." For the general value of m (19) is replaced by where R is a real finite number, and finally ........ (21) (22) The ratio of the amplitudes of successive vibrations is thus 1 :l-T.*[J, m _ l (2m)-J, m+l (2m)}l'2L, (23) in which the values of J 2?rt _ 1 (2w) J m+l (2m) can be taken from the tables (see Gray and Mathews). We have as far as m equal to 12 : m *HM-*.M - ! *-*-*-. 1 0-448 7 0-136 2 0-298 8 0-125 3 0-232 9 0-116 4 0-194 10 0-108 5 0-169 11 0-102 6 0-150 12 0-096 It appears that the damping during a single vibration diminishes as m increases, viz., the greater the number of subdivisions of the circumference. An approximate expression for the tabulated quantity when m is large may be at once derived from a formula due to Nicholson*, who shows that Phil. Mag. 1908, Vol. xvi. pp. 276. 2?7. 82 116 ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING [365 when n and * are large and nearly equal, J n (z) is related to Airy's integral. In fact, so that J^ (2m) -J M (*m)- .................. (25) If we apply this formula to m = 10, we get 0111 as compared with the tabular 01 08*. It follows from (25) that the damping in each vibration diminishes without limit as m increases. On the other hand, the damping in a given time v.aries as ra* and increases indefinitely, if slowly, with m. We proceed to examine more in detail the character at a great distance of the vibration radiated from the ring. For this purpose we choose axes of x and y in the plane of the ring, and the coordinates (x, y, z) of any point may also be expressed as r sin 6 cos <f>, r sin 8 sin </>, r cos 0. The contribution of an element ad<f>' at <f>' is given by (4). The direction cosines of this element are sin <', cos <', ; and those of the disturbance due to it are taken to be I, m, n. The direction of this disturbance is perpendicular to r and in the plane containing r and the element of arc ad<f>'. The first condition gives Ix + my + nz = 0, and the second gives I . z cos </>' + m . z sin <' n (x cos <J>' + y sin <f>') = ; so that _ I __ m _ n (z* + y 3 ) sin <' + xy cos <f> (z 1 + x 1 ) cos <>' + xy sin </>' zy cos <' zx sin <' ' ............ (26) The sum of the squares of the denominators in (26) is r* {z 1 (y sin fi + a; cos </>') 2 }. Also in (4) and thus f* . / sin x = (z 3 + y") sin <f>' + xy cos <', - r 2 . m sin ^ = (z 3 + x 3 ) cos <f> + jry sin <', ............... (28) r . n sin ^ = zy cos <' - zx sin </>'. To these quantities the components P, Q, R due to the element ad$' are proportional. * lo glo r(!)= 0-13166. 1912] ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING 117 Before we can proceed to an integration there are two other factors to be regarded. The first relates to the intensity of the source situated at ad<f>'. To represent this we must introduce cos m<j>'. Again, there is the question of phase. In e iap we have p = r a sin 6 cos (<' <) ; and in the denominator of (4) we may neglect the difference between p and r. Thus, as the components due to adfi, we have P = - with similar expressions for Q and R corresponding to the right-hand members of (28). The integrals to be considered may be temporarily denoted by 8, C, where S, C= d< / cosrac^-* cos > '-'<> ) (sin<f>', cos<'), .......... (30) being written for aa sin 0. Here S = % I + " dfie-^W-V (sin (ra + 1) $ - sin (m - 1) </>'}, J TT and in this, if we write ty for <f>' $, sin (m + 1) <' = sin (m + 1) -^ . cos (m + 1) < + cos (m + 1) >|r . sin (m + 1) <. We thus find 1 sin (m + 1 )<- m _! sin (w -!)<, ............ (31) where @ w = ctycos wf e -* C08 * ........................ (32) Jo In like manner, C = w+1 cos (m + 1) < + m _j cos (m - 1) </> ............. (33) Now n=| d-drcoswjr {cos (cosT/r) i sin(^cos<^)}. Jo When n is even, the imaginary part vanishes, and e *JM (34) cos mr On the other hand, when n is odd, the real part vanishes, and lTrJ n () /QK\ - (H) n = . -^- \dd) Thus, when w is even, m + 1 and m 1 are both odd and S and C are both pure imaginaries. But when m is odd, S and (7 are both real. As functions of direction we may take P, Q, R to be proportional to ^' ~ 118 ELECTRICAL VIBRATION'S ON A THIN ANCHOR-RING [365 Whether m be odd or even, the three components are in the. same phase. On the same scale the intensity of disturbance, represented by P 2 + Q* + R*, is in terms of 0, <f> cos*0(S*+C*) + s}n*0(Ccos<f> + S8m<f>)>, ............ (36) an expression whose sign should be changed when m is even. Introducing the values of C and S in terms of @ from (31), (33), we find that P 2 + Q* + R- is proportional to cos 5 6 [ m+l > + e m _ l a + 2e w+1 ,_: cos 2i^>j + sin 8 O cos 2 m$ {0 m+1 -f m _ 1 ', 2 . ...... (37) From this it appears that for directions lying in the plane of the ring (cos = 0) the radiation vanishes with cos ni(f>. The expression (37) may also be written W+1 ' + e,,,-, 2 + 2 m+i e,,,_ 1 cos 2m - } sin 2 (0 m+1 - m _tf (1 ...... (38) or, in terms of J's, by (34), (35), TT [J m+1 2 + J m _r - 2J m+l J m -* cos 2m<j> *- ^ sin 2 (J m+l + J m _^ (1 - cos 2m0)], ...... (39) and this whether m be odd or even. The argument of the J's is oca sin 0. Along the axis of symmetry (0 = 0) the expression (39) should be independent of <f>. That this is so is verified when we remember that J n (0) vanishes except n = 0. The expression (39) thus vanishes altogether with unless m = l, when it reduces to TT- simply*. In the neighbourhood of the axis the intensity is of the order 0- m ~ 2 , In the plane of the ring (sin = 1) the general expression reduces to 7T 2 (J m+l - </,_, ) 2 cos 2 m<f>, or 47r 2 / m ' 2 cos 2 m<j> .......... (40) It is of interest to consider also the mean value of (39) reckoned over angular space. The mean with respect to <j> is evidently 7T 2 [J m+l * + J m -S + $ sin 2 (J m+l + J^W ............. (41 ) By a known formula in Bessel's functions (Ol i -/. i (0 ................... (42) For the present purpose * = a'a 2 sin- = w 2 sin 2 ; and (41) becomes [JU^o+tWca-i/.w] .................... (43) * [June 20. Reciprocally, plane waves, travelling parallel to the axis of symmetry and incident upon the ring, excite none of the higher modes of vibration.] 1912] ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING 119 To obtain the mean over angular space we have to multiply this by sin 6 dO, and integrate from to \ir. For this purpose we require f '* J n - (m sin 0) sin 0d0, . (44) .'o an integral which does not seem to have been evaluated. By a known expansion* we have Jo (2m sin 6 sin /3) = J * (m sin 0) + 2J^ (m sin 0) cos ft + 2 Jf (m sin 0) cos 2/3 -I- , whence ri* I J (2msin0sm^)sin0d0 ,-Jir rjTT = Jo 2 (m sin 0) sin Odd + 2 cos ft \ Jj 2 (m sin 0) sin 0d0 + .0 .0 4- 2cosw/9 I V n 2 (msin0)sin0d0 (45) .'o Now I for the integral on the left 2m sin ft and thus f^ra/ a\ a IQ l [ v jo sin(2msiii*/8) J n (m sin cOsin c/aa = apcosnp ^ -. r-4r - .0 27rm .' o 2?ftsinp 1 i"*"" 7 , tt . sin (2m sin ilr) 1 f 2m , , x = d-vjr cos 2mlr ^ - *-/ = - J zn (x)dx, (46) frm'o smo/r 2?nJ as in (15). Thus the mean value of (43) is = ^ {/,,_, (2m) - J- MH . 1 (2m)}, (47) as before. In order to express fully the mean value of P 2 + Q- + R? at distance r, we have to introduce additional factors from (29). If a = -a 1 -ior 2 , e iar _ e -fa,r ^^ an( j these factors may be taken to be a 4 a 2 e 2a r /r 2 . The occurrence of the factor e 2 "*'', where or 2 is positive, has a strange appearance ; but, as Lamb has shown I, it is to be expected in such cases as the present, where the vibrations to be found at any time at a greater distance corre- spond to an earlier vibration at the nucleus. * Gray and Mathews, p. 28. t Enc. Brit. "Wave Theory of Light," Equation (43), 1888; Scientific Papers, Vol. in. p. 98. Proc. Math. Soc. 1900, Vol. xxxn. p. 208. 120 ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING [365 The calculations just effected afford an independent estimate of the dissipation. The rate at which energy is propagated outwards away from the sphere of great radius r, is dE . . *a&* TT* . T . --{J m . 1 -J 2m+1 }, ............ (48) or, since T (the period) = 2-n-a/mV, the loss of energy in one complete vibration is given by dE.r 8ir*a*aV--,, ~ dt ~~ri* - l'2m-i-./*m+,} ................... (49) With this we have to compare the total energy to be found within the sphere. The occurrence of the factor &** r is a complication from which we may emancipate ourselves by choosing r great in comparison with a, but still small enough to justify the omission of e 2 "^, conditions which are reconcilable when e is sufficiently small. The mean value of P 2 + Q 3 -f R 2 at a small distance p from the circular axis is 2ra 2 /a*p 2 . This is to be multiplied by 2-Tra . 1-n-pdp, and integrated from e to a value of p comparable with a, which need not be further specified. Thus 8m 2 7T 2 d 8mir dE.r_7T* [J^ (2m) - J- m+1 (2m}} "EdT~ -log. in agreement with (23). 366. COLOURED PHOTOMETRY. [Philosophical Magazine, Vol. xxiv. pp. 301, 302, 1912.] IN his recent paper on the Photometry of Lights of Different Colours* Mr H. Ives remarks : " No satisfactory theory of the action of the flicker photometer can be said to exist. What does it actually measure ? We may assume the existence of a ' luminosity sense ' distinct from the colour sense If, for instance, there exists a physiological process called into action both by coloured and uncoloured light, a measure of this would be a measure of a common property." Very many years ago it occurred to me that the adjustment of the iris afforded just such a "physiological process "f. The iris contracts when the eye is exposed to a bright red or to a bright green light. There must therefore be some relative brightness of the two lights which tends equally to close the iris, and this may afford the measure required. The flicker adjustment is complete when the iris has no tendency to alter under the alternating illumination. This question was brought home to me very forcibly, when in 1875 I fitted the whole area of the window of a small room with revolving sectors after the manner of Talbot. The intention was to observe, more conveniently than when the eye is at a small hole, the movements of vibrating bodies. The apparatus served this purpose well enough; but incidentally I was much struck with the remarkably disagreeable and even painful sensations experienced when at the beginning or end of operations the slits were revolving slowly so as to generate flashes at the rate of perhaps 3 or 4 per second. I soon learned in self-defence to keep my eyes closed during this phase ; and I attributed the discomfort to a vain attempt on the part of the iris to adjust itself to fluctuating conditions. * Phil. Mag. Vol. xxiv. p. 178. t If my memory serves me, I have since read somewhere a similar suggestion, perhaps in Helmholtz. 122 COLOURED PHOTOMETRY It is clear, I think, that we have here a common element in variously coloured lights, such as might serve as the basis of coloured photometry. I suppose that there would be no particular difficulty in observing the movements of an iris, and I would suggest that experiments be undertaken to ascertain whether in fact the flicker match coincides with quiescence of the iris. Should this prove to be the case, the view suggested would be amply confirmed ; otherwise, it would be necessary to turn to some of the other possibilities discussed by Mr Ives. [1913. Mr H. C. Stevens (Phil. Mag. Vol. xxvi. p. 180, 1912), in con- nexion with the above suggestion, describes an experiment in which the musculus sphincter pupillae was paralysed with atropine, without changing " in any observable particular " the appearance of flicker. This observation may prove that an actual movement of the iris is not necessary to the sensation of flicker, but it can hardly be said that the iris has no tendency to alter because it is prevented from doing so by the paralysis of the muscle. There must be more than one step between the impression upon the retina which initiates a message to close the iris and the actual closing thereof. The flicker adjustment may, so far as appears, correspond to the absence of such messages.] 367. ON SOME IRIDESCENT FILMS*. [Philosophical Magazine, Vol. xxiv. pp. 751 755, 1912.] THE experiments now to be described originated in an accidental observa- tion. Some old lantern-plates, from which the gelatine films had been cleaned off a few years before (probably with nitric acid), being required for use, were again placed in dilute nitric acid to ensure cleanliness. From these plates a gas-flame burning over the dish was seen reflected with colour, of which the cause was not obvious. On examination in daylight a dry plate was observed to be iridescent, but so slightly that the fact might easily escape attention. But when the plate was under water and suitably illuminated, the brilliancy was remarkably enhanced. Upon this question of illumination almost everything depends. The window-shutter of one of the rooms in my laboratory has an aperture about 4 inches square. In front of this the dish of water is placed and at the bottom of the dish a piece of dark-coloured glass. In the water the plate under observation is tilted, so as to separate the reflexions of the sky as given by the plate and by the glass underneath. In this way a dark background is ensured. At the corners and edges of the plate the reflected light is white, then follow dark bands, and afterwards the colours which suggest reflexion from a thin plate. On this view it is necessary to suppose that the iridescent film is thinnest at the outside and thickens towards the interior, and further, that the material constituting the film has an index intermediate between those of the glass and of the water. In this way the general behaviour is readily explained, the fact that the colours are so feeble in air being attributed to the smallness of the optical difference between the film and the glass underneath. In the water there would be a better approach to equality between the reflexions at the outer and inner surfaces of the film. From the first I formed the opinion that the films were due to the use of a silicate substratum in the original preparation, but as the history of the * Read before the British Association at Dundee. 124 ON SOME IRIDESCENT FILMS [367 plates was unknown this conjecture could not be satisfactorily confirmed. No ordinary cleaning or wiping had any effect ; to remove the films recourse must be had to hydrofluoric acid, or to a polishing operation. My friend Prof. T. W. Richards, after treating one with strong acids and other chemicals, pronounced it to be what chemists would call " very insoluble." The plates first encountered manifested (in the air) a brilliant glassy surface, but afterwards I found others showing in the water nearly or quite as good colours, but in the air presenting a smoky appearance. Desirous of obtaining the colours as perfectly as possible, I endeavoured to destroy the reflexion from the back surface of the plate, which would, I supposed, dilute the colours due to the iridescent film. But a coating of black sealing-wax, or marine glue, did not do so much good as had been expected. The most efficient procedure was to grind the back of the plate, as is very easily done with carborundum. The colours seemed now to be as good as such colours can ever be, the black also being well developed. Doubtless the success was due in great measure to the special localized character of the illumination. The substitution of strong brine for water made no perceptible improvement. At this stage I found a difficulty in understanding fully the behaviour of the unground plates. In some places the black would occasionally be good, while in others it had a washed-out appearance, a difference not easily accounted for. A difficulty had already been experienced in deciding upon which side of a plate the film was, and had been attributed to the extreme thinness of the plates. But a suspicion now arose that there were films upon both sides, and this was soon confirmed. The best proof was afforded by grinding away half the area upon one side of the plate and the other half of the area upon the other side. Whichever face was uppermost, the unground half witnessed the presence of a film by brilliant coloration. Attempts to produce silicate films on new glass were for some time an almost complete failure. I used the formula given by Abney (Instruction in Photography, llth edition, p. 342): Albumen 1 part. Water 20 parts. Silicate of Soda solution of syrupy consistency 1 part. But whether the plates (coated upon one side) were allowed to drain and dry in the cold, or were more quickly dried off over a spirit flame or before a fire, the resulting films washed away under the tap with the slightest friction or even with no friction at all. Occasionally, however, more adherent patches were observed, which could not so easily be cleaned off. Although it did not seem probable that the photographic film proper played any part, I tried without success a superposed coat of gelatine. In view of these failures 1912] ON SOME IRIDESCENT FILMS 125 I could only suppose that the formation of a permanent film was the work of time, and some chemical friends were of the same opinion. Accordingly a number of plates were prepared and set aside duly labelled. Examination at intervals proved that time acted but slowly. After six months the films seemed more stable, but nothing was obtained comparable with the old iridescent plates. It is possible that the desired result might eventually be achieved in this way, but the prospect of experimenting under such conditions is not alluring. Luckily an accidental observation came to my aid. In order to prevent the precipitation of lime in the observing-dish a few drops of nitric acid were sometimes added to the water, and I fancied that films tested in this acidified water showed an advantage. A special experiment confirmed the idea. Two plates, coated similarly with silicate and dried a few hours before, were immersed, one in ordinary tap water, the other in the same water moderately acidified with nitric acid. After some 24 hours' soaking the first film washed off easily, but the second had much greater fixity. There was now no difficulty in preparing films capable of showing as good colours as those of the old plates. The best procedure seems to be to dry off the plates before a fire after coating with recently- filtered silicate solution. In order to obtain the most suitable thickness, it is necessary to accommodate the rapidity of drying to the strength of the solution. If heat'is not employed the strength of the above given solution may be doubled. When dry the plates may be immersed for some hours in (much) diluted nitric acid. They are then fit for optical examination, but are best not rubbed at this stage. If the colours are suitable the plates may now be washed and allowed to dry. The full development of the colour effects requires that the back of the plates be treated. In rny experience grinding gives the best results when the lighting is favourable, but an opaque varnish may also be used with good effect. The comparative failure of such a treatment of the old plates was due to the existence of films upon both sides. A sufficiently opaque glass, e.g. stained with cobalt or copper, may also be employed. After the films have stood some time subsequently to the treatment with acid, they may be rubbed vigorously with a cloth even while Wet ; but one or two, which probably had been rubbed prematurely, showed scratches. The surfaces of the new films are not quite as glassy as the best of the old ones, nor so inconspicuous in the air, but there is, I suppose, no doubt that they are all composed of silica. But I am puzzled to understand how the old plates were manipulated. The films cover both sides without interruption, and are thinner at all the four corners than in the interior. The extraordinary development of the colours in water as compared with what can be seen in air led me to examine in the same way other thin films deposited on glass. A thin coat of albumen (without silicate) is inconspicuous 126 ON SOME IRIDESCENT FILMS [367 in air. As in photography it may be rendered insoluble by nitrate of silver acidified with acetic acid, and then exhibits good colours when examined under water with favourable illumination. Filtered gelatine, with which a little bichromate has been mixed beforehand, may also be employed. In this case the dry film should be well exposed to light before washing. Ready- made varnishes also answer well, provided they are capable of withstanding the action of water, at least for a time. I have used amber in chloroform, a " crystal " (benzole) varnish such as is, or was, used by photographers, and bitumen dissolved in benzole. The last is soon disintegrated under water, but the crystal varnish gives very good films. The varnish as sold may probably require dilution in order that the film may be thin enough. Another varnish which gives interesting results is celluloid in pear-oil. All these films show little in air, but display beautiful colours in water .when the reflexion from the back of the glass is got rid of as already described. The advantage from the water depends, of course, upon its mitigating the in- equality of the reflexion from the two sides of the film by diminishing the front reflexion. A similar result may be arrived at by another road if we can increase the back reflexion, with the further advantage of enhanced illumination. For this purpose we may use silvering. A glass is coated with a very thin silver film and then with celluloid varnish of suitable consistency. Magnificent colours are then seen without the aid of water, and the only difficulty is to hit off the right thickness for the silver. Other methods of obtaining similar displays are described in Wood's Physical Optics (Macmillan, 1905, p. 142). 368. BREATH FIGURES* [Nature, Vol. xc. pp. 436, 437, 1912.] AT intervals during the past year I have tried a good many experiments in the hope of throwing further light upon the origin of these figures, especially those due to the passage of a small blow-pipe flame, or of hot sulphuric acid, across the surface of a glass plate on which, before treatment, the breath deposits evenly. The even deposit consists of a multitude of small lenses easily seen with a hand magnifier. In the track of the flame or sulphuric acid the lenses are larger, often passing into flat masses which, on evaporation, show the usual colours of thin plates. When the glass is seen against a dark ground, and is so held that regularly reflected light does not reach the eye, the general surface shows bright, while the track of the flame or acid is by comparison dark or black. It will be convenient thus to speak of the deposit as bright or dark descriptive words implying no doubtful hypothesis. The question is what difference in the glass surface determines the two kinds of deposit. In Aitken's view (Proc. Ed. Soc. p. 94, 1893; Nature, June 15, 1911), the flame acts by the deposit of numerous fine particles constituting nuclei of aqueous condensation, and in like manner he attributes the effect of sulphuric (or hydrofluoric) acid to a water-attracting residue remaining in spite of washing. On the other hand, I was disposed to refer the dark deposit to a greater degree of freedom from grease or other water-repelling contamination (Nature, May 25, 1911), supposing that a clean surface of glass would everywhere attract moisture. It will be seen that the two views are sharply contrasted. My first experiments were directed to improving the washing after hot sulphuric or hydrofluoric acid. It soon appeared that rinsing and soaking prolonged over twenty-four hours failed to abolish the dark track ; but probably Mr Aitken would not regard this as at all conclusive. It was more to the point that dilute sulphuric acid (1/10) left no track, even after perfunctory washing. Rather to my surprise, I found that even strong * See p. 26 of this volume. 128 BREATH FIGURES [368 sulphuric acid fails if employed cold. A few drops were poured upon a glass (^-plate photographic from which the film had been removed), and caused to form an elongated pool, say, half an inch wide. After standing level for about five minutes longer than the time required for the treatment with hot acid the plate was rapidly washed under the tap, soaked for a few minutes, and finally rinsed with distilled water, and dried over a spirit lamp. Examined when cold by breathing, the plate showed, indeed, the form of the pool, but mainly by the darkness of the edge. The interior was, perhaps, not quite indistinguishable from the ground on which the acid had not acted, but there was no approach to darkness. This experiment may, I suppose, be taken to prove that the action of the hot acid is not attributable to a residue remaining after the washing. I have not found any other treatment which will produce a dark track without the aid of heat. Chromic acid, aqua regia, and strong potash are alike ineffective. These reagents do undoubtedly exercise a cleansing action, so that the result is not entirely in favour of the grease theory as ordinarily understood. My son, Hon. R. J. Strutt, tried for me an experiment in which part of an ordinarily cleaned glass was exposed for three hours to a stream of strongly ozonised oxygen, the remainder being protected. On examination with the breath, the difference between the protected and unprotected parts was scarcely visible. It has been mentioned that the edges of pools of strong cold sulphuric acid and of many other reagents impress themselves, even when there is little or no effect in the interior. To exhibit this action at its best, it is well to employ a minimum of liquid ; otherwise a creeping of the edge during the time of contact may somewhat obscure it. The experiment succeeds about equally well even when distilled water from a wash-bottle is substituted for powerful reagents. On the grease theory the effect maybe attributed to the cleansing action of a pure free surface, but other interpretations probably could be suggested. Very dark deposits, showing under suitable illumination the colours of thin plates, may be obtained on freshly-blown bulbs of soft glass. It is con- venient to fill the interior with water, to which a little ink may be added. From this observation no particular conclusion can be deduced, since the surface, though doubtless very clean, has been exposed to the blow-pipe flame. In my former communication, I mentioned that no satisfactory result was obtained when a glass plate was strongly heated on the back by a long Bunsen burner; but I am now able to bring forward a more successful experiment. A test-tube of thin glass, about inch in diameter, was cleaned internally until it gave an even bright deposit. The breath is introduced through 1912] BREATH FIGURES 129 a tube of smaller diameter, previously warmed slightly with the hand. The closed end of the test-tube was then heated in a gas flame urged with a foot blow-pipe until there were signs of incipient softening. After cooling, the breath deposit showed interesting features, best brought out by transmitted light under a magnifier. The greater part of the length showed, as before, the usual fine dew. As the closed end was approached the drops became gradually larger, until at about an inch from the end they disappeared, leaving the glass covered with a nearly uniform film. One advantage of the tube is that evaporation of dew, once formed, is slow, unless promoted by suction through the mouth-tube. As the film evaporated, the colours of thin plates were seen by reflected light. Since it is certain that the flame had no access to the internal surface, it seems proved that dark deposits can be obtained on surfaces treated by heat alone. In some respects a tube of thin glass, open at both ends, is more con- venient than the test-tube. It is easier to clean, and no auxiliary tube is required to introduce or abstract moisture. I have used one of 3/10 in. diameter. Heated locally over a simple spirit flame to a point short of softening, it exhibited similar effects. This easy experiment may be recom- mended to anyone interested in the subject. One of the things that I have always felt as a difficulty is the comparative permanence of the dark tracts. On flat plates they may survive in some degree rubbing by the finger, with subsequent rinsing and wiping. Practi- cally the easiest way to bring a plate back to its original condition is to rub it with soapy water. But even this does not fully succeed with the test-tube, probably on account of the less effective rubbing and wiping near the closed end. But what exactly is involved in rubbing and wiping ? I ventured to suggest before that possibly grease may penetrate the glass somewhat. From such a situation it might not easily be removed, or, on the other hand, introduced. There is another form of experiment from which I had hoped to reap decisive results. The interior of a mass of glass cannot be supposed to be greasy, so that a surface freshly obtained by fracture should be clean, and give the dark deposit. One difficulty is that the character of the deposit on the irregular surface is not so easily judged. My first trial on a piece of plate glass f in. thick, broken into two pieces with a hammer, gave anomalous results. On part of each new surface the breath was deposited in thin laminae capable of showing colours, but on another part the water masses were decidedly smaller, and the deposit could scarcely be classified as black. The black and less black parts of the two surfaces were those which had been contiguous before fracture. That there should be a well-marked difference in this respect between parts both inside a rather small piece of glass is very surprising. I have not again met with this anomaly; but K. VI. 9 130 BREATH FIGURES [368 further trials on thick glass have revealed deposits which may be considered dark, though I was not always satisfied that they were so dark as those obtained on flat surfaces with the blow-pipe or hot sulphuric acid. Similar experiments with similar results may be made upon the edges of ordinary glass plates (such as are used in photography), cut with a diamond. The breath deposit is best held pretty close to a candle-flame, and is examined with a magnifier. In conclusion, I may refer to two other related matters in which my experience differs from that of Mr Aitken. He mentions that with an alcohol flame he " could only succeed in getting very slight indications of any action." I do not at all understand this, as I have nearly always used an alcohol flame (with a mouth blow-pipe) and got black deposits. Thinking that perhaps the alcohol which I generally use was contaminated, I replaced it by pure alcohol, but without any perceptible difference in the results. Again, I had instanced the visibility of a gas flame through a dewed plate as proving that part of the surface was uncovered. I have improved the experiment by using a curved tube through which to blow upon a glass plate already in position between the flame and the eye. I have not been able to find that the flame becomes invisible (with a well-defined outline) at any stage of the deposition of dew. Mr Aitken mentions results pointing in the opposite direction. Doubtless, the highly localized light of the flame is favourable. [1913. Mr Aitken returned to the subject in a further communication to Nature, Vol. xc. p. 619, 1912, to which the reader should refer.] 369. REMARKS CONCERNING FOURIER'S THEOREM AS APPLIED TO PHYSICAL PROBLEMS. [Philosophical Magazine, Vol. xxiv. pp. 864869, 1912.] FOURIER'S theorem is of great importance in mathematical physics, but difficulties sometimes arise in practical applications which seem to have their origin in the aim at too great a precision. For example, in a series of observations extending over time we may be interested in what occurs during seconds or years, but we are not concerned with and have no materials for a remote antiquity or a distant future ; and yet these remote times deter- mine whether or not a period precisely denned shall be present. On the other hand, there may be no clearly marked limits of time indicated by the circumstances of the case, such as would suggest the other form of Fourier's theorem where everything is ultimately periodic. Neither of the usual forms of the theorem is exactly suitable. Some method of taking off the edge, as it were, appears to be called for. The considerations which follow, arising out of a physical problem, have cleared up my own ideas, and they may perhaps be useful to other physicists. A train of waves of length X, represented by ^ = gZwfcH-WA (1) advances with velocity c in the negative direction. If the medium is absolutely uniform, it is propagated without disturbance ; but if the medium is subject to small variations, a reflexion in general ensues as the waves pass any place x. Such reflexion reacts upon the original waves; but if we suppose the variations of the medium to be extremely small, we may neglect the reaction and calculate the aggregate reflexion as if the primary waves were undisturbed. The partial reflexion which takes place at x is repre- sented by } dx . e^ 1 *, (2) 132 REMARKS CONCERNING FOURIER'S THEOREM AS [369 in which the first factor expresses total reflexion supposed to originate at x=Q,<f>(x)dx expresses the actual reflecting power at x, and the last factor gives the alteration of phase incurred in traversing the distance 2#. The aggregate reflexion follows on integration with respect to x; with omission of the first factor it may be taken to be C + iS (3) f+oc i* + where C=\ <b(v)cosuvdv, S=l <j>(v)smuvdv, (4) J _ J -<*> with M=47r/X. When <j> is given, the reflexion is thus determined by (3). It is, of course, a function of \ or u. In the converse problem we regard (3) the reflexion as given for all values of u and we seek thence to determine the form of <f> as a function of x. By Fourier's theorem we have at once = ![ w J o (5) It will be seen that we require to know C and S separately. A knowledge of the intensity merely, viz. G 2 + S*, does not suffice. Although the general theory, above sketched, is simple enough, questions arise as soon as we try to introduce the approximations necessary in practice. For example, in the optical application we could find by observation the values of C and S for a finite range only of u, limited indeed in eye obser- vations to less than an octave. If we limit the integration in (5) to corre- spond with actual knowledge of C and S, the integral may not go far towards determining <f>. It may happen, however, that we have some independent knowledge of the form of <. For example, we may know that the medium is composed of strata each uniform in itself, so that within each <f) vanishes. Further, we may know that there are only two kinds of strata, occurring alternately. The value of $<f>dx at each transition is then numerically the same but affected with signs alternately opposite. This is the case of chlorate of potash crystals in which occur repeated twinnings*. Information of this kind may supplement the deficiency of (5) taken by itself. If it be for high values only of u that C and S are not known, the curve for < first obtained may be subjected to any alteration which leaves f<j>dx, taken over any small range, undisturbed, a consideration which assists materially where is known to be discontinuous. If observation indicates a large C or S for any particular value of u, we infer of course from (5) a correspondingly important periodic term in <. If the large value of C or S is limited to a very small range of u, the periodicity of < extends to a large range of x ; otherwise the interference of Phil. Mag. Vol. MVI. p. 256 (1888) ; Scientific Papers, Vol. in. p. 204. 1912] APPLIED TO PHYSICAL PROBLEMS 133 components with somewhat different values of ?/ may limit the periodicity to a comparatively small range. Conversely, a prolonged periodicity is associated with an approach to discontinuity in the values of C or 8. The complete curve representing < (x) will in general include features of various lengths reckoned along x, and a feature of any particular length is associated with values of u grouped round a corresponding centre. For some purposes we may wish to smooth the curve by eliminating small features. One way of effecting this is to substitute everywhere for <f> (#) the mean of the values of <f> (x) in the neighbourhood of x, viz. the range (2a) of integration being chosen suitably. With use of (5) we find for (6) . ....... (7) differing from the right-hand member of (5) merely by the introduction of the factor sin ua 4- ua. The effect of this factor under the integral sign is to diminish the importance of values of u which exceed -rr/a and gradually to annul the influence of still larger values. If we are content to speak very roughly, we may say that the process of averaging on the left is equivalent to the omission in Fourier's integral of the values of u which exceed 7r/2a. We may imagine the process of averaging to be repeated once or more times upon (6). At each step a new factor sin ua -=- ua is introduced under the integral sign. After a number of such operations the integral becomes practically independent of all values of u for which ua is not small. In (6) the average is taken in the simplest way with respect to x, so that every part of the range 2a contributes equally (fig. 1). Other and perhaps Fig. 1. Fig. 2. Fig. 3. better methods of smoothing may be proposed in which a preponderance is given to the central parts. For example we may take (fig. 2) a 2 Jo (a-- From (5) we find that (8) is equivalent to _f du ~ C ^ Ua {Ccosux+ Ssinux], (9) 134 REMARKS CONCERNIXQ FOURIER'S THEOREM AS [369 reducing to (5) again when a is made infinitely small. In comparison with (7) the higher values of ua are eliminated more rapidly. Other kinds of averaging over a finite range may be proposed. On the same lines as above the formula next in order is (fig. 3) r. ...(10) In the above processes for smoothing the curve representing < (x), ordinates which lie at distances exceeding a from the point under consideration are without influence. This mayor may not be an advantage. A formula in which the integration extends to infinity is -V- l + <(*+) e-? !at d% = - f due-" 4 [C cos ux + S sin ux} (11) a v^r J -x TTJQ In this case the values of ua which exceed 2 make contributions to the integral whose importance very rapidly diminishes. The intention of the operation of smoothing is to remove from the curve features whose length is small. For some purposes we may desire on the contrary to eliminate features of great length, as for example in considering the record of an instrument whose zero is liable to slow variation from some extraneous cause. In this case (to take the simplest formula) we may sub- tract 'from < (x) the uncorrected record the average over a length b relatively large, so obtaining Here, if ub is much less than TT, the corresponding part of the range of integration is approximately cancelled and features of great length are eliminated. There are cases where this operation and that of smoothing may be com- bined advantageously. Thus if we take (13, we eliminate at the same time the features whose length is small compared with a and those whose length is large compared with b. The same method may be applied to the other formulse (9), (10), (11). A related question is one proposed by Stokes*, to which it would be interesting to have had Stokes' own answer. What is in common and what * Smith's Prize Examination, Feb. 1, 1882 ; Math, and Phyt. Papers, Vol. v. p. 367. 1912] APPLIED TO PHYSICAL PROBLEMS 135 is the difference between C and S in the two cases (i) where </> (./) fluctuates between - oo and + oo and (ii) where the fluctuations are nearly the same as in (i) between finite limits + a but outside those limits tends to zero ? When x is numerically great, cos ux and sin ux fluctuate rapidly with u ; and inspection of (5) shows that < (x) is then small, unless C or & are themselves rapidly variable as functions of u. Case (i) therefore involves an approach to discontinuity in the forms of G or S. If we eliminate these discontinuities, or rapid variations, by a smoothing process, we shall annul < (x) at great distances and at the same time retain the former values near the origin. The smoothing may be effected (as before) by taking l ru+a 1 ru+a ^J Cdu, g-j Sdu in place of C and S simply. C then becomes r +0 , . , , sin aw dvd> (v) cos uv , J -oo av <j> (v) being replaced by </> (v) sin av H- av. The effect of the added factor disappears when av is small, but when av is large, it tends to annul the corresponding part of the integral. The new form for <f> (x) is thus the same as the old one near the origin but tends to vanish at great distances on either side. Case (ii) is thus deducible from case (i) by the application of a smoothing process to C and 8, whereby fluctuations of small length are removed. We may sum up by saying that a smoothing of < (x} annuls C and S for large values of u, while a smoothing of C and 8 (as functions of u) annuls < (x) for values of x which are numerically great. 370. SUR LA RESISTANCE DES SPHERES DANS L'AIR EN MOUVEMENT. [Comptes Rendus, t. CLVI. p. 109, 1913.] DANS les Comptes rendus du 30 decembre 1912, M. Eiffel donne des re'sultats tres inteVessants pour la resistance rencontree, a vitesse variable, par trois spheres de 16'2, 244 et 33 cm. de diametre. Dans la premiere figure, ces resultats sont exprimes par les valeurs d'un coefficient K, e"gal a K/SF 1 , ou R est la resistance totale, S la surface diametrale et V la vitesse. En chaque cas, il y a une vitesse critique, et M. Eiffel fait remarquer que la loi de similitude n'est pas toujours vraie; en effet, les trois spheres donnent des vitesses critiques tout a fait diffe'rentes. D'apres la loi de similitude dynamique, pr&jise'e par Stokes* et Reynolds pour les liquides visqueux, K est une fonction d'une seule variable v/VL, ou v est la viscosit^ cine'matique, constante pour un liquide donne', et L est la dimension linaire, proportionnelle a S^. Ainsi les vitesses critiques ne doivent pas e"tre les memes dans les trois cas, mais inversement proportionnelles a L. En verite, si nous changeons 1'echelle des vitesses suivant cette loi, nous trouvons les courbes de M. Eiffel presque identiques, au moins que ces vitesses ne sont pas tres petites. Je ne sais si les hearts re'siduels sont reels ou non. La theorie simple admet que les spheres sont polies, sinon que les ine'galite's sont proportionnelles aux diametres, que la compressibility de 1'air est negligeable et que la viscosite cin^matique est absolument constante. Si les resultats de I'exp&ience ne sont pas completement d'accord avec la theorie, on devra examiner ces hypotheses de plus pres. J'ai traite d'autre part et plus en detail de la question dont il s'agit icif. * [Camb. Trant. 1860 ; Math, and Phyg. Papers, Vol. in. p. 17.] t Voir Scientific Paperi, t. v. 1910, pp. 532534. 371. THE EFFECT OF JUNCTIONS ON THE PROPAGATION OF ELECTRIC WAVES ALONG CONDUCTORS. [Proceedings of the Royal Society, A, Vol. LXXXVIII. pp. 103110, 1913.] SOME interesting problems in electric wave propagation are suggested by an experiment of Hertz*. In its original form waves of the simplest kind travel in the positive direction (fig. 1), outside an infinitely thin conducting cylindrical shell, A A, which comes to an end, say, at the plane z = 0. Co-axial with the cylinder a rod or wire BB (of less diameter) extends to infinity in both directions. The conductors being supposed perfect, it is required to determine the waves propagated onwards beyond the cylinder on the positive side of z, as well as those reflected back outside the cylinder and in the annular space between the cylinder and the rod. Fig. 1. So stated, the problem, even if mathematically definite, is probably intractable ; but if we modify it by* introducing an external co-axial con- ducting sheath CC (fig. 2), extending to infinity in both directions, and if we further suppose that the diameter of this sheath is small in comparison with the wave-length (\) of the vibrations, we shall bring it within the scope of approximate methods. It is under this limitation that I propose here to * "Ueber die Fortleitung electrischer Wellen durch Drahte," Wied. Ann. 1889, Vol. p. 395. 138 THE EFFECT OF JUNCTIONS ON THE [371 consider the present and a few analogous problems. Some considerations of a more general character are prefixed. If P, Q, R be components of electromotive intensity, a, b, c those of magnetisation, Maxwell's general circuital relations* for the dielectric give rfa dQ dR and two similar equations, and dP dc db also with two similar equations, V being the velocity of propagation. From (1) and (2) we may derive da db dc dP dQ dR -=-- + -= -- h -T- = 0, ~1 -- P*j + ~J- = " 5 ............... V"/ dx dy dz dx dy dz and, further, that - V ^' ( P > & R > a > b > c ) = ' where V 2 = d*/dx 2 + d n -/df + d*fdz* ........................ (5) At any point upon the surface of a conductor, regarded as perfect, the condition to be satisfied is that the vector (P, Q, R) be there normal. In what follows we shall have to deal only with simple vibrations in which all the quantities are proportional to e ipt , so that djdt may be replaced by ip. It may be convenient to commence with some cases where the waves are in two dimensions (x, z) only^ supposing that , c, Q vanish, while 6, P, R are independent of y. From (1) and (2) we have At the surface of a conductor P, Q are proportional to the direction cosines of the normal (n) ; so that the surface condition may be expressed simply by I- ...................................... < which > with suffices to determine 6. In (7) k = p/V. It will be seen that equations (6), (7) are identical with those which apply in two dimensions to aerial vibrations executed in spaces bounded by fixed walls, 6 then denoting velocity-potential. When 6 is known, the remaining functions follow at once. * Phil. Tram. 1868 ; Maxwell's Scientific Papers, Vol. n. p. 128. 1913] PROPAGATION OF ELECTRIC WAVES ALONG CONDUCTORS 139 It may be remarked by the way that the above analogy throws light upon the question under what circumstances electric waves are guided by con- ductors. Some high authorities, it would seem, regard such guidance as ensuing in all cases as a consequence of the boundary condition fixing the direction of the electric force. But in Acoustics, though a similar condition holds good, there is no guidance of aerial waves round convex surfaces, and it follows that there is none in the two-dimensional electric vibrations under consideration. Near the concave surface of walls there is in both cases a whispering gallery effect*. The peculiar guidance of electric waves by wires depends upon the conductor being encircled by the magnetic force. No such circulation, for example, could ensue from the incidence of plane waves upon a wire which lies entirely in the plane containing the direction of propagation and that of the magnetic force. Our first special application is to the extreme form of Hertz's problem (as modified) which occurs when all the radii of the cylindrical surfaces concerned become infinite, while the differences CA, AB remain finite and indeed small in comparison with X. In fig. 2, A, B, C then represent Fig. 2. planes perpendicular to the plane of the paper and the problem is in two dimensions. The two halves, corresponding to plus and minus values of x, are isolated, and we need only consider one of them. Availing ourselves of the acoustical analogy, we may at once transfer the solution given (after Poisson) in Theory of Sound, 264. If the incident wave in CA be repre- sented by f CA and that therein reflected by F, while the waves propagated along CB, AB be denoted by /<,/, we have 2CA ,, CA f , CA^TH^J CA W and .(9) Phil. Mag. 1910, Vol. xx. p. 1001 ; Scientific Papers, Vol. v. p. 617. 140 THE EFFECT OF JUNCTIONS ON THE [371 The wave in AB is to be regarded as propagated onwards round the corner at A rather than as reflected. As was to be anticipated, the reflected wave f is smaller, the smaller is AB. It will be understood that the validity of these results depends upon the assumption that the region round A through which the waves are irregular has dimensions which are negligible in comparison with X. An even simpler example is sketched in fig. 3, where for the present the f ~ ~ I 5 JA > r Fig. 3. various lines represent planes or cylindrical surfaces perpendicular to the paper. One bounding plane C is unbroken. The other boundary consists mainly of two planes with a transition at AB, which, however, may be of any form so long as it is effected within a distance much less than X. With a notation similar to that used before, f CA may denote the incident positive wave and F the reflected wave, while that propagated onwards in CB is f CB . We obtain in like manner When AB vanishes we have, of course, f' CB =f' CA , F'=0. A little later we shall consider the problem of fig. 3 when the various surfaces are of revolution round the axis of z. Leaving the two-dimensional examples, we find that the same general method is applicable, always under the condition that the region occupied by irregular waves has dimensions which are small in comparison with X. Within this region a simplified form of the general equations avails, and thus the difficulty is turned. An increase in X means a decrease in p. When this goes far enough, it justifies the omission of dfdt in equations (1), (2), (3), (4). Thus P, Q, R become the derivatives of a simple potential function <, which itself satisfies 1913] PROPAGATION OF ELECTRIC WAVES ALONG CONDUCTORS 141 V 2 </> = ; that is, the electric forces obey the laws of electrostatics. Similarly a, b, c are derivatives of another function i/r satisfying the same equation. The only difference is that -fy may be multivalued. The magnetism is that due to steady electric currents. If several wires meet in a point, the total current is zero. This expresses itself in terms of a, 6, c as a relation between the " circulations." The method then consists in forming the solutions which apply to the parts at a distance on the two sides from the region of irregularity, and in accommodating them to one another by the conditions which hold good at the margins of this region in virtue of the fact that it is small. In the application to the problem of fig. 3 we will suppose that the conductors are of revolution round z, though this limitation is not really imposed by the method itself. The problem of the regular waves (whatever may be form of section) was considered in a former paper*. All the dependent variables expressing the electric conditions being proportional to > d 2j dt 2 in ( 4 ) compensates V z d z jdz z , so that also jR and c vanish. In the present case we have for the negative side, where there is both a direct and a reflected wave, P, Q, R = ^(H^ + K^ (^ , ^ , G) logr, ......... (13) where r is the distance of any point from the axis of symmetry, and H l , K l are arbitrary constants. Corresponding to (13), gr ...... (14) In the region of regular waves on the positive side there is supposed to be no wave propagated in the negative direction. Here accordingly P, Q,R = HJ<-*> (^, , O)logr, .............. (15) V(a, b, c) = H z e i ^- k -~ ,,ologr, ........... (16) H 2 being another constant. We have now to determine the relations between the constants H lt K l} H 2) hitherto arbitrary, in terms of the remaining data. For this purpose consider cross-sections on the two sides both near the origin and yet within the regions of regular waves. The electric force as expressed in (13), (15) is purely radial. On the positive side its integral * Phil. Mag. 1897, Vol. XLIV. p. 199; Scientific Papers, Vol. iv. p. 327. 142 THE EFFECT OF JUNCTIONS ON THE [371 between i\ the radius of the inner and r' that of the outer conductor is, with omission of e* 1 *, #,- log (r7r s ), z having the value proper to the section. On the negative side the corre- sponding integral is r, being the radius of the inner conductor at that place. But when we consider the intermediate region, where electrostatical laws prevail, we recognize that these two integrals must be equal ; and further that the exponentials may be identified with unity. Accordingly, the first relation is -fl.logCrVrO .................... (17) In like manner the magnetic force in (14), (16) is purely circumferential. And the circulations at the two sections are as H i K l and H 3 . But since these circulations, representing electric currents which may be treated as steady, are equal, we have as the second relation (18) The two relations (17), (18) determine the wave propagated onwards H and that reflected K l in terms of the incident wave HI. If = r,, we have of course, H z = J5T,, K l = 0. If we suppose i\, r 2 , r' all great and nearly equal and expand the logarithms, we fall back on the solution for the two-dimensional case already given. In the above the radius of the outer sheath is supposed uniform through- out. If in the neighbourhood of the origin the radius of the sheath changes from r,' to r a ', while (as before) that of the inner conductor changes from r^ to r z , we have instead of (17), r 1 ) = J ff 2 lo g (r 2 7r 2 ), ................. (19) while (18) remains undisturbed. In (19) the logarithmic functions are proportional to the reciprocals of the electric capacities of the system on the two sides, reckoned in each case per unit of length. From the general theory given in the paper referred to we may infer that this substitution suffices to liberate us from the restriction to symmetry round the axis hitherto imposed. The more general functions which then replace logr on the two sides must be chosen with such coefficients as will make the circulations of magnetic force equal. The generalization here indicated applies equally in the other problems of this paper. 1913] PROPAGATION OF ELECTRIC WAVES ALONG CONDUCTORS 143 In Hertz's problem, fig. 2, the method is similar. In the region of regular waves on the left in CA we may retain (13), (14), and for the regular waves on the right in CB we retain (15), (16). But now in addition for the regular waves on the left in AB, we have (20) I'-at' ) 10 *' (2D Three conditions are now required to determine K l} H 3 , K 3 in terms of Hi. We shall denote the radii taken in order, viz. %BB, \AA, %CC, by ^n ?*2> r -3 respectively. As in (17), the electric forces give (Hi + Ki) log - + K 3 log = H z log (22) r 2 r-L T! The magnetic forces yield two equations, which may be regarded as expressing that the currents are the same on the two sides along BB, and that, since the section is at a negligible distance from the insulated end, there is no current in A A. Thus TT T7- TT TT ^23^ From (22) and (23) gQogr.-logn (24) ^""logr.-logrj' g.--g.-!S r '-!S r ' (25) log r 3 log TI If r 2 exceeds r t but little, K^ tends to vanish, while H 2 and K 3 approach unity. Again, if the radii are all great, (24), (25) reduce to jr v "-I _ ^_2 M TT _ _ rr _ ^_3 [2 (26} as already found in (8), (9). The same method applies with but little variation to the more general problem where waves between one wire and sheath (r l5 r/) divide so as to pass along several wires and sheaths (r 2 , r 2 ), (r 3 , r 3 ), etc., always under the condition that the whole region of irregularity is negligible in comparison with the wave-length*. The various wires and sheaths are, of course, supposed to be continuous. With a similar notation the direct and reflected waves along the first wire are denoted by H^, K lt and those propagated * This condition will usually suffice. But extreme cases may be proposed where, in spite of the smallness of the intermediate region, its shape is such as to entail natural resonances of frequency agreeing with that of the principal waves. The method would then fail. 144 PROPAGATION OF ELECTRIC WAVES ALONG CONDUCTORS [371 onwards along the second, third, and other wires by H t , H 3 , etc. The equations are = #>g = tf s log^= ................. (27) ~ It is hardly necessary to detail obvious particular cases. The success of the method used in these problems depends upon the assumption of a great wave-length. This, of course, constitutes a limitation ; but it has the advantage of eliminating the irregular motion at the junctions. In the two-dimensional examples it might be possible to pursue the approxi- mation by determining the character of the irregular waves, at least to a certain extent, somewhat as in the question of the correction for the open end of an organ pipe. 372. THE CORRECTION TO THE LENGTH OF TERMINATED RODS IN ELECTRICAL PROBLEMS. [Philosophical Magazine, Vol. xxv. pp. 1 9, 1913.] IN a short paper " On the Electrical Vibrations associated with thin terminated Conducting Rods"* I endeavoured to show that the difference between the half wave-length of the gravest vibration and the length (I) of the rod (of uniform section) tends to vanish relatively when the section is reduced without limit, in opposition to the theory of Macdonald which makes X = 2'53 I. Understanding that the argument there put forward is not con- sidered conclusive, I have tried to treat the question more rigorously, but the difficulties in the way are rather formidable. And this is not surprising in view of the discontinuities presented at the edges where the flat ends meet the cylindrical surface. The problem assumes a shape simpler in some respects if we suppose that the rod of length I and radius a surrounded by a cylindrical coaxial con- ducting case of radius b extending to infinity in both directions. One advantage is that the vibrations are now permanently maintained, for no waves can escape to infinity along the tunnel, seeing that / is supposed great compared with 6-f. The greatness of I secures also the independence of the two ends, so that the whole correction to the length, whatever it is, may be regarded as simply the double of that due to the end of a rod infinitely long. At an interior node of an infinitely long rod the electric forces, giving rise (we may suppose) to potential energy, are a maximum, while the magnetic forces representing kinetic energy are evanescent. The end of a terminated rod corresponds, approximately at any rate, to a node. The complications * Phil. Ma( t . Vol. viii. p. 105 (1904) ; Scientific Papers, Vol. v. p. 198. t Phil. Mag. Vol. XLIII. p. 125 (1897) ; Scientific Papers, Vol. iv. p. 276. The conductors are supposed to be perfect. R. VI. 10 146 THE CORRECTION TO THE LENGTH OF [372 due to the end thus tell mainly upon the electric forces*, and the problem is reduced to the electrostatical one of finding the capacity of the terminated rod as enclosed in the infinite cylindrical case at potential zero. But this simplified form of the problem still presents difficulties. Taking cylindrical coordinates z, r, we identify the axis of symmetry with that of *, supposing also that the origin of z coincides with the flat end of the interior conducting rod which extends from oo to 0. The enclosing case on the other hand extends from - oo to + oo . At a distance from the end on the negative side the potential V, which is supposed to be unity on the rod and zero on the case, has the form logft/r and the capacity per unit length is l/(2 logft/a). On the plane z = the value of V from r = to r = a is unity. If we knew also the value of V from r = a to r b, we could treat separately the problems arising on the positive and negative sides. On the positive side we could express the solution by means of the functions appropriate to the complete cylinder r< b, and on the negative side by those appropriate to the annual cylindrical space b > r > a. If we assume an arbitrary value for V over the part in question of the plane z = 0, the criterion of its suitability may be taken to be the equality of the resulting values of dV/dz on the two sides. We may begin by supposing that (1) holds good on the negative side throughout ; and we have then to form for the positive side a function which shall agree with this at z = 0. The general expression for a function which shall vanish when r = b and when z =* + <x> , and also satisfy Laplace's equation, is ..... .................. (2) where k lt k z , &c. are the roots of J (kb) = 0; and this is to be identified when z = with (1) from a to b and with unity from to a. The coefficients A are to be found in the usual manner by multiplication with J (k n r) and integration over the area of the circle r = b. To this end we require (3) (4) flog r J (Ar) r dr = - i {6 log bJ.' (kb) - a log a/.' (ka)} - ^ J 9 (ka). ... (5) * Compare the analogous acoustical questions in Theory of Sound, 265, 317.. 1913] TERMINATED RODS IN ELECTRICAL PROBLEMS 147 Thus altogether AgL = A /V..(*r)r*- WAJ.-W ............. (6) For Jo' 2 we may write Jj 2 ; so that if in (2) we take _ 2/Q (fca) ' we shall have a function which satisfies the necessary conditions, and at z = assumes the value 1 from to a and that expressed in (1) from a to 6. But the values of dV/dz are not the same on the two sides. If we call the value, so determined on the positive as well as upon the negative side, F , we may denote the true value of V by V + V. The con- ditions for V will then be the satisfaction of Laplace's equation throughout the dielectric (except at z = 0), that on the negative side it make V = both when r = a and when r = b, and vanish at z = GO , and on the positive side y = when r = b and when z = + oo , and that when z = V assume the same value on the two sides between a and 6 and on the positive side the value zero from to a. A further condition for the exact solution is that dV/dz, or dVo/dz + dV/dz, shall be the same on the two sides from r = a to r = b when z = 0. Now whatever may be in other respects the character of V on the negative side, it can be expressed by the series V' = H l <f>(h 1 r)e h i z + H 2 <j>(h,,r)e h ' z + ..., ............... (8) where $ (f^r), &c. are the normal functions appropriate to the symmetrical vibrations of an annular membrane of radii a and 6, so that <f> (hr) vanishes for r = a, r b. In the usual notation we may write J (hr) Y (hr) with the further condition Y (ha)J (hb)-J () (ha)Y (hb) = Q, (10) determining the values of h. The function $ satisfies the same differential equation as do J and F . Considering for the present only one term of the series (8), we have to find for the positive side a function which shall satisfy the other necessary conditions and when z = make V = from to o, and V = H<f> (hr) from a to b. As before, such a function may be expressed by and the only remaining question is to find the coefficients B. For this purpose we require to evaluate '<f>(hr)J (kr)rdr. b 102 148 THE CORRECTION TO THE LENGTH OF [372 From the differential equation satisfied by J and < we get and so that (fc* A s ) I J (kr) <j> (hr) r dr = r J*-?. r -r- 2 / o L = -haJ (ka)^'(ha), (12) since here <f>(ha) = <f>(hb) = 0, and also J (kb)=Q. Thus in (11), corre- sponding to a single term of (8), D _2Aa#J (A;aH'(Aa) (13) The exact solution demands the inclusion in (8) of all the admissible values of h, with addition of (1) which in fact corresponds to a zero value of h. And each value of h contributes a part to each of the infinite series of coefficients B, needed to express the solution on the positive side. But although an exact solution would involve the whole series of values of h, approximate methods may be founded upon the use of a limited number of them. I have used this principle in calculations relating to the potential from 1870 onwards*. A potential V, given over a closed surface, makes reckoned over the whole included volume, a minimum. If an expression for V, involving a finite or infinite number of coefficients, is proposed which satisfies the surface condition and is such that it necessarily includes the true form of V, we may approximate to the value of (14), making it a minimum by variation of the coefficients, even though only a limited number be included. Every fresh coefficient that is included renders the approximation closer, and as near an approach as we please to the truth may be arrived at by continuing the process. The true value of (14) is equal by Green's theorem to the integration being over the surface, so that at all stages of the approxi- mation the calculated value of (14) exceeds the true value of (15). In the application to a condenser, whose armatures are at potentials and 1, Phil. Tram. Vol. cuu. p. 77 (1870) ; Scientific Papert, Vol. i. p. 33. Phil. Mag. Vol. xuv. p. 328 (1872); Scientific Papers, Vol. i. p. 140. Compare also Phil. Mag. Vol. XLVII. p. 568 (1899), Vol. xxn. p. 225 (1911). 1913] TERMINATED RODS IN ELECTRICAL PROBLEMS 149 (15) represents the capacity. A calculation of capacity founded upon an approximate value of V in (14) is thus always an overestimate. In the present case we may substitute (15) for (14), if we consider the positive and negative sides separately, since it is only at z = that Laplace's equation fails to receive satisfaction. The complete expression for V on the right is given by combination of (2) and (11), and the surface of integration is composed of the cylindrical wall r = b from z = to z = oo , and of the plane z = from r = to r = b*. The cylindrical wall contributes nothing, since F vanishes along it. At z F= 2 (A + B) J (kr\ - d V/dz = 2k (A + B) J (kr) ; and (15) = J6 2 2fc (A + BY Jf (kb) ................... (16) On the left the complete value of Fincludes (1) and (8). There are here two cylindrical surfaces, but r = b contributes nothing for the same reason as before. On r = a we have F = 1 and - ^r = - TTT- dr a log b/a so that this part of the surface, extending to a great distance z = I, contri- butes to (15) There remains to be considered the annular area at z = 0. Over this (19) The integrals required are b a<j>'(ha)\, .................. (20) r b \ogr<f>(hr)rdr=-h- l {b\ogb(j)'(hb)-a\oga(f>'(ha')}, ...(21) ft ! b {<t>(hr)Yrdr = 1tb*{<l>'(hb)}*-}ta*{<j>'(ha)}*; ............... (22) d and we get for this part of the surface (23) Thus for the whole surface on the left (15) = 2To 1 ^ + 2h& [b^(hb) - a^ (ha)], ......... (24) * The surface at z= + o> may evidently be disregarded. 150 THE CORRECTION TO THE LENGTH OF [372 the simplification arising from the fact that (1) is practically a member of the series <. The calculated capacity, an overestimate unless all the coefficients H are correctly assigned, is given by addition of (16) and (24-). The first approxi- mation is obtained by omitting all the quantities H, so that the B's vanish also. The additional capacity, derived entirely from (16), is then ^b' t ^ l kA t J l -(kb), or on introduction of the value of A, (25) log 2 6/a the summation extending to all the roots of J (kb) = 0. Or if we express the result in terms of the correction 81 to the length (for one end), we have - 26 - J f<*L, ...(26) as the first approximation to 81 and an overestimate. The series in (26) converges sufficiently. Jo 2 (ka) is less than unity. The wth root of J (x) = is x = (m ^)TT approximately, and J 1 t (x) = 2/'jrx, so that when m is great *-^v < 27 > The values of the reciprocals of a^J^(x) for the earlier roots can be calculated from the tables* and for the higher roots from (27). I find ffl X * (x) - --*(*) 1 . .. 2-4048 51915 2668 2 5-5201 34027 0513 3 4 8-6537 11*7915 27145 23245 0209 0113 5 14-9309 20655 0070 The next five values are '0048, '0035, '0026, '0021, '0017. Thus for any value of a the series in (26) is 2668 Jo' (2-405 a/6) + '0513 J * (5'520 a/6) + . . . ; ...... (28) it can be calculated without difficulty when a/6 is given. When a/6 is very small, the J's in (28) may be omitted, and we have simply to sum the numbers in the fourth column of the table and its continuation. The first ten roots give '3720. The remainder I estimate at -015, making in all '387. Thus in this case log 6/tt * Gray and MathewB, BeueVs Function, pp. 244, 247. (29) 1913] TERMINATED RODS IN ELECTRICAL PROBLEMS 151 It is particularly to be noticed that although (29) is an overestimate, it vanishes when a tends to zero. The next step in the approximation is the inclusion of H l corresponding to the first root /^ of </> (lib) = 0. For a given k, B has only one term, expressed by (13) when we write hi, H+ for h, H. In (16) when we expand (A + B) z , we obtain three series of which the first involving J. 2 is that already dealt with. It does not depend upon H*. Constant factors being omitted, the second series depends upon -by ........................ (3 and the third upon the summations including all admissible values of k. In (24) we have under 2 merely the single term corresponding to H l , h^. The sum of (16) and (24) is a quadratic expression in ^T^and is to be made a minimum by variation of that quantity. The application of this process to the case of a very small leads to a rather curious result. It is known (Theory of Sound, 213 a) that kf and h^ are then nearly equal, so that the first terms of (30) and (31) are relatively large, and require a special evaluation. For this purpose we must revert to (10) in which, since ha is small, so that nearly enough and fc-ft- . ......................... ( 33 > a , \ogha \ogka Thus, when a is small enough, the first terms of (30) and (31) dominate the others, and we may take simply Also <t>'(k ia ) = - r --, k.alogk.a Using these, we find from (16) and (24) _l_v __ L_ log 2 b/a W/j 2 (kb) + k*b log b/a . Y (k, 01 , . 2 log bja 4 log 2 152 THE CORRECTION TO THE LENGTH OF TERMINATED RODS, ETC. [372 as the expression for the capacity which is to be made a minimum. Com- paring the terms in H?, we see that the two last, corresponding to the negative side, vanish in comparison with the other in virtue of the large denominator log^a. Hence approximately 1 . 11 ' and (37) becomes I b v 1 _6 __ 1_ 2 log b/a ' log" 6/a ~ WJ, 2 (kb) log 8 b/a kfb* Jf (k, b)" when made a minimum by variation of H^. Thus the effect of the correction depending on the introduction of ff^ is simply to wipe out the initial term of the series which represents the first approximation to the correction. After this it may be expected that the remaining terms of the first approximation to the correction will also disappear. On examination this conjecture will be found to be verified. Under each value of k in (16) only that part of B is important for which h has the particular value which is nearly equal to k. Thus each new H annuls the corresponding member of the series in (39), so that the continuation of the process leaves us with the first term of (39) isolated. The inference is that the correction to the capacity vanishes in comparison with b + log 2 6/a, or that Bl vanishes in com- parison with b -i- log 6/a. It would seem that &l is of the order 6 -f- log 2 6/a, but it would not be easy to find the numerical coefficient by the present method. In any case the correction 81 to the length of the rod vanishes in the electrostatical problem when the radius of the rod is diminished without limit a conclusion which I extend to the vibrational problem specified in the earlier portion of this paper. 373. ON CONFORMAL REPRESENTATION FROM A MECHANICAL POINT OF VIEW. [Philosophical Magazine, Vol. xxv. pp. 698702, 1913.] IN what is called conformal representation the coordinates of one point x, y in a plane are connected with those of the corresponding point , 77 by the relation * + y =/( + **), .............................. (i) where f denotes an arbitrary function. In this transformation angles remain unaltered, and corresponding infinitesimal figures are similar, though not in general similarly situated. If we attribute to , 77 values in arithmetical progression with the same small common difference, the simple square net- work is represented by two sets of curves crossing one another at right angles so as to form what are ultimately squares when the original common differ- ence is made small enough. For example, as a special case of (1), if a? + tyadsm(f -Miy), ........................... (2) x = c sin cosh 77, y = c cos sinh 77 ; and the curves corresponding to 77 = constant are + ? =1 ...(3) c- cosh 2 77 c 2 sinh 2 77 and those corresponding to = constant are -<-. -_ = 1 (4) c 2 sin 2 c 2 cos 2 f a set of confocal ellipses and hyperbolas. It is usual to refer x, y and , 77 to separate planes and, as far as I have seen, no transition from the one position to the other is contemplated. But of course there is nothing to forbid the two sets of coordinates being taken in the same plane and measured on the same axes. We may then 154- ON CONFORMAL REPRESENTATION FROM A [373 regard the angular points of the network as moving from the one position to the other. Some fifteen or twenty years ago I had a model made for me illustrative of these relations. The curves have their material embodiment in wires of hard steel. At the angular points the wires traverse small and rather thick brass disks, bored suitably so as to impose the required perpendicularity, the Fig. 1. two sets of wires being as nearly as may be in the same plane. But some- thing more is required in order to secure that the rectangular element of the network shall be square. To this end a third set of wires (shown dotted in fig. 1) was introduced, traversing the corner pieces through borings making 45 with the previous ones. The model answered its purpose to a certain extent, but the manipulation was not convenient on account of the friction entailed as the wires slip through the closely-fitting corner pieces. Possibly with the aid of rollers an improved construction might be arrived at. The material existence of the corner pieces in the model suggests the consideration of a continuous two-dimensional medium, say a lamina, whose deformation shall represent the transformation. The lamina must be of such a character as absolutely to preclude shearing. On the other hand, it must admit of expansion and contraction equal in all (two-dimensional) directions, and if the deformation is to persist without the aid of applied forces, such expansion must be unresisted. Since the deformation is now regarded as taking place continuously, f in (1) must be supposed to be a function of the time t as well as of + iij. We may write +*y-/fcf+*t) (5) The component velocities u, v of the particle which at time t occupies the position x, y are given by dx/dt, dyjdt, so that 1913] MECHANICAL POINT OF VIEW 155 Between (5) and (6) + 177 may be eliminated; u + iv then becomes a function of t and of x + iy, say iv = F(t, x + iy) ............................ (7) The equation with which we started is of what is called in Hydro- dynamics the Lagrangian type. We follow the motion of an individual particle. On the other hand, (7) is of the Eulerian type, expressing the velocities to be found at any time at a specified place. Keeping t fixed, i.e. taking, as it were, an instantaneous view of the system, we see that u, v, as given by (7), satisfy w) = 0, ........................ (8) equations which hold also for the irrotational motion of an incompressible liquid. It is of interest to compare the present motion with that of a highly viscous two-dimensional fluid, for which the equations are* Du v dp , dd (d*u d*u\ P M =pX -^ + *dx + f *(dtf + Wr Dv ^ dp d0 (d*v d*v f. du dv where 6 = -y- + ^- . dx dy If the pressure is independent of density and if the inertia terms are neglected, these equations are satisfied provided that pX + // d0/dx = 0, p Y + p'd0/dy = 0. In the case of real viscous fluids, there is reason to think that // = /u. Impressed forces are then required so long as the fluid is moving. The supposition that p is constant being already a large departure from the case of nature, we may perhaps as well suppose jjf = 0, and then no impressed bodily forces are called for either at rest or in motion. If we suppose that the motion in (7) is steady in the hydrodynamical sense, u + iv must be independent of t, so that the elimination of g + ir} between (5) and (6) must carry with it the elimination of t This requires that df/dt in (6) be a function of / and not otherwise of t and -I- iy ; and it follows that (5) must be of the form * Stokes, Camb. Trans. 1850 ; Mathematical and Physical Papers, Vol. iv. p. 11. It does not seem to be generally known that the laws of dynamical similarity for viscous fluids were formulated in this memoir. Reynolds's important application was 30 years later. 156 CONFORMAL REPRESENTATION FROM A MECHANICAL POINT OF VIEW [373 where F v F* denote arbitrary functions. Another form of (9) is F 3 (x + iy) = t + F i ( + ir } ) (10) For an individual particle F. 2 ( + it]) is constant, say a + ib. The equation of the stream-line followed by this particle is obtained by equating to ib the imaginary part of F s (x + iy). As an example of (9), suppose that x + iy = csm{it + !; + 117} (11) so that # = csin .cosh(?7 + 0> y = c cos . sinh (77 + 1), (12) whence on elimination of t we obtain (4) as the equation of the stream-lines. It is scarcely necessary to remark that the law of flow along the stream- lines is entirely different from that with which we are familiar in the flow of incompressible liquids. In the latter case the motion is rapid at any place where neighbouring stream-lines approach one another closely. Here, on the contrary, the motion is exceptionally slow at such a place. 374. ON THE APPROXIMATE SOLUTION OF CERTAIN PROBLEMS RELATING TO THE POTENTIAL. II. [Philosophical Magazine, Vol. xxvi. pp. 195199, 1913.] THE present paper may be regarded as supplementary to one with the same title published a long while ago*. In two dimensions, if <f>, ^ be potential and stream -functions, and if (e.g.) -fy be zero along the line y=0, we may take / being a function of x so far arbitrary. These values satisfy the general conditions for the potential and stream-functions, and when y = make d(j>/dx =/ A/T = 0. Equation (2) may be regarded as determining the lines of flow (any one of which may be supposed to be the boundary) in terms of f. Conversely, if y be supposed known as a function of x and i/r be constant (say unity), we may find / by successive approximation. Thus 1 f_ d?_ (l\ ^ d?_ ( d^ fl\\ _ j*_ <fr_ /1\ ( J y 6 dx 2 \y) 36 dx* \ y dx 2 \y)} 120 dx* (y) ' We may use these equations to investigate the stream-lines for which i/r has a value intermediate between and 1. If 77 denote the corresponding value of y, we have to eliminate /between =2//-'/"+ /*- 2o and f = J/ _| / " + i ?L / ,_ fit fiv whence 77 = ^y + J ~^ (yrj 3 - n y s ) - * Proc. Lond. Math. Soc. Vol. vn. p. 75 (1876) ; Scientific Papers, Vol. i. p. 272. 158 ON THE APPROXIMATE SOLUTION OF [374 or by use of (3) The evanescence of i/r when y = may arise from this axis being itself a boundary, or from the second boundary being a symmetrical curve situated upon the other side of the axis. In the former paper expressions for the " resistance " and " conductivity " were developed. We will now suppose that \/r = along a circle of radius a, in substitution for the axis of x. Taking polar coordinates a + r and 6, we have as the general equation dr ^ dO* ~ Assuming ty = R l r + R. 2 r* + R 3 r 3 + ... , '..(6) where R lt R 2 , &c., are functions of 0, we find on substitution in (5) '0, + - St r- + ..................... (8) is the form corresponding to (2) above. If i|r = 1, (8) yields expressing 72, as a function of 0, when r is known as such. To interpolate a curve for which p takes the place of r, we have to eliminate jK t between 7? Thus p = r+ - On- - r?) + and by successive approximation with use of (9) 1913] CERTAIN PROBLEMS RELATING TO THE POTENTIAL 159 The significance of the first three terms is brought out if we suppose that r is constant (ct), so that the last term vanishes. In this case the exact solution is ......................... (11) whence in agreement with (10). In the above investigation i/r is supposed to be zero exactly upon the circle of radius a. If the circle whose centre is taken as origin of coordinates be merely the circle of curvature of the curve i/r = at the point (6 = 0) under consideration, -\fr will not vanish exactly upon it, but only when r has the approximate value c6 z , c being a constant. In (6) an initial term R must be introduced, whose approximate value is c&R^. But since R " vanishes with 6, equation (7) and its consequences remain undisturbed and (10) is still available as a formula of interpolation. In all these cases, the success of the approximation depends of course upon the degree of slowness with which y, or r, varies. Another form of the problem arises when what is given is not a pair of neighbouring curves along each of which {e.g.) the stream-function is con- stant, but one such curve together with the variation of potential along it. It is then required to construct a neighbouring stream-line and to determine the distribution of potential upon it, from which again a fresh departure may be made if desired. For this purpose we regard the rectangular coordinates x, y as functions of (potential) and 77 (stream-function), so that x + iy =/( + iri), ........................... (13) in which we are supposed to know /() corresponding to 77 = 0, i.e., x and y. are there known functions of . Take a point on 77 = 0, at which without loss of generality may be supposed also to vanish, and form the expressions for x and y in the neighbourhood. From we derive x = A + A, % - B.r, + A 9 ( 2 - 77*) - When 77 = 0, x = A n + A^+ A. 2 ? + A 3 ? + A 4 ? 4- ... , y = B, + B l S + B 2 ? + B,? + B^+.... 160 ON THE APPROXIMATE SOLUTION OF CERTAIN PROBLEMS, ETC. [374 Since a; and y are known as functions of when 77 = 0, these equations determine the A's and the B's, and the general values of x and y follow. When =0, but rj undergoes an increment, t -..., (14) + ... t (15) in which we may suppose rj = 1. The A's and B's are readily determined if we know the values of x and y for i\ = and for equidistant values of , say = 0, f = 1, = + 2. Thus, if the values of a? be called x , #_,, a?,, # 2 , #_ 2 , we find ,4 = #, and ^i - 3 (*,-*-i)- 12 fo-*-), ^3-^2- 24 24 6 The .B's are deduced from the .A's by merely writing y for x throughout. Thus from (14) when = 0, 77 = 1, 5, 1 Similarly y = y - (y, + y_, - 2y ) -f (y z + y_ 2 - 2y ) (17) By these formulae a point is found upon a new stream-line (77=!) cor- responding to a given value of . And there would be no difficulty in carrying the approximation further if desired. As an example of the kind of problem to which these results might be applied, suppose that by observation or otherwise we know the form of the upper stream-line constituting part of the free surface when liquid falls steadily over a two-dimensional weir. Since the velocity is known at every point of the free surface, we are in a position to determine along this stream-line, and thus to apply the formulae so as to find interior stream-lines in succession. Again (with interchange of and 77) we could find what forms are admissible for the second coating of a two-dimensional condenser, in order that the charge upon the first coating, given in size and shape, may have a given value at every point. [Sept. 1916. As another example permanent wave-forms may be noticed.] 375. ON THE PASSAGE OF WAVES THROUGH FINE SLITS IN THIN OPAQUE SCREENS. [Proceedings of the Royal Society, A, Vol. LXXXIX. pp. 194 219, 1913.] IN a former paper* I gave solutions applicable to the passage of light through very narrow slits in infinitely thin perfectly opaque screens, for the two principal cases where the polarisation is either parallel or perpendicular to the length of the slit. It appeared that if the width (26) of the slit is very small in comparison with the wave-length (X), there is a much more free passage when the electric vector is perpendicular to the slit than when it is parallel to the slit, so that unpolarised light incident upon the screen will, after passage, appear polarised in the former manner. This conclusion is in accordance with the observations of Fizeauf upon the very narrowest slits. Fizeau found, however, that somewhat wider slits (scratches upon silvered glass) gave the opposite polarisation ; and I have long wished to extend the calculations to slits of width comparable with X. The subject has also a practical interest in connection with observations upon the Zee man effect J. The analysis appropriate to problems of this sort would appear to be by use of elliptic coordinates; but I have not seen my way to a solution on these lines, which would, in any case, be rather complicated. In default of such a solution, I have fallen back upon the approximate methods of my former paper. Apart from the intended application, some of the problems which present themselves have an interest of their own. It will be conve- nient to repeat the general argument almost in the words formerly employed * "On the Passage of Waves through Apertures in Plane Screens and Allied Problems," Phil. Mag. 1897, Vol. XLIII. p. 259 ; Scientific Papers, Vol. iv. p. 283. t Annales de Chimie, 1861, Vol. LXIII. p. 385; Mascart's Traite d'Optique, 645. See also Phil. Mag. 1907, Vol. xiv. p. 350 ; Scientific Papers, Vol. v. p. 417. Zeeman, Amsterdam Proceedings, October, 1912. R. VI. 11 162 ON THE PASSAGE OF WAVES THROUGH [375 Plane waves of simple type impinge upon a parallel screen. The screen is supposed to be infinitely thin and to be perforated by some kind of aperture. Ultimately, one or both dimensions of the aperture will be regarded as small, or, at any rate, as not large, in comparison with the wave- length (X); and the investigation commences by adapting to the present purpose known solutions concerning the flow of incompressible fluids. The functions that we require may be regarded as velocity-potentials 0, satisfying d*<j>jdt 3 = FV 2 (1) where V* = d*/da? + d*/dy* + d?jdz\ and V is the velocity of propagation. If we assume that the vibration is everywhere proportional to e itu , (1) becomes (* + *-) = 0, (2) where & = n/F=27r/\ (3) It will conduce to brevity if we suppress the factor e int . On this under- standing the equation of waves travelling parallel to x in the positive direction, and accordingly incident upon the negative side of the screen situated at x = 0, is = 6-** (4) When the solution is complete, the factor e int is to be restored, and the imaginary part of the solution is to be rejected. The realised expression for the incident waves will therefore be = cos (nt - kx) (5) There are two cases to be considered corresponding to two alternative boundary conditions. In the first (i) d<f>/dn = over the unperforated part of the screen, and in the second (ii) = 0. In case (i) dn is drawn outwards normally, and if we take the axis of z parallel to the length of the slit, will represent the magnetic component parallel to z, usually denoted by c, so that this case refers to vibrations for which the electric vector is perpendicular to the slit. In the second case (ii) is to be identified with the component parallel to z of the electric vector R, which vanishes upon the walls, re- garded as perfectly conducting. We proceed with the further consideration of case (i). If the screen be complete, the reflected waves under condition (i) have the expression 0=e***. Let us divide the actual solution into two parts, X and -ty', the first, the solution which would obtain were the screen complete ; the second, the alteration required to take account of the aperture ; and let us distinguish by the suffixes m and p the values applicable upon the negative (minus), and upon the positive side of the screen. In the present case we have *P = (6) 1913] FINE SLITS IN THIN OPAQUE SCREENS 163 This %-solution makes d^m/dn = 0, d% p /dn = over the whole plane x = 0, and over the same plane % m = 2, % p = 0. For the supplementary solution, distinguished in like manner upon the two sides, we have aikr where r denotes the distance of the point at which ty is to be estimated from the element dS of the aperture, and the integration is extended over the whole of the area of aperture. Whatever functions of position "fy m , ~^ p may be, these values on the two sides satisfy (2), and (as is evident from symmetry) they make d^r m jdn, d-^ p /dn vanish over the wall, viz., the un- perforated part of the screen, so that the required condition over the wall for the complete solution is already satisfied. It remains to consider the further conditions that < and dfyjdx shall be continuous across the aperture. These conditions require that on the aperture 2 + * = * d+ m /dx = d+ p /dx ................ (8)* The second is satisfied ifV p = -V m ; so that (9) making the values of \|r m , ty p equal and opposite at all corresponding points, viz., points which are images of one another in the plane x = 0. In order further to satisfy the first condition, it suffices that over the area of aperture and the remainder of the problem consists in so determining ty m that this shall be the case. It should be remarked that "V in (9) is closely connected with the normal velocity at dS. In general, doc At a point (x) infinitely close to the surface, only the neighbouring elements contribute to the integral, and the factor e~ ikr may be omitted. Thus d^rjdn being the normal velocity at the point of the surface in question. * The use of dx implies that the variation is in a fixed direction, while dn may be supposed to be drawn outwards from the screen in both cases. 112 164 ON THE PASSAGE OF WAVES THROUGH [375 In the original paper these results were applied to an aperture, especially of elliptical form, whose dimensions are small in comparison with X. For our present purpose we may pass this over and proceed at once to consider the case where the aperture is an infinitely long slit with parallel edges, whose width is small, or at the most comparable with X, The velocity-potential of a point-source, viz., r ^ hp /r, is now to be replaced by that of a linear source, and this, in general, is much more complicated. If we denote it by D(kr), r being the distance from the line of the point where the potential is required, the expressions are* where 7 is Euler's constant (0'577215), and S,. = l+i + i + ... + l/m ...................... (14) Of these the first is "semi-convergent" and is applicable when kr is large; the second is fully convergent and gives the form of the function when kr is moderate. The function D may be regarded as being derived from e -ncr/ r by integration over an infinitely long and infinitely narrow strip of the surface S. As the present problem is only a particular case, equations (6) and (10) remain valid, while (9) may be written in the form dy .......... (15) the integrations extending over the width of the slit from y = - b to y = + b. It remains to determine m , so that on the aperture ifr m = 1, *, = + !. At a sufficient distance from the slit, supposed for the moment to be very narrow, D (kr) may be removed from under the integral sign and also be replaced by its limiting form given in (13). Thus If the slit be not very narrow, the partial waves arising at different parts of the width will arrive in various phases, of which due account must be taken. The disturbance is no longer circularly symmetrical as in (16) But if, as is usual in observations with the microscope, we restrict ourselves to * See Theory of Sound, 341. 1913] FINE SLITS IN THIN OPAQUE SCREENS 165 the direction of original propagation, equality of phase obtains, and (16) remains applicable even in the case of a wide slit. It only remains to determine "W m as a function of y, so that for all points upon the aperture (17) where, since kr is supposed moderate throughout, the second form in (13) may be employed. Before proceeding further it may be well to exhibit the solution, as formerly given, for the case of a very narrow slit. Interpreting <f> as the velocity-potential of aerial vibrations and having regard to the known solution for the flow of incompressible fluid through a slit in an infinite plane wall, we may infer that ^ m will be of the form A (6 2 2/ 2 )~*, where A is some constant. Thus (17) becomes In this equation the first part is obviously independent of the position of the point chosen, and if the form of W m has been rightly taken the second integral must also be independent of it. If its coordinate be rj, lying between + 6, ft J \og(rj-y)dy [ b \g (y - *)) dy -t V(& 2 -2/ 2 ) / V(& 2 -2/ 2 ) ~ must be independent of 17. To this we shall presently return ; but merely to determine A in (18) it suffices to consider the particular case of 77 = 0. Here Thus so that (16) becomes ^- ...................... (20) From this, fy p is derived by simply prefixing a negative sign. The realised solution is obtained from (20) by omitting the imaginary part after introduction of the suppressed factor e int . If the imaginary part of \og($ikb) be neglected, the result is TT \*coa(nt-kr-lir) ,,. S3 7 +log(p&) ' ' corresponding to ^ m = 2 cos nt cos kx ......................... (22) Perhaps the most remarkable feature of the solution is the very limited dependence of the transmitted vibration on the width (26) of the aperture. 166 ON THE PASSAGE OF WAVES THROUGH [375 We will now verify that (19) is independent of the special value of 17. Writing y = b cos 9, rj = b cos a, we have r* v ( ! r -to = C iog ( * 6) rf * + A" iog 2 (c s * " c s a) ^ + | * log 2 (cos a - cos 0) d# = TT log (6) + (' log J2 sin ^4 <tf + I log J2 sin ^l d0 + I* * log J2 sin ^-"| dff .'O ( ^ J .'0 ( * ) .'a ( * } rl+fr ri log (2 sin <) d</> + 2 Iog(2 J Ja JO + 2 = TT log 6 + 2 I log (2 sin <f>) d(j> + 2 I log (2 sin <) d</> +2 /r rf = 7rlogi& + 4 log (2 sin <) d<, .' o as we see by changing into TT < in the second integral. Since a has disappeared, the original integral is independent of 77. In fact* I log (2 sin <f>) d<f> = 0, and we have f* ^f% = if log H (23) as in the particular case of 77 = 0. The required condition (17) can thus be satisfied by the proposed form of ^, provided that kb be small enough. When kb is greater, the resulting value of ijr in (15) will no longer be constant over the aperture, but we may find what the actual value is as a function of 77 by carrying out the integration with inclusion of more terms in the series representing D. As a preliminary, it will be convenient to discuss certain definite integrals which present themselves. The first of the series, which has already occurred, we will call h , so that h = j ' log (2 sin 6) dB = f log (2 cos 0) d0 = l" log (2 sin 2 6} d0 log (2 sin <) d<f> = % \ log (2 sin <) d<f> = A - o * See below. 1913] FINE SLITS IN THIN OPAQUE SCREENS 167 Accordingly, h = 0. More generally we set, n being an even integer, h n = f*\m0\og(2sm0)d0, .. ...(24) Jo or, on integration by parts, h n = ! ' cos 0{(n-l) sin"- 2 6 cos log (2 sin 0) + sin w ~ 2 cos 0} dd J o = (n - 1) (A n _ 2 - h^ +/ : (sin"- 2 - sin" 6) d0. J o m , 7 n 1, In 3, n 5. ..ITT Thus *- -*~ + ii- 8[..-.4.-.:g ............. < 25 > by which the integrals h n can be calculated in turn. Thus h a = 7T/8, 6 ~ . 4 a 4'2'2" 24.2 5.3.1 TT / 1 1 6.4.22 = - 5 - 3 - 1 / * _i_ i \ Similarly k.- ^ + + + , and so on. It may be remarked that the series within brackets, being equal to approaches ultimately the limit log 2. A tabulation of the earlier members of the series of integrals will be convenient : TABLE I. 2 h /7r = 2A 2 /7r = 1/4 = 0-25 2A 4 /7r =7/32 =0-21875 2A 6 /7r = 37/192 = 0-19271 2A 8 /7r =533/3072 =017350 2h 10 /7r = 1627/10240 = 0-15889 2A 12 /7r = 18107/122880 = 014736 2A 14 /7r= ................... =013798 2A 16 /7r= ................... =013018 2/i 18 /7r= ................... =012356 .... ............... =011784 The last four have been calculated in sequence by means of (25). 168 ON THE PASSAGE OF WAVES THROUGH [375 In (24) we may, of course, replace sin by cos throughout. If both sin and cos occur, as in j *sm n 0cos w 01og(2sin0)d0, (26) where n and m are even, we may express cos m by means of sin 0, and so reduce (26) to integrals of the form (24). The particular case where m = n is worthy of notice. Here f * sin" cos n log (2 sin 0) d0 = J sin n cos" log (2 cos 0} d0 tt- .-..(27) A comparison of the two treatments gives a relation between the integrals h. Thus, if /?. = 4, h t -'2h 6 + h s = hJ2\ We now proceed to the calculation of the left-hand member of (17) with W = (b* y 2 )"*, or, as it may be written, The leading term has already been found to be ticb 7 + logf) ............................... (29) In (28) r is equal to (y 77). Taking, as before, y = b cos 0, r) = b cos a, we have | ' d0 I I 7 + log ^ + log + 2 (cos - cos o)j J {kb (cos - cos a)} A- 2 6 a (cos - cos a)' M (cos - cos ) 4 3 Wfoosfl-cosa) 6 11 _ 2' 2 2 .4 8 ~ ' 2 + 2'.4 2 .6 ' 6 ............ (30) As regards the terms which do not involve log (cos cos a), we have to deal merely with f'(cos^-coso)"^, . ...(31) Jo where n is an even integer, which, on expansion of the binomial and integration by a known formula, becomes [n 1 .n-3. n 5 ... 1 n . n 1 n 3 . n 5 ... 1 n.n-2.n-4...2 ~T^~ -2.n-4 ... 2 C n.n- l.n-2.n-3 -5.n-7 ... 1 " 1.2.3.4 -n-4.n-6...2 1913] FINE SLITS IN THIN OPAQUE SCREENS 109 Thus, if n = 2, we get TT [ + cos 2 a]. If n = 4, [O 1 A, O 1 ~| -^^ + ^^ = cos 2 a + cos 4 a , and so on. The coefficient of (31), or (32), in (30) is At the centre of the aperture where vj = 0, cos a = 0, (32) reduces to its first term. At the edges where cos a = + 1, we may obtain a simpler form directly from (31). Thus g . 2n.2n 2 ... 2 n.n - 1 .n - '2 ... 1 .- ........... (34) For example, if n 6, 11.9.7.5.3.1 2317T (34) = 7r 6.5.4.3.2.1 = IT- We have also in (30) to consider (n even) 2~" I' (10 (cos 6 - cos a)" log [ 2 (cos - cos a)} + 0-a, (. . + a . a- - sin" -- log 4 sin -- sin - f "" T/1 . + ct . cc. (. . -\- o. . ct I dd sin" sin" ^ log j 4 sin ^ sin ^ f'j/l n# + a n^~ a i O 6> + dO sin n ^ sm n = log K 2 sin ^ J a ,, /Jir+Ja d^> sin" ^) sin n (d> a) log (2 sin <) ) /Jjr-Ja + 2 d</>sin n <sin n (</> + a)log(2sin<) ^o rif 2 ^ sin n 4> {sin 71 (<^> a) + sin n (0 + a)} log (2 sin fir+Ja + 2 c?</> sin" sin" (0 - a) log (2 sin <f>) JJr - 2 I * d(f> sin" sin" (0 + a) log (2 sin <) /Jn- = 2 rf<f> sin" <f> {sin" (<^> - a) + sin" (0 + a)} log (2 sin <j>), .. . .(35) .'o 170 ON THE PASSAGE OF WAVES THROUGH [375 since the last two integrals cancel, as appears when we write TT -ty- for <, n being even. In (35) sin n (<f> + o) 4- sin n (< a) = sin" < cos n a n n 1 H 1~~9~~' sin n ~*< cos s </> sin 2 a cos n ~* a + ~ ' sin"" 4 < cos 4 < sin 4 a cos n ~ 4 a + . . . + cos n <f> sin n o, (36) and thus the result may be expressed by means of the integrals h. Thus if n = 2, rtk (35) = 4 I d< sin 2 <f> {sin 2 < cos 2 a + cos 2 <f> sin 2 a} log (2 sin <f>) Jo = 4 {(cos 2 o - sin 2 a) A 4 + sin 2 a h^} ............................... (37 ) Ifn = 4, (35) = 41 dd> sin 4 d> {sin 4 d> cos 4 a + 6 sin 2 < cos 2 <f> sin 1 a cos" a ./o + cos 4 < sin 4 a} log (2 sin <) = 4 {(cos 4 o 6 sin 2 a cos 2 a + sin 4 a) /< + ( 6 sin 2 a cos 2 a - 2 sin 4 a) h e + sin 4 a h 4 ] ............. (38) If n = 6, (35) = 4 {(cos 6 a - 15 cos 4 a sin 2 a + 15 cos 2 a sin 4 a - sin" a) A,, + (15 cos 4 a sin 2 a 30 cos 2 a sin 4 a + 3 sin 8 a) h w + (15 cos 2 o sin 4 o - 3 sin 8 o) A* + sin 8 o h,} ..................... (39) It is worthy of remark that if we neglect the small differences between the h's in (39), it reduces to 4cos 8 aA 12 , and similarly in other cases. When n is much higher than 6, the general expressions corresponding to (37), (38), (39) become complicated. If, however, cos a be either 0, or 1, (36) reduces to a single term, viz., cos n < or sin n $. Thus at the centre (cos a = 0) from either of its forms 2-. 2& n ............................... (40) On the other hand, at the edges (cos a = + 1) (35) = 4 [ '^sin 2 ^>log(2sin<^) = 4A 2n ............... (41) In (30), the object of our quest, the integral (35) occurs with the coefficient 2.4 2 .6 a ...n 1913] FINE SLITS IN THIN OPAQUE SCREENS 171 Thus, expanded in powers of kb, (28) or (30) becomes ikb\ -rrtefr ikb T 4 H (cos 2 o sin 2 a) + - 2 sin 2 a 7T ikb 3] (3 T - 2 8 irW [( + 2T# [I 2 5 2A 8 + (cos 4 a 6 cos 2 a sin 2 a + sin 4 a) 7T + *i2*! (6 cos 2 a sin 2 a - 2 sin 4 a) + - 4 sin 4 *"] 7T 7T J 5 45 15 2? 2h . -\ 12 (cos 8 a 15 cos 4 a sin 2 a + 15 cos 2 a sin 4 a sin 6 a) 7T 4- 7 '' 10 (15 cos 4 a sin 2 a - 30 cos 2 a sin 4 a + 3 sin 8 a) 7T 97 Ot O7 9A, 51 _l - 8 (15 cos 2 a sin 4 a 3sin 8 a)-| ' "sin 8 a + (43) 7T 7T J At the centre of the aperture (cos a = 0), in virtue of (40), a simpler form is available. We have 3 . 5.3.1/ ikb 11 . 5 . 3 . 1 / . ikb Similarly at the edges, by (34), (41), we have ikb -rrkW 3.1 ikb . 5 . 3 . 1 / ikb 3\ .. 2A 8 5 .9.7.5.3.1/ ikb ^ , 7 12 -- + 2 - + -- (45) 172 ON THE PASSAGE OF WAVES THROUGH [375 For the general value of a, (43) is perhaps best expressed in terms of cos a, equal to 17/6. With introduction of the values of h, we have ikb\ TT^&'IV ikb\/ l\ 1 1~1 y + } s -4- j - -gr- [(y + } s -f ) ( cos ' + 2) + 2 cos a ~ 4 J 15 37 23 159 73 (46) These expressions are the values of for the various values of 17. We now suppose that kb = 1. The values for other particular cases, such as &6 = \, may then easily be deduced. For cos a = 0, from (44) we have i\[ 1^1 1 3.1 1 5.3.1 >gJ 22 ~' + ni_ j_n _j __ _73_ i ^[2*4 2 8 .4 2 32 2 2 .4 a .6 2 192 = TT ( 7 + log ^ [1 - 0-12500 4- 0-00586 + 0-00013] + TT [0-06250 - 0-00537 + 0'00016] = TT (7 + log^ x 88073 4- TT x 0-05729 = w [-0-65528 + 1-3834 1] ......................................... (48) since 7 = 0-577215, log 2 = 0*693147, log i = iri. In like manner, if kb = , we get still with cos = 0, f7 + lo gl) [1 - 0-03125 + 0-00037] + TT [0-01562 - 0-00033] \ o/ = TT [-1-4405 + 1-5223 i] ............ (49) If & = 2, we have * (7 + lo g|) [1 - 0-5 + 0-0938 - 0-0087 + 0-0005] + TT [0-25 - 0-0859 + 0-0102 - 0'0006] = TT[+ 0-1058 + 0-9199t*] ........................ ....(50) 1913] FINE SLITS IN THIN OPAQUE SCREENS 173 If kb = 1 and cos a = + 1, we have from (45) 7T 7 + 1 _i? 1 35 231 2 2 .4 2 8 2*.4 2 .6 2 16 1 6435 1 _ __ _ ___ 19 . 17 . 6435 2 2 .4 a .6 3 .8 2 128 2 2 .4 2 .6 2 .8 2 .10 2 10.9.128 + ., 97 7303 38-084 170'64 2 2 .4 2 .6 2 960 2 2 .4 2 .6 2 .8 2 2 2 .4 2 . 6 2 .8 2 . 10 2 = TT ( 7 + log^J [1 - 0-375 + 0-068359 - 0-006266 + 0-000341 - 0-000012] - TT [0-0625 + 0-015788 - 0-003302 + 0'000258 + 0-000012] = TT[- 0-63141 + 1 -0798 i] (51) Similarly, if kb = , we have TT (7 + log I) [1 - 0-09375 + 0-00427 - O'OOOIO] - TT [0-01562 + 0-00099 - 0-00005] = TT[- 1-3842 + 1-4301 1] (52) And if kb = 2, with diminished accuracy, TT ( 7 + log I) [1 - 1-5 + 1-094 - 0-401 + 0-087 - 0'012 + O'OOl] - TT [0-25 + 0-253 - 0-211 + 0'066 - 0'012 + O'OOl] = TT [- 0-378 + 0-422 1\ (53) As an intermediate value of a. we will select cos 2 a = ^. For kb = 1, from (46) TT (7 + log ^ [1 - 0-25 + 0-03320 - 0-00222 + . . .] + TT [0 - 0-01286 + 0-001522 + . . .] = TT [-0-6432 + 1-2268 i] (54) Also, when kb = |, TT[- 1-4123 + 1-4759?:] (55) When kb = 2, only a rough value is afforded by (46), viz., TT [-0-16 + 0-61 i] (56) The accompanying table exhibits the various numerical results, the factor TT being omitted. TABLE II. it.* kb = l kb = 2 cos a = COS 2 a = COS 2 a = 1 - 1-4405 + 1-5223 t - 1-4123 + 1-4759 i - 1-3842 + 1-4301 * - 0-65528 + 1-3834 t -0-6432 +1-2268* -0-63141 + 1-0798? +0-1058 + 0-9199 i -0-16 +0-61 i -0-378 +0-422 174 ON THE PASSAGE OF WAVES THROUGH [375 As we have seen already, the tabulated quantity when kb is very small takes the form y + log (ikb/4,), or log kb- 0-8091 + 1 -57081, whatever may be the value of er. In this case the condition (17) can be completely satisfied with = ^(6* y*)" 1 , A being chosen suitably. When kb is finite, (17) can no longer be satisfied for all values of a. But when kb = A, or even when kb = 1, the tabulated number does not vary greatly with a and we may consider (17) to be approximately satisfied if we make in the first case TT(- 1-4123 + 1-4759 i)A =-l, ................... (57) and in the second, TT(- 0-6432 + 1-2268 i) 4 =-1 .................... (58) The value of ty, applicable to a point at a distance directly in front of the aperture, is then, as in (16), (59) In order to obtain a better approximation we require the aid of a second solution with a different form of ". When this is introduced, as an addition to the first solution and again with an arbitrary constant multiplier, it will enable us to satisfy (17) for two distinct values of a, that is of 77, and thus with tolerable accuracy over the whole range from cosec = to cos a = 1. Theoretically, of course, the process could be carried further so as to satisfy (17) for any number of assigned values of cos a. As the second solution we will take simply M* = 1, so that the left-hand member of (17) is rb+ri rb-i) D(kr)dr + D(kr)dr ..................... (60) Jo Jo If we omit k, which may always be restored by consideration of homo- geneity, we have . 3 2'. 4*. 5 2 + + the same expression with the sign of rj changed. The leading term in (60) is thus 26( 7 - 1 + logii) + (b + ,) log (6 + r,) + (b-7)) log (6 - T,). ...(61) 1913] FINE SLITS IN THIN OPAQUE SCREENS 175 At the centre of the aperture (77 = 0), (61) = 26 {7-1 + log $#}, and at the edges (77 = + b), (61) = 26{ 7 -l+logi6}. It may be remarked that in (61), the real part varies with 77, although the imaginary part is independent of that variable. The complete expression (60) naturally assumes specially simple forms at the centre and edges of the aperture. Thus, when 77 = 0, 3 and, similarly, when ij = b, ...... (62) ...... (63) To restore k we have merely to write kb for b in the right-hand members of (62), (63). The calculation is straightforward. For the same values as before of kb and of cos 2 a, equal to rf/V 1 , we get for (60) -r- 26 TABLE III. ,'/6* *6=i kb = 1 kb = 2 - 1 -7649 + 1 -5384 i - 1-4510+1 -4912 i -1-0007 +1-4447 i - 1 -0007 + 1 '4447 i -0-6740 + T2771 i - 0-2217 + 1-1 198 i -0-2167 + 1-1198 i -0-1079 + 0-7166 i + 0-1394+0-4024 i We now proceed to combine the two solutions, so as to secure a better satisfaction of (17) over the width of the aperture. For this purpose we determine A and B in V = A(b*-f)-* + B, (64) so that (17) may be exactly satisfied at the centre and edges (77 = 0, 17 = 6). The departure from (17) when r) 2 /b- = $ can then be found. If for any value of kb and 77 = the first tabular (complex) number is p and the second q, and for 77 = + b the first is r and the second s, the equations of condition from (17) are 7rA.p + 2bB.q = -l, -rrA . r + 2bB . s = - 1 (65) 176 ON THE PASSAGE OF WAVES THROUGH When A and B are found, we have in (16) [375 -rrA + 2bB. From (65) we get ps-qr +b ps-qr SO /:: Thus for kb = 1 we have p = -0-65528+1 -3834 *, r = - 0-63141 -I- 1-0798 i, whence IT A = + 0-60008 + 0-51828 i, ps-qr 9 8 .(66) .(67) - 1 -0007 + 1-4447?, -0-2217 + 1-1198 1, 265 = - 0-2652 + 0-1073 i, and (67) = + 0'3349 + 6256 i. The above values of irA and 265 are derived according to (17) from the values at the centre and edges of the aperture. The success of the method may be judged by substitution of the values for tf/b* = $. Using these in (17) we get - 0-9801 0'0082 i, for what should be 1, a very fair approxi- mation. In like manner, for kb = 2 (67) = + 0-259 + 1-2415 i ; and for kb (67) = + 0'3378 + 0-3526 i. As appears from (16), when k is given, the modulus of (67) may be taken to represent the amplitude of disturbance at a distant point imme- diately in front, and it is this with which we are mainly concerned. The following table gives the values of Mod. and Mod. 2 for several values of kb. The first three have been calculated from the simple formula, see (20). TABLE IV. kb Mod.2 Mod. o-oi 0-0174 0-1320 0-05 0*0590 0-2429 0-25 0-1372 0-3704 0-50 0-2384 0-4883 I'OO 0-5035 0-7096 2-00 1 -608 1 -268 | The results are applicable to the problem of aerial waves, or shallow water waves, transmitted through a slit in a thin fixed wall, and to electric 1913] FINE SLITS IN THIN OPAQUE SCREENS 177 (luminous) waves transmitted by a similar slit in a thin perfectly opaque screen, provided that the electric vector is perpendicular to the length of the slit. In curve A, fig. 1, the value of the modulus from the third column of Table IV is plotted against kb. 0-5 1-0 1-5 Fig. 1. 2 2-5 When kb is large, the limiting form of (67) may be deduced from a formula, analogous to (12), connecting M* and d<f>/dn. As in (11), in which, when x is very small, we may take D = log r. Thus d\lr f +0 xdy ]+ ao 1 rix = ^ - y = ^ tan- 1 ? = TT^, or "*F = - CW J -<*X 2 -f- y 2 #J_oo 7T Now, when && is large, dty/dn tends, except close to the edges, to assume the value ik, and ultimately r+b Sikh (67)= f Vib. -.=, (69) J -b I? of which the modulus is *2kb/7r simply, i.e. 0'637 kb. We now pass on to consider case (ii), where the boundary condition to be satisfied over the wall is < = 0. Separating from <j> the solution (%) which would obtain were the wall unperforated, we have X m =e- ikx -e ikx , XP = > C'O) giving over the whole plane (x 0), 178 ON THE PASSAGE OF WAVES THROUGH [375 The supplementary solutions y, equal to <f> x, may be written *.-/*-*. +,-!&*,* ................ (> where m , W p are functions of y, and the integrations are over the aperture. D as a function of r is given by (13), and r, denoting the distance between dy and the point (x, >/), at which i/r, ft , ^ p are estimated, is equal to V{^* + (y I?}- The form (71) secures that on the walls 1/^ = ^ = 0, so that the condition of evanescence there, already satisfied by x> is not disturbed. It remains to satisfy over the aperture (72) The first of these is satisfied if m = - p , so that ^ m and ^ p are equal at any pair of corresponding points on the two sides. The values of d-^r m /d.r, are then opposite, and the remaining condition is also satisfied if (73) At a distance, and if the slit is very narrow, dDjdx may be removed from under the integral sign, so that ,n wh,ch (74) dD ikx f T And even if kb be not small, (74) remains applicable if the distant point be directly in front of the slit, so that x = r. For such a point V p dy. ...(76) There is a simple relation, analogous to (68), between the value t M',, ;it .my point (r)) of the aperture and that of fy p at the same point. For in tho application of (71) only those elements of the integral contribute which lie infinitely near the point where i/r p is to be estimated, and for these dDjdx = ar/r 3 . The evaluation is effected by considering in the first instance a point for which x is finite and afterwards passing to the limit. Thus It remains to find, if possible, a form for V p , or ^r pt which shall make d\lr p /dx constant over the aperture, as required by (73). In my former paper, dealing with the case where kb is very small, it was shown that known 1913] FINE SLITS IN THIN OPAQUE SCREENS 179 theorems relating to the flow of incompressible fluids lead to the desired conclusion. It appeared that (74), (75) give showing that when b is small the transmission falls off greatly, much more than in case (i), see (20). The realised solution from (78) is .cos(^-^r-l7r), ............... (79) corresponding to ^ m = 2 sin nt sin kx ............................ (80) The former method arrived at a result by assuming certain hydrodynamical theorems. For the present purpose we have to go further, and it will be appropriate actually to verify the constancy of dty/dx over the aperture as resulting from the assumed form of M*, when kb is small. In this case we may take D = logr, where r 2 = x* + (y - 17 ) 2 . From (71), the suffix p being omitted, . and herein -y-r = -- j-r- = -- j (? const.). da? drf dy* ^ Thus, on integration by parts, . ... dx [_ dy\ J , b dy dy ' dD dD dr y-r) dj=fodj ss <y-rt + *' and so long as 77 is not equal to + b, it does not become infinite at the limits (y b), even though x = 0. Thus, if ^ vanish at the limits, the integrated terms in (81) disappear. We now assume for trial ^ = V(& 2 -y')> .............................. (82) which satisfies the last-mentioned condition. Writing y = b cos 6, V) = b cos a, as' = x/b, we have _ = - COB ) + cos (COB g - c^o) ...... dx Jo (cos 6 cos a) 2 + ti* Of the two parts of the integral on the right in (83) the first yields TT when ,/ = 0. For the second we have to consider cos 6 - cos i , fi a; " '" 122 180 ON THE PASSAGE OF WAVES THROUGH [375 in which cos 6 cos a passes through zero within the range of integration. It will be shown that (84) vanishes ultimately when x = 0. To this end the range of integration is divided into three parts: from to ,, where ! < a, from a, to <%, where a 2 > a, and lastly from o 2 to TT. In evaluating the first and third parts we may put x = at once. And if z = tan i# f dd 1 t( dz dz | Jcostf cosa "sinctj {tana + tan^a z}' Sin a being omitted, the first and third parts together are thus where t = tan ct, ^ = tan a 1} t^ = tan fa, and z is to be made infinite. It appears that the two parts taken together vanish, provided ^ , t 2 are so chosen that P ,,. It remains to consider the second part, viz., " d0(cos0-cosa) 0- in which we may suppose the range of integration o 2 ttj to be very small. Thus _ /*"* d6 . 2 sin %(0 + a) sin ^ (a - 0) ~ J., 4 sin 2 (0 + a) sin 2 (a - 8) + x' 9 ~~ 2 sin a sin 2 a (a a,) 2 + x" 1 ' and this also vanishes if 2 - a = a - a, , a condition consistent with the former to the required approximation. We infer that in (83) <> so that, with the aid of a suitable multiplier, (73) can be satisfied. Thus if = A^/(b 3 - f), (73) gives A = ikjir, and the introduction of this into (74) gives (78). We have now to find what departure from (86) is entailed when icb is no longer very small. Since, in general, ffiD/da? + d*D/dy* + k*D = 0, we find, as in (81), and for the present has the value defined in (82). The first term on the right of (87) may be treated in the same way as (28) of the former problem, the difference being that V(& a - y j ) occurs now in the numerator instead of 1913] FINE SLITS IN THIN OPAQUE SCREENS 181 the denominator. In (30) we are to introduce under the integral sign the additional factor k 2 b 2 sin*0. As regards the second term of (87) we have dD d = f+ b y(y-r})dyl dD dy dy y ~j _ 6 V(& 2 -2/ 2 ) r dr ' where in - -=- we are to replace r by + (y 17). We then assume as before y = b cos 6, i] = b cos a, and the same definite integrals h n suffice ; but the calculations are more complicated. We have seen already that the leading term in (87) is TT. For the next term we have n . ikr IdD k* k* ik />-7 + log T , r& = *- and thus 1 d-Jr TT/ ikb 1\ - f cos 2 + 1 cos a cos 0) log 2 (cos - cos a). . . .(88) The latter integral may be transformed into 2 f <fy {1 - f cos 2 (20 - a) + \ cos a cos (20 - a) + 1 - f cos 2 (20 + a) + cos a cos (20 + a)} log (2 sin 0), and this by means of the definite integrals h is found to be - | (1 + 2 sin 2 a). To this order of approximation the complete value is --^ = 7r + '7ryfc 2 & 2 (7-sin 2 a + logt'&&) ............. (89) For the next two terms I find + 3 sin 4 a + ^ cos 4 a + 6 sin 2 a cos 2 a] a ~ in4 " ~~ S 2 ^ sin " a ' When cos a = 0, or + 1, the calculation is simpler. Thus, when cos a = 0, 1 dty k*fr ( ikb , \ frb* r ikb -u = 1 + -4-(^ 1 ^4-- 1 )-r28( 1 - ikb 5\ ok s b s t ikb (01) 182 ON THE PASSAGE OF WAVES THROUGH and when cosa= 1, [375 ikb\ 16 - O- t 429 - 329 ikb\ 6831 -6of ..(92) the last term, deduced from h l4 , h ltt being approximate. For the values of -ir-^d^r/dx we find from (91), (90), (92) for kb = i 1, v/2, 2 : TABLE V. fcb = i kb = l fcfc = v /2 kb = 2 cosa =0 cos*a = i cos s a=l 0-8448 + 0'0974t 0-8778+0-0958 i 0'9103 + 0-0944t 0-5615+0'3807t -6998+0 -3583 t 0-8353+0-3364i 0-3123 + 0-7383 1 0-8587 + 0-5783 i 0-0102 + 1-389$; 0-518 + 1-129 i 1-020 +0-861t These numbers correspond to the value of "^ expressed in (82). We have now, in pursuance of our method, to seek a second solution with another form of ^ The first which suggests itself with " = 1 does not answer the purpose. For (81) then gives as the leading term _. 26 '*'' becoming infinite when tj= b. A like objection is encountered if = 6* y*. In this case The first part gives 46 simply when a; becomes zero. And -' \y ~ T />r -r- 5 ~ g xJ.f. so that (94) becoming infinite when T; = 6. So far as this difficulty is concerned we might take = (6 J - y a ) a , but another form seems preferable, that is (95) 1913] FIXE SLITS IN THIN OPAQUE SCREENS 183 With the same notation as was employed in the treatment of (82) we have cos (cos - cos ) d0 _of* cos 3 (cos cos a) , - - cos a) 2 + as' r J e (cos 0- cos .)* + x" 1 The first of these integrals is that already considered in (83). It yields Sir. In the second integral we replace cos 3 by {(cos 6 cos a) + cos a} 3 , and we find, much as before, that when x' = cos 3 (cos 6 - cos a) d0 Thus altogether for the leading term we get - ^ = 37r (^ - cos 2 a) = 3?r (| - 7; 2 /& 2 ). . . . . .(97) ciac This is the complete solution for a fluid regarded as incompressible. We have now to pursue the approximation, using a more accurate value of D than that (logr) hitherto employed. In calculating the next term, we have the same values of D and r~ 1 dD/dr as for (88) ; and in place of that equation we now have 1 c Sir ikb + | d0[% sin 4 - f sin 2 6 + f sin 2 6 cos cos a] log {+ 2 (cos - cos a)}. (98) Jo The integral may be transformed as before, and it becomes /i" 4 d<f> log (2 sin <ft) [4 (sin 4 26 cos 4 a + 6 sin 2 26. cos 2 26 sin 2 a cos 2 a .'o + cos 4 20 sin 4 a) - f (sin 2 20 cos 2 a + cos 2 20 sin 2 a) + f cos a cos 20 {sin 2 a cos a + sin 2 20 (cos s a 3 sin 2 a cos a)}]. (99) The evaluation could be effected by expressing the square bracket in terms of powers of sin 2 0, but it may be much facilitated by use of two lemmas. If /(sin 26, cos 2 20) denote an integral function of sin 20, cos 2 20, /*" rin- d(j> log (2 sin 0)/(sin 20, cos 2 20) = d6 log (2 cos )/(sin 20, cos 2 20) j) .'o = f * d(f> log (2 sin 20)/(sin 20, cos 2 20) = f ** d0 log (2 sin 0)/(sin 0,cos 2 0), Jo .'o .................. (100) in which the doubled angles are got rid of. 184 ON THE PASSAGE OF WAVES THROUGH [375 Again, if m be integral, J** d<l> sin 2< cos 2<j> log (2 sin </>) 4m + 2 J + C S 2m-1.2m-S...l,r 2m.2m-2...2 2 For example, if m = 0, 7T fy cos 2< log (2 sin <) = - - , (102) and(w = l) d<f>sin 2 2<f>cos2<f>log(2 sin <) = - (103) .'o ^ Using these lemmas, we find (99) = 5^ (cos 4 a 6 cos 2 a sin 2 a + sin 4 a) + h 2 ( 30 cos 2 a sin 2 a - 10 sin 4 a - 3 cos 2 a + 3 sin 2 o) - \TT cos 2 a (cos 2 o+3 sin 2 o) ; and thence, on introduction of the values of h?, h t , for the complete value to this order of approximation, (104) 1(5 cos 4 a +18 cos 2 a sin 2 a + 21 sin 4 a) 1 ....... To carry out the calculations to a sufficient approximation with the general value of a would be very tedious. I have limited myself to the extreme cases cos a = 0, cos a = + 1. For the former, we have 3 / ikb 64 6 . 256 4 3 . 256 . 8 and for the latter ir'dx'' 2~ l V 7 ' f 10g 4J | 16 16. 16 + 4. 16". 16.16 24 . 16 4 j 1069 W _ 41309W 64 16. 64. 15 + 16. 3. 70. 64. 64 16 5 .9.420 '" " '" " AJ./V u 3289n^ 8 O 1 ' /1ftft\ h ~32~ + 4Ti6.T6~2TT6' + T6T36~ ( 1913] FINE SLITS IN THIN OPAQUE SCREENS 185 From these formulae the following numbers have been calculated for the value of - ir- l d^jdx: TABLE VI kb = l ttl kb = J2 kb = 2 cosa=0 cos a = 1 l-3716+0-0732i -1-5634 + 0-07101 1-1215+0-2885& -l'6072+0-2546i 0-8824+ 0-5653 1 - 1-5693 + 0-4401 i 0-5499 + 1-08601 -l-3952 + 0-6567i They correspond to the value of *P formulated in (95). Following the same method as in case (i), we now combine the two solutions, assuming V = A V(6 s -3f) + 56-(6-y a )* (107) and determining A and B so that for cos a = and for cos a = + 1, dty/dx shall be equal to ik. The value of ty at a distance in front is given by (76), in which (108) We may take the modulus of (108) as representing the transmitted vibration, in the same way as the modulus of (67) represented the transmitted vibration in case (i). Using p, q, r, s, as before, to denote the tabulated complex numbers, we have as the equations to determine A and B, so that ik 1"^ dy = =- *SE 1 (110) J 2 ps-qr For the second fraction on the right of (110) and for its modulus we get in the various cases kb= , 1-1470- 01287 i, 1-1542, kb= 1, 1-1824 - 0-6986 i, 1-3733, kb = V2, 0-6362 - 1-0258 i, 1-2070, kb= 2, 0-1 239- 0-7303 t, 07407. And thence (on introduction of the value of kb} for the modulus of (110) representing the vibration on the same scale as in case (i) TABLE VII. kb Modulus * 0-1443 1 0-6866 V2 1 -2070 2 1-4814 186 ON THE PASSAGE OF WAVES THROUGH FINE SLITS, ETC. [375 These are the numbers used in the plot of curve B, fig. 1. When kb is much smaller than , the modulus may be taken to be ffib*. When kb is large, the modulus approaches the same limiting form as in case (i). This curve is applicable to electric, or luminous, vibrations incident upon a thin perfectly conducting screen with a linear perforation when the electric vector is parallel to the direction of the slit. It appears that if the incident light be unpolarised, vibrations perpen- dicular to the slit preponderate in the transmitted light when the width of the slit is very small, and the more the smaller this width. In the neighbourhood of kb = 1, or 26 = \/TT, the curves cross, signifying that the transmitted light is unpolarised. When kb = 1, or 2& = 3X/27r, the polarisation is reversed, vibrations parallel to the slit having the advantage, but this advantage is not very great. When kb > 2, our calculations would hardly succeed, but there seems no reason for supposing that anything distinctive would occur. It follows that if the incident light were white and if the width of the slit were about one-third of the wave-length of yellow-green, there would be distinctly marked opposite polarisations at the ends of the spectrum. These numbers are in good agreement with the estimates of Fizeau : " Une ligne polarise'e perpendiculairement a sa direction a paru etre de y^^ de millimetre; une autre, beaucoup moins lumineuse, polarisee parallelement a sa direction, a ete estimee a 7^^ de millimetre. Je dois ajouter que ces valeurs ne sont qu'une approximation ; elles peuvent etre en r^alite plus faibles encore, mais il est peu probable qu'elles soient plus fortes. Ce qu'il y a de certain, c'est que la polarisation parallele n'apparait que dans les fentes les plus fines, et alors que leur largeur est bien moindre que la longueur d'une ondulation qui est environ de ^ 5 de millimetre." It will be remembered that the " plane of polarisation " is perpendicular to the electric vector. It may be well to emphasize that the calculations of this paper relate to an aperture in an infinitely thin perfectly conducting screen. We could scarcely be sure beforehand that the conditions are sufficiently satisfied even by a scratch upon a silver deposit. The case of an ordinary spectroscope slit is quite different. It seems that here the polarisation observed with the finest practicable slits corresponds to that from the less fine scratches on silver deposits. 376. ON THE MOTION OF A VISCOUS FLUID. [Philosophical Magazine, Vol. XXVI. pp. 776 786, 1913.] IT has been proved by Helmholtz* and Kortewegf that when the velocities at the boundary are given, the slow steady motion of an incom- pressible viscous liquid satisfies the condition of making F, the dissipation, an absolute minimum. If U Q , v , w be the velocities in one motion M , and u, v, w those of another motion M satisfying the same boundary conditions, the difference of the two u', v', w', where u' = u U Q , v' = v v , w' = w w , .................. (1) will constitute a motion M' such that the boundary velocities vanish. If F , F, F' denote the dissipation-functions for the three motions M , M, M' respectively, all being of necessity positive, it is shown that F=F Q + F'- 2p(u'Vu + v"V*v + w'VX) dxdydz, ......... (2) the integration being over the whole volume. Also F' = - p I (w' W + t/W + w'W) dx dy dz These equations are purely kinematical, if we include under that head the incompressibility of the fluid. In the application of them by Helmholtz and Korteweg the motion M is supposed to be that which would be steady if small enough to allow the neglect of the terms involving the second powers of the velocities in the dynamical equations. We then have * Collected Works, Vol. i. p. 223 (1869). t Phil. Mag. Vol. xvi. p. 112 (1883). 188 ON THE MOTION OF A VISCOUS FLUID [376 where V is the potential of impressed forces. In virtue of (4) () ................... (5) if the space occupied by the fluid be simply connected, or in any case if V be single-valued. Hence F = F + F', ................................. (6) or since F' is necessarily positive, the motion M makes F an absolute minimum. It should be remarked that F' can vanish only for a motion such as can be assumed by a solid body (Stokes), and that such a motion could not make the boundary velocities vanish. The motion M Q determined by (4) is thus unique. The conclusion expressed in (6) that M makes F an absolute minimum is not limited to the supposition of a slow motion. All that is required to ensure the fulfilment of (5), on which (6) depends, is that V 2 , V 2 y , V 2 w should be the derivatives of some single-valued function. Obviously it would suffice that V 2 , V*v , V*w vanish, as will happen if the motion have a velocity-potential. Stokes* remarked long ago that when there is a velocity- potential, not only are the ordinary equations of fluid motion satisfied, but the equations obtained when friction is taken into account are satisfied likewise. A motion with a velocity-potential can always be found which shall have prescribed normal velocities at the boundary, and the tangential velocities are thereby determined. If these agree with the prescribed tangential velocities of a viscous fluid, all the conditions are satisfied by the motion in question. And since this motion makes F an absolute minimum, it cannot differ from the motion determined by (4) with the same boundary conditions. We may arrive at the same conclusion by considering the general equation of motion fdu du du du\ __ d (p V + ) P J7 + W j~ + v j~ + w j- )=/*Vtt -- x-j .......... (7) r \dt dx dy dz) dx If there be a velocity-potential </>, so that u = d<f>jdx, &c., du du du I d (fd<l>\* /c^ and then (7) and its analogues reduce practically to the form (4) if the motion be steady. Other cases where F is an absolute minimum are worthy of notice. It suflSces that Cnmb. Trans. Vol. ix. (1850) ; Math, and Phyg. Papers, Vol. HI. p. 73. 1913] ON THE MOTION OF A VISCOUS FLUID 189 where H is a single-valued function, subject to V 2 T = 0. If %, ij , f be the rotations and thus (9) requires that V 2 = 0, V^ = 0, V^ = ...................... (10) In two dimensions the dynamical equation reduces to D /Dt = Q*, so that is constant along a stream-line. Among the cases included are the motion between two planes u = A + By + Cy 2 , v = Q, w, = 0, .................. (11) and the motion in circles between two coaxal cylinders ( = constant). Also, without regard to the form of the boundary, the uniform rotation, as of a solid body, expressed by Uo = Cy, v = -Cx ............................ (12) In all these cases F is an absolute minimum. Conversely, if the conditions (9) be not satisfied, it will be possible to find a motion for which F< F . To see this choose a place as origin of coordinates where dV^/dy is not equal to dV 2 v /da;. Within a small sphere described round this point as centre let uf = Cy, v Cx, w' = 0, and let u = 0, v' = 0, w' = outside the sphere, thus satisfying the prescribed boundary conditions. Then in (2) [ (tt'VX + v'V*v + w'V 2 w ) dx dy dz = C I (y VX - #V 2 v ) dx dy dz, . . .(13) the integration being over the sphere. Within this small region we may take so that (13) reduces to Since the sign of C is at disposal, this may be made positive or negative at pleasure. Also F' in (2) may be neglected as of the second order when it', v', w' are small enough. It follows that F is not an absolute minimum for u , v , w a , unless the conditions (9) are satisfied. Korteweg has also shown that the slow motion of a viscous fluid denoted by M is stable. " When in a given region occupied by viscous * Where DjDt = d/dt + u d/dx + v djdy + w djdz. 190 ON THE MOTION OF A VISCOUS FLUID [37(j incompressible fluid there exists at a certain moment a mode of motion M which does not satisfy equation (4), then, the velocities along the boundary being maintained constant, the change which must occur in the mode of motion will be such (neglecting squares and products of velocities) that the dissipation of energy by internal friction is constantly decreasing till it reaches the value F and the mode of motion becomes identical with M ." This theorem admits of instantaneous proof. If the terms of the second order are omitted, the equations of motion, such as (7), are linear, and any two solutions may be superposed. Consider two solutions, both giving the same velocities at the boundary. Then the difference of these is also a solution representing a possible motion with zero velocities at the boundary. But such a motion necessarily comes to rest. Hence with flux of time the two original motions tend to become and to remain identical. If one of these is the steady motion, the other must tend to become coincident with it. The stability of the sloiv steady motion of a viscous fluid, or (as we may put it) the steady motion of a very viscous fluid, is thus ensured. When the circumstances are such that the terms of the second order must be retained, there is but little definite knowledge as to the character of the motion in respect of stability. Viscous fluid, contained in a vessel which rotates with uniform velocity, would be expected to acquire the same rotation and ultimately to revolve as a solid body, but the expectation is perhaps founded rather upon observation than upon theory. We might, however, argue that any other event would involve perpetual dissipation which could only be met by a driving force applied to the vessel, since the kinetic energy of the motion could not for ever diminish. And such a maintained driving couple would generate angular momentum without limit a conclusion which could not be admitted. But it may be worth while to examine this case more closely. We suppose as before that u 0t v n , w are the velocities in the steady motion M and u, v, w those of the motion M, both motions satisfying the dynamical equations, and giving the prescribed boundary velocities ; and we consider the expression for the kinetic energy T of the motion (1) which is the difference of these two, and so makes the velocities vanish at the boundary. The motion M' with velocities u', v, w' does not in general satisfy the dynamical equations. We have IdT (( ,du! ,dv ,d In equations (7) which are satisfied by the motion M we substitute u = + u, &c. ; and since the solution M is steady we have --- ............................. < 15 > 1913] OX THE MOTION OF A VISCOUS FLUID 191 We further suppose that V 2 w , V 2 v , V 2 w are derivatives of a function H, as in (9). This includes the case of uniform rotation expressed by o = y, v = -a:, w = Q, ........................ (16) as well as those where there is a velocity-potential. Thus (7) becomes du with two analogous equations, where These values of du'/dt, &c., are to be substituted in (14). In virtue of the equation of continuity to which u', v', w' are subject, the terms in tsr contribute nothing to dT'/dt, as appears at once on integration by parts. The remaining terms in dT'fdt are of the first, second, and third degree in u', v', w . Those of the first degree contribute nothing, since u , v , w satisfy equations such as du du du cfe M -; 1- V -; 1- W -j- = j dx dy dz dx ,du -f w -j- dz The terms of the third degree are f f , ( , du' , du' \\u <u ^ h v -r- .' L I dx d v ,( ,dv' ,dv ,dv'\ + v hi -j- 4 v -, h w f} ( dx dy dz } , ( , dw' , dw' , dw' which may be written -\l[ u ' d(u ' + ^ - + '- + w '* -^r- ~] and this vanishes for the same reason as the terms in CT. We are left with the terms of the second degree in u', v, w'. Of these the part involving v is v ! [u' V'V + v' v V + w'V-<v f ] dxdydz (20) So far as this part is concerned, we see from (3) that dT'/dt = -F f , (21) F' being the dissipation-function calculated from u', v', w'. 192 ON THE MOTION OF A VISCOUS FLUID [376 Of the remaining 18 terms of the second degree, 9 vanish as before when integrated, in virtue of the equation of continuity satisfied by u^, v , w . Finally we have* r- = F' I \u' \U' -J-? + V -j-2 + W -j-^\ dt ^ J L ( dx dy dz) , dv , dv , < If the motion u , v , w n be in two dimensions, so that w = Q, while u and i' are independent of z, (22) reduces to , '/ ",, , dv , , /du dv \ "1 , , Under this head comes the case of uniform rotation expressed in (16), for which du a _ dv _ du dv _ ~~i "> "i T I ~i " dx dy dy dx Here then dT' /dt = F' simply, that is T' continually diminishes until it becomes insensible. Any motion superposed upon that of uniform rotation gradually dies out. When the motion u , v , w has a velocity-potential <f>, (22) may be written + 2uV - + W - + *w'u' -dxdydz ..... (24) - + W - + *w'u' dxdy dydz So far as I am aware, no case of complete stability for all values of ft is known, other than the motion possible to a solid body above considered. It may be doubted whether such cases exist. Under the head of (24) a simple example occurs when <j> = tan -1 (y/x), the irrotational motion taking place in concentric circles. Here if r 2 = a? + y 2 , ....... (25) Compare 0. Reynolds, Phil. Tram. 1895, Part i. p. 146. In Lorentz's deduction of a similar equation (Abhandlungen, Vol. i. p. 46) the additional motion is assumed to be small. This memoir, as well as that of Orr referred to below, should be consulted by those interested. See also Lamb's Hydrodynamics, 346. 1913] ON THE MOTION OF A VISCOUS FLUID 193 If the superposed motion also be two-dimensional, it may be expressed by means of a stream-function ty. We have in terms of polar coordinates , Gty Cfyr u = -f- = -f- dy dr . 1 d& sm B + - - cos 0, d^r d^r I -f- = -f- cos 6 - - dx dr r so that a * cos 2 - sm 2 - u'v' = cos sm 6 -- - - - + r dr dB ' Thus cos 6 sin (u' z - v" 2 ) - (cos 2 6 - sin 2 0}u'v' = --f- ) ... .(26) r dr du and (25) becomes T', F', as well as the last integral, being proportional to z. We suppose the motion to take place in the space between two coaxal cylinders which revolve with appropriate velocities. If the additional motion be also symmetrical about the axis, the stream-lines are circles, and ^ is a function of r only. The integral in (27) then disappears and dT'/dt reduces to F', so that under this restriction * the original motion is stable. The experiments of Couette^ and of MallockJ, made with revolving cylinders, appear to show that when u\ v', w' are not specially restricted the motion is unstable. It may be of interest to follow a little further the indications of (27). The general value of -^ is ^ = <7 + G l cos 6 + S x sin + . . . + C n cos n0 + S n sin n0, (28) Qi> &n being functions of r, whence dCn_ Cn dSn\ (29) n being 1, 2, 3, &c. If S n , C n differ only by a constant multiplier, (29) vanishes. This corresponds to ^ = R, + R, cos (6 + e,) + . . . + R n cos n (0 + e,) + ..., (30) * We may imagine a number of thin, coaxal, freely rotating cylinders to be interposed between the extreme ones whose motion is prescribed, t Ann. d. Chimie, t. xxi. p. 433 (1890). J Proc. Roy. Soc. Vol. LIX. p. 38 (1895). K. VI. 13 194 ON THE MOTION OF A VISCOUS FLUID [376 where R , RI, &c. are functions of r, while e lf e 2 , &c. are constants. If i/r can be thus limited, dT'/dt reduces to F', and the original motion is stable. In general r -**.** s-C, .......... (31) C n , S n must be such as to give at the boundaries C n =Q, dC n /dr = Q, S n = 0, dS n /dr = Q', ............ (32) otherwise they are arbitrary functions of r. It may be noticed that the sign of any term in (29) may be altered at pleasure by interchange of C n and ^ When fj, is great, so that the influence of F preponderates, the motion is stable. On the other hand when //, is small, the motion is probably unstable, unless special restrictions can be imposed. A similar treatment applies to the problem of the uniform shearing motion of a fluid between two parallel plane walls, defined by t> = 0, w = ...................... (33) From (23) ^- = -F'- pBJfu'v'dxdy ...................... (34) If in the superposed motion v' = 0, the double integral vanishes and the original motion is stable. More generally, if the stream-function of the superposed motion be ........................ (35) where C, S are functions of y, we find Here again if the motion can be such that C and 8 differ only by a constant multiplier, the integral would vanish. When p is small and there is no special limitation upon the disturbance, instability probably prevails. The question whether /*, is to be considered great or small depends of course upon the other data of the problem. If D be the distance between the planes, we have to deal with BD>/v (Reynolds). In an important paper* Orr, starting from equation (34), has shown that if B&/V is less than 177 " every disturbance must automatically decrease, and that (for a higher value than 177) it is possible to prescribe a dis- turbance which will increase for a time." We must not infer that when Proc. Roy. Irish Acad. 1907. 1913] ON THE MOTION OF A VISCOUS FLUID 195 BD~/v > 177 the regular motion is necessarily unstable. As the fluid moves under the laws of dynamics, the initial increase of certain disturbances may after a time be exchanged for a decrease, and this decrease may be without limit. At the other extreme when v is very small, observation shows that the tangential traction on the walls, moving (say) with velocities U, tends to a statistical uniformity and to become proportional, no longer to U, but to U 2 . If we assume this law to be absolute in the region -of high velocity, the principle of dynamical similarity leads to rather remarkable conclusions. For the tangential traction, having the dimensions of a pressure, must in general be of the form .............................. < 37 > D being the distance between the walls, and f an arbitrary function. In the regular motion (z large) /(^) = 2z, and (37) is proportional to U. If (37) is proportional to U 2 ,f must be a constant and the traction becomes inde- pendent not only of /j,, but also of D. If the velocity be not quite so great as to reduce /to constancy, we may take f(z) = a + bz, where a and b are numerical constants, so that (37) becomes apUt + bpU/D ............................... (38) It could not be % assumed without further proof that b has the value (2) appropriate to a large z; nevertheless, Korteweg's equation (6) suggests that such may be the case. From data given by Couette I calculate that in c.G.S. measure a = -000027. The tangential traction is thus about a twenty thousandth part of the pressure (%pU*) due to the normal impact of the fluid moving with velocity U. Even in cases where the steady motion of a viscous fluid satisfying the dynamical equations is certainly unstable, there is a distinction to be attended to which is not without importance. It may be a question of the time during which the fluid. remains in an unstable condition. When fluid moves be- tween two coaxal cylinders, the instability has an indefinite time in which to develop itself. But it is otherwise in many important problems. Suppose that fluid has to move through a narrow place, being guided for example by hyperbolic surfaces, either in two dimensions, or in three with symmetry about an axis. If the walls have suitable tangential velocities, the motion 132 196 ON THE MOTION OF A VISCOUS FLUID [376 maybe irrotational. This irrotational motion is that which would be initiated from rest by propellent impulses acting at a distance. If the viscosity were great, the motion would be steady and stable; if the viscosity is less, it still satisfies the dynamical equations, but is (presumably) unstable. But the instability, as it affects any given portion of the fluid, has a very short duration. Only as it approaches the narrows has the fluid any considerable velocity, and as soon as the narrows are passed the velocity falls off again. Under these circumstances it would seem probable that the instability in the narrows would be of little consequence, and that the irrotational motion would practically hold its own. If this be so, the tangential movement of the walls exercises a profound influence, causing the fluid to follow the walls on the down stream side, instead of shooting onwards as a jet the behaviour usually observed when fluid is invited to follow fixed divergent walls, unless indeed the expansion is very gradual. 377. ON THE STABILITY OF THE LAMINAR MOTION OF AN INVISCID FLUID. [Philosophical Magazine, Vol. xxvi. pp. 1001 1010, 1913.] THE equations of motion of an inviscid fluid are satisfied by a motion such that U, the velocity parallel to x, is an arbitrary function of y only, while the other component velocities V and W vanish. The motion may be supposed to be limited by two fixed plane walls for each of which y has a constant value. In order to investigate the stability of the motion, we superpose upon it a two-dimensional disturbance u, v, where u and v are regarded as small. If the fluid is incompressible, ^ + ^=0; ................................. (1) dx dy and if the squares and products of small quantities are neglected, the hydro- dynamical equations give* From (1) and (2), if we assume that as functions of t and a, u and v are proportional to e i(nt+kx} , where k is real and n may be real or complex, In the paper quoted it was shown that under certain conditions n could not be complex ; and it may be convenient to repeat the argument. Let n/k = p + iq, v = a + ift, * Proceedings of London Mathematical Society, Vol. xi. p. 57 (1880) ; Scientific Papers, Vol. i. p. 485. Also Lamb's Hydrodynamics, 345. 198 ON THE STABILITY OF THE [377 where p, q, a, ft are real. Substituting in (3) and equating separately to zero the real and imaginary parts, we get <fa_j, d?U(p+ dy>~ + dy* whence if we multiply the first by ft and the second by a and subtract, A ( R d a d $\- d * U g( a + ff) dy\ P dy *dy)-'dtf (p+U)* + q*' At the limits, corresponding to finite or infinite values of y, we suppose that v, and therefore both a and ft, vanish. Hence when (4) is integrated with respect to y between these limits, the left-hand member vanishes and we infer that q also must vanish unless d^U/dy* changes sign. Thus in the motion between walls if the velocity curve, in which U is ordinate and y abscissa, be of one curvature throughout, n must be wholly real ; otherwise, so far as this argument shows, n may be complex and the disturbance exponen- tially unstable. Two special cases at once suggest themselves. If the motion be that which is possible to a viscous fluid moving steadily between two fixed walls under external pressure or impressed force, so that for example U=y* b 2 , d*U/dy* is a finite constant, and complex values of n are clearly excluded. In the case of a simple shearing motion, exemplified \>yU=y, d*U/dy 3 = Q, and no inference can be drawn from (4). But referring back to (3), we see that in this case if n be complex, would have to be satisfied over the whole range between the limits where v=0. Since such satisfaction is not possible, we infer that here too a complex n is excluded. It may appear at first sight as if real, as well as complex, values of n were excluded by this argument. But if n be such that n/k + U vanishes anywhere within the range, (5) need not there be satisfied. In other words, the arbitrary constants which enter into the solution of (5) may there change values, subject only to the condition of making v continuous. The terminal conditions can then be satisfied. Thus any value of n/k is admissible which coincides with a value of U to be found within the range. But other real values of n are excluded. Let us now examine how far the above argument applies to real values of n, when d*Ujdy* in (3) does not vanish throughout. It is easy to recognize 1913] LAMINAR MOTION OF AN INVISCID FLUID 199 that here also any value of kU is admissible, and for the same reason as before, viz., that when n + kU= 0, dv/dy may be discontinuous. Suppose, for example, that there is but one place where n 4- k U = 0. We may start from either wall with v = and with an arbitrary value of dv/dy and gradually build up the solutions inwards so as to satisfy (3)*. The process is to be continued on both sides until we come to the place where n + kU=Q. The two values there found for v and for dv/dy will presumably disagree. But by suitable choice of the relative initial values of dv/dy, v may be made con- tinuous, and (as has been said) a discontinuity in dv/dy does not interfere with the satisfaction of (3). If there are other places where U has the same value, dv/dy may there be either continuous or discontinuous. Even when there is but one place where n + kU = with the proposed value of n, it may happen that dv/dy is there continuous. The argument above employed is not interfered with even though U is such that dU/dy is here and there discontinuous, so as to make d*U/dy* infinite. At any such place the necessary condition is obtained by integrating (3) across the discontinuity. As was shown in my former paper (loc. cit.\ it is r )_ A (^)..-0 (6) \fljj \dyj A being the symbol of finite differences; and by (6) the corresponding sudden change in dv/dy is determined. It appears then that any value of k U is a possible value of n. Are other real values admissible ? If so, n + k U is of one sign throughout. It is easy to see that if d 2 U/dy' 2 has throughout the same sign as n + k U, no solution is possible. I propose to prove that no solution is possible in any case if n + kU, being real, is of one sign throughout. If U' be written for U + n/k, our equation (3) takes the form U'~-v^ = k*U'v, (7) dy* dy 2 or on integration with respect to y, rr ,dv dU' - 'vdy, .................. (8) dy dy J where K is an arbitrary constant. Assume v = U'v' ; then dv' K * Graphically, the equation directs us with what curvature to proceed at any point already reached. 200 ON THE STABILITY OF THE [377 whence, on integration and replacement of v, 'vdy .......... (10) H denoting a second arbitrary constant. In (10) we may suppose y measured from the first wall, where v 0. Hence, unless U' vanish with y, H=0. Also from (8) when y = 0, Let us now trace the course of v as a function of y, starting from the wall where y = 0, v = ; and let us suppose first that U' is everywhere positive. By (11) K has the same sign as (dv/dy) , that is the same sign as the early values of v. Whether this sign be positive or negative, v as determined by (10) cannot again come to zero. If, for example, the initial values of v are positive, both (remaining) terms in (10) necessarily continue positive; while if v begins by being negative, it must remain finitely negative. Similarly, if U' be everywhere negative, so that K has the opposite sign to that of the early values of v, it follows that v cannot again come to zero. No solution can be found unless U' somewhere vanishes, that is unless n coincides with some value of kU. In the above argument U', and therefore also n, is supposed to be real, but the formula (10) itself applies whether n be real or complex. It is of special value when k is very small, that is when the wave-length along x of the disturbance is very great ; for it then gives v explicitly in the form When k is small, but not so small as to justify (12), a second approximation might be found by substituting from (12) in the last term of (10). If we suppose in (12) that the second wall is situated at y = l, n is determined by The integrals (12), (13) must not be taken through a place where U+n/k = Q, as appears from (8). We have already seen that any value of n for which this can occur is admissible. But (13) shows that no other real value of n is admissible ; and it serves to determine any complex values of n. In (13) suppose (as before) that n/k=p + iq; then separating the real and imaginary parts, we get 1913] LAMINAR MOTION OF AN INVISCID FLUID 201 from the second of which we may infer that if q be finite, p + U must change sign, as we have already seen that it must do when q 0. In every case then, when k is small, the real part of n must equal some value of kU*. It may be of interest to show the application of (13) to a case formerly treatedf in which the velocity-curve is made up of straight portions and is anti-symmetrical with respect to the point lying midway between the two walls, now taken as origin of y. Thus on the positive side from 2/ = to y = |6', U=^-,; from y = i&' to y=W + b, U = + /i7(y -|6') ; while on the negative side U takes symmetrically the opposite values. Then if we write n/kV = n f , (13) becomes rW a y rift' + i = J (fyJb+*y + J' + same with n' reversed. Effecting the integrations, we find after reduction /2 _ n 2 _2b + b' + 2fib(b+b') + ^b*b' .. ~k*V*~ 26 + 6' in agreement with equation (23) of the paper referred to when k is there made small. Hence n, if imaginary at all, is a pure imaginary, and it is imaginary only when p lies between - 1/6 and - 1/6 - 2/6'. The regular motion is then exponentially unstable. In the only unstable cases hitherto investigated the velocity-curve is made up of straight portions meeting at finite angles, and it may perhaps be thought that the instability has its origin in this discontinuity. The method now under discussion disposes of any doubt. For obviously in (13) it can make no important difference whether dU/dy is discontinuous or not. If a motion is definitely unstable in the former case, it cannot become stable merely by easing off the finite angles in the velocity-curve. There exist, therefore, exponentially unstable motions in which both U and dU/dy are continuous. And it is further evident that any proposed velocity-curve may be replaced approximately by straight lines as in my former papers. * By the method of a former paper " On the question of the Stability of the Flow of Fluids " (Phil. Mag. Vol. xxxiv. p. 59 (1892) ; Scientific Papers, Vol. in. p. 579) the conclusion that p+U must change sign may be extended to the problem of the simple shearing motion between two parallel walls of a viscous fluid, and this whatever may be the value of k. t Proc. Land. Math. Soc. Vol. xix. p. 67 (1887); Scientific Papers, Vol. m. p. 20, figs. (3), (4), (5). 202 ON THE STABILITY OF THE [377 The fact that n in equation (15) appears only as w a is a simple conse- quence of the anti-symmetrical character of U. For if in (13) we measure y from the centre and integrate between the limits $1, we obtain in that /JJ w s/t , m I, (n'l^-U'y^ (16) in which only n 9 occurs. But it does not appear that n a is necessarily real, as happens in (15). Apart from such examples as were treated in my former papers in which d?U/dy* vanishes except at certain definite places, there are very few cases in which (3) can be solved analytically. If we suppose that v = sin (Try /I), vanishing when y = and when y = I, and seek what is then admissible for U, we get (17) in which A and B are arbitrary and n may as well be supposed to be zero. But since Ovaries with k, the solution is of no great interest. In estimating the significance of our results respecting stability, we must of course remember that the disturbance has been assumed to be and to remain infinitely small. Where stability is indicated, the magnitude of the admissible disturbance may be very restricted. It was on these lines that Kelvin proposed to explain the apparent contradiction between theoretical results for an inviscid fluid and observation of what happens in the motion of real fluids which are all more or less viscous. Prof. McF. Orr has carried this explanation further *. Taking the case of a simple shearing motion between two walls, he investigates a composite disturbance, periodic with respect to x but not with respect to t, given initially as v = B cos Ixcosmy, (18) and he finds, equation (38), that when m is large the disturbance may increase very much, though ultimately it comes to zero. Stability in the mathe- matical sense (B infinitely small) may thus be not inconsistent with a practical instability. A complete theoretical proof of instability requires not only a method capable of dealing with finite disturbances but also a definition, not easily given, of what is meant by the term. In the case of stability we are rather better situated, since by absolute stability we may understand complete recovery from disturbances of any kind however large, such as Reynolds showed to occur in the present case when viscosity is paramount f. In the absence of dissipation, stability in this sense is not to be expected. * Proc. Roy. Irith Academy, Vol. xivn. Section A, No. 2, 1907. Other related questions are also treated. t See also Orr, Proc. Boy. Irith Academy, 1907, p. 124. 1913] LAMINAR MOTION OF AN INVISC1D FLUID 203 Another manner of regarding the present problem of the shearing motion of an inviscid fluid is instructive. In the original motion the vorticity is constant throughout the whole space between the walls. The disturbance is represented by a superposed vorticity, which may be either positive or nega- tive, and this vorticity everywhere moves with the fluid. At any subsequent time the same vorticities exist as initially ; the only question is as to their distribution. And when this distribution is known, the whole motion is determined. Now it would seem that the added vorticities will produce most effect if the positive parts are brought together, and also the negative parts, as much as is consistent with the prescribed periodicity along x, and that even if this can be done the effect cannot be out of proportion to the magnitude of the additional vorticities. If this view be accepted, the temporary large increase in Prof. Orr's example would be attributed to a specially unfavourable distribution initially in which (m large) the positive and negative parts of the added vorticities are closely intermingled. We may even go further and regard the subsequent tendency to evanescence, rather than the temporary increase, as the normal phenomenon. The difficulty in reconciling the observed behaviour of actual fluids with the theory of an inviscid fluid still seems to me to be considerable, unless indeed we can admit a distinction between a fluid of infinitely small viscosity and one of none at all. At one time I thought that the instability suggested by observation might attach to the stages through which a viscous liquid must pass in order to acquire a uniform shearing motion rather than to the final state itself. Thus in order to find an explanation of " skin friction " we may suppose the fluid to be initially at rest between two infinite fixed walls, one of which is then suddenly made to move in its own plane with a uniform velocity. In the earlier stages the other wall has no effect and the problem is one considered by Fourier in connexion with the conduction of heat. The velocity U in the laminar motion satisfies generally an equation of the form dU d*U with the conditions that initially (t = 0) U = 0, and that from t = onwards U=l when y = 0, and (if we please) U = when y = I. We might employ Fourier's solution, but all that we require follows at once from the differential equation itself. It is evident that dU/dt, and therefore d*Ujdy*, is every r where positive and accordingly that a non-viscous liquid, moving laminarly as the viscous fluid moves in any of these stages, is stable. It would appeal- then that no explanation is to be found in this direction. Hitherto we have supposed that the disturbance is periodic as regards x, but a simple example, not coming under this head, may be worthy of notice. It is that of the disturbance due to a single vortex filament in which the ON THE STABILITY OF THE LAMINAR MOTION OF AN INVISCID FLUID [377 vorticity differs from the otherwise uniform vorticity of the neighbouring fluid. In the figure the lines A A, BB represent the situation of the walls and AM the velocity-curve of the original shearing motion rising from zero at A to a finite value at M. For the present purpose, however, we suppose material walls to be absent, but that the same effect (of prohibiting normal motion) is arrived at by suitable suppositions as to the fluid lying outside and now imagined infinite. It is only necessary to continue the velocity-curve in the manner shown AMCN... , the vorticities in the alternate layers of equal width being equal and opposite. Symmetry then shows that under the operation of these vorticities the fluid moves as if AA, BB, &c. were material walls. C' B' A B C D E We have now to trace the effect of an additional vorticity, supposed posi- tive, at a point P. If the wall AA were alone concerned, its effect would be imitated by the introduction of an opposite vorticity at the point Q which is the image of P in AA. Thus P would move under the influence of the original vorticities, already allowed for, and of the negative vorticity at Q. Under the latter influence it would move parallel to A A with a certain velocity, and for the same reason Q would move similarly, so that PQ would remain perpendicular to A A. To take account of both walls the more com- plicated arrangement shown in the figure is necessary, in which the points P represent equal positive vorticities and Q equal negative vorticities. The conditions at both walls are thus satisfied; and as before all the vortices P, Q move under each other's influence so as to remain upon a line perpen- dicular to AA. Thus, to go back to the original form of the problem, P moves parallel to the walls with a constant velocity, and no change ensues in the character of the motion a conclusion which will appear the more remarkable when we remember that there is no limitation upon the magnitude of the added vorticity. The same method is applicable in imagination at any rate whatever be the distribution of vorticities between the walls, and the corresponding velocity at any point is determined by quadratures on Helinholtz's principle. The new positions of all the vorticities after a short time are thus found, and then a new departure may be taken, and so on indefinitely. 378. REFLECTION OF LIGHT AT THE CONFINES OF A DIFFUSING MEDIUM. [Nature, Vol. xcii. p. 450, 1913.] I SUPPOSE that everyone is familiar with the beautifully graded illumina- tion of a paraffin candle, extending downwards from the flame to a distance of several inches. The thing is seen at its best when there is but one candle in an otherwise dark room, and when the eye is protected from the direct light of the flame. And it must often be noticed when a candle is broken across, so that the two portions are held together merely by the wick, that the part below the fracture is much darker than it would otherwise be, and the part above brighter, the contrast between the two being very marked. This effect is naturally attributed to reflection, but it does not at first appear that the cause is adequate, seeing that at perpendicular incidence the re- flection at the common surface of wax and air is only about 4 per cent. A little consideration shows that the efficacy of the reflection depends upon the incidence not being limited to the neighbourhood of the perpendicular. In consequence of diffusion* the propagation of light within the wax is not specially along the length of the candle, but somewhat approximately equal in all directions. Accordingly at a fracture there is a good deal of " total reflection." The general attenuation downwards is doubtless partly due to defect of transparency, but also, and perhaps more, to the lateral escape of light at the surface of the candle, thereby rendered visible. By hindering this escape the brightly illuminated length may be much increased. The experiment may be tried by enclosing the candle in a reflecting tubular envelope. I used a square tube composed of four rectangular pieces of mirror glass, 1 in. wide, and 4 or 5 in. long, held together by strips of * To what is the diffusion due ? Actual cavities seem improbable. Is it chemical hetero- geneity, or merely varying orientation of chemically homogeneous material operative in virtue of double refraction ? 206 REFLECTION OF LIGHT AT THE [378 pasted paper. The tube should be lowered over the candle until the whole of the flame projects, when it will be apparent that the illumination of the candle extends decidedly lower down than before. In imagination we may get quit of the lateral loss by supposing the diameter of the candle to be increased without limit, the source of light being at the same time extended over the whole of the horizontal plane. To come to a definite question, we may ask what is the proportion of light reflected when it is incident equally in all directions upon a surface of transition, such as is constituted by the candle fracture. The answer depends upon a suitable integration of Fresnel's expression for the re- flection of light of the two polarisations, viz. sin 2 (0-0') tan 2 (0-0') '' tan" '' where 0, 0' are the angles of incidence and refraction. We may take first the case where > 0', that is, when the transition is from the less to the more refractive medium. The element of solid angle is 2-Tr sin dO, and the area of cross-section corresponding to unit area of the refracting surface is cos ; so that we have to consider 2 (** sin cos (S 2 or T 2 ) d6, . . .(2) Jo the multiplier being so chosen as to make the integral equal to unity when S* or T 2 has that value throughout. The integral could be evaluated analytically, at any rate in the case of S 2 , but the result would scarcely repay the trouble. An estimate by quadratures in a particular case will suffice for our purposes, and to this we shall presently return. In (2) varies from to TT and 6' is always real. If we suppose the passage to be in the other direction, viz. from the more to the less refractive medium, S 1 and T 2 , being symmetrical in and 0', remain as before, and we have to integrate 2 sin 0' cos 0' (S* or T 2 ) d0'. The integral divides itself into two parts, the first from to o, where o is the critical angle corresponding to = TT. In this S 1 , T* have the values given in (1). The second part of the range from 6' = a. to 0' = ^TT involves " total reflection," so that S 1 and T 2 must be taken equal to unity. Thus altogether we have 2 fsin 0' cos & (S 2 or T 2 ) d6' + 2 t mn 0' cos 6'd6', ...... (3) .'O J a 1913] CONFINES OF A DIFFUSING MEDIUM 207 in which sin a = I//*, /JL (greater than unity) being the refractive index. In (3) 2 sin 6' cos 6' d& = d sin 2 6' = p-*d sin 2 6, and thus (3) = /*- x (2) + 1 - /a- 2 = - a U 2 - 1+ [ i>r sin 20 (S 2 or T 2 ) d0\, . . .(4) A*" ( Jo } expressing the proportion of the uniformly diffused incident light reflected in this case. Much the more important part is the light totally reflected. If /A = 1*5, this amounts to 5/9 or 0*5556. With the same value of /*, I find by Weddle's rule f ^ sin 20 . S 2 d0 = 0-1460, f sin 20 . T z d0 = 0-0339. Jo Jo Thus for light vibrating perpendicularly to the plane of incidence (4) = 0-5556 + 0-0649 = 0*6205 ; while for light vibrating in the plane of incidence (4) = 0-5556 + 0-0151 = 0'5707. The increased reflection due to the diffusion of the light is thus abundantly explained, by far the greater part being due to the total reflection which ensues when the incidence in the denser medium is somewhat oblique. 379. THE PRESSURE OF RADIATION AND CARNOT'S PRINCIPLE. [Nature, Vol. xcn. pp. 527, 528, 1914.] As is well known, the pressure of radiation, predicted by Maxwell, and since experimentally confirmed by Lebedew and by Nichols and Hull, plays an important part in the theory of radiation developed by Boltzmann and W. Wien. The existence of the pressure according to electromagnetic theory is easily demonstrated*, but it does not appear to be generally remembered that it could have been deduced with some confidence from thermodynamical principles, even earlier than in the time of Maxwell. Such a deduction was, in fact, made by Bartoli in 1876, and constituted the foundation of Boltz- mann's work f . Bartoli's method is quite sufficient for his purpose ; but, mainly because it employs irreversible operations, it does not lend itself to further developments. It may therefore be of service to detail the elementary argument on the lines of Carnot, by which it appears that in the absence of a pressure of radiation it would be possible to raise heat from a lower to a higher temperature. The imaginary apparatus is, as in Boltzmann's theory, a cylinder and piston formed of perfectly reflecting material, within which we may suppose the radiation to be confined. This radiation is always of the kind charac- terised as complete (or black), a requirement satisfied if we include also a very small black body with which the radiation is in equilibrium. If the operations are slow enough, the size of the black body may be reduced without limit, and then the whole energy at a given temperature is that of the radiation and proportional to the volume occupied. When we have occasion to introduce or abstract heat, the communication may be supposed * See, for example, J. J. Thomson, Elements of Electricity and Magnetism (Cambridge, 1895, 241); Rayleigh, Phil. Mag. Vol. XLV. p. 222 (1898); Scientific Papers, Vol. iv. p. 364. t Wied. Ann. Vol. XXXH. pp. 31, 291 (1884). It is only through Boltzmann that I am acquainted with Bartoli's reasoning. 1914] THE PRESSURE OF RADIATION AND CARNOT's PRINCIPLE 209 in the first instance to be with the black body. The operations are of two kinds: (1) compression (or rarefaction) of the kind called adiabatic, that is, without communication of heat. If the volume increases, the temperature must fall, even though in the absence of pressure upon the piston no work is done, since the same energy of complete radiation now occupies a larger space. Similarly a rise of temperature accompanies adiabatic contraction. In the second kind of operation (2) the expansions and contractions are isothermal that is, without change of temperature. In this case heat must pass, into the black body when' the volume expands and out of it when the volume contracts, and at a given temperature the amount of heat which must pass is proportional to the change of volume. The cycle of operations to be considered is the same as in Carnot's theory, the only difference being that here, in the absence of pressure, there is no question of external work. Begin by isothermal expansion at the lower temperature during which heat is taken in. Then compress adiabatically until a higher temperature is reached. Next continue the compression iso- thermally until the same amount of heat is given out as was taken in during the first expansion. Lastly, restore the original volume adiabatically. Since no heat has passed upon the whole in either direction, the final state is identical with the initial state, the temperature being recovered as well ap the volume. The sole result of the cycle is that heat is raised from a lower to a higher temperature. Since this is assumed to be impossible, the sup- position that the operations can be performed without external work is to be rejected in other words, we must regard the radiation as exercising a pressure upon the moving piston. Carnot's principle and the absence of a pressure are incompatible. For a further discussion it is, of course, desirable to employ the general formulation of Carnot's principle, as in a former paper*. If p be the pressure, 6 the absolute temperature, where M dv represents the heat that must be communicated, while the volume alters by dv and dd = 0. In the application to radiation M cannot vanish, and therefore p cannot. In this case clearly M=U + p .................................. (30) where U denotes the volume-density of the energy a function of 8 only. Hence < 31 > * "On the Pressure of Vibrations," Phil. Mag. Vol. in. p. 338, 1902; Scientific Papers, Vol. v. p. 47. K. VI. H 210 THE PRESSURE OF RADIATION AND CARNOT'S PRINCIPLE [379 If we assume from electromagnetic theory that P = W, (32) it follows at once that tfoctf*, (33) the well-known law of Stefan. In (31) if p be known as a function of 6, U as a function of 6 follows immediately. If, on the other hand, U be known, we have and thence 380. FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF HIGH ORDER TO THE WHISPERING GALLERY AND ALLIED PROBLEMS. [Philosophical Magazine, Vol. xxvn. pp. 100109, 1914.] IN the problem of the Whispering Gallery* waves in two dimensions, of length small in comparison with the circumference, were shown to run round the concave side of a wall with but little tendency to spread themselves inwards. The wall was supposed to be perfectly reflecting for all kinds of waves. But the question presents itself whether the sensibly perfect re- flexion postulated may not be attained on the principle of so-called "total reflexion," the wall being merely the transition between two uniform media of which the outer is the less refracting. It is not to be expected that absolutely no energy should penetrate and ultimately escape to an infinite distance. The analogy is rather with the problem treated by Stokes f of the communication of vibrations from a vibrating solid, such as a bell or wire, to a surrounding gas, when the wave-length in the gas is somewhat large compared with the dimensions of the vibrating segments. The energy radiated to a distance may then be extremely small, though not mathe- matically evanescent. A comparison with the simple case where the surface of the vibrating body is plane (x = 0) is interesting, especially as showing how the partial * Phil. Mag. Vol. xx. p. 1001 (1910); Scientific Papers, Vol. v. p. 619. But the numbers there given require some correction owing to a slip in Nicholson's paper from which they were derived, as was first pointed out to me by Prof. Macdonald. Nicholson's table should be inter- preted as relating to the values, not of 2-1123 (n -z)jz*, but of 1*3447 (*-*)/!*, see Nicholson, Phil. Mag. Vol. xxv. p. 200 (1913). Accordingly, in my equation (5) 1*1814*' should read 1*8558 *, and in elation (8) -51342 * should read -8065 n*. [1916. Another error should be f noticed. In (3), = I cos n (w- sin u) dujir must be omitted, the integrand being periodic. See Watson, Phil. Mag. Vol. xxxn. p. 233, 1916.] t Phil. Tram. 1868. See Ttieory of Sound, Vol. n. 324. 142 212 FURTHER APPLICATIONS OF BESSEI/S FUNCTIONS OF [380 escape of energy is connected with the curvature of the surface. If V be the velocity of propagation, and Zir/k the wave-length of plane waves of the given period, the time-factor is e ikvt , and the equation for the velocity- potential in two dimensions is If be also proportional to cos my, (1) reduces to ** + (*- w)$ = 0, ........................... (2) of which the solution changes its form when m passes through the value k. For our purpose 7/1 is to be supposed greater than k, viz. the wave-length of plane waves is to be greater than the linear period along y. That solution of (1) on the positive side which does not become infinite with x is propor- tional to g- >/(*-**), so that we may take <f> = coskVt.cosmy.e-*^ m > l -' fl) ...................... (3) However the vibration may be generated at x = 0, provided only that the linear period along y be that assigned, it is limited to relatively small values of x and, since no energy can escape, no work is done on the whole at x = 0. And this is true by however little m may exceed k. The reason of the difference which ensues when the vibrating surface is curved is now easily seen. Suppose, for example, that in two dimensions < is proportional to cos nff, where 6 is a vectorial angle. Near the surface of a cylindrical vibrator the conditions may be such that (3) is approximately applicable, and <j> rapidly diminishes as we go outwards. But when we reach a radius vector r which is sensibly different from the initial one, the con- ditions may change. In effect the linear dimension of the vibrating compartment increases proportionally to r, and ultimately the equation (2) changes its form and <f> oscillates, instead of continuing an exponential decrease. Some energy always escapes, but the amount must be very small if there is a sufficient margin to begin with between m and k. It may be well before proceeding further to follow a little more closely what happens when there is a transition at a plane surface x = from a more to a less refractive medium. The problem is that of total reflexion when the incidence is grazing, in which case the usual formulas* become nugatory. It will be convenient to fix ideas upon the case of sonorous waves, but the results are of wider application. The general differential equation is of the form ( } * See for example Theory of Sound, Vol. n. 270. 1914] HIGH ORDER TO THE WHISPERING GALLERY 213 which we will suppose to be adapted to the region where x is negative. On the right (x positive) V is to be replaced by V lt where V l > V, and </> by <f> 1 . In optical notation Fj/F=/x, where //, (greater than unity) is the refractive index. We suppose < and fa to be proportional to e i(by+ct> , b and c being the same in both media. Further, on the left we suppose b and c to be related as they would be for simple plane waves propagated parallel to y. Thus (4) becomes, with omission of e i{by+et >, -O, **-(, -I)/,', .................. ...(5) da? da? of which the solutions are A, B, G denoting constants so far arbitrary. The boundary conditions require that when #=0, d<f>/dx = d<j) 1 /da; and that p^ p^i, p, pi being the densities. Hence discarding the imaginary part, and taking -4 = 1, we get finally <}>=\l- pbX ^~ l) \cos(by + ct\ (7) (8) PI It appears that while nothing can escape on the positive side, the amplitude on the negative side increases rapidly as we pass away from the surface of transition. If p, < 1, a wave of the ordinary kind is propagated into the second medium, and energy is conveyed away. In proceeding to consider the effect of curvature it will be convenient to begin with Stokes' problem, taking advantage of formulae relating to Bessel's and allied functions of high order developed by Lorenz, Nicholson, and Macdonald*. The motion is supposed to take place in two dimensions, and ideas may be fixed upon the case of aerial vibrations. The velocity- potential < is expressed by means of polar coordinates r, 0, and will be assumed to be proportional to cos nd, attention being concentrated upon the case where n is a large integer. The problem is to determine the motion at a distance due to the normal vibration of a cylindrical surface at r = a, and it turns upon the character of the function of.r which represents a disturbance propagated outwards. If D n (kr) denote this function, we have <f> = e ik cosn0.D n (kr), ........................ (9) and D n (z) satisfies Bessel's equation (10) * Compare also Debye, Math. Ann. Vol. LXVII. (1909). 214 FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF [380 It may be expressed in the form - in ', , ...(11) which, however, requires a special evaluation when n is an integer. Using Schlafli's formula n being positive or negative, and z positive, we find D n (*)--(" e n *-* 8inh de + ^-^ f X e-'-* 8illh d0 T.'O 7T .'0 -- I' sm(zsin0-n0)d0-- I" cos(zsin0 - n0)d0, ...... (13) TjQ TTJo the imaginary part being iJ n (z) simply. This holds good for any integral value of n. The present problem requires the examination of the form assumed by D n when n is very great and the ratio z/n decidedly greater, or decidedly less, than unity. In the former case we set n = z sin a, and the important part of D n arises from the two integrals last written. It appears* that (14) 1TZ COS a/ where p = \ir + z {cos a (TT a) sin a}, .................. (15) or when z is extremely large (a = 0) (16) At a great distance the value of <f> in (9) thus reduces to from which finally the imaginary part may be omitted. When on the other hand z/n is decidedly less than unity, the most important part of (13) arises from the first and last integrals. We set n = .zcoshy9, and then, n being very great, where t = n (tanh ft - ft) ............................ (19) ' Nicholson, B. A. Report, Dublin, 1908, p. 595 ; Phil. Mag. Vol. xix. p. 240 (1910); Mac- donald, Phil. Tram. Vol. ccx. p. 135 (1909). . 1914] HIGH ORDER TO THE WHISPERING GALLERY 215 Also, the most important part of the real and imaginary terms being retained, The application is now simple. From (9) with introduction of an arbitrary coefficient (21) If we suppose that the normal velocity of the vibrating cylindrical surface (r = a) is represented by e ikvt cosn0, we have kAD n '(ka) = I, .............................. (22) and thus at distance r or when r is very great / 2 \*e*{*<- *)-!} A = cosw0(--) , _ ,,. , ................... (24) \irkr) kDn(ka) We may now, following Stokes, compare the actual motion at a distance with that which would ensue were lateral motion prevented, as by the insertion of a large number of thin plane walls radiating outwards along the lines 6 = constant, the normal velocity at r = a being the same in both cases. In the altered problem we have merely in (23) to replace D n , D n ' by DO, DQ. When z is great enough, D n (z) has the value given in (16), independently of the particular value of n. Accordingly the ratio of velocity-potentials at a distance in the two cases is represented by the symbolic fraction in which I) / (ka) = -i-e- i ^+ k ^ ................... (26) We have now to introduce the value of D n ' (ka). When n is very great, and ka/n decidedly less than unity, t is negative in (20), and e* is negligible in comparison with er*. The modulus of (25) is therefore -n(g-tanh sinh* ft For example, if n = 2ka, so that the linear period along the circumference of the vibrating cylinder (2ira/w) is half the wave-length, cosh ^ = 2, =1-317, sinh/8 = 1-7321, tanh ft = '8660, and the numerical value of (27) is e --ion j. ^(1-732). 216 FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF [380 When n is great, the vibration at a distance is extraordinarily small in com- parison with what it would have been were lateral motion prevented. As another example, let n=Mo, Then (27) = e- w 4- V('4587). Here n would need to be about 17 times larger for the same sort of effect. The extension of Stokes' analysis to large values of n only emphasizes his conclusion as to the insignificance of the effect propagated to a distance when the vibrating segments are decidedly smaller than the wave-length. We now proceed to the case of the whispering gallery supposed to act by " total reflexion." From the results already given, we may infer that when the refractive index is moderate, the escape of energy must be very small, and accordingly that the vibrations inside have long persistence. There is, however, something to be said upon the other side. On account of the con- centration near the reflecting wall, the store of energy to be drawn upon is diminished. At all events the problem is worthy of a more detailed examination. Outside the surface of transition (r = a) we have the same expression (9) as before for the velocity-potential, k and V having values proper to the outer medium. Inside k and V are different, but the product kV is the same. We will denote the altered k by h. In accordance with our sup- positions h > k, and h/k represents the refractive index (/LI) of the inside medium relatively to that outside. On account of the damping k and h are complex, though their ratio is real ; but the imaginary part is relatively small. Thus, omitting the factors e ikvt cos n0, we have (? > a) <f> = AD n (kr), (28) and inside (r < a) (f> = BJ n (hr) (29) The boundary conditions to be satisfied when r = a are easily expressed. The equality of normal motions requires that kAD n '(ka) = hBJn(ha); (30) and the equality of pressures requires that <rAD n (ka) = pJ n (ha), (31) a-, p being the densities of the outer and inner media respectively. The equation for determining the values of ha, ka (in addition to h/k = p) is accordingly kD n '(ka) hJ n '(ha) <rD n (ka) pJ n (ha)' Equation (32) cannot be satisfied exactly by real values of h and k ; for, although JnjJ n is then real, D n '/D n includes an imaginary part. But since the imaginary part is relatively small, we may conclude that approximately h and k are real, and the first step is to determine these real values. 1914] HIGH ORDER TO THE WHISPERING GALLERY 217 Since ka is supposed to be decidedly less than n, D n and D n ' are given by (18), (20); and, if we neglect the imaginary part, D n ' (ka) D n (ka) -sinh/3 (33) Thus (32) becomes = -sinh/3, ...(34) J n (ha) <rh the right-hand member being real and negative. Of this a solution can always be found in which ha = n very nearly. For* J n (z) increases with z from zero until z = n + '8065 w , when J n '(z) = Q, and then decreases until it vanishes when z = n + 1 '8558 /A Between these limits for z, J n '/J n assumes all possible negative values. Substituting n for ha on the right in (34), we get _ s i n h/3, or -tanh/3, ....(35) an a while cosh $ = JJL. The approximate real value of ha is thus n simply, while that of ka is n/jA. These results, though stated for aerial vibrations, have as in all such (two-dimensional) cases a wider application, for example to electrical vibra- tions, whether the electric force be in or perpendicular to the plane of r, 6. For ordinary gases, of which the compressibility is the same, Hitherto we have neglected the small imaginary part of D n '{D n . By (18), (20), when z is real, approximately, with cosh ft n/z. We have now to determine what small imaginary additions must be made to ha, ka in order to satisfy the complete equation. Let us assume ha = x + iy, where x and y are real, and y is small. Then approximately Jn (X + Jy) Jn QP) J n (x + iy) Jn (x) + iy J n (x) ' and J n " (X) = - - J n ' (X) - (l - ~\ J n (x). X \ / Since the approximate value of x is n, Jn" is small compared with J n or 7 n ', and we may take - >(37) See paper quoted on p. 211 and correction. 218 FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF [380 Similarly, if we write ka = x' + iy' , where x' = x/p, y' = yjfi, D n ' (x' + iy') D n ' (of) + iy D n " (x') Dn (*' + iy') I> <0 + # A/ 00 ' and in virtue of (10) D n ( X '} = - ^S. D n ' (x') + sinh' ft D n (x), where cosh ft = nja/. Thus Accordingly with use of (36) Equation (32) asserts the equality of the expressions on the two sides of (38) with h<rJ n '(x) kp J n (x) If we neglect the imaginary terms in (38), (37), we fall back on (34). The imaginary terms themselves give a second equation. In forming this we notice that the terms in y' vanish in comparison with that in y. For in the coefficient of y' the first part, viz. n,- 1 cosh ft, vanishes when n is made infinite, while the second and third parts compensate one another in virtue of (33). Accordingly (32) gives with regard to (34) ffh ** *"*** ... (39 ) sinh/3 ' ' in which coshft = jj, (40) In (39) iy is the imaginary increment of ha, of which the principal real part is n. In the time-factor e ikrt , the exponent 7, TTf "'""JTT In one complete period T, nVt/fjta undergoes the increment 2?r. The ex- ponential factor giving the decrement in one period is thus or with regard to the smallness of (39) "^ sinh/S This is the factor by which the amplitude is reduced after each complete period. 1914] HIGH ORDER TO THE WHISPERING GALLERY 219 In the case of ordinary gases p/<r = /* 2 . As an example, take ft = cosh (3 1*3 ; then (42) gives e- 236n . ........................... (43) When n rises beyond 10, the damping according to (43) becomes small ; and when n is at all large, the vibrations have very great persistence. In the derivation of (42) we have spoken of stationary vibrations. But the damping is, of course, the same for vibrations which progress round the circumference, since these may be regarded as compounded of two sets of stationary vibrations which differ in phase by 90. Calculation thus confirms the expectation that the whispering gallery effect does not require a perfectly reflecting wall, but that the main features are reproduced in transparent media, provided that the velocity of waves is moderately larger outside than inside the surface of transition. And further, the less the curvature of this surface, the smaller is the refractive index (greater than unity) which suffices. 381. ON THE DIFFRACTION OF LIGHT BY SPHERES OF SMALL* RELATIVE INDEX. [Proceedings of the Royal Society, A, Vol. xc. pp. 219225, 1914.] IN a short paper " On the Diffraction of Light by Particles Comparable with the Wave-length f," Keen and Porter describe curious observations upon the intensity and colour of the light transmitted through small particles of precipitated sulphur, while still in a state of suspension, when the size of the particles is comparable with, or decidedly larger than, the wave-length of the light. The particles principally concerned in their experiments appear to have decidedly exceeded those dealt with in a recent paperj, where the calculations were pushed only to the point where the circumference of the sphere is 2*25 \. The authors cited give as the size of the particles, when the intensity of the light passing through was a minimum, 6 fj, to 10 p, that is over 10 wave-lengths of yellow light, and they point out the desirability of extending the theory to larger spheres. The calculations referred to related to the particular case where the (relative) refractive index of the spherical obstacles is 1*5. This value was chosen in order to bring out the peculiar polarisation phenomena observed in the diffracted light at angles in the neighbourhood of 90, and as not inappro- priate to experiments upon particles of high index suspended in water. I remarked that the extension of the calculations to greater particles would be of interest, but that the arithmetical work would rapidly become heavy. There is, however, another particular case of a more tractable character, viz., when the relative refractive index is small*; and although it may not be the one we should prefer, its discussion is of interest and would be expected * [1914. It would have been in better accordance with usage to have said "of Relative Index differing little from Unity."] t Roy. Soc. Proc. A, Vol. LXXXIX. p. 370 (1918). J Roy. Soc. Proc. A, Vol. LXXXIV. p. 25 (1910) ; Scientific Papers, Vol. v. p. 547. 1914] DIFFRACTION OF LIGHT BY SPHERES OF SMALL RELATIVE INDEX 221 to throw some light upon the general course of the phenomenon. It has already been treated up to a certain point, both in the paper cited and the earlier one * in which experiments upon precipitated sulphur were first described. It is now proposed to develop the matter further. The specific inductive capacity of the general medium being unity, that of the sphere of radius R is supposed to be K, where K 1 is very small. Denoting electric displacements by/, g, h, the primary wave is taken to be so that the direction of propagation is along x (negatively), and that of vibration parallel to z. The electric displacements (f I} g 1} Aj) in the scattered wave, so far as they depend upon the first power of (K 1), have at a great distance the values in which P = -(tf-l).e<e*^><&dy<fc ................... (3) In these equations r denotes the distance between the point (a, 0, ry) where the disturbance is required to be estimated, and the element of volume (dx dy dz) of the obstacle. The centre of the sphere R will be taken as the origin of coordinates. It is evident that, so far as the secondary ray is concerned, P depends only upon the angle (^) which this ray makes with the primary ray. We will suppose that % = in the direction backwards along the primary ray, and that % = TT along the primary ray continued. The integral in (3) may then be found in the form t * J Jo - cos cos2 k cos %x r now denoting the distance of the point of observation from the centre of the sphere. Expanding the Bessel's function, we get 4,7rR s (K-l)e i(nt -^ ~ 2.4.5.7 2.4.6.5.7.9 2.4.6.8.5.7.9.11 in which m is written for ZkRcosfa- It is to be observed that in this solution there is no limitation upon the value of R if {K I) 2 is neglected absolutely. In practice it will suffice that (K-l) R/\ be small, X (equal to 2-7T/&) being the wave-length. * Phil. Mag. Vol. xn. .p. 81 (1881) ; Scientific Papers, Vol. i. p. 518. 222 ON THE DIFFRACTION OF LIGHT BY [381 These are the formulae previously given. I had not then noticed that the integral in (4) can be expressed in terms of circular functions. By a general theorem due to Hobson * J r ** T t , j, If TT \ , , sin m cos w t J, <m cos*) cos' -M-M^gW /,(>> = ,- __, ...... (6) so that P = -(K-l).^R>. --*> (!_<5) ..... (7) 3 * in agreement with (5). The secondary disturbance vanishes with P, viz., when tan m = m, or 7r(r4303, 2-4590, 3*4709, 4-4774, 5 "4818, etc.)f. ...(8) The smallest value of kR for which P vanishes occurs when % = 0, i.e. in the direction backwards along the primary ray. In terms of \ the diameter is 2# = 0'715\. ................................. (9) In directions nearly along the primary ray forwards, cos %x ^ s small, and evanescence of P requires much larger ratios of R to X. As was formerly fully discussed, the secondary disturbance vanishes, independently of P, in the direction of primary vibration (o = 0, $ = 0). In general, the intensity of the secondary disturbance is given by in which P denotes P with the factor e i (nt ~ kr) omitted, and is a function of x, the angle between the secondary ray and the axis of x. If we take polar coordinates (x, <f>) round the axis of x, 1 - ^ = 1 -sin a x cos a <J>; ........................ (11) and the intensity at distance r and direction (^, </>) may be expressed in terms of these quantities. In order to find the effect upon the transmitted light, we have to integrate (10) over the whole surface of the sphere r. Thus f f J Js -f h*) = TT ^ sin x d x \j) (1 + cos 2 x ) (sin m m cos m) a (m a -l)cos2m-2msin2m} ....... (12) * Land. Math. Soc. Proc. Vol. xxv. p. 71 (1893). t See Theory of Sound, Vol. n. 207. 1914] SPHERES OF SMALL RELATIVE INDEX 223 The integral may be expressed by means of functions regarded as known. Thus on integration by parts \ m (1 + m 2 + (m 2 - 1) cos 2m - 2m sin 2m} ^ 1 cos 2m sin 2m 1 1 4m 4 " ' 2m 3 ~~ 2m 2 + 2 ' I m [I + m 2 + (m 2 - 1) cos 2m - 2m sin 2m} -^ Jo wi I [ m 1 cos 2m cos 2m sin 2m t m (1 + m 2 + (m 2 - 1) cos 2m - 2m sin 1m] Jo m [ m l cos 2m 7 m 2 m sin 2m 5 cos 2m 5 _ I _ fifVYi .1. __ I __ _ J __ \AjUl ~f ~|~ . ~r . Jo m 22 44 Accordingly, if m now stand for *2kR, we get - 1 ) 2 f 7(1- cos 2m ) r 2 sm f 7(1- / 4 . \ f m 1 cos 2m , ) 5+m*+( -4 dm\ ....... (13) Vm 2 /7 * J m If m is small, the { } in (13) reduces to -f x m 2 4- ^ m 4 , so that ultimately l) 2 , ........................ (14) in agreement with the result which may be obtained more simply from (5). If we include another term, we get As regards the definite integral, still written as such, in (13), we have where 7 is Euler's constant (O5772156) and Ci is the cosine-integral, defined by [ x COS U 7 /I >7\ Ci(#)= I ^-du ............................ (17) As in (16), when x is moderate, we may use + i-... 1 ............ (18) 224 ON THE DIFFRACTION OF LIGHT BY [381 which is always convergent. When x is great, we have the semi-convergent series 11.2 1.2.3.4 -... (19) l 1.2.3 1.2.3.4.5 Fairly complete tables of Ci (#), as well as of related integrals, have been given by Glaisher*. When m is large, Ci (2m) tends to vanish, so that ultimately f m 1 cos 2m 7 dm = 7 + log (2m). Hence, when kR is large, (13) tends to the form .(20) Glaisher's Table XII gives the maxima and minima values of the cosine- integral, which occur when the argument is an odd multiple of TT. Thus : n Ci (n7r/2) |j n i: Ci (iw/2) 1 +0-4720007 3 -0-1984076 5 +0-1237723 : s 11 - 0-0895640 + 0-0700653 -0-0575011 These values allow us to calculate the { } in (13), viz., 7(1 cos 2m) sin 2m 2m 2 4- 5 + m 2 + - 4) [ 7 + log 2m - Ci (2m)], (21) when 2m = n?r/2, and n is an odd integer. In this case cos 2m = and sin 2m = 1, so that (21) reduces to fi4 \ - *) [7 + log(r/2) - Ci ( (22) We find (22) n (22) 1 0-0530 7 23-440 3 2-718 9 42-382 5 10-534 11 65-958 Phil. Trans. Vol. CLX. p. 367 (1870). 1914] SPHERES OF SMALL RELATIVE INDEX 225 For values of n much greater, (22) is sufficiently represented by nV 2 /16, or m"- : simply. It appears that there is no tendency to a falling-off in the scattering, such as would allow an increased transmission. In order to make sure that the special choice of values for m has not masked a periodicity, I have calculated also the results when n is even. Here sin 2m = and cos 2m = ] , so that (21) reduces to The following are required : n Ci (nir/2) n Ci (BT/2) 2 4 6 + 0-0738 8 -0-0224 10 + 0-0106 -0-0061 +0-0040 of which the first is obtained by interpolation from Glaisher's Table VI, and the remainder directly from (19). Thus: n (23) n (23) 2 0-7097 8 32-336 4 6-1077 10 53-477 6 16-156 The better to exhibit the course of the calculation, the actual values of the several terms of (23) when n = 10 may be given. We have -=-0-11348, ^ = 6V685, 22 16 4 4 - ~~ = 4 - 0-06485 = 3-93515, 7 + log (-7T/2) + log n - Ci (mr/2) = 0-57722 + 0*45158 + 2'30259 - 0'0040 = 13-094, so that 4 - - {7 + log (rwr/2) - Ci (mr/2)} = 13'094. It will be seen that from this onwards the term ?i 2 7T 2 /16, viz., m 2 , greatly preponderates ; and this is the term which leads to the limiting form (20). The values of 2R/X concerned in the above are very moderate. Thus, n = 10, making m = 47rR/\ = 10?r/4, gives 2R/\ = 5/4 only. Neither below R. VI. 15 226 DIFFRACTION OF LIGHT BY SPHERES OF SMALL RELATIVE INDEX [381 this point, nor beyond it, is there anything but a steady rise in the value of (13) as X diminishes when R is constant. A fortiori is this the case when R increases and X is constant. An increase in the light scattered from a single spherical particle implies, of course, a decrease in the light directly transmitted through a suspension containing a given number of particles in the cubic centimetre. The calculation is detailed in my paper " On the Transmission of Light through an Atmosphere containing Small Particles in Suspension*," and need not be repeated. It will be seen that no explanation is here arrived at of the augmentation of transparency at a certain stage observed by Keen and Porter. The discrepancy may perhaps be attributed to the fundamental supposition of the present paper, that the relative index is very small [or rather very near unity], a supposition not realised when sulphur and water are in question. But I confess that I should not have expected so wide a difference, and, indeed, the occurrence of anything special at so great diameters as 10 wave-lengths is surprising. One other matter may be alluded to. It is not clear from the description that the light observed was truly transmitted in the technical sense. This light was much attenuated down to only 5 per cent. Is it certain that it contained no sensible component of scattered light, but slightly diverted from its original course ? If such admixture occurred, the question would be much complicated. * Phil. Mag. Vol. XLVII. p. 375 (1899) ; Scientific Papers, Vol. iv. p. 397. 382. SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM. [Proceedings of the Royal Society, A, Vol. xc. pp. 318323, 1914] ACCORDING to Fourier's theorem a curve whose ordinate is arbitrary over the whole range of abscissae from x = oo to # = + oo can be compounded of harmonic curves of various wave-lengths. If the original curve contain a discontinuity, infinitely small wave-lengths must be included, but if the discontinuity be eased off, infinitely small wave-lengths may not be necessary. In order to illustrate this question I commenced several years ago calcula- tions relating to a very simple case. These I have recently resumed, and although the results include no novelty of principle they may be worth putting upon record. The case is that where the ordinate is constant (TT) between the limits + 1 for x and outside those limits vanishes. In general 6(x)=-f dkl +X> dv6(v)cosk(v-ao) ............... (1) TTJO J-oc, Here I dv </> (v) cos k (v - x) = 2?r cos koc I dv cos kv = 2?r cos kx . J -co JQ K = {smk(x + 1) - sin&(# - 1)}, and As is well known, each of the integrals in (2) is equal to -TT; so that, as was required, < (#) vanishes outside the limits 1 and between those limits takes the value TT. It is proposed to consider what values are assumed by <(#) when in (2) we omit that part of the range of integration in which k exceeds a finite value k\. 152 228 SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM [382 The integrals in (2) are at once expressible by what is called the sine- integral, defined by Jo Thus < O) = Si j (# + l)-Si,(#- 1), (4) and if the sine-integral were thoroughly known there would be scarcely anything more to do. For moderate values of the integral may be calcu- lated from an ascending series which is always convergent. For larger values this series becomes useless ; we may then fall back upon a descending series of the semi-convergent class, viz., 1.2.3.4_ ) -sintf ^- 1 1.2.3 1.2.3.4.5 -.. (5) Dr Glaisher* has given very complete tables extending from 6 = to = 1, and also from 1 to 5 at intervals of 0*1. Beyond this point he gives the function for integer values of 6 from 5 to 15 inclusive, and afterwards only at intervals of 5 for 20, 25, 30, 35, &c. For my purpose these do not suffice, and I have calculated from (5) the values for the missing integers up to 6 = 60. The results are recorded in the Table below. In each case, except those quoted from Glaisher, the last figure is subject to a small For the further calculation, involving merely subtractions, I have selected the special cases &, = 1, 2, 10. For ^ = 1, we have Si (* + !)- Si (#-1) (6) e 8i(0) e Si(0) e Si(0) e Bi<) 16 T63130 28 1 -60474 39 1-56334 50 1-55162 17 1-59013 29 1 -59731 40 1-58699 51 1-55600 18 1-53662 30 1 -56676 41 1 -59494 52 1 -57357 19 1-51863 31 1-54177 42 1-58083 53 1-58798 20 1-54824 32 1-54424 43 1 -55836 54 1-58634 21 1-59490 33 1-57028 44 1-54808 55 1-57072 22 1-61609 34 1 -59525 45 1 -55871 56 1 -55574 23 1-59546 35 1-59692 46 1-57976 57 1 -55490 24 1-55474 36 1-57512 47 1-59184 58 1-56845 25 1-53148 37 1-54861 48 1-58445 59 1-58368 26 1 -54487 38 1 -54549 49 1 '66507 60 1-58675 27 1-58029 In every case <(#) is an even function, so that it suffices to consider x positive. * Phil. Tram. Vol. CLX. p. 367 (1870). 1914] SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM 229 X + () X *(*) x <f>(x) o-o + 1-8922 2-5 ' +0-5084 6-0 -0-0953 0-5 1-8178 3-0 + 0-1528 7-0 + 0-1495 1-0 1 -6054 3-5 -0-1244 8-0 +0-2104 1-5 1-2854 4-0 -0-2987 9-0 +0-0842 2-0 0-9026 5-0 -0-3335 10-0 -0-0867 When k, and we find 2, <f> ( X <f>(x) X *(*) X *() o-o + 3-2108 0-9 + 1 -9929 3-0 -0-1840 o-i 3-1934 1-0 1-7582 3-5 +0-1151 0-2 3-1417 1-1 1-5188 4-0 + 0-2337 0-3 3-0566 1-2 1 -2794 4-5 + 0-1237 0-4 2-9401 1-3 1 -0443 5-0 -0-0692 0-5 2-7947 1-4 0-8179 5-5 -0-1657 0-6 2-6235 1-5 + 0-6038 6-0 -0-1021 0-7 2-4300 2-0 -0-1807 0-8 2-2184 2-5 -0-3940 Both for &! = 1 and for ^ = 2 all that is required for the above values of <f> (x) is given in Glaisher's tables. -5 230 SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM [382 When^ = 10, <(#) = Si(10# + 10) - Si(10#- 10) (8) We find k\ = 10. X *(*) X *(*) X +(x) o-o + 3-3167 1-7 +0-1257 3-4 -0-0067 O'l 3-2433 1-8 +0-0305 3-5 + 0-0272 0-2 3-0792 1-9 -0-0677 3-6 + 0-0349 0'3 2-9540 2-0 -0-0916 3-7 +0-0115 0-4 2-9809 2-1 -0-0365 3-8 -0-0203 0-5 3-1681 2-2 +0-0393 3-9 -0-0322 0-6 3-3895 2-3 +0-0709 4-0 -0-0151 0-7 3-4388 2-4 + 0-0390 4-1 +0-0142 0-8 3-1420 2-5 -0-0213 4-2 +0-0293 0-9 2-4647 2-6 -0-0562 4-3 +0-0178 ro 1-5482 2-7 -0-0415 4 -0-0089 1-1 0-6488 2-8 +0-0089 5 -0-0262 1-2 +0-0107 2-9 +0-0447 6 -0-0194 1-3 -0-2532 3-0 +0-0387 7 +0-0063 1-4 -0-2035 3-1 +0-0000 8 +0-0230 1-5 -0-0184 3-2 -0-0353 9 + 0-0203 1-6 +0-1202 3-3 -0-0371 5-0 -0-0002 8 The same set of values of Si up to Si (60) would serve also for the calculation of <f> (x) for jfc, = 20 and from x = to a; = 2 at intervals of O'Oo. It is hardly necessary to set this out in detail. 1914] SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM 231 An inspection of the curves plotted from the above tables shows the approximation towards discontinuity as ^ increases. That the curve remains undulatory is a consequence of the sudden stoppage of the integration at k^k^ If we are content with a partial suppression only of the shorter wave-lengths, a much simpler solution is open to us. We have only to introduce into (1) the factor e~ ak , where a is positive, and to continue the integration up to x = x . In place of (2), we have ,/aj+lN , fx-l = tan -1 1 1 tan f dkp~ ak <j>(x)= (sin k (x + 1) - sin k (x - 1)} JO K \ a (9) The discontinuous expression corresponds, of course, to a = 0. If a is merely small, the discontinuity is eased off. The following are values of 4>(as), calculated from (9) for a = 1, 0'5, 05 : ami. X <f>(x) x t(x) x *(*) o-o 1-571 2-0 0-464 4-0 0-124 0-5 1 -446 2-5 0-309 5-0 0-080 1-0 1-107 3-0 0-219 6-0 0-055 1-5 0-727 1 a = 0-5. X <t>(x) x *(*) x *(*) o-oo 2-214 1-00 1-326 2-00 0-298 0-25 2-J73 1-25 0-888 2-50 0-180 0-50 2-111 1-50 0-588 3-00 0-120 0-75 1-756 1-75 0-408 3-50 0-087 a = 0-05. X *(*) x *(*) x *(*) o-oo 3-041 0-90 2-652 1-20 0-222 0-20 3-037 0-95 2-331 1-40 0-103 0-40 3-023 1-00 1-546 1 1-60 0-064 0-60 2-986 1-05 0-761 ! 1-80 0-045 0-80 2-869 1-10 0-440 2-00 0-033 As is evident from the form of (9), <f> (x) falls continuously as x increases whatever may be the value of a. 383. FURTHER CALCULATIONS CONCERNING THE MOMENTUM OF PROGRESSIVE WAVES. [Philosophical Magazine, Vol. xxvu. pp. 436 440, 1914.] THE question of the momentum of waves in fluid is of interest and has given rise to some difference of opinion. In a paper published several years ago* I gave an approximate treatment of some problems of this kind. For a fluid moving in one dimension for which the relation between pressure and density is expressed by P=f(p), (1) it appeared that the momentum of a progressive wave of mean density equal to that of the undisturbed fluid is given by (2) in which p is the undisturbed density and a the velocity of propagation. The momentum is reckoned positive when it is in the direction of wave- propagation. For the " adiabatic " law, viz. : .............................. (3) f -S In the case of Boyle's law we have merely to make 7 = 1 in (5). For ordinary gases 7 > 1 and the momentum is positive ; but the above argument applies to all positive values of 7. If 7 be negative, the pressure would increase as the density decreases, and the fluid would be essentially unstable. Phil. Mag. Vol. x. p. 364 (1905) ; Scientific Papers, Vol. v. p. 265. 1914] CALCULATIONS CONCERNING THE MOMENTUM OF PROGRESSIVE WAVES 233 However, a slightly modified form of (3) allows the exponent to be negative. If we take .............................. (6) with /3 positive, we get as above /<*)_&_,, f (f .).-(f>U ............. (7) Po Po and accordingly *^Q*> + 1 = 1=4 ............................ (8) If /3 = 1, the law of pressure is that under which waves can be propagated without a change of type, and we see that the momentum is zero. In general, the momentum is positive or negative according as @ is less or greater than 1. In the above formula (2) the calculation is approximate only, powers of the disturbance above the second being neglected. In the present note it is proposed to determine the sign of the momentum under the laws (3) and (6) more generally and further to extend the calculations to waves in a liquid moving in two dimensions under gravity. It should be clearly understood that the discussion relates to progressive waves. If this restriction be dispensed with, it would always be possible to have a disturbance (limited if we please to a finite length) without momentum, as could be effected very simply by beginning with displace- ments unaccompanied by velocities. And the disturbance, considered as a whole, can never acquire (or lose) momentum. In order that a wave may be progressive in one direction only, a relation must subsist between the velocity and density at every point. In the case of Boyle's law this relation, first given by De Morgan*, is u = a log (p/p ), .............................. (9) and more generally f ........ - ................... < Wherever this relation is violated, a wave emerges travelling in the negative direction. For the adiabatic law (3), (10) gives po * Airy, Phil. Mag. Vol. xxxiv. p. 402 (1849). + Earnshaw, Phil. Trans. 1859, p. 146. 234 FURTHER CALCULATIONS CONCERNING THE [383 a being the velocity of infinitely small disturbances, and this reduces to (9) when 7 = 1. Whether 7 be greater or less than 1, u is positive when p exceeds p . Similarly if the law of pressure be that expressed in (6), Since 13 is positive, values of p greater than p are here also accompanied by positive values of u. By definition the momentum of the wave, whose length may be supposed to be limited, is per unit of cross-section jpudx, ................................. (13) the integration extending over the whole length of the wave. If we intro- duce the value of u given in (11), we get and the question to be examined is the sign of (14). For brevity we may write unity in place of p , and we suppose that the wave is such that its mean density is equal to that of the undisturbed fluid, so that \pdx=l, where I is the length of the wave. If I be divided into n equal parts, then when n is great enough the integral may be represented by the sum in which all the p's are positive. Now it is a proposition in Algebra that l+i Ii . j! pi 2 +p 2 2 +... ...\ * J when (7 -i- 1) is negative, or positive and greater than unity; but that the reverse holds when (7 + !) is positive and less than unity. Of course the inequality becomes an equality when all the n quantities are equal. In the present application the sum of the p's is n, and under the adiabatic law (3), 7 and (7+ 1) are positive. Hence (15) is positive or negative according as (7 + !) is greater or less than unity, viz., according as 7 is greater or less than unity. In either case the momentum represented by (13) is positive, and the conclusion is not limited to the supposition of small disturbances. In like manner if the law of pressure be that expressed in (6), we get from (12) (13) 1914] MOMENTUM OF PROGRESSIVE WAVES 235 from which we deduce almost exactly as before that the momentum (13) is positive if @ (being positive) is less than 1 and negative if is greater than 1. If /3=1, the momentum vanishes. The conclusions formerly obtained on the supposition of small disturbances are thus extended. We will now discuss the momentum in certain cases of fluid motion under gravity. The simplest is that of long waves in a uniform canal. If ij be the (small) elevation at any point x measured in the direction of the length of the canal and u the corresponding fluid velocity parallel to x, which is uniform over the section, the dynamical equation is* As is well known, long waves of small elevation are propagated without change of form. If c be the velocity of propagation, a positive wave may be represented by 77 = F (ct - x}, .............................. (18) where F denotes an arbitrary function, and c is related to the depth A according to c 2 = #A .................................. (19) From (17), (18) is the relation obtaining between the velocity and elevation at any place in a positive progressive wave of small elevation. Equation (20), however, does not suffice for our present purpose. We may extend it by the consideration that in a long wave of finite disturbance the elevation and velocity may be taken as relative to the neighbouring parts of the wave. Thus, writing du for u and k for h , so that ij = dh, we have and on integration The arbitrary constant of integration is determined by the fact that outside the wave u = when h = h , whence and replacing h by h + 17, we get as the generalized form of (20). It is equivalent to a relation given first in another notation by De M organ -f, and it may be regarded as the condition * Lamb's Hydrodynamics, 168. t Airy, Phil. Mag. Vol. xxxiv. p. 402 (1849). 236 CALCULATIONS CONCERNING THE MOMENTUM OF PROGRESSIVE WAVES [383 which must be satisfied if the emergence of a negative wave is to be obviated. We are now prepared to calculate the momentum. For a wave in which the mean elevation is zero, the momentum corresponding to unit horizontal breadth is (22) when we omit cubes and higher powers of 77. We may write (22) also in the form ,, 3 Total Energy Momentum^-: -&-, (23) 4 c c being the velocity of propagation of waves of small elevation. As in (14), with 7 equal to 2, we may prove that the momentum is positive without restriction upon the value of 77. As another example, periodic waves moving on the surface of deep water may also be referred to. The momentum of such waves has been calculated by Lamb*, on the basis of Stokes' second approximation. It appears that the momentum per wave-length -and per unit width perpendicular to the plane of motion is 7rpa 2 c, (24) where c is the velocity of propagation of the waves in question and the wave form is approximately 77 = a cos (ct x) (25) The forward velocity of the surface layers was remarked by Stokes. For a simple view of the matter reference may be made also to Phil. Mag. Vol. I. p. 257 (1876) ; Scientific Papers, Vol. i. p. 263. * Hydrodynamics, 246. 384. FLUID MOTIONS. [Proc. Roy. Inst. March, 1914; Nature, Vol. xcm. p. 364, 1914.] THE subject of this lecture has received the attention of several gene- rations of mathematicians and experimenters. Over a part of the field their labours have been rewarded with a considerable degree of success. In all that concerns small vibrations, whether of air, as in sound, or of water, as in waves and tides, we have a large body of systematized knowledge, though in the case of the tides the question is seriously complicated by the fact that the rotation of the globe is actual and not merely relative to the sun and moon, as well as by the irregular outlines and depths of the various oceans. And even when the disturbance constituting the vibration is not small, some progress has been made, as in the theory of sound waves in one dimension, and of the tidal bores, which are such a remarkable feature of certain estuaries and rivers. The general equations of fluid motion, when friction or viscosity is neg- lected, were laid down in quite early days by Euler and Lagrange, and in a sense they should contain the whole theory. But, as Whevvell remarked, it soon appeared that these equations by themselves take us a surprisingly little way, and much mathematical and physical talent had to be expended before the truths hidden in them could be brought to light and exhibited in a practical shape. What was still more disconcerting, some of the general propositions so arrived at were found to be in flagrant contradiction with observation, even in cases where at first sight it would not seem that viscosity was likely to be important. Thus a solid body, submerged to a sufficient depth, should experience no resistance to its motion through water. On this principle the screw of a submerged boat would be useless, but, on the other hand, its services would not be needed. It is little wonder that practical men should declare that theoretical hydrodynamics has nothing at all to do with real fluids. Later we will return to some of these difficulties, not yet fully surmounted, but for the moment I will call your attention to simple phenomena of which theory can give a satisfactory account. FLUID MOTIONS [384 Considerable simplification attends the supposition that the motion is always the same at the same place is steady, as we say and fortunately this covers many problems of importance. Consider the flow of water along a pipe whose section varies. If the section were uniform, the pressure would vary along the length only in consequence of friction, which now we are neglecting. In the proposed pipe how will the pressure vary ? I will not prophesy as to a Royal Institution audience, but I believe that most un- sophisticated people suppose that a contracted place would give rise to an increased pressure. As was known to the initiated long ago, nothing can be further from the fact. The experiment is easily tried, either with air or water, so soon as we are provided with the right sort of tube. A suitable shape is shown in fig. 1, but it is rather troublesome to construct in metal. W. Froude found paraffin-wax the most convenient material for ship models, and I have followed him in the experiment now shown. A brass tube is filled with candle-wax and bored out to the desired shape, as is easily done with templates of tin plate. When I blow through, a suction is developed at the narrows, as is witnessed by the rise of liquid in a manometer connected laterally. In the laboratory, where dry air from an acoustic bellows or a gas-holder is available, I have employed successfully tubes built up of cardboard, for a circular cross-section is not necessary. Three or more precisely similar pieces, cut for example to the shape shown in fig. 2 and joined together Fig. 2. 1914] FLUID MOTIONS closely along the edges, give the right kind of tube, and may be made air- tight with pasted paper or with sealing-wax. Perhaps a square section requiring four pieces is best. It is worth while to remark that there is no stretching of the cardboard, each side being merely bent in one dimension. A model is before you, and a study of it forms a simple and useful exercise in solid geometry. Another form of the experiment is perhaps better known, though rather more difficult to think about. A tube (fig. 3) ends in a flange. If I blow through the tube, a card presented to the flange is drawn up pretty closely, instead of being blown away as might be expected. When we consider the J I Fig. 3. Fig. 4. matter, we recognize that the channel between the flange and the card through which the air flows after leaving the tube is really an expanding one, and thus that the inner part may fairly be considered as a contracted place. The suction here developed holds the card up. A slight modification enhances the effect. It is obvious that immediately opposite the tube there will be pressure upon the card and not suction. To neutralize this a sort of cap is provided, attached to the flange, upon which the objectionable pressure is taken (fig. 4). By blowing smartly from the mouth through this little apparatus it is easy to lift and hold up a penny for a short time. The facts then are plain enough, but what is the explanation ? It is really quite simple. In steady motion the quantity of fluid per second passing any section of the tube is everywhere the same. If the fluid be incom- pressible, and air in these experiments behaves pretty much as if it were, this means that the product of the velocity and area of cross-section is constant, so that at a narrow place the velocity of flow is necessarily increased. And when we enquire how the additional velocity in passing from a wider to a narrower place is to be acquired, we are compelled to recognize that it can only be in consequence of a fall of pressure. The section at the narrows is the only result consistent with the great principle of conservation of energy ; 240 FLUID MOTIONS [384 but it remains rather an inversion of ordinary ideas that we should have to deduce the forces from the motion, rather than the motion from the forces. The application of the principle is not always quite straightforward. Consider a tube of slightly conical form, open at both ends, and suppose that we direct upon the narrower end a jet of air from a tube having the same (narrower) section (fig. 5). We might expect this jet to enter the Fig. 5. conical tube without much complication. But if we examine more closely a difficulty arises. The stream in the conical tube would have different velocities at the two ends, and therefore different pressures. The pressures at the ends could not both be atmospheric. Since at any rate the pressure at the wider delivery end must be very nearly atmospheric, that at the narrower end must be decidedly below that standard. The course of the events at the inlet is not so simple as supposed, and the apparent contra- diction is evaded by an inflow of air from outside, in addition to the jet, which assumes at entry a narrower section. If the space surrounding the free jet is enclosed (fig. 6), suction is there developed and ultimately when the motion has become steady the jet enters the conical tube without contraction. A model shows the effect, and the pnnciple is employed in a well-known laboratory instrument arranged for working off the water-mains. nm Fig. 6. I have hitherto dealt with air rather than water, not only because air makes no mess, but also because it is easier to ignore gravitation. But there is another and more difficult question. You will have noticed that in our expanding tubes the section changes only gradually. What happens when the expansion is more sudden in the extreme case when the diameter of a previously uniform tube suddenly becomes infinite ? (fig. 3) without 1914] FLUID MOTIONS 241 card. Ordinary experience teaches that in such a case the flow does not follow the walls round the corner, but shoots across as a jet, which for a time preserves its individuality and something like its original section. Since the velocity is not lost, the pressure which would replace it is not developed. It is instructive to compare this x;ase with another, experimented on by Savart* and W. Froude -f*, in which a free jet is projected through a snort cone, or a mere hole in a thin wall, into a vessel under a higher pressure. The apparatus consists of two precisely similar vessels with apertures, in which the fluid (water) may be at different levels (fig. 7, copied from Froude). Savart found that not a single drop of liquid was spilt so long as the pressure in the recipient vessel did not exceed one-sixth of that under which the jet issues. And Froude reports that so long as the head in the discharge cistern is maintained at a moderate height above that in the Fig. 7. recipient cistern, the whole of the stream enters the recipient orifice, and there is " no waste, except the small sprinkling which is occasioned by in- exactness of aim, and by want of exact circularity in the orifices." I am disposed to attach more importance to the small spill, at any rate when the conoids are absent or very short. For if there is no spill, the jet (it would seem) might as well be completely enclosed ; and then it would propagate itself into the recipient cistern without sudden expansion and consequent recovery of pressure. In fact, the pressure at the narrows would never fall below that of the recipient cistern, and the discharge would be correspondingly lessened. When a decided spill occurs, Froude explains it as due to the retardation by friction of the outer layers which are thus unable to force themselves against the pressure in front. Evidently it is the behaviour of these outer layers, especially at narrow places, which determines the character of the flow in a large variety of cases. * Ann. de Chimie, Vol. LV. p. 257, 1833. t Nature, Vol. xni. p. 93, 1875. 242 FLUID MOTIONS [384 They are held back, as Froude pointed out, by friction acting from the walls ; but, on the other hand, when they lag, they are pulled forward by layers farther in which still retain their velocity. If the latter prevail, the motion in the end may not be very different from what would occur in the absence of friction ; otherwise an entirely altered motion may ensue. The situation as regards the rest of the fluid is much easier when the layers upon which the friction tells most are allowed to escape. This happens in instruments of the injector class, but I have sometimes wondered whether full advantage is taken of it. The long gradually expanding cones are overdone, perhaps, and the friction which they entail must have a bad effect. Similar considerations enter when we discuss the passage of a solid body through a large mass of fluid otherwise at rest, as in the case of an airship or submarine boat. I say a submarine, because when a ship moves upon the surface of the water the formation of waves constitutes a complication, and one of great importance when the speed is high. In order that the water in its relative motion may close in properly behind, the after-part of the ship must be suitably shaped, fine lines being more necessary at the stern than at the bow, as fish found out before men interested themselves in the problem. In a well-designed ship the whole resistance (apart from wave- making) may be ascribed to skin friction, of the same nature as that which is encountered when the ship is replaced by a thin plane moving edgeways. At the other extreme we may consider the motion of a thin disk or blade flatways through the water. Here the actual motion differs altogether from that prescribed by the classical hydrodynamics, according to which the character of the motion should be the same behind as in front. The liquid refuses to close in behind, and a region of more or less "dead water" is developed, entailing a greatly increased resistance. To meet this Helmholtz, Kirchhoff, and their followers have given calculations in which the fluid behind is supposed to move strictly with the advancing solid, and to be separated from the remainder of the mass by a surface at which a finite slip takes place. Although some difficulties remain, there can be no doubt that this theory constitutes a great advance. But the surface of separation is unstable, and in consequence of fluid friction it soon loses its sharpness, breaking up into more or less periodic eddies, described in some detail by Mallock (fig. 8). It is these eddies which cause the whistling of the wind in trees and the more musical notes of the aeolian harp. The obstacle to the closing-in of the lines of flow behind the disk is doubtless, as before, the layer of liquid in close proximity to the disk, which at the edge has insufficient velocity for what is required of it. It would be an interesting experiment to try what would be the effect of allowing a small "spill." For this purpose the disk or blade would be made double, with a suction applied to the narrow interspace. Relieved of the slowly 1914] FLUID MOTIONS 243 moving layer, the liquid might then be able to close in behind, and success would be witnessed by a greatly diminished resistance. Fig. 8. When a tolerably fair-shaped body moves through fluid, the relative velocity is greatest at the maximum section of the solid which is the minimum section for the fluid, and consequently the pressure is there least. Thus the water-level is depressed at and near the midship section of an advancing steamer, as is very evident in travelling along a canal. On the same principle may be explained the stability of a ball sustained on a vertical jet as in a Avell-known toy (shown). If the ball deviate to one side, the jet in bending round the surface develops a suction pulling the ball back. As Mr Lanchester has remarked, the effect is aided by the rotation of the ball. That a convex surface is attracted by a jet playing obliquely upon it was demonstrated by T. Young more than 100 years ago by means of a model, of which a copy is before you (fig. 9). D Fig. 9. A plate, bent into the form ABC, turning on centre B, is impelled by a stream of air D in the direction shown. It has been impossible in dealing with experiments to keep quite clear of friction, but I wish now for a moment to revert to the ideal fluid of hydro- dynamics, in which pressure and inertia alone come into account. The possible motions of such a fluid fall into two great classes those which do and those which do not involve rotation. What exactly is meant by rotation is best explained after the manner of Stokes. If we imagine any spherical 162 244 FLUID MOTIONS [384 portion of the fluid in its motion to be suddenly solidified, the resulting solid may be found to be rotating. If so, the original fluid is considered to possess rotation. If a mass of fluid moves irrotationally, no spherical portion would revolve on solidification. The importance of the distinction depends mainly upon the theorem, due to Lagrange and Cauchy, that the irrotational character is permanent, so that any portion of fluid at any time destitute of rotation will always remain so. Under this condition fluid motion is com- paratively simple, and has been well studied. Unfortunately many of the results are very unpractical. As regards the other class of motions, the first great step was taken in 1858, by Helmholtz, who gave the theory of the vortex-ring. In a perfect fluid a vortex-ring has a certain permanence and individuality, which so much impressed Kelvin that he made it the foundation of a speculation as to the nature of matter. To him we owe also many further developments in pure theory. On the experimental side, the first description of vortex-rings that I have come across is that by W. B. Rogers*, who instances their production during the bursting of bubbles of phosphuretted hydrogen, or the escape of smoke from cannon and from the lips of expert tobacconists. For private obser- vation nothing is simpler than Helmholtz's method of drawing a partially immersed spoon along the surface, for example, of a cup of tea. Here half a ring only is developed, and the places where it meets the surface are shown as dimples, indicative of diminished pressure. The experiment, made on a larger scale, is now projected upon the screen, the surface of the liquid and its motion being made more evident by powder of lycopodium or sulphur scattered over it. In this case the ring is generated by the motion of a half-immersed circular disk, withdrawn after a travel of two or three inches. In a modified experiment the disk is replaced by a circular or semi-circular aperture cut in a larger plate, the level of the water coinciding with the horizontal diameter of the aperture. It may be noticed that while the first forward motion of the plate occasions a ring behind, the stoppage of the plate gives rise to a second ring in front. As was observed by Reuschf, the same thing occurs in the more usual method of projecting smoke-rings from a box ; but in order to see it the box must be transparent. In a lecture given here in 1877, Reynolds showed that a Helmholtz ring can push the parent disk before it, so that for a time there appears to be little resistance to its motion. For an explanation of the origin of these rings we must appeal to friction, for in a perfect fluid no rotation can develop. It is easy to recognize that friction against the wall in which the aperture is perforated, or against the * Amer. J. Set. Vol. MVI. p. 246, 1858. t Fogg. Ann. Vol. ex. p. 309, 1860. 1914] FLUID MOTIONS 245 face of the disk in the other form of experiment, will start a rotation which, in a viscous fluid, such as air or water actually is, propagates itself to a finite distance inwards. But although a general explanation is easy, many of the details remain obscure. It is apparent that in dealing with a large and interesting class of fluid motions we cannot go far without including fluid friction, or viscosity as it is generally called, in order to distinguish it from the very different sort of friction encountered by solids, unless well lubricated. In order to define it, we may consider the simplest case where fluid is included between two parallel walls, at unit distance apart, which move steadily, each in its own plane, with velocities which differ by unity. On the supposition that the fluid also moves in plane strata, the viscosity is measured by the tangential force per unit of area exercised by each stratum upon its neighbours. When we are concerned with internal motions only, we have to do rather with the so-called " kinematic viscosity," found by dividing the quantity above defined by the density of the fluid. On this system the viscosity of water is much less than that of air. Viscosity varies with temperature ; and it is well to remember that the viscosity of air increases while that of water decreases as the temperature rises. Also that the viscosity of water may be greatly increased by admixture with alcohol. I used these methods in 1879 during investigations respecting the influence of viscosity upon the behaviour of such fluid jets as are sensitive to sound and vibration. Experimentally the simplest case of motion in which viscosity is para- mount is the flow of fluid through capillary tubes. The laws of such motion are simple, and were well investigated by Poiseuille. This is the method employed in practice to determine viscosities. The apparatus before you is arranged to show the diminution of viscosity with rising temperature. In the cold the flow of water through the capillary tube- is slow, and it requires sixty seconds to fill a small measuring vessel. When, however, the tube is heated by passing steam through the jacket surrounding it, the flow under the same head is much increased, and the measure is filled in twenty-six seconds. Another case of great practical importance, where viscosity is the leading consideration, relates to lubrication. In admirably conducted ex- periments Tower showed that the solid surfaces moving over one another should be separated by a complete film of oil, and that when this is attended to there is no wear. On this basis a fairly complete theory of lubrication has been developed, mainly by O. Reynolds. But the capillary nature of the fluid also enters to some extent, and it is not yet certain that the whole character of a lubricant can be expressed even in terms of both surface tension and viscosity. It appears that in the extreme cases, when viscosity can be neglected and again when it is paramount, we are able to give a pretty good account of 246 FLUID MOTIONS [384 what passes. It is in the intermediate region, where both inertia and viscosity are of influence, that the difficulty is greatest. But even here we are not wholly without guidance. There is a general law, called the law of dynamical similarity, which is often of great service. In the past this law has been unaccountably neglected, and not only in the present field. It allows us to infer what will happen upon one scale of operations from what has been observed at another. On the present occasion I must limit myself to viscous fluids, for which the law of similarity was laid down in all its completeness by Stokes as long ago as 1850. It appears that similar motions may take place provided a certain condition be satisfied, viz. that the product of the linear dimension and the velocity, divided by the kinematic viscosity of the fluid, remain unchanged. Geometrical similarity is presupposed. An example will make this clearer. If we are dealing with a single fluid, say air under given conditions, the kinematic viscosity remains of course the same. When a solid sphere moves uniformly through air, the character of the motion of the fluid round it may depend upon the size of the sphere and upon the velocity with which it travels. But we may infer that the motions remain similar, if only the product of diameter and velocity be given. Thus, if we know the motion for a particular diameter and velocity of the sphere, we can infer what it will be when the velocity is halved and the diameter doubled. The fluid velocities also will eve^where be halved at the corresponding places. M. Eiffel found that for any sphere there is a velocity which may be regarded as critical, i.e. a velocity at which the law of resistance changes its character somewhat suddenly. It follows from the rule that these critical velocities should be inversely proportional to the diameters of the spheres, a conclusion in pretty good agreement with M. Eiffel's observations*. But the principle is at least equally important in effecting a comparison between different fluids. If we know what happens on a certain scale and at a certain velocity in water, we can infer what will happen in air on any other scale, provided the velocity is chosen suitably. It is assumed here that the compressibility of the air does not come into account, an assumption which is admissible so long as the velocities are small in comparison with that of sound. But although the principle of similarity is well established on the theoretical side and has met with some confirmation in experiment, there has been much hesitation in applying it, due perhaps to certain discrepancies with observation which stand recorded. And there is another reason. It is rather difficult to understand how viscosity can play so large a part as it seems to do, especially when we introduce numbers, which make it appear that the viscosity of air, or water, is very small in relation to the other data occurring in practice. In order to remove these doubts it is very desirable to experiment with different viscosities, but this is not easy to do on a Comptet Rendiu, Dec. 30, 1912, Jan. 13, 1913. [This volume, p. 136.] 1914] FLUID MOTIONS 247 moderately large scale, as in the wind channels used for aeronautical purposes. I am therefore desirous of bringing before you some observations that I have recently made with very simple apparatus. When liquid flows from one reservoir to another through a channel in which there is a contracted place, we can compare what we may call the head or driving pressure, i.e. the difference of the pressures in the two reservoirs, with the suction, i.e. the difference between the pressure in the recipient vessel and that lesser pressure to be found at the narrow place. The ratio of head to suction is a purely numerical quantity, and according to the principle of similarity it should for a given channel remain unchanged, provided the velocity be taken proportional to the kinematic viscosity of the fluid. The use of the same material channel throughout has the advantage that no question can arise as to geometrical similarity, which in principle should extend to any roughnesses upon the surface, while the necessary changes of velocity are easily attained by altering the head and those of viscosity by altering the temperature. The apparatus consisted of two aspirator bottles (fig. 10) containing water and connected below by a passage bored in a cylinder of lead, 7 cm. Fig. 10. long, fitted water-tight with rubber corks. The form of channel actually employed is shown in fig. 11. On the up-stream side it contracts pretty suddenly from full bore (8 mm.) to the narrowest place, where the diameter is 2'75 mm. On the down-stream side the expansion takes place in four or five steps, corresponding to the drills available. It had at first been intended to use a smooth curve, but preliminary trials showed that this was un- necessary, and the expansion by steps has the advantage of bringing before the mind the dragging action of the jets upon the thin layers of fluid 24S FLUID MOTIONS [384 between them and the walls. The three pressures concerned are indicated on manometer tubes as shown, and the two differences of level representing head and suction can be taken off with compasses and referred to a milli- metre scale. In starting an observation the water is drawn up in the discharge vessel, as far as may be required, with the aid of an air-pump. The rubber cork at the top of the discharge vessel necessary for this purpose is not shown. As the head falls during the flow of the water, the ratio of head to suction increases. For most of the observations I contented myself with recording the head for which the ratio of head to suction was exactly 2 : 1, as indicated by proportional compasses. Thus on January 23, when the temperature of the water was 9 C., the 2 : 1 ratio occurred on four trials at 120, 130, 123, 126, mean 125 mm. head. The temperature was then raised with precaution by pouring in warm water with passages backwards and forwards. The occurrence of the 2 : 1 ratio was now much retarded, the mean head being only 35 mm., corresponding to a mean temperature of 37 C. The ratio of Fig. 11. head to suction is thus dependent upon the head or velocity, but when the velocity is altered the original ratio may be recovered if at the same time we make a suitable alteration of viscosity. And the required alteration of viscosity is about what might have ben expected. From Landolt's tables I find that for .9 C. the viscosity of water is 01368, while for 37 C. it is -00704. The ratio of viscosities is accordingly 1-943. The ratio of heads is 125 : 35. The ratio of velocities is the square- root of this or T890, in sufficiently good agreement with the ratio of viscosities. In some other trials the ratio of velocities exceeded a little the ratio of viscosities. It is not pretended that the method would be an accurate one for the comparison of viscosities. The change in the ratio of head to suction is rather slow, and the measurement is usually somewhat prejudiced by unsteadiness in the suction manometer. Possibly better results would be obtained in more elaborate observations by several persons, the head and suction being recorded separately and referred to a time scale so as to facilitate interpolation. But as they stand the results suffice for my purpose, showing directly and conclusively the influence of viscosity as compensating H change in the velocity. 1914] FLUID MOTIONS 249 In conclusion, I must touch briefly upon a part of the subject where theory is still at fault, and I will limit myself to the simplest case of all the uniform shearing motion of a viscous fluid between two parallel walls, one of which is at rest, while the other moves tangentially with uniform velocity. It is easy to prove that a uniform shearing motion of the fluid satisfies the dynamical equations, but the question remains : Is this motion stable ? Does a small departure from the simple motion tend of itself to die out ? In the case where the viscosity is relatively great, observation suggests an affirmative answer; and O. Reynolds, whose illness and com- paratively early death were so great a loss to science, was able to deduce the same conclusion from theory. Reynolds' method has been improved, more especially by Professor Orr of Dublin. The simple motion is thoroughly stable if the viscosity exceed a certain specified value relative to the velocity of the moving plane and the distance between the planes ; while if the viscosity is less than this, it is possible to propose a kind of departure from the original motion which will increase for a time. It is on this side of the question that there is a deficiency. When the viscosity is very small, obser- vation appears to show that the simple motion is unstable, and we ought to be able to derive this result from theory. But even if we omit viscosity altogether, it does not appear possible to prove instability a priori, at least so long as we regard the walls as mathematically plane. We must confess that at the present we are unable to give a satisfactory account of skin- friction, in order to overcome which millions of horse-power are expended in our ships. Even in the older subjects there are plenty of problems left ! 385. ON THE THEORY OF LONG WAVES. AND BORES. [Proceedings of the Royal Society, A, Vol. xc. pp. 324328, 1914.] IN the theory of long waves in two dimensions, which we may suppose to be reduced to a " steady " motion, it is assumed that the length is so great in proportion to the depth of the water that the velocity in a vertical direction can be neglected, and that the horizontal velocity is uniform across each section of the canal. This, it should be observed, is perfectly distinct from any supposition as to the height of the wave. If I be the undisturbed depth, and h the elevation of the water at any point of the wave, w , u the velocities corresponding to I, I + h respectively, we have, as the equation of continuity, By the principles of hydrodynamics, the increase of pressure due to retardation will be On the other hand, the loss of pressure (at the surface) due to height will be gph ; and therefore the total gain of pressure over the undisturbed parts is (3 > If. now, the ratio h/l be very small, the coefficient of h becomes pMl-9) .................................. (4) and we conclude that the condition of a free surface is satisfied, provided u? = gl. This determines the rate of flow u^, in order that a stationary wave may be possible, and gives, of course, at the same time the velocity of a wave in still water. 1914] ON THE THEORY OF LONG WAVES AND BORES 251 Unless A* can be neglected, it is impossible to satisfy the condition of a free surface for a stationary long wave which is the same as saying that it is impossible for a long wave of finite height to be propagated in still water without change of type. Although a constant gravity is not adequate to compensate the changes of pressure due to acceleration and retardation in a long wave of finite height, it is evident that complete compensation is attainable if gravity be made a suitable function of height ; and it is worth while to enquire what the law of force must be in order that long waves of unlimited height may travel with type unchanged. If f be the force at height h, the condition of constant surface pressure is whence /= _ | . ^ _JL_ = M , ................... (6) which shows that the force must vary inversely as the cube of the distance from the bottom of the canal. Under this law the waves may be of any height, and they will be propagated unchanged with the velocity V(/iO> where /i is the force at the undisturbed level *. It may be remarked that we are concerned only with the values of f at water-levels which actually occur. A change in f below the lowest water- level would have no effect upon the motion, and thus no difficulty arises from the law of inverse cube making the force infinite at the bottom of the canal. When a wave is limited in length, we may speak of its velocity relatively to the undisturbed water lying beyond it on the two sides, and it is implied that the uniform levels on the two sides are the same. But the theory of long waves is not thus limited, and we may apply it to the case where the uniform levels on the two sides of the variable region are different, as, for example, to bores. This is a problem which I considered briefly on a former occasion f, when it appeared that the condition of conservation of energy could not be satisfied with a constant gravity. But in the calculation of the loss of energy a term was omitted, rendering the result erroneous, although the general conclusions are not affected. The error became apparent in applying the method to the case above considered of a gravity varying as the inverse cube of the depth. But, before proceeding to the calculation of energy, it may be well to give the generalised form of the relation between velocity and height which must be satisfied in a progressive wave}, whether or not the type be permanent. * Phil. Mag. Vol. i. p. 257 (1876); Scientific Papers, Vol. i. p. 254. t Roy. Soc. Proc. A, Vol. LXXXI. p. 448 (1908) ; Scientific Papers, Vol. v. p. 495. J Compare Scientific Papers, Vol. i. p. 253 (1899). 252 ON THE THEORY OF LONG WAVES AND BORES [385 In a small positive progressive wave, the relation between the particle- velocity u at any point (now reckoned relatively to the parts outside the wave) and the elevation h is tt-^(//J).A (7) If this relation be violated anywhere, a wave will emerge, travelling in the negative direction. In applying (7) to a wave of finite height, the appropriate form of (7) is where f is a known function of I + h t or on integration dh (9) To this particle-velocity is to be added the wave-velocity V{(Z+A)/}, (10) making altogether for the velocity of, e.g., the crest of a wave relative to still water Thus iff be constant, say g, (9) gives De Morgan's formula - 2 vty ((*+*)*-*}, ........................ (12) and (11) becomes (13) (11) gives as the velocity of a crest which is independent of h, thus confirming what was found before for this law of force. As regards the question of a bore, we consider it as the transition from a uniform velocity u and depth I to a uniform velocity u and depth I', I' being greater than L The first relation between these four quantities is that given by continuity, viz., lu = l'u' .................................. (16) The second relation arises from a consideration of momentum. It may be convenient to take first the usual case of a constant gravity g. The mean pressures at the two sections are $gl, ^gl', and thus the equation of momentum is *') ............................ (17) 1914] ON THE THEORY OF LONG WAVES AND BORES 253 By these equations u and u' are determined in terms of I, I' : = *$r (I + ?)*'/*, * = iflr (* + ').//*' ............. (18) We have now to consider the question of energy. The difference of work done by the pressure at the two ends (reckoned per unit of time and per unit of breadth) is lu (%gl %gl'). And the difference between the kinetic energies entering and leaving the region is lutyu* ^w' 2 ), the density being taken as unity. But this is not all. The potential energies of the liquid leaving and entering the region are different. The centre of gravity rises through a height W l\ and the gain of potential energy is therefore lu.^g(l'-l). The whole loss of energy is accordingly This is much smaller than the value formerly given, but it remains of the same sign. " That there should be a loss of energy constitutes no difficulty, at least in the presence of viscosity ; but the impossibility of a gain of energy shows that the motions here contemplated cannot be reversed." We now suppose that the constant gravity is replaced by a force f, which is a function of y, the distance from the bottom. The pressures p, p' at the two sections are also functions of y, such that P'= fdy ...................... (20) - y The equation of momentum replacing (17) is now .(21) the integrated terms vanishing at the limits. This includes, of course, all special cases, such as f= constant, or f<x y~ s . As regards the reckoning of energy, the first two terms on the left of ( 1 9) are replaced by lu\}\ l pdy-] t \ l 'p'dy\ ...................... (22) (I Jo I J o j The third and fourth terms representing kinetic energy remain as before. For the potential energy we have to consider that a length u and depth I is converted into a length u' and depth I'. If we reckon from the bottom, the potential energy is in the first case ri rv u dy fdy, Jo Jo 254 ON THE THEORY OF LONG WAVES AND BORES [385 in which !/ dy= l/ dy ~ f s d y = p-p> p denoting the pressure at the bottom, so that the potential energy is id \Pt-\l\pdy\. The difference of potential energies, corresponding to the fifth and sixth terms of (19), is thus (23) The integrals in (23) compensate those of (22), and we have finally as the loss of energy to bo - Po'+ i 2 -K 2 } = *" j**-i*-j] /<fr| ....... (24) It should be remarked that it is only for values of y between I and V that / is effectively involved. In the special case where f=fj.y~ 3 , equations (16), (21) give uH*=(jt,, u*l' 2 =fjL, ........................ (25) the introduction of which into (24) shows that, in this case, the loss of energy vanishes ; all the conditions can be satisfied, even though there be no dissipation. The reversed motion is then equally admissible. Experimental. The formation of bores is illustrated by a very ordinary observation, probably not often thought of in this connection. Something of the kind may usually be seen whenever a stream of water from a tap strikes a horizontal surface [or when water from a can is poured into a flat bath], The experiment is best made by directing a vertically falling stream into a flat and shallow dish from which the water overflows*. The effective depth may be varied by holding a glass plate in a horizontal position under the water surface. Where the jet strikes, it expands into a thin sheet which diverges for a certain distance, and this distance diminishes as the natural depth of the water over the plate is made greater. The circular boundary where the transition from a small to a greater depth takes place constitutes a bore on a small scale. The flow may be made two-dimensional by limiting it with two battens held in contact with the glass. I have not attempted measures. On the smallest scale surface-tension doubtless plays a considerable part, but this maybe minimised by increasing the stream, and correspondingly the depth of the water over the plate, so far as may be convenient. * The tap that I employed gives a jet whose diameter is 6 mm. A much larger tap may need to be fitted with a special nozzle. May 14, [1914]. 386. THE SAND-BLAST. [Nature, Vol. xcin. p. 188, 1914.] AMONG the many remarkable anticipations contained in T. Young's Lectures on Natural Philosophy (1807) is that in which he explains the effect of what is now commonly known as the sand-blast. On p. 144 he writes : " There is, however, a limit beyond which the velocity of a body striking another cannot be increased without overcoming its resilience, and breaking it, however small the bulk of the first body may be, and this limit depends on the inertia of the parts of the second body, which must not be disregarded when they are impelled with a considerable velocity. For it is demonstrable that there is a certain velocity, dependent on the nature of a substance, with which the effect of any impulse or pressure is transmitted through it ; a certain portion of time, which is shorter accordingly as the body is more elastic, being required for the propagation of the force through any part of it ; and if the actual velocity of any impulse be in a greater proportion to this velocity than the extension or compression, of which the substance is capable, is to its whole length, it is obvious that a separation must be pro- duced, since no parts can be extended or compressed which are not yet affected by the impulse, and the length of the portion affected at any instant is not sufficient to allow the required extension or compression. Thus if the velocity with which an impression is transmitted by a certain kind of wood be 15,000 ft. in a second, and it be susceptible of compression to the extent of 1/200 of its length, the greatest velocity that it can resist will be 75 ft. in a second, which is equal to that of a body falling from a height of about 90 ft." Doubtless this passage was unknown to O. Reynolds when, with customary penetration, in his paper on the sand-blast (Phil. Mag. Vol. XLVI. p. 337, 1873) he emphasises that "the intensity of the pressure between bodies on first impact is independent of the size of the bodies." After his manner, Young was over-concise, and it is not clear precisely what circumstances he had in contemplation. Probably it was the longitudinal impact of bars, and at any rate this affords a convenient example. We may 256 THE SAND-BLAST [386 begin by supposing the bars to be of the same length, material, and section, and before impact to be moving with equal and opposite velocities v. At impact, the impinging faces are reduced to rest, and remain at rest so long as the bars are in contact at all. This condition of rest is propagated in each bar as a wave moving with a velocity a, characteristic of the material. In such a progressive wave there is a general relation between the particle- velocity (estimated relatively to the parts outside the wave) and the com- pression (e), viz., that the velocity is equal to ae. In the present case the relative particle- velocity is v, so that v = ae. The limit of the strength of the material is reached when e has a certain value, and from this the greatest value of v (half the original relative velocity) which the bars can bear is immediately inferred. But the importance of the conclusion depends upon an extension now to be considered. It will be seen that the length of the bars does not enter into the question. Neither does the equality of the lengths. However short one of them may be, we may contemplate an interval after first impact so short that the wave will not have reached the further end, and then the argument remains unaffected. However short one of the impinging bars, the above calculated relative velocity is the highest which the material can bear without undergoing disruption. As more closely related to practice, the case of two spheres of radii r, r', impinging directly with relative velocity v, is worthy of consideration. According to ordinary elastic theory the only remaining data of the problem are the densities p, p, and the elasticities. The latter may be taken to be the Young's moduli q, q', and the Poisson's ratios, <T, a', of which the two last are purely numerical. The same may be said of the ratios q'/q, p'/p, and r'/r. So far as dimensional quantities are concerned, any maximum strain e may be regarded as a function of r, v, q, and p. The two last can occur only in the combination q/p, since strain is of no dimensions. Moreover, q/p = a*, where a is a velocity. Regarding e as a function of r, v, and a, we see that v and a can occur only as the ratio v/a, and that r cannot appear at all. The maximum strain then is independent of the linear scale ; and if the rupture depends only on the maximum strain, it is as likely to occur with small spheres as with large ones. The most interesting case occurs when one sphere is very large relatively to the other, as when a grain of sand impinges upon a glass surface. If the velocity of impact be given, the glass is as likely to be broken by a small grain as by a much larger one. It may be remarked that this conclusion would be upset if rupture depends upon the duration of a strain as well as upon its magnitude. The general argument from dynamical similarity that the maximum strain during impact is independent of linear scale, is, of course, not limited to the case of spheres, which has been chosen merely for convenience of statement. 387. THE EQUILIBRIUM OF REVOLVING LIQUID UNDER CAPILLARY FORCE. [Philosophical Magazine, Vol. XXVIIL pp. 161170, 1914.] THE problem of a mass of homogeneous incompressible fluid revolving with uniform angular velocity (w) and held together by capillary tension (T) is suggested by well-known experiments of Plateau. If there is no rotation, the mass assumes a spherical form. Under the influence of rotation the sphere flattens at the poles, and the oblateness increases with the angular velocity. At higher rotations Plateau's experiments suggest that an annular form may be one of equilibrium. The earlier forms, where the liquid still meets the axis of rotation, have been considered in some detail by Beer*, but little attention seems to have been given to the equilibrium in the form of a ring. A general treatment of this case involves difficulties, but if we assume that the ring is thin, viz. that the diameter of the section is small compared with the diameter of the circular axis, we may prove that the form of the section is approximately circular and investigate the small departures from that figure. It is assumed that in the cases considered the surface is one of revolution about the axis of rotation. Fig. 1 represents a section by a plane through the axis Oy, being the point where the axis meets the equatorial plane. One of the principal y Q Fig. 1. * Pogg. Ann. Vol. xcvi. p. 210 (1855) ; compare Poincar^'s Capillarity 1895. R. VI. 17 258 THE EQUILIBRIUM OF REVOLVING LIQUID [387' curvatures of the surface at P is that of the meridianal curve, the radius of the other principal curvature is PQ the -normal as terminated on the axis. The pressure due to the curvature is thus T { - + \P PQJ' and the equation of equilibrium may be written where p is the pressure at points lying upon the axis, and <r is the density of the fluid. The curvatures may most simply be expressed by means of s, the length of the arc of the curve measured say from A. Thus J__ldy 'l_*yjd* PQ'xds* p~~dx^ > so that (1) becomes dy dx ^ ^ d*y _ capo? dx ^ pgX dx or on integration ds* ds* 2f~ ds* T ds' dy Thus dy/ds is a function of x of known form, say X, and we get for y in terms of x as given by Beer. * If, as in fig. 1, the curve meets the axis, (3) must be satisfied by x = 0, dy/ds = 0. The constant accordingly disappears, and we have the much simplified form ds = 8T + 2T '^' At the point A on the equator dy/ds = 1. If OA = a, whence eliminating p and writing W we get 1914] UNDER CAPILLARY FORCE In terms of y and x from (7) -- n ^- n )T or if we write (9) . V{1 + 2 (1 - when we neglect higher powers of fl than ft 2 . Reverting to x, we find for the integral of (10) no constant being added since y = when x = a. If we stop at ft, we have a , f representing an ellipse whose minor axis OB is a (1 ft). When ft 2 is retained, 05 = (1 -n + fl 2 )a (13) The approximation in powers of fl could of course be continued if desired. So long as H < 1, p is positive and the (equal) curvatures at B are convex. When ft = 1, p = and the surface at B is flat. In this case (8) gives or if we set x = a sin <j>, Here # = a corresponds to </> = TT, and # = corresponds to <f> = 0. Hence if The integral in (16) may be expressed in 'terms of gamma functions and we get (17) When H > 1, the curvature at B is concave and p is negative, as is quite permissible. 172 260 THE EQUILIBRIUM OF REVOLVING LIQUID [387 In order to trace the various curves we may calculate by quadratures from (4) the position of a sufficient number of points. This, as I understand, was the procedure adopted by Beer. An alternative method is to trace the curves by direct use of the radius of curvature at the point arrived at. Starting from (7) we find ds* V a* a / ds ' and thence From (18) we see at once that H = makes p = a throughout, and that when ft = 1, x = makes p = oo . In tracing a curve we start from the point A in a known direction and with p = a/(2H + 1), and at every point arrived at we know with what curvature to proceed. If, as has been assumed, the curve meets the axis, it must do so at right angles, and a solution is then obtained. The method is readily applied to the case fl = 1 with the advantage that we know where the curve should meet the axis of y. From (18) with O = 1 and a = 5, Starting from x 5 we draw small portions of the curve corresponding to decrements of x equal to '2, thus arriving in succession at the points for which x = 4*8, 4'G, 4*4, &c. For these portions we employ the mean curvatures, corresponding to x = 4'9, 4'7, &c. calculated from (19). It is convenient to use squared paper and fair results may be obtained with the ordinary ruler and compasses. There is no need actually to draw the normals. But for such work the procedure recommended by Boys* offers great advantages. The ruler and compasses are replaced by a straight scale divided upon a strip of semi-transparent celluloid. At one point on the scale a fine pencil point protrudes through a small hole and describes the diminutive circular arc. Another point of the scale at the required distance occupies the centre of the circle and is held temporarily at rest with the aid of a small brass tripod standing on sharp needle points. After each step the celluloid is held firmly to the paper and the tripod is moved to the point of the scale required to give the next value of the curvature. The ordinates of the curve so drawn are given in the second and fifth columns of the annexed table. It will be seen that from x = to x = 2 the curve is very flat. Fig. (1). * I'hil. Mag. Vol. xxzvi. p. 75 (1893). I am much indebted to Mr Boys for the loan of suitable instruments. The use is easy after a little practice. 1914] UNDER CAPILLARY FORCE 261 Another case of special interest is the last figure reaching the axis of symmetry at all, which occurs at the point x = 0. We do not know before- hand to what value of 1 this corresponds, and curves must be drawn tentatively. It appears that fl = 2'4 approximately, and the values of y obtained from this curve are given in columns 3 and 6 of the table. Fig. (2)*. Fig. (1). X *1f y' x y iy o-o 2-16 o-oo 2-6 2-06 0-75 0-2 2-16 o-oi 2-8 2-03 0-83 0'4 2-16 0-03 3-0 1-99 0-90 0-6 2-16 0-06 3'2 1-95 0-95 0-8 2-16 o-io 3-4 1-89 0-99 1-0 2-15 0-14 3-6 1-81 1-01 1-2 2-15 0-20 3-8 1-72 1-02 1-4 2-15 0-27 4-0 1-61 1-00 1-6 2-15 0-34 4-2 1-49 0-98 1-8 2-14 0-42 4.4 1-32 0-89 2-0 2-12 0-50 4-6 I'll 0-78 2-2 2-11 0-58 4-8 0-80 0-67 2-4 2-09 0-65 4-9 0-59 0-41 5-0 o-oo o-oo There is a little difficulty in drawing the curve through the point of zero curvature. I found it best to begin at both ends (x = 0, y = 0) and (x = 5, y = 0) with an assumed value of fl and examine whether the two parts could be made to fit. * [1916. These figures were omitted in the original memoir.] THE EQUILIBRIUM OF REVOLVING LIQUID [387 When ft > 2'4 and the curve does not meet the axis at all, the constant in (3) must be retained, and the difficulty is much increased. If we suppose that dy/ds = + 1 when x = a* and dy/ds = 1 when a? = Oj, we can determine p as well as the constant of integration, and (3) becomes .(20) We may imagine a curve to be traced by means of this equation. We start from the point A where y = 0, x = a., and in the direction perpendicular to OA, and (as before) we are told in what direction to proceed at any point reached. When # = c^, the tangent must again be parallel to the axis, but there is nothing to ensure that this occurs when y = 0. To secure this end and so obtain an annular form of equilibrium, (rtf/T must be chosen suitably, but there is no means apparent of doing this beforehand. The process of curve tracing can only be tentative. If we form the expression for the curvature as before, we obtain by means of which the curves may be traced tentatively. If we retain the normal PQ, as we may conveniently do in using Boys' method, we have the simpler expression 1 . 1 <reo 2 /0 , Oa-0, ...(22) When the radius CP of the section is very small in comparison with the radius of the ring OC, the conditions are approximately satisfied by a circular y form. We write CP r, OC = a, PC A = 6. Then, r being supposed constant, the principal radii of curvature are r and a sec + r, so that the equation of equilibrium is 1914] UNDER CAPILLARY FORCE in which p should be constant as 6 varies. In this cos 6 a + rcos8 / r V r 2 2r \ a J 2o* a Thus approximately The term in cos# will vanish if we take o> so that -^) (25) The coefficient of cos 26 then becomes + cubes of - (26) If we are content to neglect r/a in comparison with unity, the condition of equilibrium is satisfied by the circular form ; otherwise there is an inequality of pressure of this order in the term proportional to cos 20. From (25) it is seen that if a and T be given, the necessary angular velocity increases as the radius of the section decreases. In order to secure a better fulfilment of the pressure equation it is necessary to suppose r variable, and this of course complicates the expressions for the curvatures. For that in the rneridianal plane we have P or with sufficient approximation p r For the curvature in the perpendicular plane we have to substitute PQ[, measured along the normal, for PQ, whose expression remains as before (fig. 3). Now W = slnir = C S P ~ tan e Sin P in which 264 THE EQUILIBRIUM OF REVOLVING LIQUID approximately, [387 Thus 1 COS0 f _ J_ PQ'~a + rcos0\ 2r* a + r cos r 2r 2 J j * .(28) Fig. 3. It will be found that it is unnecessary to retain (drfdO) 2 , and thus the pressure equation becomes ?-HS acos# a sin 1 dr &> 2 a 3 a + r* cos a + r cos (29) It is proposed to satisfy this equation so far as terms of the order r*/a 2 inclusive. As a function of 6, r may be taken to be r = r + 8r = r + r t cos 6 + r z cos 20 + .(30) where r,, r 2 , &c. are constants small relatively to r . It will appear that to our order of approximation (8r/r ) 2 may be neglected and that it is unnecessary to include the r's beyond r 3 inclusive. We have acostf a + r cos 5 + 5 + JL + 5Ql + cos 39 |r> + r 4 + pql , * * 2 2 1914] UNDER CAPILLARY FORCE 265 ~ S -J&. = - 2 | r i cos + 4r 2 cos 20 + 9r s cos 301 , aff r o ( ) asin0 Irfr^ r, + r 2 + ^ Q (r^ _ r, + jjr,) {2r ~2r J ' |ro~4aj' Thus altogether for the coefficient of cos on the right of (29) we get 3r 2 r l r 2 aPa? J2r r a ) + 4a?~2a~r ~~2T (a, + aj ' This will be made to vanish if we take &> such that , 3r 2 r, 3r a The coefficient of cos 20 is 3ar 2 _ ^ _j_ _ 3rs _ &) 2 , , r 2 2a 2r 2r 2T |a a 2a or when we introduce the value of &> from (31) 3ar 2 3r 2r 8 r ft 2 4a r, .(32) The coefficient of cos 30 is in like manner ^TsT ~*~ T^i + oIT ("") These coefficients are annulled and o^o/^ 7 is rendered constant so far as the second order of r /a inclusive, when we take r 4 , r s , &c. equal to zero and r 2 /r = r 2 /4a 2 , r 3 /r = - 3r 8 /64a 3 ................ (34) We may also suppose that r x = 0. The solution of the problem is accordingly that ............... (35) gives the figure of equilibrium, provided &> be such that (36) The form of a thin ring of equilibrium is thus determined ; but it seems probable that the equilibrium would be unstable for disturbances involving a departure from symmetry round the axis of revolution. 388. FURTHER REMARKS ON THE STABILITY OF VISCOUS FLUID MOTION. [Philosophical Magazine, Vol. xxvm. pp. 609619, 1914.] AT an early date my attention was called to the problem of the stability of fluid motion in connexion with the acoustical phenomena of sensitive jets, which may be ignited or unignited. In the former case they are usually referred to as sensitive flames. These are naturally the more conspicuous experimentally, but the theoretical conditions are simpler when the jets are unignited, or at any rate not ignited until the question of stability has been decided. The instability of a surface of separation in a non-viscous liquid, i.e. of a surface where the velocity is discontinuous, had already been remarked by Helmholtz, and in 1879 I applied a method, due to Kelvin, to investigate the character of the instability more precisely. But nothing very practical can be arrived at so long as the original steady motion is treated as discontinuous, for in consequence of viscosity such a discontinuity in a real fluid must instantly disappear. A nearer approach to actuality is to suppose that while the velocity in a laminated steady motion is continuous, the rotation or vorticity changes suddenly in passing from one layer of finite thickness to another. Several problems of this sort have been treated in various papers*. The most general conclusion may be thus stated. The steady motion of a non-viscous liquid in two dimensions between fixed parallel plane walls is stable provided that the velocity U, everywhere parallel to the walls and a function of y only, is such that cPU/dy 1 is of one sign throughout, y being the coordinate measured perpendicularly to the walls. It is here assumed that the disturbance is in two dimensions and infinitesimal. It involves * Proc. Lond. Math. Soc. Vol. x. p. 4 (1879) ; xi. p. 57 (1880) ; MX. p. 67 (1887) ; xxvn. p. 6 (1895) ; Phil. Mag. Vol. xxxiv. p. 59 (1892) ; xxvi. p. 1001 (1913) ; Scientific Paper*, Arts. 58, 66, 144, 216, 194. [See also Art. 377.] 1914] ON THE STABILITY OF VISCOUS FLUID MOTION 267 a slipping at the walls, but this presents no inconsistency so long as the fluid is regarded as absolutely non- viscous. The steady motions for which stability in a non-viscous fluid may be inferred include those assumed by a viscous fluid in two important cases, (i) the simple shearing motion between two planes for which d?U/dy* = 0, and (ii) the flow (under suitable forces) between two fixed plane walls for which d*U/dy 2 is a finite constant. And the question presented itself whether the effect of viscosity upon the disturbance could be to introduce instability. An affirmative answer, though suggested by common experience and the special investigations of 0. Reynolds*, seemed difficult to reconcile with the undoubted fact that great viscosity makes for stability. It was under these circumstances that " the Criterion of the Stability and Instability of the Motion of a Viscous Fluid," with special reference to cases (i) and (ii) above, was proposed as the subject of an Adams Prize essayf, and shortly afterwards the matter was taken up by Kelvin J in papers which form the foundation of much that has since been written upon the subject. His conclusion was that in both cases the steady motion is wholly stable for infinitesimal disturbances, whatever may be the value of the viscosity (yu.) ; but that when the disturbances are finite, the limits of stability become narrower and narrower as /j, diminishes. Two methods are employed : the first a special method applicable only to case (i) of a simple shear, the second (ii) more general and applicable to both cases. In 1892 (I.e.) I had occasion to take exception to the proof of stability by the second method, and Orr has since shown that the same objection applies to the special method. Accordingly Kelvin's proof of stability cannot be considered sufficient, even in case (i). That Kelvin himself (partially) recognized this is shown by the following interesting and characteristic letter, which I venture to give in full. July 10 (?1895). " On Saturday I saw a splendid illustration by Arnulf Mallock of our ideas regarding instability of water between two parallel planes, one kept moving and the other fixed. (Fig. 1) Coaxal cylinders, nearly enough planes for our illustration. The rotation of the outer can was kept very accurately uniform at whatever speed the governor was set for, when left to itself. At one of the speeds he shewed me, the water came to regular regime, quite smooth. I dipped a disturbing rod an inch or two down into the water and immediately the torque increased largely. Smooth regime could only be * Phil. Trans. 1883, Part HI. p. 935. t Phil. Mag. Vol. xxiv. p. 142 (1887). The suggestion came from me, but the notice was (I think) drawn up by Stokes. J PhiL Mag. Vol. xxiv. pp. 188, 272 (1887) ; Collected Papers, Vol. iv. p. 321. Orr, Proc. Roy. Irish Acad. Vol. XXVH. (1907). 268 ON THE STABILITY OF VISCOUS FLUID MOTION [388 re-established by slowing down and bringing up to speed again, gradually enough. " Without the disturbing rod at all, I found that by resisting the outer can by hand somewhat suddenly, but not very much so, the torque increased suddenly and the motion became visibly turbulent at the lower speed and remained so. " I have no doubt we should find with higher and higher speeds, very gradually reached, stability of laminar or non-turbulent motion, but with narrower and narrower limits as to magnitude of disturbance ; and so find through a large range of velocity, a confirmation of Phil Mag. 1887, 2, pp. 191 196. The experiment would, at high velocities, fail to prove the stability which the mathematical investigation proves for every velocity however high. mercury- -* hung torsiona/ly to measure torque rotating rotating fixed Fig. 1. " As to Phil. Mag. 1887, 2, pp. 272278, I admit that the mathematical proof is not complete, and withdraw [temporarily ?] the words ' virtually inclusive ' (p. 273, line 3). I still think it probable that the laminar motion is stable for this case also. In your (Phil. Mag. July 1892, pp. 67, 68) refusal to admit that stability is proved you don't distinguish the case in which my proof was complete from the case in which it seems, and therefore is, not complete. " Your equation (24) of p. 68 is only valid for infinitely small motion, in which the squares of the total velocities are everywhere negligible ; and in this case the motion is manifestly periodic, for any stated periodic con- ditions of the boundary, and comes to rest according to the logarithmic law if the boundary is brought to rest at any time. 1914] ON THE STABILITY OF VISCOUS FLUID MOTION 269 "In your p. 62, lines 11 and 12 are 'inaccurate.' Stokes limits his investigation to the case in which the squares of the velocities can be neglected . . radius of globe x velocity (i.e. . * . . -- * very small), diffusivity in which it is manifest that the steady motion is the same whatever the viscosity ; but it is manifest that when the squares cannot be neglected, the steady motion is very different (and horribly difficult to find) for different degrees of viscosity. " In your p. 62, near the foot, it is not explained what V is ; and it disappears henceforth. Great want of explanation here Did you not want your paper to be understandable without Basset in hand ? I find your two papers of July/92, pp. 6170, and Oct./93,pp. 355372, very difficult reading, in every page, and in some oc ly difficult. " Pp. 366, 367 very mysterious. The elastic problem is not defined. It is impossible that there can be the rectilineal motion of the fluid asserted in p. 367, lines 17 19 from foot, in circumstances of motion, quite undefined, but of some kind making the lines of motion on the right side different from those on the left. The conditions are not explained for either the elastic- solid *, or the hydraulic case. " See p. 361, lines 19, 20, 21 from foot. The formation of a backwater depends essentially on the non-negligibility of squares of velocities ; and your p. 367, lines 1 4, and line 17 from foot, are not right. " If you come to the R. S. Library Committee on Thursday we may come to agreement on some of these questions." Although the main purpose in Kelvin's papers of 1887 was not attained, his special solution for a disturbed vorticity in case (i) is not without interest. The general dynamical equation for the vorticity in two dimensions is where v(=^jp) is the kinematic viscosity and V 2 = d^fda? + d 2 /dy 2 . In this hydrodynamical equation is itself a feature of the motion, being connected with the velocities u, v by the relation du dv while u, v themselves satisfy the " equation of continuity " du dv * I think Kelvin did not understand that the analogous elastic problem referred to is that of a thin plate. See words following equation (5) of my paper. 270 ON THE STABILITY OF VISCOUS FLUID MOTION [388 In other applications of (1), e.g. to the diffusion of heat or dissolved matter in a moving fluid, f is a new dependent variable, not subject to (2), and representing temperature or salinity. We may then regard the motion as known while % remains to be determined. In any case D^/Dt = v f V a If the fluid move within fixed boundaries, or extend to infinity under suitable conditions, and we integrate over the area included, so that ......... (4) by Green's theorem. The boundary integral disappears, if either or d/dn there vanishes, and then the integral on the left necessarily diminishes as time progresses*. The same conclusion follows if f and d^/dn have all along the boundary contrary signs. Under these conditions tends to zero over the whole of the area concerned. The case where at the boundary is required to have a constant finite value Z is virtually included, since if we write Z + ' for , Z disappears from (1), and f everywhere tends to the value Z. In the hydrodynamical problem of the simple shearing motion, is a constant, say Z, u is a linear function of y, say U, and v = 0. If in the disturbed motion the vorticity be Z + and the components of velocity be U + u and v, equation (1) becomes in which f, u, and v relate to the disturbance. If the disturbance be treated as infinitesimal, the terms of the second order are to be omitted and we get simply s+ *-'** .............................. <> In (6) the motion of the fluid, represented by U simply, is given independently of f, and the equation is the same as would apply if denoted the tempera- ture, or salinity, of the fluid moving with velocity U. Any conclusions that we may draw have thus a widened interest. In Kelvin's solution of (6) the disturbance is supposed to be periodic in oc, proportional to e ikx , and U is taken equal to /3y. He assumes for trial Compare Orr, I.e. p. 115. 1914] ON THE STABILITY OF VISCOUS FLUID MOTION 271 where T is a function of t On substitution in (6) he finds t?T a = , v {k* + (n- W] T, whence T = Ce-^+^-n^+iW}, ........................ (8) and comes ultimately to zero. Equations (7) and (8) determine and so suffice for the heat and salinity problems in an infinitely extended fluid. As an example, if we suppose n = and take the real part of (7), (9) reducing to =Ccoskx simply when = 0. At this stage the lines of constant are parallel to y. As time advances, T diminishes with increasing rapidity, and the lines of constant " tend to become parallel to x. If x be constant, varies more and more rapidly with y. This solution gives a good idea of the course of events when a liquid of unequal salinity is stirred. In the hydrodynamical problem we have further to deduce the small velocities u, v corresponding to From (2) and (3), if u and v are pro- portional to e***, Thus, corresponding to (9), No complementary terms satisfying cfty/cfa/ 2 k z v = are admissible, on account of the assumed periodicity with x. It should be mentioned that in Kelvin's treatment the disturbance is not limited to be two-dimensional. Another remarkable solution for an unlimited fluid of Kelvin's equation (6) with U '= fty has been given by Oseen*. In this case the initial value of is concentrated at one point (, rj), and the problem may naturally be regarded as an extension of one of Fourier relating to the conduction of heat. Oseen finds }' _ (n-y)* where (7 = f(, T,, 0) ddr, ; ......................... (13) and the result may be verified by substitution. * Arkivfor Matematik, Astronomi och Fysik, Upsala, Bd. vn. No. 15 (1911). 272 ON THE STABILITY OF VISCOUS FLUID MOTION [388 "The curves = const, constitute a system of coaxal and similar ellipses, whose centre at t = coincides with the point , 77, and then moves with the velocity /3i) parallel to the ar-axis. For very small values of t the eccen- tricity of the ellipse is very small and the angle which the major axis makes with the tf-axis is about 45. With increasing t this angle becomes smaller. At the same time the eccentricity becomes larger. For infinitely great values of t, the angle becomes infinitely small and the eccentricity infinitely great." When = in (12), we fall back on Fourier's solution. Without loss of generality we may suppose = 0, 77 = 0, and then (r 2 representing the diffusion of heat, or vorticity, in two dimensions. It may be worth while to notice the corresponding tangential velocity in the hydro- dynamical problem. If ^r be the stream-function, so that the constant of integration being determined from the known value of d^/dr when r= oc . When r is small (15) gives becoming finite when r = so soon as t is finite. At time t the greatest value of d-^/dr occurs when r = 1-256 x4irf ............................ (17) On the basis of his solution Oseen treats the problem of the stability of the shearing motion between two parallel planes and he ^arrives at the conclusion, in accordance with Kelvin, that the motion is stable for infinitesimal disturbances. For this purpose he considers " the specially unfavourable case " where the distance between the planes is infinitely great. I cannot see myself that Oseen has proved his point. It is doubtless true that a great distance between the planes is unfavourable to stability, but to arrive at a sure conclusion there must be no limitation upon the character of the infinitesimal disturbance, whereas (as it appears to me) Oseen assumes that the disturbance does not sensibly reach the walls. The simultaneous evanescence at the walls of both velocity-components of an otherwise sensible disturbance would seem to be of the essence of the question. 1914] ON THE STABILITY OF VISCOUS FLUID MOTION 273 It may be added that Oseen is disposed to refer the instability observed in practice not merely to the square of the disturbance neglected in (6), but also to the inevitable unevenness of the walls. We may perhaps convince ourselves that the infinitesimal disturbances of (6), with U '= fiy, tend to die out by an argument on the following lines, in which it may suffice to consider the operation of a single wall. The argument could, I think, be extended to both walls, but the statement is more complicated. When there is but one wall, we may as well fix ideas by supposing that the wall is at rest (at y = 0). The difficulty of the problem arises largely from the circumstance that the operation of the wall cannot be imitated by the introduction of imaginary vorticities on the further side, allowing the fluid to be treated as uninterrupted. We may indeed in this way satisfy one of the necessary conditions. Thus if corresponding to every real vorticity at a point on the positive side we introduce the opposite vorticity at the image of the point in the plane y = 0, we secure the annulment in an unlimited fluid of the velocity-component v parallel to y, but the component u, parallel to the flow, remains finite. In order further to annul u, it is in general necessary to introduce new vorticity at y = 0. The vorticities on the positive side are not wholly arbitrary. Let us suppose that initially the only (additional) vorticity in the interior of the fluid is at A, and that this vorticity is clockwise, or positive, like that of the undisturbed motion (fig. 2). If this existed alone, there would be of necessity a finite velocity u along the wall in its neighbourhood. In order y=p Fig. 2. Fig. 3. to satisfy the condition u 0, there must be instantaneously introduced at the wall a negative vorticity of an amount sufficient to give compensation. To this end the local intensity must be inversely as the distance from A and as the sine of the angle between this distance and the wall (Helmholtz). As we have seen these vorticities tend to diffuse and in addition to move with the velocity of the fluid, those near the wall slowly and those arising from A more quickly. As A is carried on, new negative vorticities are developed at those parts of the wall which are being approached. At the other end the vorticities near the wall become excessive and must be com- pensated. To effect this, new positive vorticity must be developed at the wall, whose diffusion over short distances rapidly annuls the negative so far K. vi. 18 274 ON THE STABILITY OF VISCOUS FLUID MOTION [388 as may be required. After a time, dependent upon its distance, the vorticity arising from A loses its integrity by coming into contact with the negative diffusing from the wall and thus suffers diminution. It seems evident that the end can only be the annulment of all the additional vorticity and restoration of the undisturbed condition. So long as we adhere to the suppositions of equation (6), the argument applies equally well to' an original negative vorticity at A, and indeed to any combination of positive and negative vorticities, however distributed. It is interesting to inquire how this argument would be affected by the retention in (5) of the additional velocities u, v, which are omitted in (6), though a definite conclusion is hardly to be expected. In fig. 2 the negative vorticity which diffuses inwards is subject to a backward motion due to the vorticity at A in opposition to the slow forward motion previously spoken of. And as A passes on, this negative vorticity in addition to the diffusion is also convected inwards in virtue of the component velocity v due to A. The effect is thus a continued passage inwards behind A of negative vorticity, which tends to neutralize in this region the original constant vorticity (Z). When the additional vorticity at A is negative (fig. 3), the convection behind A acts in opposition to diffusion, and thus the positive developed near the wall remains closer to it, and is more easily absorbed as A passes on. It is true that in front of A there is a convection of positive inwards ; but it would seem that this would lead to a more rapid annulment of A itself; and that upon the whole the tendency is for the effect of fig. 2 to preponderate. If this be admitted, we may perhaps see in it an explanation of the diminution of vorticity as we recede from a wall observed in certain circumstances. But we are not in a position to decide whether or not a disturbance dies down. By other reasoning (Reynolds, Orr) we know that it will do so if /9 be small enough in relation to the other elements of the problem, viz. the distance between the walls and the kinematic viscosity v. A precise formulation of the problem for free infinitesimal disturbances was made by Orr (1907). We suppose that and v are proportional to e int e iftx f w here n =p + iq. If V0 = S, we have from (6) and (10) (18) and fi<- **-.' ................................. (19) with the boundary conditions that v = 0, dvjdy = at the walls. Orr easily shows that the period-equation takes the form .......... (20) 1914] ON THE STABILITY OF VISCOUS FLUID MOTION 275 where S lt S 2 are any two independent solutions of (18), and the integrations are extended over the interval between the walls. An equivalent equation was given a little later (1908) independently by Sommerfeld*. Stability requires that for no value of k shall any of the q's determined by (20) be negative. In his discussion Orr arrives at the conclusion that this condition is satisfied, though he does not claim that his method is rigorous. Another of Orr's results may be mentioned here. He shows that p + kfiy necessarily changes sign in the interval between the walls. The stability problem has further been skilfully treated by v. Misesf and by Hopf J, the latter of whom worked at the suggestion of Sommerfeld, with the result of confirming the conclusions of Kelvin and Orr. Doubtless the reasoning employed was sufficient for the writers themselves, but the statements of it put forward hardly carry conviction to the mere reader. The problem is indeed one of no ordinary difficulty. It may, however, be simplified in one respect, as has been shown by v. Mises. It suffices to prove that q can never be zero, inasmuch as it is certain that in some cases (0 = 0) q is positive. In this direction it may be possible to go further. When /8=0, it is easy to show that not merely q, but q k*v, is positive. According to Hopf, this is true generally. Hence it should suffice to omit k* q/v in (18), and then to prove that the S-solutions obtained from the equation so simplified cannot satisfy (20). The functions Si and S 2 , satisfying the simplified equation where 77 is real, being a linear function of y with real coefficients, could be completely tabulated by the combined use of ascending and descending series, as explained by Stokes in his paper of 1857 1|. At the walls 77 takes opposite signs. Although a simpler demonstration is desirable, there can remain (I suppose) little doubt but that the shearing motion is stable for infinitesimal dis- turbances. It has not yet been proved theoretically that the stability can fail for finite disturbances on the supposition of perfectly smooth walls ; but such failure seems probable. We know from the work of Reynolds, Lorentz, and Orr that no failure of stability can occur unless @D*/v > 177, where D is the distance between the walls, so that j3D represents their relative motion. * Atti del IV. Congr. intern, dei Math. Roma (1909). t Festschrift H. Weber, Leipzig (1912), p. 252; Jahresber. d. Deutschen Math. Ver. Bd. xxi. p. 241 (1913). The mathematics has a very wide scope. J Ann. der Physik, Bd. XLIV. p. 1 (1914). Phil. Mag. Vol. xxxiv. p. 69 (1892) ; Scientific Papers, Vol. in. p. 583. || Camb. Phil. Trans. Vol. x. p. 106 ; Math, and Phyt. Papers, Vol. iv. p. 77. This appears to have long preceded the work of Hankel. I may perhaps pursue the line of inquiry here suggested. 182 389. NOTE ON THE FORMULA FOR THE GRADIENT WIND. [Advisory Committee for Aeronautics. Reports and Memoranda. No. 147. January, 1915.] AN instantaneous derivation of the formula for the " gradient wind " has been given by Gold*. " For the steady horizontal motion of air along a path whose radius of curvature is r, we may write directly the equation (cor sin X + vf _ 1 dp (cor sin X)* r p dr r expressing the fact that the part of the centrifugal force arising from the motion of the wind is balanced by the effective gradient of pressure. "In the equation p is atmospheric pressure, p density, v velocity of moving air, X is latitude, and o> is the angular velocity of the earth about its axis." Gold deduces interesting consequences relating to the motion and pressure of air in anti-cyclonic regions f . But the equation itself is hardly obvious without further explanations, unless we limit it to the case where sin X = 1 (at the pole) and whore the relative motion of the air takes place about the same centre as the earth's rotation. I have thought that it may be worth while to take the problem avowedly in two dimensions, but without further restriction upon the character of the relative steady motion. The axis of rotation is chosen as axis of z. The axes of x and y being supposed to rotate in their own plane with angular velocity co, we denote by u, v, the velocities at time t, relative to these axes, of the particle which then occupies the position x, y. The actual velocities of the same particle, parallel to the instantaneous positions of the axes, will be u coy, v + cox, and the accelerations in the same directions will be du du du -ji + w j- + *> j 2cov a>*x dt dx dy * Proc. Roy. Soc. Vol. LXXX A. p. 436 (1908). t See also Shaw's Forecasting Weather, Chapter u. 1915] NOTE ON THE FORMULA. FOR THE GRADIENT WIND 277 and dv dv dv -T7 + 1* -T- + v- r - + 2ow o) 2 y*. at ax dy Since the relative motion is supposed to be steady, du/dt, dv/dt disappear, and the dynamical equations are i *-.*.+ 2 --,*!,. ...(1) p dx dx dy - ---. p dy dx dy The velocities u, v may be expressed by means of the relative stream- function 1/r : u = dty/dy, v = - d-^/dx. Equations (1), (2) then become - ..... P dx dx 2 dx \\dx ) \ dy ) j dx ' I d d I and on integration, if we leave out the part of p independent of the relative motion, in which by a known theorem V 2 \/r is a function of ^r only. If &> be omitted, (5) coincides with the equation given long ago by Stokes f . It expresses p in terms of ty ; but it does not directly allow of the expression of >|r in terms of p, as is required if the data relate to a barometric chart. We may revert to the more usual form, if in (1) or (3) we take the axis of x perpendicular to the direction of (relative) motion at any point. Then u = 0, and \f = Zmv + ^^ ......................... (6) p dx dx dy* But since d-^/dy = 0, the curvature at this place of the stream-line (ty = const.) is 1 and thus -^ = 2ft,v + -, ....... (7) p dx ~ r * Lamb's Hydrodynamics, 206. f Camb. Phil. Trans. Vol. vu. 1842 ; Math, and Phys. Papers, Vol. i. p. 9. 278 NOTE ON THE FORMULA FOR THE GRADIENT WIND giving the velocity v in terms of the barometric gradient dp/dx\>y means of a quadratic. As is evident from the case at = 0, the positive sign in the alternative is to be taken when x and r are drawn in opposite directions. In (7) r is not derivable from the barometric chart, nor can -fy be deter- mined strictly by means of p. But in many cases it appears that the more important part of p, at any rate in moderate latitudes, is that which depends upon a>, so that approximately from (5) (8) Substituting this value of -^ in the smaller terms, we get as a second approximation With like approximation we may identify r in (7) with the radius of curvature of the isobaric curve which passes through the point in question. The interest of these formulae depends largely upon the fact that the velocity calculated as above from the barometric gradient represents fairly well the wind actually found at a moderate elevation. At the surface the discrepancy is larger, especially over the land, owing doubtless to friction. 390. SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF RESONATORS EXPOSED TO PRIMARY PLANE WAVES. [Philosophical Magazine, Vol. xxix. pp. 209222, 1915.] RECENT investigations, especially the beautiful work of Wood on " Radia- tion of Gas Molecules excited by Light"*, have raised questions as to the behaviour of a cloud of resonators under the influence of plane waves of their own period. Such questions are indeed of fundamental importance. Until they are answered we can hardly approach the consideration of absorp- tion, viz. the conversion of radiant into thermal energy. The first action is upon the molecule. We may ask whether this can involve on the average an increase of translatory energy. It does not seem likely. If not, the transformation into thermal energy must await collisions. The difficulties in the way of answering the questions which naturally arise are formidable. In the first place we do not understand what kind of vibration is assumed by the molecule. But it seems desirable that a be- ginning should be made ; and for this purpose I here consider the case of the simple aerial resonator vibrating symmetrically. The results cannot be regarded as even roughly applicable in a quantitative sense to radiation, inasmuch as this type is inadmissible for transverse vibrations. Nevertheless they may afford suggestions. The action of a simple resonator under the influence of suitably tuned primary aerial waves was considered in Theory of Sound, 319 (1878). The primary waves were supposed to issue from a simple source at a finite distance c from the resonator. With suppression of the time-factor, and at a distance r from their source, they are represented! by the potential * A convenient summary of many of the more important results is given in the Guthrie Lecture, Proc. Phy*. Soc. Vol. xxvi. p. 185 (1914). t A slight change of notation is introduced. 280 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390 in which k = 2-rr/X, and X is the wave-length ; and it appeared that the potential of the secondary waves diverging from the resonator is so that 47rr' a Mod 2 i/r = 47r/ s c 1 ......................... (3) The left-hand member of (3) may be considered to represent the energy dispersed. At the distance of the resonator If we inquire what area S of primary wave-front propagates the same energy as is dispersed by the resonator, we have or S = 4,7r/J<? = \*/'jr ............................... (4) Equation (4) applies of course to plane primary waves, and is then a particular case of a more general theorem established by Lamb*. It will be convenient for our present purpose to start de novo with plane primary waves, still supposing that the resonator is simple, so that we are concerned only with symmetrical terms, of zero order in spherical harmonics. Taking the place of the resonator as origin and the direction of pro- pagation as initial line, we may represent the primary potential by (f> = C rco8 _ 1 + ifo cos _ fcs r 2 CQS 2 Q + ............. (5) The potential of the symmetrical waves issuing from the resonator may be taken to be Since the resonator is supposed to be an ideal resonator, concentrated in a point, r is to be treated as infinitesimal in considering the conditions to be there satisfied. The first of these is that no work shall be done at the resonator, and it requires that total pressure and total radial velocity shall be in quadrature. The total pressure is proportional to d (<j> + ^/dt, or to i($ + ^), and the total radial velocity is d (0 + ^r)/dr. Thus (<j> + >/r) and d (<j> + ty) / dr must be in the same (or opposite) phases, in other words their ratio must be real. Now, with sufficient approximation, so that a -1 ik=xe&\ ............................... (7) * Camb. Trans. Vol. xvm. p. 348 (1899) ; Proc. Math. Soc. Vol. xxxn. p. 11 (1900). The resonator is no longer limited to be simple. See also Rayleigh, Phil. Mag. Vol. m. p. 97 (1902) ; Scientific Papers, Vol. v. p. 8. 1915] RESONATORS EXPOSED TO PRIMARY PLANE WAVES 281 If we write l/a = A- 1 e-'*, ........................ (8) then 4= Ar'sina ............................... (9) So far a is arbitrary, since we have used no other condition than that no work is being done at the resonator. For instance, (9) applies when the source of disturbance is merely the presence at the origin of a small quantity of gas of varied character. The peculiar action of a resonator is to make A a maximum, so that sin a = + 1, say 1. Then A = l/k, a = -i/k, ........................ (10) tgOtr and ^ = -- ............................... (11) As in (3), 47rr 2 Mod 2 ^ = 47r/fc 2 = ;\ 2 /7r, ..................... (12) and the whole energy dispersed corresponds to an area of primary wave- front equal to X 2 /7r. The condition of resonance implies a definite relation between (<f> + ty) and d (<f) + ty) / dr. If we introduce the value of a from (10), we see that this is <*> + * = l/a + l/r-ft d(<f> + +)/dr -1/r* and this is the relation which must hold at a resonator so tuned as to respond to the primary waves, when isolated from all other influences. The above calculation relates to the case of a single resonator. For many purposes, especially in Optics, it would be desirable to understand the operation of a company of resonators. A strict investigation of this question requires us to consider each resonator as under the influence, not only of the primary waves, but also of the secondary waves dispersed by its neighbours, and in this many difficulties are encountered. If, however, the resonators are not too near one another, or too numerous, they may be supposed to act independently. From (11) it will be seen that the standard of distance is the wave-length. The action of a number (n) of similar and irregularly situated centres of secondary disturbance has been considered in various papers on the light from the sky*. The phase of the disturbance from a single centre, as it reaches a distant point, depends of course upon this distance and upon the situation of the centre along the primary rays. If all the circumstances are accurately prescribed, we can calculate the aggregate effect at a distant point, and the resultant intensity may be anything between and that corresponding to complete agreement of phase among all the components. But such a calculation would have little significance for our present purpose. * Compare also "Wave Theory of Light," Enc. Brit. Vol. xxrv. (1888), 4; Scientific Papers, Vol. in. pp. 53, 54. 282 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390 Owing to various departures from ideal simplicity, e.g. want of homogeneity in the primary vibrations, movement of the disturbing centres, the impossi- bility of observing what takes place at a mathematical point, we are in effect only concerned with the average, and the average intensity is n times that due to a single centre. In the application to a cloud of acoustic resonators the restriction was necessary that the resonators must not be close compared with X; otherwise they would react upon one another too much. This restriction may appear to exclude the case of the light from the sky, regarded as due mainly to the molecules of air; but these molecules are not resonators at any rate as regards visible radiations. We can most easily argue about an otherwise unifonn medium disturbed by numerous small obstacles composed of a medium of different quality. There is then no difficulty in supposing the obstacles so small that their mutual reaction may be neglected, even although the average distance of immediate neighbours is much less than a wave- length. When the obstacles are small enough, the whole energy dispersed may be trifling, but it is well to observe that there must be some. No medium can be fully transparent in all directions to plane waves, which is not itself quite uniform. Partial exceptions may occur, e.g. when the want of uniformity is a stratification in plane strata. The dispersal then becomes a regular reflexion, and this may vanish in certain cases, even though the changes of quality are sudden (black in Newton's rings)*. But such trans- parency is limited to certain directions of propagation. To return to resonators : when they may be close together, we have to consider their mutual reaction. For simplicity we will suppose that they all lie on the same primary wave-front, so that as before in the neighbourhood of each resonator we may take </>=!, d<f>/dr = ............................ (14) Further, we suppose that all the resonators are similarly situated as regards their neighbours, e.g., that they lie at the angular points of a regular polygon. The waves diverging from each have then the same expression, and altogether where r 1( r 2 , ... are the distances of the point where yjr is measured from the various resonators, and a is a coefficient to be determined. The whole potential is <f> + -^r, and it suffices to consider the state of things at the first resonator. With sufficient approximation .................. (16) * See Proe. Roy. Soc. Vol. LXXXVI A, p. 207 (1912) ; [This volume, p. 77]. 1915] RESONATORS EXPOSED TO PRIMARY PLANE WAVES 283 R being the distance of any other resonator from the first, while (as before) d(<f> + W_ a 7 , ~dT~ ~n 2 ............................ ( We have now to distinguish two cases. In the first, which is the more important, the tuning of the resonators is such that each singly would respond as much as possible to the primary waves. The ratio of (16) to (17) must then, as we have seen, be equal to r lf when r^ is indefinitely diminished. Accordingly 1 p-ikR which, of course, includes (10). If we write a = Ae ia , then The other case arises when the resonators are so tuned that the aggregate responds as much as possible to the primary waves. We may then proceed as in the investigation for a single resonator. In order that no work may be done at the disturbing centres, ($ + *$) and d((f> + -^r)fdr must be in the same phase, and this requires that 1 1 p-ikR Jl X M V , - H --- ik + 2, ==- = real, a T! R H .(20) The condition of maximum resonance is that the real part in (20) shall vanish, so that a r ,, ^J'JLJJj (22> The present value of A 2 is greater than that in (19), as was of course to be expected. In either case the disturbance is given by (15) with the value of a determined by (18), or (21). The simplest example is when there are only two resonators and the sign of summation may be omitted in (18). In order to reckon the energy dispersed, we may proceed by either of two methods. In the first we con- sider the value of i/r and its modulus at a great distance r from the resonators. It is evident that \jr is symmetrical with respect to the line R joining the resonators, and if 6 be the angle between r and R, r, r a = R cos 0. Thus r 2 . Mod 2 i/r = A 2 {2 + 2 cos (kR cos 0)} ; 284 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390 and on integration over angular space, (23) Introducing the value of A 3 from (19), we have finally / sin kR\ (: ~TI Mod 8 yr. sin 6 d0 sin kR ' .(24) If we suppose that kR is large, but still so that R is small compared with r, (24) reduces to 87rfc~ 2 or 2\ a /7r. The energy dispersed is then the double of that which would be dispersed by each resonator acting alone ; otherwise the mutual reaction complicates the expression. The greatest interference naturally occurs when kR is small. (24) then becomes 2&IR 2 . 2\ 2 /7r, or 167T.R 2 , in agreement with Theory of Sound, 321. The whole energy dispersed is then much less than if there were only one resonator. It is of interest to trace the influence of distance more closely. If we put kR = 2-Trm, so that R = mX, we may write (24) S = (<2\*/7r).F, (25) where S is the area of primary wave-front which carries the same energy as is dispersed by the two resonators and 2-Trm + sin (2?rm) p = 27T7/1 + (27rm)- 1 + 2 sin (2irm) If 2m is an integer, the sine vanishes and 1 .(26) .(27) l+(27rm)- 2 ' not differing much from unity even when 2m = 1 ; and whenever 2m is great, F approaches unity. The following table gives the values of F for values of 2m not greater than 2 : 2m F 2m F 2m F 0'05 0-0459 0-70 0-7042 1-40 1-266 o-io 0-1514 0-80 0-7588 1-50 1-269 0-20 0-3582 0-90 0-8256 1-60 1-226 0-30 0-4836 1-00 0-9080 1-70 1-159 0-40 0-5583 110 I -006 1-80 1-088 0-50 0-6110 T20 I -1 13 1-90 1 -026 0-60 0-6569 I -30 1-208 2-00 0-975 1915] RESONATORS EXPOSED TO PRIMARY PLANE WAVES 285 In the case of two resonators the integration in (23) presents no difficulty ; but when there are a larger number, it is preferable to calculate the emission of energy in the dispersed waves from the work which would have to be done to generate them at the resonators (in the absence of primary waves) a method which entails no integration. We continue to suppose that all the resonators are similarly situated, so that it suffices to consider the work done at one of them say the first. From (15) (l-ik -ikr ^e- ikR ) d+ a T -i ?i r < ~j~ = . dr r 2 The pressure is proportional to ity, and the part of it which is in the same phase as dty/dr is proportional to Accordingly the work done at each source is proportional to Hence altogether by (19) the energy dispersed by n resonators is that carried by an area 8 of primary wave-front, where ^ sin kR o!^: 2 _ kR _ ( _ r ~^ n i~ D the constant factor being determined most simply by a comparison with the case of a single resonator, for which n = 1 and the S's vanish. We fall back on (24) by merely putting n = 2, and dropping the signs of summation, as there is then only one R. If the tuning is such as to make the effect of the aggregate of resonators a maximum, the cosines in (29) are to be dropped, and we have a- " xv ' ............................ (30) sin kR As an example of (29), we may take 4 resonators at the angular points of a square whose side is b. There are then 3 R's to be included in the sum- mation, of which two are equal to b and one to b \/2, so that (28) becomes (31) 286 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390 A similar result may be arrived at from the value of -^ at an infinite distance, by use of the definite integral* f*V,( sin 0) sin 6 dd = . . . .(32) .'o x As an example where the company of resonators extends to infinity, we may suppose that there is a row of them, equally spaced at distance R. By (18) 1 - -l\R -ScR (33) _ The series may be summed. If we write he-** h'e-* ix 2 = e- fa + 2~ + +..., .................. (34) where h is real and less than unity, we have and 2 = -~log(l-/ie- ia! ) ......................... (35) ft no constant of integration being required, since 2 = - A- 1 log (1- A) when x = 0. If now we put h = 1, 2 = - log (1 - e-**) = - log (2 sin |) + \i (x-ir) + 2i mr ....... (36) Thus ^ = i : - ^ j- log ^2 sin ^ + \i (kR - TT) + 2imr| ....... (37) If kR = 2w7r, or R = m\, where m is an integer, the logarithm becomes infinite and a tends to vanish^. When R is very small, a is also very small, tending to a = R -=- 2 log (kR) ............................ (38) The longitudinal density of the now approximately linear source may be considered to be a/R, and this tends to vanish. The multiplication of resonators ultimately annuls the effect at a distance. It must be remembered that the tuning of each resonator is supposed to be as for itself alone. In connexion with this we inay consider for a moment the problem in two dimensions of a linear resonator parallel to the primary waves, which responds symmetrically. As before, we may take at the resonator * Enc. Brit. 1. c. equation (43) ; Scientific Papert, Vol. in. p. 98. t Phil. Mag. Vol. xrv. p. 60 (1907) ; Scientific Papers, Vol. v. p. 409. 1915] RESONATORS EXPOSED TO PRIMARY PLANE WAVES 287 As regards -v/r, the potential of the waves diverging in two dimensions, we must use different forms when r is small (compared with X) and when r is large*. When r is small -" ; ......... (39) and when r is large, By the same argument as for a point resonator we find, as the condition that no work is done at ?' = 0, that the imaginary part of I/a is ITT/ 2. For maximum resonance a = 2i/7r, ................................. (41) so that at a distance -Jr approximates to o\ Thus 27rr.Mod 2 T/r= , ........................... (43) which expresses the width of primary wave-front carrying the same energy as is dispersed by the linear resonator tuned to maximum resonance. A subject which naturally presents itself for treatment is the effect of a distribution of point resonators over the whole plane of the primary wave- front. Such a distribution may be either regular or haphazard. A regular distribution, e.g. in square order, has the advantage that all the resonators are similarly situated. The whole energy dispersed is then expressed by (29), though the interpretation presents difficulties in general. But even this would not cover all that it is desirable to know. Unless the side of the square (6) is smaller than A,, the waves directly reflected back are accom- " panied by lateral " spectra " whose directions may be very various. When b < X, it seems that these are got rid of. For then not only the infinite lines forming sides of the squares which may be drawn through the points, but a fortiori lines drawn obliquely, such as those forming the diagonals, are too close to give spectra. The whole of the effect is then represented by the specular reflexion. In some respects a haphazard distribution forms a more practical problem, especially in connexion with resonating vapours. But a precise calculation of the averages then involved is probably not easy. * Theory of Sound, 341. 288 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390 If we suppose that the scale (fc) of the regular structure is very small compared with \, we can proceed further in the calculation of the regularly reflected wave. Let Q be one of the resonators, the point in the plane of the resonators opposite to P, at which ty is required ; OP = x, OQ = y, PQ = r. Then if m be the number of resonators per unit area, / e -*r \Jr = 27T7nci I y dy - , Jo v or since ydy = r dr, i/r = 27rma I tr** dr. J X The integral, as written, is not convergent ; but as in the theory of diffraction we may omit the integral at the upper limit, if we exclude the case of a nearly circular boundary. Thus (44) > and Mod^ = ^ .......... , .................... (4p) The value of A 1 is given by (19). We find, with the same limitation as above, ? = 27rw (" cos kR dR = 0, Jo = 27TW (* sin kRdR = 2-irm/k. Jo Thus A*=l/(lc+27rmlk)* and Mo **- ......................... (46) When the structure is very fine compared with \, k? in the denominator may be omitted, and then Mod'^r = 1, that is the regular reflexion becomes total. The above calculation is applicable in strictness only to resonators arranged in regular order and very closely distributed. It seems not unlikely that a similar result, viz. a nearly total specular reflexion, would ensue even when there are only a few resonators to the square wave-length, and these are in motion, after the manner of gaseous molecules; but this requires further examination. In the foregoing investigation we have been dealing solely with forced vibrations, executed in synchronism with primary waves incident upon the resonators, and it has not been necessary to enter into details respecting the constitution of the resonators. All that is required is a suitable adjustment to one another of the virtual mass and spring. But it is also of interest to 1915] RESONATORS EXPOSED TO PRIMARY PLANE WAVES 289 consider free vibrations. These are of necessity subject to damping, owing to the communication of energy to the medium, forthwith propagated away; and their persistence depends upon the nature of the resonator as regards mass and spring, and not merely upon the ratio of these quantities. Taking first the case of a single resonator, regarded as bounded at the surface of a small sphere, we have to establish the connexion between the motion of this surface and the aerial pressure operative upon it as the result of vibration. We suppose that the vibrations have such a high degree of persistence that we may calculate the pressure as if they were permanent. Thus if t/r be the velocity-potential, we have as before with sufficient approxi- mation l-ikr 1 er 1 so that, if p be the radial displacement of the spherical surface, dp/dt = a/r*, and ^ = -r(l-ikr)dp/dt ......................... (47) Again, if a- be the density of the fluid and 8p the variable part of the pressure, ............... (48) which gives the pressure in terms of the displacement p at the surface of a sphere of small radius r. Under the circumstances contemplated we may use (48) although the vibration slowly dies down according to the law of e int , where n is not wholly real. "If M denotes the " mass " and /* the coefficient of restitution applicable to p, the equation of motion is ^) = 0, ............... (49) or if we introduce e int and write M' for M + 4 < 7T(rr 3 , n * (_ M' + 4-Tro-yfcr 4 . t) + ^ = ...................... (50) Approximately, n = J(fi/M') .{l+i. 27r<rAr 4 /^'} ; and if we write n = p -f iq, p^JdifM'), q = p.2'jr<rkr t /M' ................... (51) If T be the time in which vibrations die down in the ratio of e : 1, T=l/q. If there be a second precisely similar vibrator at a distance R from the first, we have for the potential 19 290 MUTUAL INFLUENCE OF RESONATORS [390 and for the pressure due to it at the surface of the first vibrator fc -?,-*** ............................ (53) The equation of motion for p t is accordingly and that for p s differs only by the interchange of p, and p 2 . Assuming that both p l and p 3 are as functions of the time proportional to e int , we get to determine n n* [M 1 - 47r<7r 8 . ikr] -fji=n*. faer^R- 1 e~ ikR , or approximately (54) If, as before, we take n = p + iq, (55 > (56) We may observe that the reaction of the neighbour does not disturb the frequency if cos Ar.fi = 0, or the damping if sinfc.R = 0. When kR is small, the damping in one alternative disappears. The two vibrators then execute their movements in opposite phases and nothing is propagated to a distance. The importance of the disturbance of frequency in (55) cannot be estimated without regard to the damping. The question is whether the two vibrations get out of step while they still remain considerable. Let us suppose that there is a relative gain or loss of half a period while the vibration dies down in the ratio of e : 1, viz. in the time denoted previously by T, so that Calling the undisturbed values of p and q respectively P and Q, and supposing kR to be small, we have P 4<7ror*_ Q RM r ~ 7r ' in which Q/ P = 2ir<rki A /M'. According to this standard the disturbance of frequency becomes important only when kR< I/TT, or R less than X/TT*. It has been assumed throughout that r is much less than R. 391. ON THE WIDENING OF SPECTRUM LINES. [Philosophical Magazine, Vol. xxix. pp. 274284, 1915.] MODERN improvements in optical methods lend additional interest to an examination of the causes which interfere with the absolute homogeneity of spectrum lines. So far as we know these may be considered under five heads, and it appears probable that the list is exhaustive : (i) The translatory motion of the radiating particles in the line of sight, operating in accordance with Doppler's principle. (ii) A possible effect of the rotation of the particles. (iii) Disturbance depending on collision with other particles either of the same or of another kind. (iv) Gradual dying down of the luminous vibrations as energy is radiated away. (v) Complications arising from the multiplicity of sources in the line of sight. Thus if the light from a flame be observed through a similar one, the increase of illumination near the centre of the spectrum line is not so great as towards the edges, in accordance with the principles laid down by Stewart and Kirchhoff ; and the line is effectively widened. It will be seen that this cause of widening cannot act alone, but merely aggravates the effect of other causes. There is reason to think that in many cases, especially when vapours in a highly rarefied condition are excited electrically, the first cause is the most important. It was first considered by Lippich* and somewhat later inde- pendently by myself f. Subsequently, in reply to Ebert, who claimed to have discovered that the high interference actually observed was inconsistent with Doppler's principle and the theory of gases, I gave a more complete * Pogg. Ann. Vol. cxxxix. p. 465 (1870). t Nature, Vol. vni. p. 474 (1873) ; Scientific Papers, Vol. i. p. 188. 192 292 ON THE WIDENING OF SPECTRUM LINES [391 calculation*, taking into account the variable velocity of the molecules as defined by Maxwell's law, from which it appeared that there was really no dis- agreement with observation. Michelson compared these theoretical results with those of his important observations upon light from vacuum-tubes and found an agreement which was thought sufficient, although there remained some points of uncertainty. The same ground was traversed by Schonrockf, who made the notable remark that while the agreement was good for the monatomic gases it failed for diatomic hydrogen, oxygen, and nitrogen ; and he put forward the sugges- tion that in these cases the chemical atom, rather than the usual molecule, was to be regarded as the carrier of the emission-centres. By this substitution, entailing an increase of velocity in the ratio \/2: 1, the agreement was much improved. While I do not doubt that Schonrock's comparison is substantially correct, I think that his presentation of the theory is confused and unnecessarily com- plicated by the introduction (in two senses) of the " width of the spectrum line," a quantity not usually susceptible of direct observation. Unless I misunderstand, what he calls the observed width is a quantity not itself observed at all but deduced from the visibility of interference bands by arguments which already assume Doppler's principle and the theory of gases. I do not see what is gained by introducing this quantity. Given the nature of the radiating gas and its temperature, we can calculate from known data the distribution of light in the bands corresponding to any given retardation, and from photometric experience we can form a pretty good judgment as to the maximum retardation at which they should still be visible. This theoretical result can then be compared with a purely experimental one, and an agree- ment will confirm the principles on which the calculation was founded. I think it desirable to include here a sketch of this treatment of the question on the lines followed in 1889, but with a few slight changes of notation. The phenomenon of interference in its simplest form occurs when two equal trains of waves are superposed, both trains having the same frequency and one being retarded relatively to the other by a linear retardation X*. Then if \ denote the wave-length, the aggregate may be represented by cos nt + cos (nt - 27rZ/X) = 2 cos (wZ/X) . cos (nt - 7rX/\) (1) The intensity is given by / = 4cos 2 (7rZ/\)=2{l+cos(27rZ/X)j (2) If we regard X as gradually increasing from zero, / is periodic, the maxima (4) occurring when X is a multiple of \ and the minima (0) when X is an odd * "On the limits to interference when light is radiated from moving molecules," 1'liiL Mag. Vol. xxvii. p. 298 (1889) ; Scientific Papers, Vol. in. p. 258. t Ann. der Phyiik, Vol. xx. p. 995 (1906). J Iu the paper of 1889 the retardation was denoted by 2A. 1915] ON THE WIDENING OF SPECTRUM LINES 293 multiple of ^X. If bands are visible corresponding to various values of X, the darkest places are absolutely devoid of light, and this remains true how- ever great X may be, that is however high the order of interference. The above conclusion requires that the light (duplicated by reflexion or otherwise) should have an absolutely definite frequency, i.e. should be abso- lutely homogeneous. Such light is not at our disposal ; and a defect of homogeneity will usually entail a limit to interference, as X increases. We are now to consider the particular defect arising in accordance with Doppler's principle from the motion of the radiating particles in the line of sight. Maxwell showed that for gases in temperature equilibrium the number of molecules whose velocities resolved in three rectangular directions lie within the range dgdrjd must be proportional to If be the direction of the line of sight, the component velocities 77, are without influence in the present problem. All that we require to know is that the number of molecules for which the component lies between f and 4- dj; is proportional to e-*?d% ..................................... (3) The relation of ft to the mean (resultant) velocity v is 2 ..(4) It was in terms of v that my (1889) results were expressed, but it was pointed out that v needs to be distinguished from the velocity of mean square with which the pressure is more directly connected. If this be called v', v'=J(~ so that v /( 8 \ /R . ?~v%) (6> Again, the relation between the original wave-length A and the actual wave- length X, as disturbed by the motion, is /v i/ c denoting the velocity of light. The intensity of the light in the inter- ference bands, so far as dependent upon the molecules moving with velocity f, is by (2) + 008^(1 +}\g-K t d&.. ...(8) 294 ON THE WIDENING OF SPECTRUM LINES [391 and this is now to be integrated with respect to between the limits 00 . The bracket in (8) is 1 + cos cos > sin sin - . A Ac A Ac The third term, being uneven in , contributes nothing. The remaining integrals are included in the well-known formula ( + V 01 * 1 cos (2nr) dx= <?-"/'. J - a Z = 1 + co 8 . Exp -- ................ (9) The intensity ^ at the darkest part of the bands is found by making X an odd multiple of \, and I z the maximum brightness by making X a multiple where V denotes the " visibility " according to Michelson's definition. Equa- tion (10) is the result arrived at in my former paper, and # can be expressed in terms of either the mean velocity v, or preferably of the velocity of mean square v'*. The next question is what is the smallest value of V for which the bands are recognizable. Relying on photometric experience, I estimated that a rela- tive difference of 5 per cent, between I 1 and I z would be about the limit in the case of high interference bands, and I took V = '025. Shortly afterwardsf I made special experiments upon bands well under control, obtained by means of double refraction, and I found that in this very favourable case the bands were still just distinctly seen when the relative difference between I 1 and / 2 was reduced to 4 per cent. It would seem then that the estimate F= - 025 can hardly be improved upon. On this basis (10) gives in terms of v -690, ..................... (11) as before. In terms of v' by (6) As an example of (12), let us apply it to hydrogen molecules at 0C. Here v' = 1839 x 10 a cm./sec.J, and c = 3 x IO 10 . Thus X/A = 1-222 x 10' ............................ (13) * See also Proc. Roy. Soc. Vol. LXXVI A. p. 440 (1905) ; Scientific Papers, Vol. v. p. 261. t Phil. Mag. Vol. xxvii. p. 484 (1889); Scientific Papers, Vol. ni. p. 277. It seems to be often forgotten that tbe first published calculation of molecular velocities was that of Joale (Manchester Memoirs, Oct. 1848, Phil. Mag. ser. 4, Vol. xiv. p. 211). 1915] ON THE WIDENING OF SPECTRUM LINES 295 This is for the hydrogen molecule. For the hydrogen atom (13) must be divided by \/2. Thus for absolute temperature T and for radiating centres whose mass is m times that of the hydrogen atom, we have In Buisson and Fabry's corresponding formula, which appears to be derived from Schdnrock, T427 is replaced by the appreciably different number 1'22*. The above value of X is the retardation corresponding to the limit of visi- bility, taken to be represented by V= '025. In Schonrock's calculation the retardation X lt corresponding to V='5, is considered. In (12), V(log e 40) would then be replaced by \f(\og e 2), and instead of (14) we should have = 6-186 xlO- ......................... (15) But I do not understand how V= '5 could be recognized in practice with any precision. Although it is not needed in connexion with high interference, we can of course calculate the width of a spectrum line according to any conventional definition. Mathematically speaking, the width is infinite ; but if we dis- regard the outer parts where the intensity is less than one-half the maximum the limiting value of f by (3) is given by /3f = log e 2, .............................. (16) and the corresponding value of X by X-A_g_V(Iog e 2) A ~c~ cV Thus, if S\ denote the half-width of the line according to the above definition, = VC6931) = 3 . 57xlo _ /,rv ............... A c\/P V \ m / T denoting absolute temperature and m the mass of the particles in terms of that of the hydrogen atom, in agreement with Schonrock. In the application to particular cases the question at once arises as to what we are to understand by T and m. In dealing with a flame it is natural to take the temperature of the flame as ordinarily understood, but when we pass to the rare vapour of a vacuum-tube electrically excited, the matter is not so simple. Michelson assumed from the beginning that the temperature with which we are concerned is that of the tube itself or not much higher. This view is amply confirmed by the beautiful experiments of Buisson and Fabry-f-, * [1916. I understand from M. Fabry that the difference between oar numbers has its origin in a somewhat different estimate of the minimum value of V. The French authors admit an allowance for the more difficult conditions under which high interference is observed.] t Journ. de Physique, t. n. p. 442 (1912). 296 ON THE WIDENING OF SPECTRUM LINES [391 who observed the limit of interference when tubes containing helium, neon, and krypton were cooled in liquid air. Under these conditions bands which had already disappeared at room temperature again became distinct, and the ratios of maximum retardations in the two cases (1'66, 1'60, 1'58) were not much less than the theoretical 173 calculated on the supposition that the temperature of the gas is that of the tube. The highest value of X/A., in their notation N, hitherto observed is 950,000, obtained from krypton in liquid air. With all three gases the agreement at room temperature between the observed and calculated values of N is extremely good, but as already remarked their theoretical numbers are a little lower than mine (14). We may say not only that the observed effects are accounted for almost completely by Doppler's principle and the theory of gases, but that the temperature of the emitting gas is not much higher than that of the containing tube. As regards m, no question arises for the inert monatomic gases. In the case of hydrogen Buisson and Fabry follow Schonrock in taking the atom rather than the molecule as the moving source, so that m = 1 ; and further they find that this value suits not only the lines of the first spectrum of hydrogen but equally those of the second spectrum whose origin has some- times been attributed to impurities or aggregations. In the case of sodium, employed in a vacuum-tube, Schonrock found a fair agreement with the observations of Michelson, on the assumption that the atom is in question. It may be worth while to make an estimate for the D lines from soda in a Bunsen flame. Here m = 23, and we may perhaps take T at 2500. These data give in (14) as the maximum number of bands Z/A = 137,000. The number of bands actually seen is very dependent upon the amount of soda present. By reducing this Fizeau was able to count 50,000 bands, and it would seem that this number cannot be much increased*, so that observation falls very distinctly behind calculation f. With a large supply of soda the number of bands may drop to two or three thousand, or even further. The second of the possible causes of loss of homogeneity enumerated above, viz. rotation of the emitting centres, was briefly discussed many years ago in a letter to Michelson J, where it appeared that according to the views then * "Interference Bauds and their Applications," Nature, Vol. XLVIII. p. 212 (1893); Scientific Paper*, Vol. IT. p. 59. The parallel plate was a layer of water superposed upon mercury. An enhanced illumination may be obtained by substituting nitre-benzol for water, and the reflexions from the mercury and oil may be balanced by staining the latter with aniline blue. But a thin layer of nitro-benzol takes a surprisingly long time to become level. t Smithells (Phil. Mag. Vol. xxxvn. p. 245, 1894) argues with much force that the actually operative parts of the flame may be at a much higher temperature (if the word may be admitted) than is usually supposed, but it would need an almost impossible allowance to meet the dis- crepancy. The chemical questions involved are very obscure. The coloration with soda appears to require the presence of oxygen (Mitcherlich, Smithells). J Phil. Mag. Vol. xxxiv. p. 407 (1892) ; Scientific Papert, Vol. iv. p. 15. 1915] ON THE WIDENING OF SPECTRUM LINES 297 widely held this cause should be more potent than (i). The transverse vibra- tions emitted from a luminous source cannot be uniform in all directions, and the effect perceived in a fixed direction from a rotating source cannot in general be simple harmonic. In illustration it may suffice to mention the case of a bell vibrating in four segments and rotating about the axis of symmetry. The sound received by a stationary observer is intermittent and therefore not homogeneous. On the principle of equipartition of energy between translatory and rotatory motions, and from the circumstance that the dimensions of molecules are much less than optical wave-lengths, it followed that the loss of homogeneity from (ii) was much greater than from (i). I had in view diatomic molecules for at that time mercury vapour was the only known exception ; and the specific heats at ordinary temperatures showed that two of the possible three rotations actually occurred in accordance with equi- partition of energy. It is now abundantly clear that the widening of spectrum lines at present under consideration does not in fact occur ; and the difficulty that might be felt is largely met when we accept Schonrock's supposition that the radiating centres are in all cases monatomic. Still there are questions remaining behind. Do the atoms' rotate, and if not, why not ? I suppose that the quantum theory would help here, but it may be noticed that the question is not merely of acquiring rotation. A permanent rotation, not susceptible of alteration, should apparently make itself felt. These are problems relating to the constitution of the atom and the nature of radiation, which I do not venture further to touch upon. The third cause of widening is the disturbance of free vibration due to encounters with other bodies. That something of this kind is to be expected has long been recognized, and it would seem that the widening of the 1) lines when more than a very little soda is present in a Bunsen flame can hardly be accounted for otherwise. The simplest supposition open to us is that an entirely fresh start is made at each collision, so that we have to deal with a series of regular vibrations limited at both ends. The problem thus arising has been treated by Godfrey* and by Schonrock-f*. The Fourier analysis of the limited train of waves of length r gives for the intensity of various parts of the spectrum line A;- 2 sin 2 (7rr&), (19) where k is the reciprocal of the wave-length, measured from the centre of the line. In the application to radiating vapours, integrations are required with respect to r. Calculations of this kind serve as illustrations ; but it is not to be sup- posed that they can represent the facts at all completely. There must surely * Phil. Trans. A. Vol. cxcv. p. 346 (1899). See also Proc. Roy. Soc. Vol. LXXVI. A. p. 440 (1905) ; Scientific Papers, Vol. v. p. 257. t Ann. der Physik, Vol. xxn. p. 209 (1907). 298 ON THE WIDENING OF SPECTRUM LINES [391 be encounters of a milder kind where the free vibrations are influenced but yet not in such a degree that the vibrations after the encounter have no rela- tion to the previous ones. And in the case of flames there is another question to be faced : Is there no distinction in kind between encounters first of two sodium atoms and secondly of one sodium atom and an atom say of nitrogen ? The behaviour of soda flames shows that there is. Otherwise it seems im- possible to explain the great effect of relatively very small additions of soda in presence of large quantities of other gases. The phenomena suggest that the failure of the least coloured flames to give so high an interference as is calculated from Doppler's principle may be due to encounters with other gases, but that the rapid falling off when the supply of soda is increased is due to something special. This might be of a quasi-chemical character, e.g. tem- porary associations of atoms ; or again to vibrators in close proximity putting one another out of tune. In illustration of such effects a calculation has been given in the previous paper*. It is in accordance with this view that, as Gouy found, the emission of light tends to increase as the square root of the amount of soda present. We come now to cause (iv). Although it is certain that this cause must operate, we are not able at the present time to point to any experimental verification of its influence. As a theoretical illustration "we may consider the analysis by Fourier's theorem of a vibration in which the amplitude follows an exponential law, rising from zero to a maximum and afterwards falling again to zero. It is easily proved that = ^y- f du cos ux { 6 -<-r>'/ + e -<+r) w}, . . .(20) 2a v TT J o in which the second member expresses an aggregate of trains of waves, each individual train being absolutely homogeneous. If a be small in comparison with r, as will happen when the amplitude on the left varies but slowly, e -<+r)*/4a mav b e neglected, and e - <-*>'/*'' i s sensible only when u is very nearly equal to r"f. An analogous problem, in which the vibration is represented by e~ at sin bt, has been treated by GarbassoJ. I presume that the form quoted relates to positive values of t and that for negative values of t it is to be replaced by zero. But I am not able to confirm Garbasso's formula. As regards the fifth cause of (additional) widening enumerated at the beginning of this paper, the case is somewhat similar to that of the fourth. It must certainly operate, and yet it does not appear to be important in prac- tice. In such rather rough observations as I have made, it seems to make no * Phil. Mag. supra, p. 209. [This volume, Art. 390.] t Phil. Mag. Vol. xxxiv. p. 407 (1892) ; Scientific Papers, Vol. iv. p. 16. t Ann. der Physik, Vol. xx. p. 848 (1906). Possibly the sign of a is supposed to change when t passes through zero. But even then what are perhaps misprints would need correction. 1915] ON THE WIDENING OF SPECTRUM LINES 299 great difference whether two surfaces of a Bunsen soda flame (front and back) are in action or only one. If the supply of soda to each be insufficient to cause dilatation, the multiplication of flames in line (3 or 4) has no important effect either upon the brightness or the width of the lines. Actual measures, in which no high accuracy is needed, would here be of service. The observations referred to led me many years ago to make a very rough comparison between the light actually obtained from a nearly undilated soda line and that of the corresponding part of the spectrum from a black body at the same temperature as the flame. I quote it here rather as a suggestion to be developed than as having much value in itself. Doubtless, better data are now available. How does the intrinsic brightness of a just undilated soda flame compare with the total brightness of a black body at the temperature of the flame ? As a source of light Violle's standard, viz. one sq. cm. of just melting platinum, is equal to about 20 candles. The candle presents about 2 sq. cm. of area, so that the radiating platinum is about 40 times as bright. Now platinum is not a black body and the Bunsen flame is a good deal hotter than the melting metal. I estimated (and perhaps under estimated) that a factor of 5 might therefore be introduced, making the black body at flame temperature 200 times as bright as the candle. To compare with a candle a soda flame of which the D-lines were just beginning to dilate, I reflected the former nearly perpendicularly from a single glass surface. The soda flame seemed about half as bright. At this rate the intrinsic brightness of the flame was ^ x ^- = of that of the candle, and 2t "_) 50 accordingly of that of the black body. The black body gives a continuous spectrum. What would its brightness be when cut down to the narrow regions occupied by the D-lines ? According to Abney's measures the brightness of that part of sunlight which lies between the D's would be about ^^ of the whole. We may perhaps estimate the ^oU region actually covered by the soda lines as ~^ of this. At this rate we should get JL l i 25 X 250~6250' as the fraction of the whole radiation of the black body which has the wave- lengths of the soda lines. The actual brightness of a soda flame is thus of the same order of magnitude as that calculated for a black body when its spectrum is cut down to that of the flame, and we may infer that the light of a powerful soda flame is due much more to the widening of the spectrum lines than to an increased brightness of their central parts. 392. THE PRINCIPLE OF SIMILITUDE. [Nature, Vol. xcv. pp. 6668, March, 1915.] I HAVE often been impressed by the scanty attention paid even by original workers in phystcs to the great principle of similitude. It happens not infre- quently that results in the form of " laws " are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes' consideration. However useful verification may be, whether to solve doubts or to exercise students, this seems to be an inversion of the natural order. One reason for the neglect of the principle may be that, at any rate in its applications to particular cases, it does not much interest mathematicians. On the other hand, engineers, who might make much more use of it than they have done, employ a notation which tends to obscure it. I refer to the manner in which gravity is treated. When the question under consideration depends essentially upon gravity, the symbol of gravity (g) makes no appearance, but when gravity does not enter into the question at all, g obtrudes itself conspicuously. I have thought that a few examples, chosen almost at random from various fields, may help to direct the attention of workers and teachers "to the great importance of the principle. The statement made is brief and in some cases inadequate, but may perhaps suffice for the purpose. Some foreign considera- tions of a more or less obvious character have been invoked in aid. In using the method practically, two cautions should be borne in mind. First, there is no prospect of determining a numerical coefficient from the principle of similarity alone ; it must be found, if at all, by further calculation, or experi- mentally. Secondly, it is necessary as a preliminary step to specify clearly all the quantities on which the desired result may reasonably be supposed to depend, after which it may be possible to drop one or more if further considera- tion shows that in the circumstances they cannot enter. The following, then, are some conclusions, which may be arrived at by this method : Geometrical similarity being presupposed here as always, how does the strength of a bridge depend upon the linear dimension and the force of gravity ? 1915] THE PRINCIPLE OF SIMILITUDE 301 In order to entail the same strains, the force of gravity must be inversely as the linear dimension. Under a given gravity the larger structure is the weaker. The velocity of propagation of periodic waves on the surface of deep water is as the square root of the wave-length. The periodic time of liquid vibration under gravity in a deep cylindrical vessel of any section is as the square root of the linear dimension. The periodic time of a tuning-fork, or of a Helmholtz resonator, is directly as the linear dimension. The intensity of light scattered in an otherwise uniform medium from a small particle of different refractive index is inversely as the fourth power of the wave-length. The resolving power of an object-glass, measured by the reciprocal of the angle with which it can deal, is directly as the diameter and inversely as the wave-length of the light. The frequency of vibration of a globe of liquid, vibrating in any of its modes under its own gravitation, is independent of the diameter and directly as the square root of the density. The frequency of vibration of a drop of liquid, vibrating under capillary force, is directly as the square root of the capillary tension and inversely as the square root of the density and as the 1 power of the diameter. The time-constant (i.e. the time in which a current falls in the ratio e: 1) of a linear conducting electric circuit is directly as the inductance and inversely as the resistance, measured in electro-magnetic measure. The time-constant of circumferential electric currents in an infinite con- ducting cylinder is as the square of the diameter. In a gaseous medium, of which the particles repel one another with a force inversely as the nth power of the distance, the viscosity is as the (n + 3)/(2n 2) power of the absolute temperature. Thus, if n = 5, the viscosity is proportional to temperature. Eiffel found that the resistance to a sphere moving through air changes its character somewhat suddenly at a certain velocity. The consideration of viscosity shows that the critical velocity is inversely proportional to the diameter of the sphere. If viscosity may be neglected, the mass (M) of a drop of liquid, delivered slowly from a tube of diameter (a), depends further upon (T) the capillary tension, the density (a-), and the acceleration of gravity (g). If these data suffice, it follows from similarity that 302 THE PRINCIPLE OF SIMILITUDE [392 where F denotes an arbitrary function. Experiment shows that F varies but little and that within somewhat wide limits it may be taken to be 3'8. Within these limits Tate's law that M varies as a holds good. In the ^Eolian harp, if we may put out of account the compressibility and the viscosity of the air, the pitch (n) is a function of the velocity of the wind (v) and the diameter (d) of the wire. It then follows from similarity that the pitch is directly as v and inversely as d, as was found experimentally by Strouhal. If we include viscosity (v), the form is n = v/d.f(v/vd), where / is arbitrary. As a last example let us consider, somewhat in detail, Boussinesq's problem of the steady passage of heat from a good conductor immersed in a stream of fluid moving (at a distance from the solid) with velocity v. The fluid is treated as incompressible and for the present as inviscid, while the solid has always the same shape and presentation to the stream. In these circum- stances the total heat (A) passing in unit time is a function of the linear dimension of the solid (a), the temperature-difference (0), the stream-velocity (v), the capacity for heat of the fluid per unit volume (c), and the conductivity (/c). The density of the fluid clearly does not enter into the question. We have now to consider the " dimensions " of the various symbols. Those of a are (Length) 1 , v (Length) 1 (Time)- 1 , 6 (Temperature) 1 , c (Heat) 1 (Length)" 8 (Temp.)- 1 , K (Heat) 1 (Length)- 1 (Temp.)" 1 (Time)- 1 , h (Heat) 1 (Time)- 1 . Hence if we assume we have by heat l = u + v, by temperature = y u v, by length Q = x + z 3u v, by time 1 = - z v ; so that '-or- Since z is undetermined, any number of terms of this form may be com- bined, and all that we can conclude is that 1915] THE PRINCIPLE OF SIMILITUDE 303 where F is an arbitrary function of the one variable avc/tc. An important particular case arises when the solid takes the form of a cylindrical wire of any section, the length of which is perpendicular to the stream. In strictness similarity requires that the length I be proportional to the linear dimension of the section b ; but when I is relatively very great h must become proportional to I and a under the functional symbol may be replaced by b. Thus h = Kl6.F(bvc/ic). We see that in all cases h is proportional to 0, and that for a given fluid F is constant provided v be taken inversely as a or b. In an important class of cases Boussinesq has shown that it is possible to go further and actually to determine the form of F. When the layer of fluid which receives heat during its passage is very thin, the flow of heat is practically in one dimension and the circumstances are the same as when the plane boundary of a uniform conductor is suddenly raised in temperature and so maintained. From these considerations it follows that F varies as v^, so that in the case of the wire h oc 19 . V(6t>c/), the remaining constant factor being dependent upon the shape and purely numerical. But this development scarcely belongs to my present subject. It will be remarked that since viscosity is neglected, the fluid is regarded as flowing past the surface of the solid with finite velocity, a serious departure from what happens in practice. If we include viscosity in our discussion, the question is of course complicated, but perhaps not so much as might be ex- pected. We have merely to include another factor, v w , where v is the kine- matic viscosity of dimensions (Length) 2 (Time)" 1 , and we find by the same process as before , ,, favc\ z /cv\ w *-"'-(TJ-U-)- Here z and w are both undetermined, and the conclusion is that h = Kdd . where F is an arbitrary function of the two variables avc/tc and CV/K. The latter of these, being the ratio of the two diffusivities (for momentum and for temperature), is of no dimensions ; it appears to be constant for a given kind of gas, and to vary only moderately from one gas to another. If we may assume the accuracy and universality of this law, CV/K is a merely numerical constant, the same for all gases, and may be omitted, so that h reduces to the forms already given when viscosity is neglected altogether, F being again a function of a single variable, avc/tc or bvc/x. In any case F is constant for a given fluid, provided v be taken inversely as a or 6. 304 THE PRINCIPLE OF SIMILITUDE [392 [Nature, Vol. xcv. p. 644, Aug. 1915.] The question raised by Dr Riabouchinsky (Nature, July 29, p. 105)* belongs rather to the logic than to the use of the principle 9f similitude, with which I was mainly concerned. It would be well worthy of further discussion. The conclusion that I gave follows on the basis of the usual Fourier equation for the conduction of heat, in which heat and temperature are regarded as sui generis. It would indeed be a paradox if further knowledge of the nature of heat afforded by molecular theory put us in a worse position than before in dealing with a particular problem. The solution would seem to be that the Fourier equations embody something as to the nature of heat and tempera- ture which is ignored in the alternative argument of Dr Riabouchinsky. [1917. Reference may be made also to a letter signed J. L. in the same number of Nat we, and to Nature, April 22, 1915. See further Buckingham, Nature, Vol. xcvi. p. 396, Dec. 1915. Mr Buckingham had at an earlier date (Oct. 1914) given a valuable discussion of the whole theory (Physical Review, Vol. IV. p. 345), and further questions have been raised in the same Review by Tolman. As a variation of the last example, we may consider the passage of heat between two infinite parallel plane surfaces maintained at fixed temperatures differing by 0, when the intervening space is occupied by a stream of incom- pressible viscous fluid (e.g. water) of mean velocity v. In a uniform regime the heat passing across is proportional to the time and to the area considered ; but in many cases the uniformity is not absolute and it is necessary to take the mean passage over either a large enough area or a long enough time. On this -understanding there is a definite quantity h', representing the passage of heat per unit area and per unit time. If there be no stream (v = 0), or in any case if the kinematic viscosity (v) is infinite, we have h' = K0/a, a being the distance between the surfaces, since then the motion, if any, takes place in plane strata. But when the velocity is high enough, or the viscosity low enough, the motion becomes turbulent, and the flow of heat may be greatly augmented. With the same reasoning and with the same notation as before we have * "In Nature of March 18, Lord Rayleigh gives this formula h = ita9 . F(avc/K), considering heat, temperature, length, and time as four ' independent ' units. If we suppose that only three of these quantities are really independent, we obtain a different result. For example, if the temperature is defined as the mean kinetic energy of the molecules, the principle of similarity allows us only to affirm that h naO . F(r/*a 2 , ca 3 )." 1915] THE PRINCIPLE OF SIMILITUDE 305 or which comes to the same h , = *0 ,av cj,\ a \ v K I F, F l being arbitrary functions of two variables. And, as we have seen, ^(0, CV/K) = 1. For a given fluid CV/K is constant and may be omitted. Dynamical similarity is attained when av is kept constant, so that a complete determi- nation of F, experimentally or otherwise, does not require a variation of both a and v. There is advantage in retaining a constant ; for if a varies, geo- metrical similarity demands that any roughnesses shall be in proportion. It should not be overlooked that in the above argument, c, K, v are treated as constants, whereas they would really vary with the temperature. The assumption is completely justified only when the temperature difference is very small. Another point calls for attention. The regime ultimately established may in some cases depend upon the initial condition. Reynolds' observations suggest that with certain values of av/v the simple stratified motion once established may persist ; but that the introduction of disturbances exceeding a certain amount may lead to an entirely different (turbulent) regime. Over part of the range F would have double values. It would be of interest to know what F becomes when av tends to infinity. It seems probable that F too becomes infinite, but perhaps very slowly.] 20 393. DEEP WATER WAVES, PROGRESSIVE OR STATIONARY, TO THE THIRD ORDER OF APPROXIMATION. [Proceedings of the 'Royal Society, A, Vol. xci. pp. 345353, 1915.] As is well known, the form of periodic waves progressing over deep water urithout change of type was determined by Stokes* to a high degree of approxi- mation. The wave-length (X) in the direction of x being 2?r and the velocity of propagation unity, the form of the surface is given by y = a cos (x - t) - % a 2 cos 2 (x - t) + f a 3 cos 3 (x - t}, (1) and the corresponding gravity necessary to maintain the motion by <7 = l- 2 (2) .The generalisation to other wave-lengths and velocities follows by "dimen- sions." These and further results for progressive waves of permanent type are most easily arrived at by use of the stream-function on the supposition that the waves are reduced to rest by an opposite motion of the water as a whole, when the problem becomes one of steady motion f. My object at present is to extend the scope of the investigation by abandoning the initial restriction to progressive waves of permanent type. The more general equations may then be applied to progressive waves as a particular case, or to stationary waves in which the principal motion is proportional to a simple circular function of the time, and further to ascertain what occurs when the conditions necessary for the particular cases are not satisfied. Under these circumstances the use of the stream-function loses much of its advantage, and the method followed is akin to that originally adopted by Stokes. * Camb. Phil. Trant. Vol. vni. p. 441 (1847) ; Math, and Phys. Papers, Vol. i. p. 197. t Phil. Mag. Vol. i. p. 257 (1876) ; Scientific Paper, Vol. t. p. 262. Also Phil. Mag. Vol. xxi. p. 183 (1911) ; [This volume, p. 11]. 1915] DEEP WATER WAVES, PROGRESSIVE OR STATIONARY 307 The velocity-potential <, being periodic in x, may be expressed by the series (f>=ae~y sin x ct'e-v cos x + j3e~*y sin 2# - &e-* cos 2a> + 7 e-* sin 3# - 7'^-* cos 3# + . . . , ... (3) where a, a', /3, etc., are functions of the time only, and y is measured down- wards from mean level. In accordance with (3) the component velocities are given by u = d<f>/dac = e~ y (a. cos x + a' sin x) + 2e~ 2 *' (/3 cos 2# + /8' sin - y = d</cfy = e" 3 ' (a sin as - a' cos a?) + 2e-^ (/3 sin 2# - #' cos 2a?) + . . .. The density being taken as unity, the pressure equation is p = -d<t>/dt + F + gy-$(u* + tf), .................. (4) in which F is a function of the time. In applying (4) we will regard a, a', as small quantities of the first order, while /3, /?', 7, 7', are small quantities of the second order at most ; and for the present we retain only quantities of the second order. & etc., will then not appear in the expression for M 2 + v 2 . In fact and *) + * ...(5) The surface conditions are (i) that p be there zero, and (ii) that Dp dp dp dp A -=- = - + it -f-+v-f- = ...................... (6) Dt dt doc dy The first is already virtually expressed in (5). For the second do. da' dQ r- e~ y sin x + -=- e~ y cos x 5- e~ 2y sin 2x+ ... dt dt dt - = - -r- -j- dx dt dt dy dt dt In forming equation (6) to the second order of small quantities we need to include only the principal term of u, but v must be taken correct to the second order. As the equation of the free surface we assume y = a cos x + a sin x + b cos 2x -f 6' sin 2x + c cos 3* + c' sin 3# 4- ...... (7) 202 308 DEEP WATER WAVES, PROGRESSIVE OR in which b, b', c, c', are small compared with a, a'. Thus (6) gives (1 a cos * a' sin x) ( -^ sin x + -j- cos x J -^? sin 2# , . . /da. da' . \ f/ , / \ (a cos x + a sin x) ( -j cos x + j- sm a; 1 {(1 - a cos x a sin a;) x (a sin x a cos x) + 2 sin 2# 2' cos 2# + 87 sin 3# - 87' cos 3a?} x sr+sin*- cosa;l = ........................... ' ............. (8) This equation is to hold good to the second order for all values of x, and therefore for each Fourier component separately. The terms in sin a; and cos a; give The term in sin 2# gives f^= and, similarly, that in cos 2# gives ^' + 2<7/9' = ......... ................... (11) In like manner ^ + 3^ = 0, ^' + -W = ................... (12) and so on. These are the results of the surface condition Dp/Dt = 0. From the other surface condition (p = 0) we find in the same way , d& a dd a da. -w + iir-8S ado? From equations (9) to (16) we see that a, a' satisfy the same equations (9) as do a, of, and also that c, c satisfy the same equations (12) as do 7, 7' ; but that b, b' are not quite so simply related to /3, ft*. 1915] STATIONARY, TO THE THIRD ORDER OF APPROXIMATION 309 Let us now suppose that the principal terms represent a progressive wave. In accordance with (9) we may take a = A cos t', a' = A sin t', (17) where t' = Jg.t. Then if ft, ft', 7, 7', do not appear, c, c', are zero, and b = \ A 2 (sin 2 1' - cos 2 tf), b' = -A* cos t' sin t' ; so that y = A cos (as -t')-%A* cos 2 (x-t'\ (18) representing a permanent wave-form propagated with velocity *Jg. So far as it goes, this agrees with (1). But now in addition to these terms we may have others, for which b, b' need only to satisfy (d 2 jdt' 2 +2)(b,b') = 0,. (19) and c, c' need only to satisfy (d*/dt' 2 + 3)(c, (0 = (20) The corresponding terms in y represent merely such waves, propagated in either direction, and of wave-lengths equal to an aliquot part of the principal wave-length, as might exist alone of infinitesimal height, when there is no primary wave at all. When these are included, the aggregate, even though it be all propagated in the same direction, loses its character of possessing a permanent wave-shape, and further it has no tendency to acquire such a character as time advances. If the principal wave is stationary we may take a = Acost', a' = (21) If ft, ft', 7, 7', vanish, b = -$a?, 6'=0, c = 0, c' = 0, and y = A cos # . cos t' \A? cos 2# . cos 2 1' (22) According to (22) the surface comes to its zero position everywhere when cos t' = 0, and the displacement is a maximum when cos t' 1. Then y = Acosx-kA*cos2a;, (23) so that at this moment the wave-form is the same as for the progressive wave (18). Since y is measured downwards, the maximum elevation above the mean level exceeds numerically the maximum depression below it. In the more general case (still with ft, etc., evanescent) we may write a = A cos t' + B sin t', a' = A' cos t' + B' sin t', with b' = -aa', 6=(a' 2 -a 2 ), c' = 0, c = 0. When ft, ft', 7, 7', are finite, waves such as might exist alone, of lengths equal to aliquot parts of the principal wave-length and of corresponding frequencies, are superposed. In these waves the amplitude and phase are arbitrary. 310 DEEP WATER WAVES, PROGRESSIVE OR [393 When we retain the third order of small quantities, the equations naturally become more complicated. We now assume that in (3) & f t are small quantities of the second order, and 7, 7', small quantities of the third order. For p, as an extension of (5), we get / do. da \ / d/3 dff \ p = - ( -T sin a; -- -j- cos x\ 4- e"* 1 ( -'-jr. sin 2# + -5- cos 2#J + <r* (- sin 3a; + cos 3a? + gy + F - ^e~^ (o? + a' 2 ) ................... (24) This is to be made to vanish at the surface. Also we find, on reduction, + 4 cos a; ^ (a/9 7 + a'yS) + (a 2 + a /2 ) (a sin x - a' cos #) ; ...... (25). and at the surface DpjDt = for all values of x. In (25) y is of the form (7), where 6, 6', are of the second order, c, c', of the third order. Considering the coefficients of sin x, cos x, in (25) when reduced to Fourier's form, we see that d*a/dt* + ga, d*a?/dt* + ga!, are both of the third order of small quantities, so that in the first line the factor (1 y + ^y 2 ) may be re- placed by unity. Again, from the coefficients of sin 2x, cos 2x, we see that to the third order inclusive (26) and from the coefficients of sin 3x, cos 3# that to the third order inclusive (27) And now returning to the coefficients of sin x, cos x, we get = 0, ...(28) + ga' + 2a (a 2 + ') - 4 (a# + a'#) + a' (* 2 + a'') = 0. (29) 1915] STATIONARY, TO THE THIRD ORDER OF APPROXIMATION 311 Passing next to the condition p = 0, we see from (24), by considering the coefficients of sin x, cos x, that -T- + gaf + terms of 3rd order = 0, -- + ga + terms of 3rd order = 0. The coefficients of sin 2#, cos 2#, require, as in (14), (15), that -a 3 (30) Again, the coefficients of sin 3#, cos 3#, give c' = - ^ - f (a'b + ab') + f a' (a' 2 - 3a 2 ) When /3, /3', 7, 7', vanish, these results are much simplified. We have b' = -aa, b = ^(a' 2 -a 2 ), ..................... (33) (34) If the principal terms represent a purely progressive wave, we may take, as in (17), a = A cosnt, a! A sin?&, ..................... (35) where n is for the moment undetermined. Accordingly c' = | A 3 sin 3nt, c = A S cos 3nt ; so that y = A cos (a; - nt) - | J. 2 cos 2 (x - nt) +%A* cos 3 (x - nt), ...... (36) representing a progressive wave of permanent type, as found by Stokes. To determine n we utilize (28), (29), in the small terms of which we may take a-p ja'<fe-- cocn*. a'=-gjadt= -~ sinnt, so that a 2 + a' 2 = M 2 w 2 . and n 2 = <7+<7M 2 /n 2 =<jr(l-f,l 2 ), ..................... (37) 312 DEEP WATER WAVES, PROGRESSIVE OR [393 or, if we restore homogeneity by introduction of k (= 27r/\), (38) Let us next suppose that the principal terms represent a stationary, instead of a progressive, wave and take a = Acosnt, a = ......................... (39) Then by (33), (34), &' = 0, b=-lA*ca&nt, c' = 0, c= |A 3 cos s r?<; and y = A cos nt cos x - \A* cos 8 nt cos 2x + %A 3 cos 8 nt cos 3#. . . .(40) When cos nt = 0, y = throughout ; when cos nt = 1, y = A cos x \A* cos 2# + f^l 8 cos 3#, so that at this moment of maximum displacement the form is the same as for the progressive wave (36). We have still to determine n so as to satisfy (28), (29), with evanescent &, '. The first is satisfied by a = 0, since a' = 0. The second becomes that In the small terms we may take a = g ladt = sin nt, so *" + go.' + $*- 3 (sin nt + 5 sin 3n<) = 0. To satisfy this we assume a' = H sin nt + K sin 3nt. Then H(g-n*)+ = 0, K (g - 9n') + from the first of which *-'+-'-? ......................... <> or, if we restore homogeneity by introduction of k, n* = glk.(l-lfrA 3 ) ............ , ............... (42) With this value of n the stationary vibration y = A cos nt cos kx - $kA* cos 8 nt cos 2kx + f A*A* cos 3 nt cos 3&r,. . .(43) satisfies all the conditions. It may be remarked that according to (42) the frequency of vibration is diminished by increase of amplitude. The special cases above considered of purely progressive or purely stationary waves piossess an exceptional simplicity. In general, with omission of $, $', equations (28), (29), become -* ............. <*> 1915] STATIONARY, TO THE THIRD ORDER OF APPROXIMATION 313 and a like equation in which a and ' are interchanged. In the terms of the third order, we take a = P cos nt + Q sin nt, a' = P cos nt + Q' sin nt, ... ...... (45) so that a 2 + a' 2 = $ (P 2 + Q 2 + P* + Q' 2 ) + H^ 2 + P' 2 ~ Q 2 ~ Q" 2 ) cos 2nt The third order terms in (44) are $ (P 2 + P 2 + Q 2 + Q' 2 ) (P cos nt + Q sin nt) + 2 cos nt cos 2?i JJP (P 2 + P' 2 - Q 2 - Q' 2 ) - ^ (PQ + P'Q')1 \ y . ) + 2 sin nt sin 2nt JQ (PQ + P'Q') - W ' P (P 2 + P' 2 - Q 2 - Q /2 )j 9<M 2 P + 2 sin nt cos 2n* UQ (P 2 + P' 2 - Q 2 - Q' 2 ) + " (PQ + P'Q') + 2 cos sin 2n^ j^P (PQ + P'Q') + ^ (P 2 + P' 2 - Q 2 - Q' 2 )| , V c/ ' of which the part in sin nt has the coefficient Q (i OP 1 + P /i! ) + ! (Q 2 + Q' 2 )} + iP (PQ + ^V) + n 2 /^ . {Q (P 2 + P' 2 - Q 2 - Q' 2 ) - 2P (PQ + P'Q')) or, since n- = g approximately, Q {| ( p + P' 2) - HQ 2 + Q /2 )l - 1 P (PQ + P'Q')- ......... (46) In like manner the coefficient of cos nt is P{|(Q 2 + Q /2 )-HP 2 + -P' 2 )}-lQ(PQ + P / Q / X ......... (47) differing merely by the interchange of P and Q. But when these values are employed in (44), it is not, in general, possible, with constant values of P, Q, P', Q', to annul the terms in sin nt, cos nt. We -obtain from the first and from the second w2 = ^ + |(Q 2 + Q' 2 )-i(P 2 + P' 2 )-||(PQ + P'Q'); ...... (49) and these are inconsistent, unless (PP' + QQ')(PQ'-P'Q) = ...................... (50) The latter condition is unaltered by interchange of dashed and undashed letters, and thus it serves equally for the equation in a'. 314 DEEP WATER WAVES, ETC. The two alternatives indicated in (50) correspond to the particular cases already considered. In the first (PP t + QQ* = 0) we have a purely progressive wave and in the second a purely stationary one. When the condition (50) does not hold good, it is impossible to satisfy our equations as before with constant values of n, P, Q, P', Q[ ; and it is perhaps hardly worth while to pursue the more complicated questions which then arise. It may suffice to remark that an approximately stationary wave can never pass into an approximately progressive wave, nor vice versd. The progressive wave has momentum, while the stationary wave has none, and momentum is necessarily conserved. When y9, ff, 7, 7', are not zero, additional terms enter. Equations (26), (30), show that the additions to b, b', vary as the sine and cosine of <v/(2#) . t, and represent waves which might exist in the complete absence of the principal wave. The additions to c, c', are more complicated. As regards the parts depend- ing in (31), (32), on dy/dt, dy'/dt, they are proportional to the sine and cosine of \'(3g) . t, and represent waves which might exist alone. But besides these there are other parts, analogous to the combination-tones of Acoustics, result- ing from the interaction of the /9-waves with the principal wave. These vary as the sine and cosine of \/<jr. {V2 1} t, thus possessing frequencies differing from the former frequencies. Similar terms will enter into the expression for /i 2 as determined from (28), (29). In the particular case of $, yS', vanishing, even though 7, 7' (assumed still to be of the third order) remain, we recover most of the former simplicity, the only difference being the occurrence in c, c, of terms in V(3#) . t, such as might exist alone. 394 AEOLIAN TONES. [Philosophical Magazine, Vol. xxix. pp. 433444, 195, 1915.] IN what has long been known as the ^olian Harp, a stretched string, such as a pianoforte wire or a violin string, is caused to vibrate in one of its possible modes by the impact of wind ; and it was usually supposed that the action was analogous to that of a violin bow, so that the vibrations were executed in the plane containing the direction of the wind. A closer examina- tion showed, however, that this opinion was erroneous and that in fact the vibrations are transverse to the wind*. It is not essential to the production of sound that the string should take part in the vibration, and the general phenomenon, exemplified in the whistling of wind among trees, has been investigated by Strouhalf under the name of Reibungstone. In Strouhal's experiments a vertical wire or rod attached to a suitable frame was caused to revolve with uniform velocity about a parallel axis. The pitch of the seolian tone generated by the relative motion of the wire and of the air was found to be independent of the length and of the tension of the w.ire, but to vary with the diameter (D) and with the speed (F) of the motion. Within certain limits the relation between the frequency of vibration (N) and these data was expressible by N=-185VfD, (1){ the centimetre and the second being units. When the speed is such that the seolian tone coincides with one of the proper tones of the wire, supported so as to be capable of free independent vibration, the sound is greatly reinforced, and with this advantage Strouhal found it possible to extend the range of his observations. Under the more extreme conditions then practicable the observed pitch deviated considerably * Phil. Mag. Vol. vn. p. 149 (1879) ; Scientific Papers, Vol. i. p. 413. t Wied. Ann. Vol. v. p. 216 (1878). t In (1) V is the velocity of the wire relatively to the walls of the laboratory. 316 ^SOLIAN TONES [394 from the value given by (1). He further showed that with a given diameter and a given speed a rise of temperature was attended by a fall in pitch. If, as appears probable, the compressibility of the fluid may be left out of account, we may regard N as a function of the relative velocity V, D, and v the kinematic coefficient of viscosity. In this case N is necessarily of the form N=V/D.f( l ,/VD), (2) where f represents an arbitrary function ; and there is dynamical similarity, if v oc VD. In observations upon air at one temperature v is constant ; and if D vary inversely as V, ND/V should be constant, a result fairly in harmony with the observations of Strouhal. Again, if the temperature rises, v increases, and in order to accord with observation, we must suppose that the function f diminishes with increasing argument. "An examination of the actual values in Strouhal's experiments shows that v/VD was always small; and we are thus led to represent / by a few terms of MacLaurin's series. If we take /O) = a + bx + ca?, w e get y-o+fcil + c (3) " If the third term in (3) may be neglected, the relation between N and V is linear. This law was formulated by Strouhal, and his diagrams show that the coefficient b is negative, as is also required to express the observed effect of a rise of temperature. Further, D W= a -v?i? <*> so that D.dNjdV is very nearly constant, a result also given by Strouhal on the basis of his measurements. " On the whole it would appear that the phenomena are satisfactorily represented by (2) or (3), but a dynamical theory has yet to be given. It would be of interest to extend the experiments to liquids*." Before the above paragraphs were written I had commenced a systematic deduction of the form of f from Strouhal's observations by plotting ND/V against VD. Lately I have returned to the subject, and I find that nearly all his results are fairly well represented by two terms of (3). In C.G.S. measure -*6(l Although the agreement is fairly good, there are signs that a change of wire introduces greater discrepancies than a change in V a circumstance Theory of Sound, 2nd ed. Vol. n. 372 (1896). 1915] 4TOLIAX TONES 317 which may possibly be attributed to alterations in the character of the surface. The simple form (2) assumes that the wires are smooth, or else that the roughnesses are in proportion to D, so as to secure geometrical similarity. The completion of (5) from the theoretical point of view requires the introduction of v. The temperature for the experiments in which v would enter most was about 20 C., and for this temperature u, 1806 x 10~ 7 V = -00120 = The generalized form of (5) is accordingly VD applicable now to any fluid when the appropriate value of v is introduced. For water at 15 C., v - '0115, much less than for air. Strouhal's observations have recently been discussed by Krtiger and Lanth*, who appear not to be acquainted with my theory. Although they do not introduce viscosity, they recognize that there is probably some cause for the observed deviations from the simplest formula (1), other than the complication arising from the circulation of the air set in motion by the revolving parts of the apparatus. Undoubtedly this circulation marks a weak place in the method, and it is one not easy to deal with. On this account the numerical quantities in (6) may probably require some correction in order to express the true formula when V denotes the velocity of the wire through otherwise undisturbed fluid. We may find confirmation of the view that viscosity enters into the question, much as in (6), from some observations of Strouhal on the effect of temperature. Changes in v will tell most when VD is small, and therefore I take Strouhal's table XX., where -D = '0l79 cm. In this there appears 2 =31, F 2 = 381, Introducing these into (6), we get -195 / 20-1 *A 195 / 201 *,\ = -~ I 1 " - ~~ l " or with sufficient approximation Theorie der Hiebtone," Ann. d. Physik, Vol. XLIV. p. 801 (1914). 318 AEOLIAN TONES [394 We may now compare this with the known values of v for the temperatures in question. We have ^ = 1853 x 10- 7 , p sl = -001161, H U = 1765 x 10- 7 , Pll = -001243 ; so that v 2 = -1596, Vl = '1420, and *> 2 - vi = '018. The difference in the values of v at the two temperatures thus accounts in (6) for the change of frequency both in sign and in order of magnitude. As regards dynamical explanation it was evident all along that the origin of vibration was connected with the instability of the vortex sheets which tend to form on the two sides of the obstacle, and that, at any rate when a wire is maintained in transverse vibration, the phenomenon must be unsym- metrical. The alternate formation in water of detached vortices on the two sides is clearly described by H. Benard*. "Pour une vitesse suffisante, au-dessous de laquelle il n'y a pas de tourbillons (cette vitesse limite croit avec la viscosite et decroit quand 1'epaisseur transversale des obstacles aug- mente), les tourbillons produits periodiquement se detachent alternativement d droite et a gauche du remous d'arriere qui suit le solide ; Us gagnent presque immediatement leur emplacement definitif, de sorte qua I'arriere de I'obstacle se forme une double rangde alternee d'entonnoirs stationnaires, ceux de droite dextrogyres, ceux de gauche levogyres, sipares par des intervaUes egaux" The symmetrical and unsymmetrical processions of vortices were also figured by Mallockf from direct observation. In a remarkable theoretical investigation \ Karman has examined the question of the stability of such processions. The fluid is supposed to be incompressible, to be devoid of viscosity, and to move in two dimensions. The vortices are concentrated in points and are disposed at equal intervals (I) along two parallel lines distant h. Numerically the vortices are all equal, but those on different lines have opposite signs. Apart from stability, steady motion is possible in two arrangements (a) and (6), fig. 1, of which (a) is symmetrical. Karman shows that (a) is always unstable, whatever may be the ratio of h to I ; and further that (6) is usually unstable also. The single exception occurs when cosh (irk/l) = \/2, or h/l = 0'283. With this ratio of h/l, (6) is stable for every kind of displacement except one, for which there is neutrality. The only procession which can possess a practical permanence is thus defined. C. R. t. 147, p. 839 (1908). t Proc. Roy. Soc. Vol. LXXXIV. A, p. 490 (1910). t GSttingen Nachrichten, 1912, Heft 5, 8. 547; Karman aud Bubach, Pliyiik. Zeittchrift, 1912, p. 49. I have verified the more important results. 1915] .EOLIAN TOXES 319 The corresponding motion is expressed by the complex potential (</> potential, >/r stream-function) ?. 1. in which denotes the strength of a vortex, z = a; + iy, z = \ I + ih. The #-axis is drawn midway between the two lines of vortices and the y-axis halves the distance between neighbouring vortices with opposite rotation. Karman gives a drawing of the stream-lines thus defined. The constant velocity of the processions is given by irh .(8) = i tenh T=^ 8 This velocity is relative to the fluid at a distance. The observers who have experimented upon water seem all to have used obstacles not susceptible of vibration. For many years I have had it in my mind to repeat the seolian harp effect with water*, but only recently have brought the matter to a test. The water was contained in a basin, about 36 cm. in diameter, which stood upon a sort of turn-table. The upper part, however, was not properly a table, but was formed of two horizontal beams crossing one another at right angles, so that the whole apparatus resembled rather a turn- stile, with four spokes. It had been intended to drive from a small water-engine, but ultimately it was found that all that was needed could more conveniently be done by hand after a little practice. A metro- nome beat approximate half seconds, and the spokes (which projected beyond the basin) were pushed gently by one or both hands until the rotation was uniform with passage of one or two spokes in correspondence with an assigned number of beats. It was necessary to allow several minutes in order to * From an old note-book. "Bath, Jan. 1884. I find in the baths here that if the spread fingers be drawn pretty quickly through the water (palm foremost was best), they are thrown into transverse vibration and strike one another. This seems like ajolian string.... The blade of a flesh-brush about 1 inch broad seemed to vibrate transversely in its own plane when moved through water broadways forward. It is pretty certain that with proper apparatus these vibrations might be developed and observed. " * 320 -EOLIAN TONES [394 make sure that the water had attained its ultimate velocity. The axis of rotation was indicated by a pointer affixed to a small stand resting on the bottom of the basin and rising slightly above the level of the water. The pendulum (fig. 2), of which the lower part was immersed, was supported on two points (A, B) so that the possible vibrations were limited to one vertical plane. In the usual arrangement the vibrations of the rod would be radial, i.e. transverse to the motion of the water, but it was easy to turn the pendulum round when it was desired to test whether a circumferential vibration could be maintained. The rod C itself was of brass tube 8 mm. in diameter, and to it was clamped a hollow cylinder of lead D. The time Fig. 2. of complete vibration (T) was about half a second. When it was desired to change the diameter of the immersed part, the rod C was drawn up higher and prolonged below by an additional piece a change which did not much affect the period T. In all cases the length of the part immersed was about 6 cm. Preliminary observations showed that in no case were vibrations generated when the pendulum was so mounted that the motion of the rod would be circumferential, viz. in the direction of the stream, agreeably to what had been found for the aeolian harp. In what follows the vibrations, if any, are radial, that is transverse to the stream. In conducting a set of observations it was found convenient to begin with the highest speed, passing after a sufficient time to the next lower, and so on, 1915] .EOLIAN TONES 321 with the minimum of intermission. I will take an example relating to the main rod, whose diameter (D) is 8i mm., r = 60/106 sec., beats of metronome 62 in 30 sec. The speed is recorded by the number of beats corresponding to the passage of two spokes, and the vibration of the pendulum (after the lapse of a sufficient time) is described as small, fair, good, and so on. Thus on Dec. 21, 1914 : 2 spokes to 4 beats gave fair vibration, ....... 5 good 6 rather more . . . 7 good 8 ....... fair from which we may conclude that the maximum effect corresponds to 6 beats, or to a time (T) of revolution of the turn-table equal to 2 x 6 x 30/62 sec. The distance (r) of the rod from the axis of rotation was 116 mm., and the speed of the water, supposed to move with the basin, is 27rr/T. The result of the observations may intelligibly be expressed by the ratio of the distance travelled by the water during one complete vibration of the pendulum to the diameter of the latter, viz. r . 27rr/T_ ZTT x 116 x 62 D 8-5 x 6 x 106 ~ Concordant numbers were obtained on other occasions. In the above calculation the speed of the water is taken as if it were rigidly connected with the basin, and must be an over estimate. When the pendulum is away, the water may be observed to move as a solid body after the rotation has been continued for two or three minutes. For this purpose the otherwise clean surface may be lightly dusted over with sulphur. But when the pendulum is immersed, the rotation is evidently hindered, and that not merely in the neighbourhood of the pendulum .itself. The difficulty thence arising has already been referred to in connexion with Strouhal's experiments and it cannot easily be met in its entirety. It may be mitigated by increasing r, or by diminishing D. The latter remedy is easily applied up to a certain point, and I have experimented with rods 5 mm. and 3 mm. in diameter. With a 2 mm. rod no vibration could be observed. The final results were thus tabulated : Diameter Ratio 8*5 mm. 8-35 5'0 mm. 7'5 3*5 mm. 7-8 from which it would appear that the disturbance is not very serious. The difference between the ratios for the 5'0 mm. and 3'5 mm. rods is hardly out- side the limits of error; and the prospect of reducing the ratio much below 7 seemed remote. The instinct of an experimenter is to try to get rid of a disturbance, even though only partially; but it is often equally instructive to increase it. The K. vi. 21 322 AEOLIAN TONES [394 observations of Dec. 21 were made with this object in view ; besides those already given they included others in which the disturbance due to the vibrating pendulum was augmented by the addition of a similar rod (8 mm.) immersed to the same depth and situated symmetrically on the same diameter of the basin. The anomalous effect would thus be doubled. The record was as follows : 2 spokes to 3 beats gave little or no vibration, 4 fair 5 ...'... large 6 less 7 little or no As the result of this and another day's similar observations it was concluded that the 5 beats with additional obstruction corresponded with 6 beats with- out it. An approximate correction for the disturbance due to improper action of the pendulum may thus be arrived at by decreasing the calculated ratio in the proportion of 6 : 5; thus t(8-35) = 70 is the ratio to be expected in a uniform stream. It would seem that this cannot be far from the mark, as representing the travel at a distance from the pendulum in an othenvise uniform stream during the time of one com- plete vibration of the latter. Since the correction for the other diameters will be decidedly less, the above number may be considered to apply to all three diameters experimented on. In order to compare with results obtained from air, we must know the value of v/VD. For water at 15 C. v = //, = '0115 c.a.s.; and for the 8'5 mm. pendulum v/VD = '0011. Thus from (6) it appears that NDjV should have nearly the full value, say "190. The reciprocal of this, or 5'3, should agree with the ratio found above as 7*0 ; and the discrepancy is larger than it should be. An experiment to try whether a change of viscosity had appreciable influence may be briefly mentioned. Observations were made upon water heated to about 60 C. and at 12 C. No difference of behaviour was detected. At 60 C. fji = -0049, and at 1 2 C. /z = '0124. I have described the simple pendulum apparatus in some detail, as apart from any question of measurements it demonstrates easily the general prin- ciple that the vibrations are transverse to the stream, and when in good action it exhibits very well the double row of vortices as witnessed by dimples upon the surface of the water. The discrepancy found between the number from water (7'0) and that derived from Strouhal's experiments on air (5'3) raises the question whether 1915] ^SOLIAN TONES 323 the latter can be in error. So far as I know, Strouhal's work has not been repeated ; but the error most to be feared, that arising from the circulation of the air, acts in the wrong direction. In the hope of further light I have remounted my apparatus of 1879. The draught is obtained from a chimney. A structure of wood and paper is fitted to the fire-place, which may prevent all access of air to the chimney except through an elongated horizontal aperture in the front (vertical) wall. The length of the aperture is 26 inches (66 cm.), and the width 4 inches (10'2 cm.); and along its middle a gut string is stretched over bridges. The draught is regulated mainly by the amount of fire. It is well to have a margin, as it is easy to shunt a part through an aperture at the top of the enclosure, which can be closed partially or almost wholly by a superposed card. An adjustment can sometimes be got by opening a door or window. A piece of paper thrown on the fire increases the draught considerably for about half a minute. The string employed had a diameter of '95 mm., and it could readily be made to vibrate (in 3 segments) in unison with a fork of pitch 256. The octave, not difficult to mistake, was verified by a resonator brought up close to the string. That the vibration is transverse to the wind is confirmed by the behaviour of the resonator, which goes out of action when held symmetri- cally. The sound, as heard in the open without assistance, was usually feeble, but became loud when the ear was held close to the wooden frame. The difficulty of the experiment is to determine the velocity of the wind, where it acts upon the string. I have attempted to do this by a pendulum arrange- ment designed to determine the wind by its action upon an elongated piece of mirror (10' 1 cm. x 1'6 cm.) held perpendicularly and just in front of the string. The pendulum is supported on two points in this respect like the one used for the water experiments; the mirror is above, and there is a counter- weight below. An arm projects horizontally forward on which a rider can be placed. In commencing observations the wind is cut off by a large card inserted across the aperture and just behind the string. The pendulum then assumes a sighted position, determined in the usual way by reflexion. When the wind operates the mirror is carried with it, but is brought back to the sighted position by use of a rider of mass equal to '485 gm. Observations have been taken on several occasions, but it will suffice to record one set whose result is about equal to the average. The (horizontal) distance of the rider from the axis of rotation was 62 mm., and the vertical distance of the centre line of the mirror from the same axis is 77 mm. The force of the wind upon the mirror was thus 62 x '485 -r 77 gms. weight. The mean pressure P is 62 x "485 x 981 _ dynes 77 x 16-2 cm. 2 212 324 AEOLIAN TONES [394 The formula connecting the velocity of the wind V with the pressure P may be written where p is the density ; but there is some uncertainty as to the constancy of C. It appears that for large plates C = '62, but for a plate 2 inches square Stanton found C = - 52. Taking the latter value*, we have F2 _237 = 23-7 ~ 52p -52 x -001 23' on introduction of the value of p appropriate to the circumstances of the experiment. Accordingly V= 192 cm./sec. The frequency of vibration (r~ l ) was nearly enough 256 ; so that Yi= 192 _ 7 .q D 256 x -095 In comparing this with Strouhal, we must introduce the appropriate value of VD, that is 19, into (5). Thus V Vr Whether judged from the experiments with water or from those just detailed upon air, this (Strouhal's) number would seem to be too low ; but the uncer- tainty in the value of C above referred to precludes any very confident conclusion. It is highly desirable that Strouhal's number should be further checked by some method justifying complete confidence. When a wire or string exposed to wind does not itself enter into vibration, the sound produced is uncertain and difficult to estimate. No doubt the wind is often different at different parts of the string, and even at the same part it may fluctuate rapidly. A remedy for the first named cause of unsteadiness is to listen through a tube, whose open end is brought pretty close to the obstacle. This method is specially advantageous if we take advantage of our knowledge respecting the mode of action, by using a tube drawn out to a narrow bore (say 1 or 2 mm.) and placed so as to face the processions of vortices behind the wire. In connexion with the fire-place arrangement the drawn out glass tube is conveniently bent round through 180 and continued to the ear by a rubber prolongation. In the wake of the obstacle the sound is well heard, even at some distance (50 mm.) behind ; but little or nothing reaches the ear when the aperture is in front or at the side, even though quite close up, unless the wire is itself vibrating. But the special arrangement for * Bat I confess that I feel doubts as to the diminution of C with the linear dimension. [ 1917. See next paper.] 1915] TONES 325 a draught, where the observer is on the high pressure side, is not necessary ; in a few minutes any one may prepare a little apparatus competent to show the effect. Fig. 3 almost explains itself. A is the drawn out glass tube B the loop of iron or brass wire (say 1 mm. in diameter), attached to the tube with the aid of a cork C. The rubber prolongation is not shown. Held in the crack of a slightly opened door or window, the arrangement yields a sound which is often pure and fairly steady. 395. ON THE RESISTANCE EXPERIENCED BY SMALL PLATES EXPOSED TO A STREAM OF FLUID. [Philosophical Magazine, Vol. xxx. pp. 179181, 1915.] IN a recent paper on JSolian Tones* I had occasion to determine the velocity of wind from its action upon a narrow strip of mirror (lO'l cm. x I'Gcm.), the incidence being normal. But there was some doubt as to the coefficient to be employed in deducing the velocity from the density of the air and the force per unit area. Observations both by Eiffel and by Stanton had indicated that the resultant pressure (force reckoned per unit area) is less on small plane areas than on larger ones; and although I used provisionally a diminished value of C in the equation P = CpV 2 in view of the narrowness of the strip, it was not without hesitation f. I had in fact already commenced experiments which appeared to show that no variation in C was to be detected. Subse- quently the matter was carried a little further ; and I think it worth while to describe briefly the method employed. In any case I could hardly hope to attain finality, which would almost certainly require the aid of a proper wind channel, but this is now of less consequence as I learn that the matter is engaging attention at the National Physical Laboratory. According to the principle of similitude a departure from the simple law would be most apparent when the kinematic viscosity is large and the stream velocity small. Thus, if the delicacy can be made adequate, the use of air resistance and such low speeds as can be reached by walking through a still atmosphere should be favourable. The principle of the method consists in balancing the two areas to be compared by mounting them upon a vertical axis, situated in their common plane, and capable of turning with the minimum of friction. If the areas are equal, their centres must be at the same distance (on opposite sides) from the axis. When the apparatus is carried forward through the air, equality of mean pressures is witnessed by the plane of the obstacles assuming a position of perpendicularity to the line of motion. If in Phil. Mag. Vol. xxix. p. 442 (1915). [Art. 394.] t See footnote on p. [324]. 1915] RESISTANCE EXPERIENCED BY SMALL PLATES, ETC. 327 this position the mean pressure on one side is somewhat deficient, the plane on that side advances against the relative stream, until a stable balance is attained in an oblique position, in virtue of the displacement (forwards) of the centres of pressure from the centres of figure. The plates under test can be cut from thin card and of course must be accurately measured. In my experiments the axis of rotation was a sewing- needle held in a U-shaped strip of brass provided with conical indentations. The longitudinal pressure upon the needle, dependent upon the spring of the brass, should be no more than is necessary to obviate shift. The arms con- necting the plates with the needle are as slender as possible consistent with the necessary rigidity, not merely in order to save weight but to minimise their resistance. They may be made of wood, provided it be accurately shaped, or of wire, preferably of aluminium. Regard must be paid to the proper balancing of the resistances of these arms, and this may require otherwise superfluous additions. It would seem that a practical solution may be attained, though it must remain deficient in mathematical exactness. The junctions of the various pieces can be effected quite satisfactorily with sealing-wax used sparingly. The brass U itself is mounted at the end of a rod held horizontally in front of the observer and parallel to the direction of motion. I found it best to work indoors in a long room or gallery. Although in use the needle is approximately vertical, it is necessary to eliminate the possible effect of gravity more completely than can thus be attained. When the apparatus is otherwise complete, it is turned so as to make the needle horizontal, and small balance weights (finally of wax) adjusted behind the plates until equilibrium is neutral. In this process a good opinion can be formed respecting the freedom of movement. In an experiment, suggested by the case of the mirror above referred to, the comparison was between a rectangular plate 2 inches x 1 inches and an elongated strip '51 inch broad, the length of the strip being parallel to v the needle, i.e. vertical in use. At first this length was a little in excess, but was cut down until the resistance balance was attained. For this purpose it seemed that equal areas were required to an accuracy of about one per cent., nearly on the limit set by the delicacy of the apparatus. According to the principle of similitude the influence of linear scale (I) upon the mean pressure should enter only as a function of vf VI, where v is the kinematic viscosity of air and V the velocity of travel. In the present case v = '1505, V(4, miles per hour) = 180, and I, identified with the width of the strip, = 1'27, all in c.G.s. measure. Thus vjVl = -00066. In view of the smallness of this quantity, it is not surprising that the influence of linear scale should fail to manifest itself. 328 RESISTANCE EXPERIENCED BY SMALL PLATES, ETC. [395 In virtue of the more complete symmetry realizable when the plates to be compared are not merely equal in area but also similar in shape, this method would be specially advantageous for the investigation of the possible influence of thickness and of the smoothness of the surfaces. When the areas to be compared are unequal, so that their centres need to be at different distances from the axis, the resistance balance of the auxiliary parts demands* special attention. I have experimented upon circular disks whose areas are as 2:1. When there was but one smaller disk (6 cm. in diameter) the arms of the lever had to be also as 2 : 1 (fig. 1). In another Fig. l. experiment two small disks (each 4 cm. in diameter) were balanced against a larger one of equal total area (fig. 2). Probably this arrangement is the better. In neither case was any difference of mean pressures detected. Fig. 2. In the figures AA represents the needle, B and C the large and small disks respectively, D the extra attachments needed for the resistance balance of the auxiliary parts. 396. HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES. [Proceedings of the Royal Society, A, Vol. xci. pp. 503 511, 1915.] THE general use of Pitot's tubes for measuring the velocity of streams suggests hydrodynamical problems. It can hardly be said that these are of practical importance, since the action to be observed depends simply upon Bernoulli's law. In the interior of a long tube of any section, closed at the further end and facing the stream, the pressure must be that due to the velocity (v) of the stream, i.e. ^pv 2 , p being the density. At least, this must be the case if viscosity can be neglected. I am not aware that the influence of viscosity here has been detected, and it does not seem likely that it can be sensible under ordinary conditions. It would enter in the combination vjvl, where v is the kinematic viscosity and I represents the linear dimension of the tube. Experiments directed to show it would therefore be made with small tubes and low velocities. In practice a tube of circular section is employed. But, even when viscosity is ignored, the problem of determining the motion in the neighbourhood of a circular tube is beyond our powers. In what follows, not only is the fluid supposed frictionless, but the circular tube is replaced by its two-dimensional analogue, i.e. the channel between parallel plane walls. Under this head two problems naturally present themselves. The first problem proposed for consideration may be defined to be the flow of electricity in two dimensions, when the uniformity is disturbed by the presence of a channel whose infinitely thin non-conducting walls are parallel to the flow. By themselves these walls, whether finite or infinite, would , cause no disturbanqe ; but the channel, though open at the finite end, is sup- posed to be closed at an infinite distance away, so that, on the whole, there is no stream through it. If we suppose the flow to be of liquid instead of electricity, the arrangement may be regarded as an idealized Pitot's tube, 330 HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES [396 although we know that, in consequence of the sharp edges, the electrical law would be widely departed from. In the recesses of the tube there is no motion, and the pressure developed is simply that due to the velocity of the stream. The problem itself may be treated as a modification of that of Helmholtz*, where flow is imagined to take place within the channel and to come to evanescence outside at a distance from the mouth. If in the usual notation^ z = x + iy, and ; = </> + tX/r be the complex potential, the solution of Helm- holtz's problem is expressed by z = w + e w , ................................. (1) or x = < + & cos i/r, y = ty + ^ sin ^ ................... (2) The walls correspond to ^ = TT, where y takes the same values, and they extend from # = oo to x = 1. Also the stream-line i/r = makes y = 0, which is a line of symmetry. In the recesses of the channel <f> is negative and large, and the motion becomes a uniform stream. To annul the internal stream we must superpose upon this motion, ex- pressed say by fa + ty lt another of the form <f> 2 + ifa where = - x - y. In the resultant motion, </> = fa + < 2 = fa - x, ^ = -^i so that fa = $ + x, ^fi and we get = <f> + e* +x costy + y), = -/r + et +x sin (>/r + y), ...... (3) whence x = - < + log V(< 2 + >P), y = - ^ + tan- 1 W</>) ......... (4) or, as it may also be written, z = w + log w ............................... (5) It is easy to verify that these expressions, no matter how arrived at, satisfy the necessary conditions. Since x is an even function of -^r, and y an odd function, the line y = is an axis of symmetry. When i/r = 0, we see from (3) that sin y = 0, so that y = or TT, and that cos y and <j> have opposite signs. Thus when < is negative, y = ; and when </> is positive, y = TT. Again, when <f> is negative, a; ranges from +00 to oo ; and when <f> is positive x ranges from oo to 1, the extreme value at the limit of the wall, as appears from the equation dx/d<f>=-l + !/< = 0, making <f>= 1, x = 1. . The central stream-line may thus be considered to pass along y = from x= oo to x = oc . At a; = oo it divides into two * Berlin Monat$ber. 1868; Phil. Mag. Vol. xxxvi. p. 337 (1868). In this paper a new path was opened. t See Lamb's Hydrodynamics, 66. 1915] HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT's TUBES 331 branches along y=-jr. From x = -co to x = 1, the flow is along the inner side of the walls, and from x = I to # = oo back again along the outer side. At the turn the velocity is of course infinite. We see from (4) that when -fy is given the difference in the final values of y, corresponding to infinite positive and negative values of </>, amounts to tr, and that the smaller is ty the more rapid is the change in y. The corresponding values of x and y for various values of <f>, and for the stream-lines i/r = 1, |, , are given in Table I, and the more important parfcs are exhibited in the accompanying plots (fig. 1). TABLE I. , ,~i #--* ,-1 x y x y X y -10 12-303 0-2750 12-30 0-550 12-31 1-100 - 5 6-610 0-3000 6-614 0-600 6-63 1-198 - 3 4-102 0-3333 4-112 0-665 4-15 1-322 - 2 2-701 0-3745 2-723 0-745 2-80 1-464 - 1 1-030 0-495 1-111 0-964 1-35 1-785 - 0-50 0-081 0-714 0-153 1-285 - 0-25 -0-790 1-035 o-oo -1-386 1-821 - 0-693 2-071 o-oo 2-571 0-25 -1-290 2-606 0-50 -1-081 2-928 -0-847 2-881 - 0-388 3-035 1-0 -0-970 3-147 -0-888 3-178 - 0-653 3-356 2-0 -1-299 3-267 -1-277 3-397 - 1-195 3-678 3-0 -1-898 3-308 - 1 -888 3-477 4-0 - 2-584 3-897 5-0 -3-389 3-342 -3-386 3-542 10-0 -7-697 3-367 - 7-692 4-042 20-0 -17-00 4-092 In the second form of the problem we suppose, after Helmholtz and Kirchhoff, that the infinite velocity at the edge, encountered when the fluid adheres to the wall, is obviated by the formation of a surface of discontinuity where the condition to be satisfied is that of constant pressure and velocity. It is, in fact, a particular case of one treated many years ago by Prof. Love, entitled "Liquid flowing against a disc with an elevated rim," when the height of the rim is made infinite*. I am indebted to Prof. Love for the form into which the solution then degrades. The origin 0' (fig. 2) of x + iy or z is taken at one edge. The central stream-line (>/r = 0) follows the line of symmetry AB from y = + cctoy = oo. At y = oo it divides, one half following the inner side of the wall CO' from y oo to y = 0, then becomes a free surface &D from y = Q to y = oo. The connexion between * Camb. Phil. Proc. Vol. vn. p. 185 (1891). 332 HYDBODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES [396 1915] HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES 333 z and w (=</> + ity) is expressed with the aid of an auxiliary variable 6. Thus z = tan 6 6 {i tan 2 6 - i log cos 6, .................. (6) w=4sec 2 ....... . .......................... (7) If we put tan + iy, we get sothat $ = iO + P-^). * = ifr ...................... (8) We find further (Love), ......... (9) sothat ^ = ^ + ^ + |tan^|-+itan- 2 _ 4 _ > . ...(10) (11) The stream-lines, corresponding to a constant ^r, may be plotted from (10), (11), if we substitute 2^/f for rj and regard as the variable parameter. Since by (8) there is no occasion to consider negative values of , and < and f vary always in the same direction. As regards the fractions under the sign of tan" 1 , we see that both vanish when f = 0, and also when =oo. The former, viz., 2 -r- (4>p/ 2 + f 2 1), at first + when f is very small, rises to oc when f 2 = {1 + V(l - 1^^ 2 )}* which happens when ^r < \, but not otherwise. In the latter case the fraction is always positive. When ty < {, the fraction passes through oc , there changing sign. The numerically least negative value is reached when f 2= i {V(l + 48-\p) 1}. The fraction then retraces its entire course, until it becomes zero again when = oo . On the other hand the second fraction, at first positive, rises to infinity in all cases when 2 = (V(l + 16i/r 2 ) - 1}, after which it becomes negative and decreases numerically to zero, no part of its course being retraced. As regards the ambiguities in the resulting angles, it will suffice to suppose both angles to start from zero with This choice amounts to taking the origin of x at 0, instead of 0'. When i/r is very small the march of the functions is peculiar. The first fraction becomes infinite when a = 4i/r 2 , that is when is still small. The turn occurs when 2 =12'\/r 2 , and the corresponding least negative value is also small. The first tan" 1 thus passes from to TT while is still small. The second fraction also becomes infinite when a =4i/r 2 , there changing sign, and again approaches zero while is of the same order of magnitude. 334 HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES [396 The second tan" 1 thus passes from to TT, thereby completing its course, while is still small. When -ty = absolutely, either or 77, or both, must vanish, but we must still have regard to the relative values of ^ and Thus when is small enough, x = 0, and this part of the stream-line coincides with the axis of symmetry. But while is still small, x changes from to TT, the new value representing the inner face of the wall. The transition occurs when = 2>Jr, 77 = 1, making in (11) ^ = 00. The point 0' at the edge of the wall (a? = TT, y = 0) corresponds to = 0, 77 = 0. For the free part of the stream-line we may put 77 = 0, so that n->+=-tan-if + 7r, ............ (12) where tan" 1 f is to be taken between and TT. Also y=-ie- + ilog(l+p) ......................... (13) When is very great, * = +>, y = -W, ..................... (14) and the curve approximates to a parabola. When is small, -T = ip, y = iP, ........................ (15) so that the ratio (x - ir)/y starts from zero, as was to be expected. The upward movement of y is of but short duration. It may be observed that, while dxjd^ is always positive, ft.eoHD df 2(l + 2 )'" ";" which is positive only so long as f < 1. And when = 1, a;-7r = l- i7r = 0-2146, y= - + log 2 = 0'097. Some values of x and y calculated from (12), (13) are given in Table II and the corresponding curve is shown in fig. 3. TABLE II. T=O. o-o 0-5 ro 1-5 2-0 3-142 3-178 3-356 3-659 4-034 +0-050 + 0-097 + 0-027 -0-195 2-5 3-0 4-0 5-0 20-0 4-451 4-892 5-816 6-768 21 -621 - 0-571 - 1-098 - 2-583 - 4-62 -9700 1915] HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES 335 It is easy to verify that the velocity is constant along the curve denned by (12), (13). We have dx I + d<f> ' f l 2 and when IY O' 00 00 B C Fig. 2. Thus dx and .(17) The square root of the expression on the left of (17) represents the reciprocal of the resultant velocity. TABLE 1 X y g X y 00 0-40 2-9667 + 0-076 0-05 0-1667 9-098 0-50 3-0467 0-130 o-io 0-2995 3-008 0-60 3-1089 0-162 0-13 0-4668 1-535 0-80 3-2239 0-198 0-15 0-6725 0-766 1-00 3-3454 0-207 0-17 1-0368 + 0-109 1-50 3-6947 +0-125 0-18 1-2977 -0-143 2-00 4-0936 -0-112 0-19 1-5907 -0-304 2-50 4-5234 -0-501 0-20 1 -8708 - 0-370 3-00 4*9725 - 1 -032 JO -22 2-2828 -0-331 4-00 5-9039 -2-536 0-25 2-5954 -0-195 6-00 7-8305 -7-161 0-30 2-8036 -0-047 li 336 HYDRODYNAMICAL PROBLEMS SUGGESTED BY PITOT'S TUBES [396 Fig. 3. When ty differs from zero, the calculations are naturally more complicated. The most interesting and instructive cases occur when i/r is small. I have chosen ty = 1/10. The corresponding values of , ac, and y are given in Table III, calculated from equations (10), (11), and a plot is shown in fig. 3. As in the former problem, where the liquid is supposed to adhere to the walls notwithstanding the sharp edges, the pressure in the recesses of the tube is simply that due to the velocity at a distance. At other places the pressure can be deduced from the stream-function in the usual way. 397. ON THE CHARACTER OF THE "8" SOUND. [Nature, Vol. xcv. pp. 645, 646, 1915.] SOME two years ago I asked for suggestions as to the formation of an artificial hiss, and I remarked that the best I had then been able to do was by blowing through a rubber tube nipped at about half an inch from the open end with a screw clamp, but that the sound so obtained was perhaps more like an /than an s. " There is reason to think that the ear, at any rate of elderly people, tires rapidly to a maintained hiss. The pitch is of the order of 10,000 per second *." The last remark was founded upon experiments already briefly described f under the head " Pitch of Sibilants." " Doubtless this may vary over a considerable range. In my experiments the method was that of nodes and loops (Phil. Mag. Vol. vn. p. 149 (1879) ; Scientific Papers, Vol. I. p. 406), executed with a sensitive flame and sliding reflector. A hiss given by Mr Enock, which to me seemed very high and not over audible, gave a wave-length (A.) equal to 25 mm., with good agreement on repetition. A hiss which I gave was graver and less definite, corresponding to X = 32 mm. The frequency would be of the order of 10,000 per second, more than 5 octaves above middle C." Among the replies, publicly or privately given, with which I was favoured, was one from Prof. E. B. Titchener, of Cornell University J, who wrote : " Lord Rayleigh's sound more like an / than an s is due, according to Kohler's observations, to a slightly too high pitch. A Galton whistle, set for a tone of 8400 v.d., will give a pure s." It was partly in connexion with this that I remarked later that I doubted whether any pure tone gives the full impression of an s, having often experi- mented with bird-calls of about the right pitch. In my published papers I * Nature. Vol. xci. p. 319, 1913. t Phil. May. Vol. xvr. p. 235, 1908 ; Scientific Papers, Vol. v. p. 486. Nature, Vol. xci. p. 451, 1913. Nature, Vol. xci. p. 558, 1913. R. vi. 22 338 ON THE CHARACTER OF THE "S" SOUND [397 find references to wave-lengths 31 '2 mm., 1 -304in. = 331 mm., 1 '28 in. = 32*5 mm.* It is true that these are of a pitch too high for Kohler's optimum, which at ordinary temperatures corresponds to a wave-length of 40'6 mm., or T60 inches; but they agree pretty well with the pitch found for actual hisses in my obser- vations with Knock. Prof. Titchener has lately returned to the subject. In a communication to the American Philosophical Societyf he writes : "It occurred to me that the question might be put to the test of experiment. The sound of a Galton's whistle set for 8400 v.d. might be imitated by the mouth, and a series of observations might be taken upon material composed partly of the natural (mouth) sounds and partly of the artificial (whistle) tones. If a listening observer were unable to distinguish between the two stimuli, and if the mouth sound were shown, phonetically, to be a true hiss, then it would be proved that the whistle also gives an s, and Lord Rayleigh would be answered. " The experiment was more troublesome than I had anticipated ; but I may say at once that it has been carried out, and with affirmative result." A whistle of Edelmann's pattern (symmetrical, like a steam whistle) was used, actuated by a rubber bulb ; and it appears clear that a practised operator was able to imitate the whistle so successfully that the observer could not say with any certainty which was which. More doubt may be felt as to whether the sound was really a fully developed hiss. Reliance seems to have been placed almost exclusively upon the position of the lips and tongue of the operator. I confess I should prefer the opinion of unsophisticated observers judging of the result simply by ear. The only evidence of this kind mentioned is in a footnote (p. 328) : " Mr Stephens' use of the word ' hiss ' was spontane- ous, not due to suggestion." I have noticed that sometimes a hiss passes momentarily into what may almost be described as a whistle, but I do not think this can be regarded as a normal s. Since reading Prof. Titchener's paper I have made further experiments with results that I propose to describe. The pitch of the sounds was deter- mined by the sensitive flame and sliding reflector method, which is abundantly sensitive for the purpose. The reflector is gradually drawn back from the burner, and the positions noted in which the flame is unaffected. This phase occurs when the burner occupies a node of the stationary waves. It is a place where there is no to and fro motion. The places of recovery are thus at distances from the reflector which are (odd or even) multiples of the half wave-length. The reflector was usually drawn back until there had been five Scientific Papert, Vol. i. p. 407; Vol. n. p. 100. t Proceedings, Vol. Lm. August December, 1914, p. 323. 1915] ON THE CHARACTER OF THE " S " SOUND 339 recoveries, indicating that the distance from the burner was now 5 x \, and this distance was then measured. The first observations were upon a whistle on Edelmann's pattern of my own construction. The flame and reflector gave A, = 1-7 in., about a semi-tone flat on Kb'hler's optimum. As regards the character of the sound, it seemed to me and others to bear some resemblance to an s, but still to be lacking in something essential. I should say that since my own hearing for s's is now distinctly bad, I have always confirmed my opinion by that of other listeners whose hearing is good. That there should be some resemblance to an s at a pitch which is certainly the predominant pitch of an s is not surprising ; and it is difficult to describe exactly in what the deficiency consisted. My own impression was that the sound was too nearly a pure tone, and that if it had been quite a pure tone the resemblance to an s would have been less. In subsequent observations the pitch was raised through A. = 1*6 in., but without modifying the above impressions. Wishing to try other sources which I thought more likely to give pure tones, I fell back on bird-calls. A new one, with adjustable distance between the perforated plates, gave on different trials A. = 1*8 in., \ = 1*6 in. In neither case was the sound judged. to be at all a proper s, though perhaps some resemblance remained. The effect was simply that of a high note, like the squeak of a bird or insect. Further trials on another day gave confirmatory results. The next observations were made with the highest pipe from an organ, gradually raised in pitch by cutting away at the open end. There was some difficulty in getting quite high enough, but measures were taken giving X = 2'2 in., A, = T9 in., and eventually X. = 1'6 in. In no case was there more than the slightest suggestion of an s. As I was not satisfied that at the highest pitch the organ-pipe was speaking properly, I made another from lead tube, which could be blown from an adjustable wind nozzle. Tuned to give A,= 1'6 in., it sounded faint to my ear, and conveyed no s. Other observers, who heard it well, said it was no s. In all these experiments the sounds were maintained, the various instru- ments being blown from a loaded bag, charged beforehand with a foot blower. In this respect they are not fully comparable with those of Prof. Titchener, whose whistle was actuated by squeezing a rubber bulb. However, I have also tried a glass tube, 10-4 in. long, supported at the middle and rubbed with a resined leather. This should be of the right pitch, but the squeak heard did not suggest an s. I ought perhaps to add that the thing did not work particularly well. It will be seen that my conclusions differ a good deal from those of Prof. Titchener, but since these estimates depend upon individual judgment, perhaps 222 340 ON THE CHARACTER OF THE " S" SOUND [397 not uninfluenced by prepossessions, they are not fully satisfactory. Further independent aural observations are desirable. I fear a record, or ocular obser- vation, of vibrations at so high a pitch is hardly feasible. I may perhaps be asked if a characteristic 8, having a dominant pitch, is not a pure tone, what is it ? I am disposed to think that the vibration is irregular. A fairly defined pitch does not necessitate regular sequences of more than a few (say 3 10) vibrations. What is the state of affairs in an organ- pipe which does not speak well, or in a violin string badly bowed ? An example more amenable to observation is afforded by the procession of drops into which a liquid jet breaks up. If the jet is well protected from outside influences, the procession is irregular, and yet there is a dominant interval between consecutive drops, giving rise under suitable conditions to a sound having a dominant pitch. Vibrations of this sort deserve more attention than they have received. In the case of the s the pitch is so high that there would be opportunity for interruptions so frequent that they would not be separately audible, and yet not so many as to preclude a fairly defined dominant pitch. I have an impression, too, that the s includes subordinate components de- cidedly graver than the dominant pitch. Similar questions naturally arise over the character of the sh, f, and th sounds. 398. ON THE STABILITY OF THE SIMPLE SHEARING MOTION OF A VISCOUS INCOMPRESSIBLE FLUID. [Philosophical Magazine, Vol. xxx. pp. 329338, 1915.] A PRECISE formulation of the problem for free infinitesimal disturbances was made by Orr (1907)*. It is supposed that (the vorticity) and v (the velocity perpendicular to the walls) are proportional to e int e ikx , where n =p + iq. = S, we have and d*v/dy*-k*v = S, .............................. (2) with the boundary conditions that v = 0, dv/dy = Q at the walls where y is constant. Here v is the kinematic viscosity, and is proportional to the initial constant vorticity. Orr easily shows that the period-equation takes the form (3) where S l} S 2 are any two independent solutions of (1) and the integrations are extended over the interval between the walls. An equivalent equation was given a little later (1908) independently by Sommerfeld. Stability requires that for no value of k shall any of the q's determined by (3) be negative. In his discussion Orr arrives at the conclusion that this condition is satisfied. Another of Orr's results may be mentioned. He shows that p + kfty necessarily changes sign in the interval between the walls t. In the paper quoted reference was made also to the work of v. Mises and Hopf, and it was suggested that the problem might be simplified if it could be shown that q vk* cannot vanish. If so, it will follow that q is always * Proc. Roy. Irith Acad. Vol. xxvu. t Phil. Mag. Vol. xxvui. p. 618 (1914). 342 ON THE STABILITY OF THE SIMPLE SHEARING positive and indeed greater than vk*, inasmuch as this is certainly the case when /9 = 0*. The assumption that q = vk a , by which the real part of the { } in (1) disappears, is indeed a considerable simplification, but my hope that it would lead to an easy solution of the stability problem has been disappointed. Nevertheless, a certain amount of progress has been made which it may be desirable to record, especially as the preliminary results may have other applications. If we take a real rj such that ), ........................... (4) we obtain ~ = -9ir,S. .......................... (5) arj' This is the equation discussed by Stokes in several papers f, if we take x in his equation (18) to be the pure imaginary irj. The boundary equation (3) retains the same form with ^ drj for e**' dy, where \* = 9vfrlP .................................. (6) In (5), (6) 77 and \ are non-dimensional. Stokes exhibits the general solution of the equation -*" ................................. m in two forms. In ascending series which are always convergent, Qx 3 9V 9V 9V 9V The alternative semi-convergent form, suitable for calculation when x is large, is j 2x !J 1.6 1.5.7.11 1.5.7.11.13.17 . O \jX e - 5 T 1.144#* 1.2.144 a ar 1 1 . 2 . 3. 1.144^* 1.2.144V 1.2. 3. 144 s x* in which, however, the constants C and D are liable to a discontinuity. When x is real the case in which Stokes was mainly interested or a pure imaginary, the calculations are of course simplified. * Phil. Mag. Vol. zxxiv. p. 69 (1892) ; Scientific Papers, Vol. ni. p. 583. t Especially Camb. Phil. Trans. Vol. x. p. 106 (1857) ; Collected Papers, Vol. iv. p. 77. 1915] MOTION OF A VISCOUS INCOMPRESSIBLE FLUID 343 If we take as S l and S 2 the two series in (8), the real and imaginary parts of each are readily separated. Thus if & = ,+& S 2 = s, + it,, (10) we have on introduction of irj 9V 9V 2. 3. 5. 6 + 2. 3. 5. 6. 8. 9. 11. 12 Q-rtS ( i;; : / = _^_ j. - ^ (12^ 2.32.3.5.6.8.9 _ 977' 9V -3T4~3.4;6.7.9.10 + 9V 9V 3.4.6.7 " 3.4.6.7.9.10.12.13 in which it will be seen that s lt s 2 are even in 77, while ti, t^ are odd. When 77 < 2, these ascending series are suitable. When 77 > 2, it is better to use the descending series, but for this purpgse it is necessary to know the connexion between the constants A, B and C, D. For a? = 117 these are (Stokes) A = 7r-*r(|){C'+Z)e-*' r / 6 }, # = 37r- i r(|){-C + Z)e i ' r/6 ;. ...(15) Thus for the first series $ (A = I, B = in (8)) logD = 1-5820516, = De iir / 6 ; (16) and for S z (A = 0,5=1) log D' = 1-4012366, - C'= DV iir / 6 , (17) so that if the two functions in (9) be called 2j and 2 2 , o /^ "C 1 i r\ "^ o s^ f *? i 7VK 1 /I R"\ o x = C 2*i + JJ 2. 2 , O 2 = 2,j + JJ 2< 2 V 10 / These values may be confirmed by a comparison of results calculated first from the ascending series and secondly from the descending series when 77 = 2. Much of the necessary arithmetic has been given already by Stokes*. Thus from the ascending series *, (2) = - 13-33010, *, (2) = 11-62838 ; * a (2) = - 2-25237, * 2 (2) = - H'44664. In calculating from the descending series the more important part is 2i, since For 77 = 2 Stokes finds S x = - 14-98520 + 43-81046i, of which the log. modulus is 1-6656036, and the phase + 108 52' 58"'99. When the multiplier C or C' is introduced, there will be an addition of 30 to this phase. Towards the value of , I find -13-32487 + 1 1-63096 i; * Loc. cit. Appendix. It was to take advantage of this that the " 9 " was introduced in (5). 344 ON THE STABILITY OF THE SIMPLE SHEARING and towards that of S t -2-24892-1 l-44495t'. For the other part involving D or D' we get in like manner - -00523- -00258 i, and - -00345- -001 70 i. TABLE I. [398 r> i h *a ** o-o + 1-0000 - -oooo + -oooo + -oooo O'l + i-oooo - -0015 + -0001 + -1000 0-2 + i-oooo - -0120 + -0012 + -2000 0-3 + '9997 - -0405 + -0061 + -3000 0'4 + -9982 0960 + -0192 + '3997 0-5 + -9930 - '1874 + -0469 + -4987 0-6 + -9790 - -3234 + -0971 + '5955 0-7 + -9393 5485 + -1969 + -6845 0-8 + -8825 . - '7605 + -3055 + -7663 0-9 + '7619 - 1-0717 + -4865 + -8234 1-0 + -554 - 1-444 + '734 + '840 1-1 + -215 - 2-007 + 1-057 + -790 1-2 310 - 2-304 + 1-456 + -634 1-3 - 1-083 - 2-707 + 1-923 + -320 1-4 - 2-173 - 2-979 + 2-424 221 1-5 - 3-635 - 2-972 + 2-893 - 1-067 1-6 - 5-493 - 2-466 + 3-212 - 2-303 1-7 - 7-694 - 1-161 + 3-191 - 3-998 1-8 -10-057 + 1-325 + 2-550 - 6-173 1-9 -12-177 + 5-441 + -899 - 8-745 2-0 - 13-330 + 11-628 - 2-252 -11-447 2-1 -12-34 + 20-19 - 7-46 -13-70 2-2 - 7-49 + 31-01 -15-24 -14-50 2-3 + 3-54 +43-20 -25-84 -12-22 2-4 + 23-55 + 54-54 -38-90 - 4-53 2-5 + 55-20 + 60-44 -52-70 + 11-59 It appears that with the values of 0, D, C', D' defined by (16), (17) the calculations from the ascending and descending series lead to the same results when T; = 2. What is more, and it is for this reason principally that I have detailed the numbers, the second part involving 2 2 loses its importance when 77 exceeds 2. Beyond this point the numbers given in the table are calculated from 2, only. Thus (77 > 2) 1.144(ii,)' 1.2. 1.5 1.5.7.11 xil- ( 1.144 (it;) 7 1.2. 144* to)' ,P '"I (20, 1915] MOTION OF A VISCOUS INCOMPRESSIBLE FLUID 345 the only difference being the change from D to D' and the reversal of sign in 7T/6, equivalent to the introduction of a constant (complex) factor. When 77 exceeds 2'5, the second term of the series within { } in 2j is less than 10~ 2 , so that for rough purposes the { } may be omitted altogether. We then have Sl = 77 -V 2 '"* cos (V2.77*-7r/24), .................. (21) ^--Di,-*^ 2 '" 1 sin(V2.77 f -7r/24), .................. (22) T7i sin(v / 2.77 f -7r/24-7r/6), ......... (23) . * cos (V2.77*--7r/24-7r/6) .......... (24) Here D and D' are both positive the logarithms have already been given and we see that s lt t 2 are somewhat approximately in the same phase, and t lt s 2 in approximately opposite phases. When 77 exceeds a small integer, the functions fluctuate with great rapidity and with correspondingly in- creasing maxima and minima. When in one period \/2 . 77 increases by 2-Tr, the exponential factor is multiplied by e 2ir , viz. 535*4. From the approximate expressions applicable when 77 exceeds a small integer it appears that s lt ti are in quadrature, as also s 2 , t 2 . For some purposes it may be more convenient to take 2j, S 2 , or (expressed more correctly) the functions which identify themselves with 2 1? 2 2 when 77 is great, rather than 8 lt S 2 , as fundamental solutions. When 77 is small, these functions must be calculated from the ascending series. Thus by (15) (0- 1,4-0) ^-ir-irtt^-sir-ircf)^, ..................... (25) and(C=0, D=l) ^^-ir^e-^Si + Sir^r ()*-/& ............. (26) Some general properties of the solutions of (5) are worthy of notice. If S = s + it, we have rfsldif = 977*, dHl dvf = - 9775. Let R = (s 2 + V) ; then dR_ds dt di) dtj dv) ' d*R /ds\* fdt\ 2 d*s , dH j-V "" ( j~ ) +j- + s ^o+tT-,> drj* \dr)/ \dr)/ drf drj 2 of which the two last terms cancel, so that d^R/dr)* is always positive. In the case of S lf when 77 = 0, ^(0) = 1, t 1 (0) = Q, 5/(0) = 0, so that /(()) = , R' (0) = 0. Again, when 77 = 0, s 3 (0) = 0, , (0) = 0, so that .R (0) = 0, R' (0) = 0. In neither case can R vanish for a finite (real) value of 77, and the same is true of S 1 and'$ 2 . 346 ON THE STABILITY OF THE SIMPLE SHEARING [398 Since (5) is a differential equation of the second order, its solutions are connected in a well-known manner. Thus and on integration ^ a e^ 1 = congtan ..................... as appears from the value assumed when ij = 0. Thus MS ............................... <> which defines /Sj in terms of S^ A similar relation holds for any two particular solutions. For example, The difficulty of the stability problem lies in the treatment of the boundary condition . ( * S 2 e~>"> drj - [ % S, e~^ dy . I * 8 2 e^ dy = 0, . . .(31)* J rit J i), J T|, in which T; 2 , r} l} and X are arbitrary, except that we may suppose T; 2 and X to be positive, and 77! negative. In (31) we may replace ^, e~ Ar) by cosh XT;, sinh XT; respectively, and the substitution is especially useful when the limits of integration are such that ij, = - rj 2 . For in this case I S cosh XT; di) = 2 I s cosh XT; drj, J -n\ Jo I S sinh XT; dij = 2i I * t sinh XT; drj ; J* Jo and the equation reduces to I S } cosh XT; drj . * t. 2 sinh XT; dr) .'o . f o s 2 cosh XT; drj . I , sinh XT; drj = 0, (32) Jo Jo thus assuming a real form, derived, however, from the imaginary term in (31). In general with separation of real and imaginary parts we have by (31) from the real part ft Ft )*, e^idvj . ISye'^dr) It^^dr). [t t *~**w| \8ie~^dr) . Isy^dij + ltte'^dt) . \t>^drj =0, (33) * Rather to my surprise I find this condition already laid down in private papers of Jan. 1893. 1915] MOTION OF A VISCOUS INCOMPRESSIBLE FLUID 347 and from the imaginary part | s 2 e~ ^ dr) . fa ^ drj . I tz e* drj - L ^ drj . h e~^ drj = ....... (34) If we introduce the notation of double integrals, these equations become sinh X (T, - 77') {* fo) S 2 (r/) - t, (77) . t, (77')] dr,dr)'=0, ...... (35) I Js JJ sinh \(ri- 77') { 8l (77) . t, (T/) - s 2 (77) . t, M} dr,dr)' = 0, ...... (36) the limits for 77 and rj' being in both cases ^ and r; 2 . In these we see that the parts for which 77 and 77' are nearly equal contribute little to the result. A case admitting of comparatively simple treatment occurs when \ is so large that the exponential terms e Ar? , e~ Al? dominate the integrals. As we may see by integration by parts, (31) then reduces to StM.&M-StM. &(*,,) = 0, .................. (37) or with use of (29) ...................... (38) We have already seen that $1(77) cannot vanish; and it only remains to prove that neither can the integral do so. Owing to the character of S lt only moderate values of 77 contribute sensibly to its value. For further examination it conduces to clearness to write r) z = a, ^ = - 6, where a and b are positive. Thus drj = f" ^77 f 6 drj S ' . i> ( and it suffices to show that I -- 1 2 cannot vanish. A short table makes this apparent [see p. 348]. The fifth column represents the sums up to various values of 77. The ap- proximate value of f yr^lf 17 is thus ' 2 x 2 ' 8 34or '567. The true value Jo ( s i +ti) of this integral is (D'jD) sin 60 or '571, as we see from (30) and (19), (20). We conclude that (37) cannot be satisfied with any values of 772 and 77,. When the value of \ is not sufficiently great to justify the substitution of (37) for (31) in the general case, we may still apply the argument in a rough manner to the special case (773 + 77! = 0) of (32), at any rate when 772 348 ON THE STABILITY OF THE SIMPLE SHEARING [398 is moderately great. For, although capable of evanescence, the functions *n ^i> s *> t* increase in amplitude so rapidly with 77 that the extreme value of i\ may be said to dominate the integrals. The hyperbolic functions then disappear and the equation reduces* to (ih)-0 ...................... (40) TABLE II. 1) tf-V (8,2 + t,2)2 i'-*i 2 ' Sums of fourth column (i 2 + *i 2 ) 2 1 + i-ooo 1-000 + 1-000 1-000 3 + 0-997 1-002 + -995 1-995 5 + 0-951 1-042 + -913 2-908 7 + 0-681 1-399 + -415 3-323 9 0-569 2-989 - -191 3-132 1-1 - 3-982 16-60 - -240 2-892 1-3 - 6-155 72-25 - -085 2-807 1-5 + 4-38 485-8 + -009 2-816 1-7 + 57-9 3660-0 + -016 2-832 1-9 + 119-0 31700-0 + -004 2-836 2-1 - 255-0 314000-0 - -001 2-835 2-3 -1854-0 353xlO~ 4 - -001 2-834 2-5 - 616-0 45 x 10~ 6 - -ooo 2-834 which cannot be satisfied by a moderately large value of r) 2 . For it appears from the appropriate expressions (21)... (24) that the left-hand member of (40) is then a positive and rapidly increasing quantity. Again, it is evident from Table I that the left-hand member of (32) remains positive for all values of t] 2 from zero up to some value which must exceed I'l, since up to that point the functions s lt * 2 , ^ are positive while , is negative. Even without further examination it seems fairly safe to conclude that (32) cannot be satisfied by any values of rj 2 and X. Another case admitting of simple treatment occurs when ij 2 and i;, are both small, although A, may be great. We have approximately the next terms being in each case of 6 higher degrees in rj. Thus with omission of terms in rf under the integral sign, (31) becomes (41) * Regard being paid to the character of the functions. Needless to say, it is no general proposition that the value of an integral is determined by the greatest value, however excessive, of the integrand. 1915] MOTION OF A VISCOUS INCOMPRESSIBLE FLUID 349 or on effecting the integrations It is easy to show that (42) cannot be satisfied. For, writing X (7/3 tjj) = x, a; sinh x 2.3^2.3.4.5 every term of the first series exceeding the corresponding term of the second series. The left-hand member of (42) is accordingly always positive. This disposes of the whole question when 7? 2 and rj^ are small enough (numerically), say distinctly less than unity. 399. ON THE THEORY OF THE CAPILLARY TUBE. [Proceedings of the Royal Society, A, Vol. xcn. pp. 184195, Oct. 1915.] A RECENT paper by Richards and Coombs* discusses in some detail the determination of surface-tension by the rise of the liquid in capillary tubes, and reflects mildly upon the inadequate assistance afforded by mathematics. It is true that no complete analytical solution of the problem can be obtained, even when the tube is accurately cylindrical. We may have recourse to graphical constructions, or to numerical calculations by the method of Rungef, who took an example from this very problem. But for experimental pur- poses all that is really needed is a sufficiently approximate treatment of the two extreme cases of a narrow and of a wide tube. The former question was successfully ^attacked by Poisson, whose final formula [(18) below] would meet all ordinary requirements. Unfortunately doubts have been thrown upon the correctness of Poisson's results, especially by MathieuJ, who rejects them altogether in the only case of much importance, i.e. when the liquid wets the walls of the tube a matter which will be further considered later on. Mathieu also reproaches Poisson's investigation as implying two different values of h, of which the second is really only an improvement upon the first, arising from a further approximation. It must be admitted, however, that the problem is a delicate one, and that Poisson's explanation at a critical point leaves something to be desired. In the investigation which follows I hope to have succeeded in carrying the approximation a stage beyond that reached by Poisson. In the theory of narrow tubes the lower level from which the height of the meniscus is reckoned is the free plane level. In experiment, the lower level is usually that of the liquid in a wide tube connected below with the narrow one, and the question arises how wide this tube needs to be in order that the inner part of the meniscus may be nearly enough plane. Careful * Journ. Amer. Chem. Soc. No. 7, July, 1915. t Math. Ann. Vol. XLVI. p. 175 (1895). t Thtarie de la Capillarite, Paris, 1883, pp. 4649. 1915] ON THE THEORY OF THE CAPILLARY TUBE 351 experiments by Richards and Coombs led to the conclusion that in the case of water the diameter of the wide tube should exceed 33 mm., and that probably 38 mm. suffices. Such smaller diameters as are* often employed (20 mm.) involve very appreciable error. Here, again, we should naturally look to mathematics to supply the desired information. The case of a straight wall, making the problem two-dimensional, is easy*, but that of the circular wall is much more complicated. Some drawings (from theory) given by Kelvin, figs. 24, 26, 28 f, indicate clearly that diameters of 1'8 cm. and 2*6 cm. are quite inadequate. I have attempted below an analytical solution, based upon the assumption that the necessary diameter is large, as it will be, if the prescribed error at the axis is small enough. Although this assumption is scarcely justified in practice, the calculation indicates that a diameter of 47 cm. may not be too large. As Richards and Coombs remark, the observed curvature of the lower part of the meniscus may be used as a test. Theory shows that there should be no sensible departure from straightness over a length of about 1 cm. The Narrow T'ube. For the surface of liquid standing in a vertical tube of circular section, we have xdzldx 1 f* #sm-ur= - ' = zxdx, ............... (I) * a 2 J in which z is the vertical co-ordinate measured upwards from the free plane level, x is the horizontal co-ordinate measured from the axis, -fy is the angle the tangent at any point makes with the horizontal, and tf=Tgp\, where T is the surface-tension,^ the acceleration of gravity, and p the density of the fluid. The equation expresses the equilibrium of the cylinder of liquid of radius #. At the wall, where x = r, ty assumes a given value (^TT i), and (1) becomes a?rcosi.= l zxdx ............................ (2) Jo If the radius (r) of the tube is small, the total curvature is nearly con- stant, that is, the surface is nearly spherical. We take z = I - x /(c 2 - a; 2 ) + u, ......... .................. (3) where I is the height of the centre and c the radius of the sphere, while u represents the correction required for a closer approximation. If we omit u altogether, (2) gives ^lr 2 + ^{(c i -r 2 ^-c a } ................... (4) * Compare Phil. Mag. Vol. xxxiv. p. 509, Appendix, 1892 ; Scientific Papers, Vol. iv. p. 13. t The reference is given below. J It may be remarked that a 2 is sometimes taken to denote the double of the above quantity. 352 ON THE THEORY OF THE CAPILLARY TUBE [399 Also, if A be the height at the lowest point of the meniscus, the quantity directly measured in experiment, h=l-c ..................................... (5) In this approximation r/c = cos t, and thus in terms of c a'rVc = *r 8 (/i + c)+Hc 8 -r') l -ic' ................... (6) When the angle of contact (t) is zero, c = r, and a* = r(h+$r) .................... ........... (7) the well-known formula. When we include u, it becomes a question whether we should retain the value of c, i.e. r sec i, appropriate when the surface is supposed to be exactly spherical. It appears, however, to be desirable, if not necessary, to leave the precise value of c open. Substituting the value of z from (3) in (1), we get, with neglect of ( 20* uxdx \ - Tor the purposes of the next approximation we may omit (dujdx)* and the integral, which is to be divided by a 2 . Thus dx W ' (c 2 - off to? x (c 2 - a*)* 3a*a; and on integration We suppose with Poissori and Mathieu that so that u = 2 log{c + v/(c 2 -^)}+0, .................. (12) corresponding to ^.|.^^_? ......................... (13) To determine c we have the boundary condition dz r du ~ c 8 C- which gives c in terms of i and r. Explicitly _ r r 8 si " cost 3a 2 (H-si These latter equations are given by Mathieu. _ r r 8 sinH' " cost 3a 2 (H-siht)cost 1915] ON THE THEOEY OF THE CAPILLARY TUBE 353 We have now to find the value of a 2 to the corresponding approximation. For the observed height of the meniscus h = l-c + u^ = l-c + C+^\og(2c); ............ (16) and ar cos i = ^ zxdx = (I + C) + ^ f(c 2 - r 8 )* - c 3 } + T (u - C) xdx Jo * & Jo In the important case where i = 0, the liquid wetting the walls of the tube, c = r simply, and - 01288 r 2 / A) ...................... (18) This is the formula given long since by Poisson*, the only difference being that his a 2 is the double of the quantity here so denoted. It is remarkable that Mathieu rejects the above equations as applicable to the case i = Q, c = r, on the ground that then du/dx in (13) becomes infinite when# = r. But-d \/(r 2 ac 2 )/dx, with which du/dix comes into comparison, is infinite at the same time ; and, in fact, both in equation (8) vanish when x = r. It is this circumstance which really determines the choice of I in (11). We may now proceed to a yet closer approximation, introducing approxi- mate values of the terms previously neglected altogether. From (13) and from (12) \* uxdx= %Cx 2 + - [a? log {c + V(c 2 - tf 2 )} + c 3 - c ^(c 2 - x*) + $ (c 8 - .' Ott * Nouvelle TMorie de V Action Capillaire, 1831, p. 112. 23 354 ON THE THEORY OF THE CAPILLARY TUBE [399 Thus = _i +* 2a 2 - f 2 - faz* d 1 -**) 2 , ...... (19) _c_ 6a' log {c + V(c 2 a?)} + constant. We have now to choose /, or rather (I + C), and it may appear at first sight as though we might take it almost at pleasure. But this is not the case, at any rate if we wish our results to be applicable when c = r. For this purpose it is necessary that (dujdx\ x (r 2 - of) be a small quantity, and only a particular choice of (I + C) will make it so. For when x = c = r, ,du\ r 2 -* 2 _r __ (r_(J + (7r 2L 4. ^ (] n 4. IV - - r - \das) r r> ~V(r 2 -^)I "2a 2 3a 2 "*" Qa*\ * 2jl 6a< terms vanishing when x = r. We must therefore take > ................ < 20 > making It should be noticed that u so determined does not become infinite when c = r and x = r. For we have Also with the general value of c 1 "-j&( 1 -) h * 2+0 ' ...................... (22 > As before h=l c+u , and 1915] ON THE THEORY OF THE CAPILLARY TUBE 355 The integral in (23) can be expressed. We find + 2c 2 (log2-l)| ..................................... (24) The expression for ra?cosi in terms of c is complicated, and so is the relation between c and i demanded by the boundary condition (25) But in the particular case of greatest interest (i = 0) much simplification ensues. It follows easily from (25) that c = r. When we introduce this condition into (24), we get ............ (26) and accordingly Hence by successive approximations = r {h + ir - 0'1288r 2 //i + 01312r 3 /A 2 } ................... (28) If the ratio of r to h is at all such as should be employed in experiment, this formula will yield a 2 , viz., T/gp, with abundant accuracy. Our equations give for the whole height of the meniscus in the case t = 0, c = r, (29) Another method of calculating the correction for a small tube, originating apparently with Hagen and Desains, is to assume an elliptical form of surface in place of the circular, the minor axis of the ellipse being vertical. In any case this should allow of a closer approximation, and drawings made for Kelvin* by Prof. Perry suggest that the representation is really a good one. * Proc. Roy. Inst. 1886; " Popular Lectures and Addresses," I. p. 40. 232 356 ON THE THEORY OF THE CAPILLARY TUBE [399 If the semi-axis minor of the ellipse be ft, the curvature at the end of this axis is 0/r 3 , and in our previous notation /9 = Ar a /2a a . Also, t being equal to 0, a-r- and a 2 = Ar(/i+J/3) = ^r(lH-r 2 /6a 2 ) ................... (30) This yields a quadratic in a? ; hence hr hr = |r{/i + Jr- 0-11 11 r/A + 0-0741 r*/h*} .................. (31) approximately. It will be seen that this differs but little numerically from (28), which, however, professes to be the accurate result so far as the term in r*/A a inclusive. The Wide Tube. The equation of the second order for the surface of the liquid, assumed to be of revolution about the axis of z, is 'well known and may be derived from (1) by differentiation. It is dz (32) If dzjdx be small, (32) becomes approximately d*z l<fr_ = 3^Y__,_ ^dx. In the interior part of the surface under consideration (dzjdx)* may be neglected, and the approximate solution is + 2^ + 22> ^ 4 +...j, ..-(34) J denoting, as usual, the Bessel's, or rather Fourier's, function of zero order and h being the elevation at the axis above the free absolutely plane level. For the present purpose A is to be so small as to be negligible in experiment, and the question is how large must r be. When A is small enough, xla may be large while dzjdx still remains small. Eventually dzfdx increases so that the formula fails. But when x is large enough before this occurs, we may if necessary carry on with the two- dimensional solution properly adjusted to fit, as will be further explained later. In the meantime it will be convenient to give some numerical examples of the increase in dzjdx. In the usual notation --/(-),. -(35) dx a \al and the values of /,, up to as/a = 6, are tabulated*. * Brit. Aisoc. Rep. for 1889 ; or Gray and Mathews' BetteVs Function*, Table VI. 1915] OX THE THEORY OF THE CAPILLARY TUBE 357 In the case of water a = 0'27 cm. If we take h /a = O'Ol, and x/a = 4, we have dz/dx = 0*098, so that (dz/dx)* is still fairly small. Here for water /? = 0*0027 cm. and 2# = 2*2 cm. A diameter of 2'2 cm. is thus quite in- sufficient, unless an error exceeding 0*003 cm. be admissible. Again, suppose h /a = 0-001, and take x/a = 6. Then dzfdx = 0'061, again small. For water A =0'00027 cm., and 2#=3'2 cm. This last value of h is about that (0'003mm.) given by Richards and Coombs as the maximum admissible error of reading, and we may conclude that a diameter of 3*2 cm. is quite inadequate to take advantage of this degree of refinement. We may go further in this example without too great a loss of accuracy. Retaining 7< /a=0'001, let us make #/a=7. I find 7^7) = 156 about, so that the extreme value of dzjdx is 0'156, still moderately small. Here 2x = 3'8 cm., which is thus shown to be inadequate in the case of water. But apart from the question of the necessary diameter of tube, information sufficient for experimental purposes can be derived in another manner. The initial value of z .(on the axis) is h ; and z= 2A when I (x/a) = 2, i.e. when x=I'8a. For the best work h should be on the limit of what can be detected and then h Q and 2h could just be distinguished. The observer may be satisfied if no difference of level can be seen over the range x = l'8a; in the case of water this range is 2 x 1*8 x 0'27 = 0'97 cm., or say 1 cm. It has already been remarked that when ^ is small enough xja may become great within the limits of application of (35). To shorten our ex- pressions we will take a temporarily as the unit of length. Then when x is very great, 'W-'-W-Tfe ......................... <36) Thus if >Jr be the angle the tangent to the curve makes with the horizontal, an equation which may be employed when h Q is so small that a large x is consistent with a small ^r. In order to follow the curve further, up to -^r = ^ir, we may employ the two-dimensional solution, the assumption being that the region of moderate i/r occupies a range of x small in comparison with its actual value, i.e. a value not much less than r, the radius of the tube. On account of the magnitude of x we have only the one curvature to deal with. For this curvature so that ** = C - cos i/r = 1 - cos 358 ON THE THEORY OF THE CAPILLARY TUBE since when -^ = 0, z 1 is exceedingly small. Accordingly (39) Also dx=-^-r = -4 and #=logtan(i-f)+2cos^ + C' ................... (40) The constant is determined by the consideration that at the wall (x = r), T/rrs^Tr; thus r - x = log tan (ir/8) + \/2 - log tan (\^r) - 2 cos ( >/r) = log tan (TT/S) + V2 - 2 + 2 log 2 - log -f, ............ (41) since >/r is small. The value of x is supposed to be the same here as in (37), so that *=logf + log(27)-logA , ................... (42) whence on elimination of >/r and restoration of a, r/a = - log (V2 + 1) + x/2 - 2 + 2 log 2 -I- log (2/a) - log(A /a). ...(43) With sufficient approximation, when h is small enough, we may here substitute r for x, and thus r/a - \ log (r/a) = - log (x/2 + 1) + V2 - 2 + 2 log 2 + $ log (2ir) - log (/t /a) = 0-8381+ log (a/A,) ........................................ (44) This formula should give the relation between r/a and /< /a when h /a is small enough, but it is only roughly applicable to the case of greatest interest, where a/h = 1000, corresponding to the accuracy of reading found by Richards and Coombs. In this case 0-8381 + log (a/// ) = 7 746. For this value of r/a we should have log (r/a) = 1 '024. It is true that according to (44) r/a will be somewhat greater, but on the other hand the proper value of x (replaced by r) is less than r. We may fairly take r/a = 7-746 + 1-024 = 8-770, making with a = 0'27 cm. 2r = 4-74cm. This calculation indicates that a diameter greater even than those con- templated by Richards and Coombs may be necessary to reduce h 9 to negligibility, but it must be admitted that it is too rough to inspire great confidence in the close accuracy of the final number. Probably it would be feasible to continue the approximation, employing an approximate value for the second curvature in place of neglecting it altogether. But although the integration can be effected, the work is rather long. 1915] ON THE THEOKY OF THE CAPILLARY TUBE 359 [Added November 17. Since this paper was communicated, I have been surprised to find that the problem of the last paragraphs was treated long ago by Laplace in the Mecanique Celeste* by a similar method, and with a result equivalent to that (44) arrived at above for the relation between the radius of a wide tube and the small elevation at the axis. Laplace uses the definite integral expression for /, and obtains the approximate form appro- priate to large arguments. In view of Laplace's result, I have been tempted to carry the approximation further, as suggested already. In the previous notation, the differential equation of the surface may be written sin i/r d'Jr sin In the first approximation, where the second curvature on the left is omitted, we get -, z being the elevation at the axis, where \/r = 0. For the present purpose z is to be regarded as exceedingly small, so that we may take at this stage, as in (39), . .................... (46) We now introduce an approximate value for the second curvature in (45), writing x = r, where r is the radius of the tube, and making, according to (46), if.. -/(!-*.) ......................... (47) On integration & 4a ' s 2 \* z 2 4>a , -f a- C os^._ +5: |l__)._H._rf| ....... (48) on substitution in the small terra of the approximate value of z. When ^r=0, z 2 is very small, so that (7 = 1 + 4a/3r, and .............. (49) 2 3r smTr is the second approximation to z. From (49) 1 dz i|r We are now in a position to find x by the relation x= | cot^(d*/cty)cty, ........................ (51) * Supplement au X e Livre, pp. 6064, 1806. 360 OX THE THEORY OF THE CAPILLARY TUBE [399 the constant of integration being determined by the correspondence of x = r, + = ATT. Thus l_-co^| smj^r )' giving when i/r is small (53) where a= log(V2- 1) + x/2 + log 4- 2 = -0'0809, (54) - J/8 = log 2 + i log (v/2 - 1) + $ V2 - 7/12 = - 0-0952. . . .(55) The other equation, derived from the flat part of the surface, is A.i.r/^.V-.fV, M (07) in which xja is regarded as large ; or x a 2irx 3a In equations (53), (57) x and ty are to be identified. On elimination o r a a aB/Sr r x ITTX 3a (58) in which we may put 27r# 2?rr /, r x\ 2?rr r x lo s -- - lo + log 1 ~ - = lo - ^ in which, since a; is nearly equal to r, a(i x)/Sr* may usually be neglected. Also, in view of the smallness of a and #, it is scarcely necessary to retain the denominator 1 + o/2r, so that we may write __ i g = _ 0-0809 + 0-2798 " + ^ log ~ = 0-8381 + 0-2798 a/r + $ log (r/o) ............. (60) The effect of the second approximation is the introduction of the second term on the right of (60). 1915] ON THE THEORY OF THE CAPILLARY TUBE 361 To take an example, let us suppose as before that a/h = 1000, so that log (a/ ho) = 6'908. By successive approximation we find from (60) r/a = 8-869, (61) so that if a = 0'27 cm. (as for water), 2r = 4-79cm (62) The correction to Laplace's formula is here unimportant. The above is the diameter of tube required to render h negligible according to the standard adopted. It may sometimes be convenient to invert the calculation, and deduce the value of h from the diameter of the tube (not much less than 4 cm.) and an approximate value of a. For this purpose we may use (60), or preferably (59), taking x = \r for instance. The calculated value of h would then be used as a correction. The accompanying small Table may be useful for this purpose. rja - logic (h la) Difference h la 6 1-8275 0-0149 7 2-2319 0-4044 0-0059 8 2-6399 0-4080 0-0023 9 3-0508 0-4109 0-00089 10 3-4639 0-4131 0-00034 We have supposed throughout that the liquid surface is symmetrical about the axis, as happens when the section of the containing tube is circular. It may be worth remarking that without any restriction to symmetry the differential equation of the nearly flat parts of a large surface may be taken to be .(63) so that z may be expressed by the series z = AJ (r/a) + (A l cos d + B, sin 9) I, (r/a) (64) r, 6 denoting the usual polar co-ordinates in the horizontal plane.] 400. THE CONE AS A COLLECTOR OF SOUND. [Advisory Committee for Aeronautics, T. 618, 1915.] THE action of a cone in collecting sound coming in the direction of the axis may be investigated theoretically. If the diameter of the mouth be small compared with the wave-length (A,) of the sound, the cone may operate as a resonator, and the effect will vary greatly with the precise relation between X and the length of the cone. On the other hand, the effect will depend very little upon the direction of the sound. t It is probably more useful to consider the opposite extreme, where the diameter of the mouth is a large, or at any rate a moderate, multiple of \, when the effect may be expected to fall off with rapidity as the obliquity of the sound increases. A simple way of regarding the matter is to suppose the sound, incident axially, to be a pulse, e.g. a condensation confined to a narrow stratum bounded by parallel planes. If the angle of the cone be small, the pulse may be sup- posed to enter without much modification and afterwards to be propagated along. As the area diminishes, the condensation within the pulse must be supposed to increase. Finally the pulse would be reflected, and after emer- gence from the mouth would retrace its course. But the argument is not satisfactory, seeing that the condition for a progressive wave, i.e. of a wave propagated without reflection, is different in a cylindrical and in a conical tube. The usual condition in a cylindrical tube, or in plane waves where there is no tube, viz. u = as, where u is the particle velocity, a that of sound, and s the condensation, is replaced in spherical waves by showing that a pulse of condensation alone cannot be propagated without undergoing some reflection. If there is to be no reflection at all, the integral taken over the thickness of the pulse must vanish, and this it cannot do unless the pulse include also a rarefaction. 1915] THE CONE AS A COLLECTOR OF SOUND 363 Apart from what may happen afterwards, there is a preliminary question at the mouth. In the passage from plane to spherical waves there is a phase- disturbance (between the centre and the edge) to be reckoned with, repre- sented by R (1 - cos 6) = ZR6 x 0, where R is the length of the cone, and 6 the semi-vertical angle. That this may be a small fraction of X, itself a small fraction of the diameter of the mouth (2RB), it is evident that 6 must be very small. We may now consider the incidence along the axis (x) of plane waves of simple type. Within the cone, supposed to be complete up to the vertex, the vibrations are stationary, and since no energy passes into the cone, the same must be true of the plane waves just outside at any rate over the greater part of the mouth. The velocity potential just outside may therefore be denoted by ty = cos kat . cos (kx + e), making at the mouth (x = 0) ^r = cos kat . cos e, d^fr/dx = k cos kat . sin e. On the other hand, in the cone . sin kr , ty = A r cos kat, making at the mouth (r = R) . sinkR c . (coskR sinkR Equating the two values at the mouth of ^r and d^Jrfdx or dty/dr, we get . sinkR . . (coskR sin kR} and 1 = kR r JTL When kR is considerable, the second and third terms may be neglected, what- ever may be the particular value of kR, so that for a long enough cone A = kR simply, in which k = 2-Tr/X. Here A is the maximum value of ^r at the vertex of the cone, and the maximum value of -^r in the stationary waves outside the mouth is unity, the particular place where this maximum occurs being variable with the precise value of kR. The increase of ^r, or of the condensation, at the vertex of the cone as compared with that obtained by simple reflection at a wall is represented by the factor kR, which, under our suppositions, is a large number. 364 THE CONE AS A COLLECTOR OF SOUND [400 Although the complete fulfilment of the conditions above laid down is hardly realisable in practice with sounds of moderate pitch, one would certainly expect the use of a cone to be of more advantage than appears from the observations at the Royal Aircraft Factory (Report, T. 577). In the year 1875, I experimented with a zinc cone 10 inches wide at the mouth and about 9 feet long, but I cannot find any record of the observations. My recollection, however, is that I was disappointed with the results. Perhaps I may find opportunity for further trial, when I propose to use wave-lengths of about 3 inches. 401. THE THEORY OF THE HELMHOLTZ RESONATOR. [Proceedings of the Royal Society, A, Vol. xcn. pp. 265275, 1915.] THE ideal form of Helmholtz resonator is a cavernous space, almost enclosed by a thin, immovable wall, in which there is a small perforation establishing a communication between the interior and exterior gas. An approximate theory, based upon the supposition that the perforation is small, and con- sequently that the wave-length of the aerial vibration is great, is due to Helmholtz*, who arrived at definite results for perforations whose outline is circular or elliptic. A simplified, and in some respects generalised, treatment was given in my paper on " Resonance f." In the extreme case of a wave- length sufficiently great, the kinetic energy of the vibration is that of the gas near the mouth as it moves in and out, much as an incompressible fluid might do, and the potential energy is that of the almost uniform compressions and rarefactions of the gas in the interior. The latter is a question merely of the volume S of the cavity and of the quantity of gas which has passed, but the calculation of the kinetic energy presents difficulties which have been only partially overcome. In the case of simple apertures in the thin wall (regarded as plane), only circular and elliptic forms admit of complete treat- ment. The mathematical problem is the same as that of finding the electro- static capacity of a thin conducting plate having the form of the aperture, and supposed to be situated in the open. The project of a stricter treatment of the problem, in the case of a spherical wall and ah aperture of circular outline, has been in my mind more than 40 years, partly with the hope of reaching a closer approximation, and partly because some mathematicians have found the former method unsatis- factory, or, at any rate, difficult to follow. The present paper is on ordinary lines, using the appropriate spherical (Legendre's) functions, much as in a former one, "On the Acoustic Shadow of a Sphere J." * Crelle Journ. Math. Vol. LVII. (1860). t Phil. Trans. Vol. CLXI. p. 77 (1870) ; Scientific Papers, Vol. i. p. 33. Also Theory of Sound, ch. xvi. Phil. Trans. A, Vol. ccin. p. 87 (1904) ; Scientific Papers, Vol. v. p. 149. 366 THE THEORY OF THE HELMHOLTZ RESONATOR [401 The first step is to find the velocity-potential (i/r) due to a normal motion at the surface of the sphere localised at a single point, the normal motion being zero at every other point. This problem must be solved both for the exterior and for the interior of the sphere, but in the end the potential is required only for points lying infinitely near the spherical surface. Then if we assume a normal motion given at every point on the aperture, that is on the portion of the spherical surface not occupied by the walls, we are in a position to calculate -^ upon the two sides of the aperture. If these values are equal at every point of the aperture, it will be a proof that the normal velocity has been rightly assumed, and a solution is arrived at. If the agreement is not sufficiently good there is no question of more than an approximation some other distribution of normal velocities must be tried. In what follows, the preliminary work is the same as in the paper last referred to, and the same notation is employed. The general differential equation satisfied by i/r, and corresponding to a simple vibration, is ^ |t + *2 ......................... da? dy* dz* where k = 2ir/\, and A. denotes the length of plane waves of the same pitch. For brevity we may omit k; it can always be restored on paying attention to " dimensions." The solution in polar co-ordinates applicable to a wave of the nth order in Laplace's series may be written (with omission of the time-factor) *n=S n r X n(r) ............................... (2) The differential equation satisfied by % n is The solution of (3) applicable to a wave diverging outwards is *.-(-)' Putting n = and n = 1, we have e~ ir (1 + t'r) e~ ir Xo(r) = , X*(r)= -- ;r -- It is easy to verify that (4) satisfies (3). For if ^ n satisfies (3), r~^n satisfies the corresponding equation for % n+l . And r~ l e~ ir satisfies (3) when n = 0. From (3) and (4) the following sequence formulas may be verified : (7) (8) 1915] THE THEORY OF THE HELMHOLTZ RESONATOR 367 By means of the last, ^ 2 . X*> e ^ c -> mav be built up in succession from %o and xi- From (2) d+ Jdr = S n (wr-i Xn + r n X n), or with use of (7) n { Xn -, - (n + 1) %n } ................ (9) Thus if U n be the nth component of the normal velocity at the surface of the sphere (r = c) U^C^SnlXn-^-^l+Vx^c}} ................... (10) When n = 0, ) ............................... (11) The introduction of S n from (10), (11) into (2) gives i/r n in terms of U n supposed known. When r is very great in comparison with the wave-length, we get from (4) (12) so that ^n = S n - ............................ (13) We have now to apply these formulae to the particular case where U is sensible over an infinitesimal area do; but vanishes over the remainder of the surface of the sphere. If //, be the cosine of the angle (0) between da and the point at which Z7is expressed, P n (/*) Legendre's function, we have (14) and accordingly for the velocity-potential at the surface of the spJiere, Uda n.n - When n = 0, XH-I (n + 1) % n is to be replaced by c z xi- Equation (15) gives the value of ijr at a point whose angular distance (6) from da- is cos" 1 /u. If XH has the form given by (4), the result applies to the exterior surface of the sphere. We have also to consider the corresponding problem for the interior. The only change required is to replace % n as given in (4) by the form appropriate to the interior. For this purpose we might take simply the imaginary part of (4), but since a constant multiplier has no significance, it suffices to make 368 THE THEORY OF THE HELMHOLTZ RESONATOR [401 With this alteration (15) holds good for the interior, U denoting the localised normal velocity at the surface still measured outwards, since U-d+jdr. We have now to introduce approximate values of x-i( c ) * Xn(c) in (15), having regard to the assumed smallness of c, or rather kc. For this purpose we expand the sine and cosine of c* : cosc_ 1 c_ c* _ c 4 ~c~~c 172 4~! 61 _ 1 A / cos c \ _ 1 1 _ 3c 5c? _ 7c ~c dc \c~J~ ? 1.2c 4! 6! "8! .1 dycosc 3 !__ J__5^3 1 c 7.5^c_ c dc7 c c 8 1.2C 3 c.4! 6! 8! and so on ; sin c _ c 2 c 4 _ c 6 1 d sin c 2 4c 2 Gc 4 r dc c 2.3 5! + 7! '"' (_ 1 _d y sinc _ 4 - 2 _ 6 . 4 . c 2 8 . 6.c 4 " " V cdcj c ' 5! 7! " 9! and so on. Thus for the outside 1.3.5...(2w-l) For general values of n, we may take Xn-i-X=2^n ( 18 > For n = 1 "jfiT" 1 ""*- '"" For n = 2 2&-^+ terms in c 4 (20) X* * 1917. In the expansions for the derivative of cos c/c terms (now inserted) were accidentally omitted, as has been pointed out by Mr F. P. White (Proc. Roy. Soc. Vol. xcn. p. 549). Equation (17) as originally given was accordingly erroneous. Corresponding corrections have been intro- duced in (19), (23), (24), (36), (38) which however do not affect the approximation employed in (39). Mr White's main object was to carry the approximation further than is attained in (57) and (60). 1915] THE THEORY OF THE HELMHOLTZ RESONATOR 369 Thus in general by (18) -^+J_ _ 2+ _l C2n + l)c- Xn-i/Xn-n-l +n + l (n + l) 2 (2n-l)'' while for n = 1 __?__=_ 2 + | -c 2 + terras in c 3 , ............... (22) in accordance with (21). When n = _^ =- 1 + c 2 + ic + terms in c 3 . .................. (23) /& Using these values in (15), we see that, so far as c 2 inclusive, 2 (outside) = (- 1 + c 2 + ic) P 4 This suffices for n = 1 and onwards. When n = Xo 3 j c 2 c 4 j = ~ ~ Accordingly, so far as c 2 inclusive, 2 (inside) = 2 {P 0*) + P, 00 + . . . + P 0*)! (24) In like manner for the form of % n appropriate to the inside Y ..(c)= Jl - I ...(25) x w 1 .3.5...(2w + l) ( 2(2/1+3)]' so that in general P () (29) 5 175 T 2 n W> R. vi. 24 370 THE THEORY OF THE HELMHOLTZ RESONATOR [401 The first two series of P's on the right of (24) and (29) become divergent when /x = 1, or 6 = 0. To evaluate them we have sothat 1 + P! + P 2 + ... = ^ onr^ = o^T rz ( 31 ) Again, by integration of (30), = log [o cos 6 + V(l - 2a cos 6 + a 2 }] - log [1 cos 6] *, sothat 1 +P,+ P 2 + ... = log(l+sin0)-logsin0 .......... (32) In much the same way we may sum the third series 2,n~ l P n . We have , a -ati + a a , - f - . 2 } J a We denote the right-hand member of this equation by/ and differentiate it with respect to yu,. Thus dl ' a cfo a A d/z o (o- or when a = 1 ~P On integration / = log tan (?r - #) - log sin 6 + C ................ (34) The constant is to be found by putting /x = 0, 6 = \ir. In this case Thus C = log - - log tan = log 2, * If we integrate this equation again with respect to a between the limits and 1, we find O + A + + (TTRT2j = 1 ~ 2 8in *' + 2 8in ' [log (1 + 8ln i<?) " log 8in in When is small, the more important part is 1915J THE THEORY OF THE HELMHOLTZ RESONATOR 371 and accordingly A + P 2 + JP 3 + ... = log tan \ (IT - 0) - log ( sin 6) ....... (35) For the values of 2 in (15) we now have with restoration of k 2 (outside) = = r^ ~ lg sin + log (1 + sin $0) S1H " v 2 (inside) = -^ r^ - log (i sin 0) + log tan J (IT - 6) These equations give the value of T/T at any point of the sphere, either inside or outside, due to a normal velocity at a single point, so far as k?c~ inclusive. The inside value is dominated by the term 3/A^c 2 , except when is small. As to the sums in & 2 c 2 not evaluated, we may remark that they cannot exceed the values assumed when = and P n (p) = 1. Approximate calculation of the limiting values is easy. Thus = - 0-79040 + 1-64493 - 1-20206 + 1*62348 = 1-2759 *. In like manner 2 3 /o +1 ox = - 0-9485 + I (n- 2 - n~ 3 + f n~*} = M 178 f. i n*(2n -+ 6) i * Chrystal's Algebra, Part n. p. 343. t 1917. Mr White has shown that the accurate value of the first sum is and that of the second sum so that for the two taken together as in (38), we have The coefficient of fcV 2 in (38) is then Further in this equation 6 f 0\ f 0\ log cos -^ - log I 1 + sin - I = - 2 log I 1 + sm x ) . 9 & \ &J \ if 24-2 ~-l + 2-39292 = 372 THE THEORY OF THE HELMHOLTZ RESONATOR [401 Our special purpose is concerned with the difference in the values of ^ on the two sides of the surface r = c, and thus only with the difference of S's. We have 2 (inside) - 2 (outside) = JL _ log cos ? + log l 39., -7r^---*fo (38) In the application we have to deal only with small values of 6 and we shall omit A^c 2 , so that we take W -x ( rt ).-__-a,j ......... (39) it will indeed appear later that we do not need even the term in 6, since it is of the order k*c?. In pursuance of our plan we have now to assume a form for U over the circular aperture and examine how far it leads to agreement in the values of ^r on the inside and on the outside. For this purpose we avail ourselves of in- formation derived from the first approxi- mation. If C, fig. 1, be the centre and CA the angular radius of the spherical segment constituting the aperture, P any other point on it, we assume that U at P is proportional to {CA 3 - (7P 2 }-*, and we require to examine the con- sequences at another arbitrary point 0. Writing CA = a, CO = b, PO = 6, POA = <j>, we have from the spherical triangle cos CP = cos b cos 6 + sin 6 sin 6 cos <f>, or when we neglect higher powers than the cube of the small angles, Thus CA-- CP* = a 2 - 6 2 - 6* - 260 cos <f> and we wish to make (40) a 2 - b 2 sin 2 - (6 + b cos </>)", . . .(41) f j-sin0d0<fr[2(in) - 2 (out)] _ JJ V{a 8 -6 a -^-26^o7 as far as possible for all values of b, the integration covering the whole area of aperture. We may write 6 for sin B*, since we are content to neglect terms * [Except as regards the product of sin 6 and the first term on the right of (39), since tin- term in P is in point of fact retained in.the calculation. W. F. S.] 1915] THE THEORY OF THE HELMHOLTZ RESONATOR 373 of order 6* in comparison with the principal term. Reference to (39) shows that as regards the numerator of the integrand we have to deal with terms in 0, 0\ and 0-. For the principal term we have iff a ^0dj XT f d6 [d (0 + b cos d>) . + bcos<j) Now 771 1 = ^ = sm " Vf } For a given </> the lower limit of is and the upper limit 6^ is such as to make a 2 = 6 2 + 0* + 2b0, cos 0, or 6 l + b cos < = V( 2 - & 2 sin. 2 </>) (44) m, [ 6l dO TT . 6cos<f> Thus r = - - sm" 1 -77 .. T . , v (45) 2 - 22 .'V| } 2 When this is integrated with respect to <f>, the second part disappears, and we are left with 7r 2 simply, so that the principal term (43) is 47r 2 . That this should turn out independent of b, that is the same at all points of the aperture, is only what was to be expected from the known theory respecting the motion of an incompressible fluid. The term in 0, corresponding to the constant part of 2 (in)- 2 (out), is represented by .(46) //: Here dO d(f> is merely the polar element of area, and the integral is, of course, independent of 6. To find its value we may take the centre G as the pole of 0. We get at once . so that this part of (42) is < 48) For the third part (in 2 ), we write 0* = - (a 2 - 6 2 - 260 cos <j> - 0-)-<2b cos (0 + b cos 0) + a j - 6 2 + 26 2 cos 2 </>, giving rise to three integrals in 0, of which the first is -fd0 V{ 2 - ^ - 2b0 cos <j> - 0*} = -i(0 + bcos<f>) V( 2 - & sin 2 $ - (0 + b cos </>) 8 } a 2 - 6 2 sin 2 <f> . + b cos < - *""->*-.&. ...................... (49) The second integral is - 26 cos ( ^ 374 THE THEORY OF THE HELMHOLTZ RESONATOR [401 and the third is, as for the principal term, Thus altogether, when the three integrals are taken between the limits and lt we get - f b cos V(* - & a ) + [i a + & (2 cos 2 <f> + sin 2 0-1)] [7T . _j 6 COS 2~ -/(rf^Psirf^Xj ' and finally after integration with respect to <j> i7T 2 (a 2 -H&') ............................... (52) Thus altogether the integral on the left of (42) becomes + ) ..... ........ (53)* In consequence of the occurrence of b 2 , this expression cannot be made to vanish at all points of the aperture, a sign that the assumed form of U is imperfect. If, however, we neglect the last term, arising from - B in 2 (in) 2 (out), our expression vanishes provided showing that a is of the order k*c", so that this equation gives the relation between a and kc to a sufficient approximation. Helmholtz's solution corre- sponds to the neglect of the second and third terms on the left of (54), making 3 2?r 2-n-c Ev-T-TT' ........................... (o5) . where R denotes the linear radius of the circular aperture. If we introduce (56) S denoting the capacity of the sphere, the known approximate value. The third term on the left of (54) represents the decay of the vibration due to the propagation of energy away from the resonator. Omitting this for the moment, we have as the corrected value of \, X = 7T Let us now consider the term representing decay of the vibrations. The time factor, hitherto omitted, is e* r , or if we take A; = ^+t/fc 2 , e~ k ^ e** vt . If t = r, the period, A,FT-2w, and e -V>= e -*r* 8 /*,. This is the factor by which the amplitude of vibration is reduced in one period. Now from (55) i T a "Sir 2 " * [For read , and three lines below read " arising from - Iff 1 in sin 6 [2 (in) - 2 (out)] " : tee footnote on p. 372. W. F. 8.] 1915] THE THEORY OF THE HELMHOLTZ RESONATOR 375 so that (54) becomes 3 //372\ 27rc -jr. (58) < 60 > This gives the reduction of amplitude after one vibration. The decay is least when R is small relatively to c, although it is then estimated for a longer time. The value found in (60) differs a little from that given in Theory of Sound, 311, where the aperture is supposed to be surrounded by an infinite flange, the effect of which is to favour the propagation of energy away from the resonator. So far we have supposed the boundary of the aperture to be circular. A comparison with the corresponding process in Theory of Sound, 306 (after Helmholtz), shows that to the degree of approximation here attained the results may be extended to an elliptic aperture provided we replace R by where R l denotes the semi-axis major of the ellipse, e the eccentricity, and F the symbol of the complete elliptic function of the first order. It is there further shown that for any form of aperture not too elongated, the truth is approximately represented if we take \/(cr/7r) instead of the radius R of the circle, where <r denotes the area of aperture. It would be of interest to ascertain the electric capacity of a disc of nearly circular outline to the next approximation involving the square of 8R, the deviation of the radius in direction <w from the mean value. If 8R = a n cos n<u, cfj, would not appear, and the effect of 2 is known from the solution for the ellipse. For other values of n further investigation is required. In the case of the ellipse elongated apertures are not excluded, provided of course that the longer diameter is small enough in comparison with the diameter of the sphere. When e is nearly equal to unity, (62) R 2 being the semi-axis minor. The pitch of the resonator is now compara- tively independent of the small diameter of the ellipse, the large diameter being given. 402. ON THE PROPAGATION OF SOUND IN NARROW TUBES OF VARIABLE SECTION. [Philosophical Magazine, Vol. xxxi. pp. 8996, 1916.] UNDER this head there are two opposite extreme cases fairly. amenable to analytical treatment, (i) when the changes of section are so slow that but little alteration occurs within a wave-length of the sound propagated and (ii) when any change that may occur is complete within a distance small in comparison with a wave-length. In the first case we suppose the tube to be of revolution. A very similar analysis would apply to the corresponding problem in two dimensions, but this is of less interest. If the velocity-potential < of the simple sound be proportional to e ikat , the equation governing <f> is where # is measured along the axis of symmetry and r perpendicular to it. Since there are no sources of sound along the axis, the appropriate solution (2) in which F, a function of x only, is the value of < when r = 0. At the wall of the tube r = y, a known function of x ; and the boundary condition, that the motion shall there be tangential, is expressed by in which * Compare Proc. Lond. Math. Soc. Vol. vn. p. 70 (1876); Scientific Papers, Vol. i. p. 275. 1916] PROPAGATION OF SOUND IN NARROW TUBES OF VARIABLE SECTION 377 Using these in (3), we obtain an equation which may be put into the form As a first approximation we may neglect all the terms on the right of (6), so that the solution is t/ where A and B are constants. To the same approximation, . (8) y x x For a second approximation we retain on the right of (6) all terms of the order fri/fda?, or (dy/dx)*. By means of (8) we find sufficiently for our purpose ._ dx 2 J dx y dx dx 2 ' ^1+^^=0 (*- + ix- J dx \dx 2 Our equation thus becomes in which on the right the first approximation (7) suffices. Thus (10) where F = (11) In (10) the lower limit of the integrals is undetermined; if we introduce arbitrary constants, we may take the integration from oc to x. In order to attack a more definite problem, let us suppose that d^y/dx 2 , and therefore Y, vanishes everywhere except over the finite range from x = to x = b, b being positive. When x is negative the integrals disappear, only the arbitrary constants remaining ; and when x is positive the integrals may 378 ON THE PROPAGATION OF SOUND IN [402 be taken from to x. As regards the values of the constants of integration (10) may be supposed to identify itself with (7) on the negative side. Thus - 7 (A ...(12) The integrals disappear when a; is negative, and when x exceeds 6 they assume constant values. Let us now further suppose that when x exceeds b there is no negative wave, i.e. no wave travelling in the negative direction. The negative wave on the negative side may then be regarded as the reflexion of the there travelling positive wave. The condition is giving the reflected wave (B) in terms of the incident wave (A). There is no reflexion if [6 Y<r**da; = 0; (14) Jo and then the transmitted wave (x > b) is given by Even when there is reflexion, it is at most of the second order of small- ness, since Y is of that order. For the transmitted wave our equations give (x > b) Ar** I 1 1+ZTT, ; (16) but if we stop at the second order of smallness the last part is to be omitted, and (16) reduces to (15). It appears that to this order of approximation the intensity of the transmitted sound is equal to that of the incident sound, at least if the tube recovers its original diameter. If the final value of y differs from the initial value, the intensity is changed so as to secure an equal pro- pagation of energy. The effect of Fin (15) is upon the phase of the transmitted wave. It appears, rather unexpectedly, that there is a linear acceleration amounting to 1916] NARROW TUBES OF VARIABLE SECTION 379 or, since the ends of the disturbed region at and b are cylindrical, -**"* ..................... <>> from which the term in k^y* may be dropped. That the reflected wave should be very small when the changes are sufficiently gradual is what might have been expected. We may take (13) in the form (19) vyx 2 As an example let us suppose that from x = to x = b y = y + r) (1 - cos mx), ........................ (20) where y is the constant value of y outside the region of disturbance, and m = 27r/6. If we suppose further that 77 is small, we may remove 1/t/ from under the sign of integration, so that 2^]. ...(21) Independently of the last factor (which may vanish in certain cases) B is very small in virtue of the factors m?/k 2 and ij/y . In the second problem proposed we consider the passage of waves pro- ceeding in the positive direction through a tube (not necessarily of revolution) of uniform section o-j and impinging on a region of irregularity, whose length is small compared with the wave-length (X). Beyond this region the tube again becomes regular of section <7 2 (fig. 1). It is convenient to imagine the X 1 Fig. 1. axes of the initial and final portions to be coincident, but our principal results will remain valid even when the irregularity includes a bend. \\\- seek to determine the transmitted and reflected waves as proportional to the given incident wave. The velocity-potentials of the incident and reflected waves on the left of the irregularity and of the transmitted wave on the right are represented respectively by - (22) 380 ON THE PROPAGATION OF SOUND IN [402 so that at x 1 and # 2 we have <, = A e~ ik *> + Be ik *> , <f>,= Ce- ik *>, ............... (23) dfa/dx = tjfc (- A e~ ik *< + Be* f >), dfafdx = - ikCe-* 1 *. . . .(24) When \ is sufficiently great we may ignore altogether the space between x l and a-j, that is we may suppose that the pressures are the same at these two places and that the total flow is also the same, as if the fluid were incompressible. As there is now no need to distinguish between x l and x, we may as well suppose both to be zero. The condition fa = </> 2 gives A+B = C, ................................. (25) and the condition a-^fa/dx = a-^fa/dx gives -<T 2 C. ........................ (26) Thus = <T L -*, = -^. ...(27) A a l + cr, A (TI +<r 2 These are Poisson's formulae*. If o-j and a- 2 are equal, we have of course 5 = 0, C=A. Our task is now to proceed to a closer approximation, still supposing that the region of irregularity is small. For this purpose both of the conditions just now employed need cor- rection. Since the volume V of the irregular region is to be regarded as sensible and the fluid is really susceptible of condensation (s), we have K * *_ *b dt dx l " dx 2 and since in general s = -a~^d<f)fdt, we may take ds rf 2 <f>! d-fa -j- = - a~ 2 -^- or a~ 2 -jg , dt dt 3 dt 2 the distinction being negligible in this approximation in virtue of the smallness of V. Thus dfa dfa Vffifr ., ^-^"o^^ 1 * 2 ................ (28) In like manner, ^assimilating the flow to that of an incompressible fluid, we have for the second condition (29) where R may be defined in electrical language as the resistance between x l and x 2 , when the material supposed to be bounded by non-conducting walls coincident with the walls of the tube is of unit specific resistance. * Compare Theory of Sound, 264. 1916] NARROW TUBES OF VARIABLE SECTION 381 In substituting the values of < and dfyjdx from (23), (24) it will shorten our expressions if for the time we merge the exponentials in the constants, writing A' = Ae- ikx <, B' = Be**>, C' = Ce~**< (30) Thus <r 1 (-A' + B') + <r z C' = -ikVC', (31) A' + B'-C' = ik<r z RC' (32) We may check these equations by applying them to the case where there is really no break in the regularity of the tube, so that Then (31), (32) give B' = 0, or 5 = 0, and _ = pikfrixj A' 1 + tfr to -,)"" with sufficient approximation. Thus C' e ^ = A ' e ikx ly or c=A. The undisturbed propagation of the waves is thus verified. In general, ID' i '7 / L> TT\ > &l CT 2 + l/C \(T l (T^ It V ) A'I ~ -ii (Ti & 2T m v A' <r, + o- 2 + ijfc (o- l0 - 2J R + F) * ' When o-j <7 2 is finite, the effect of the new terms is only upon the phases of the reflected and transmitted waves. In order to investigate changes of intensity we should need to consider terms of still higher order. When o-j = cr 2 , we have _ ^ ( a *R + 7)1 = A > e -wx+ v ')/-2* ) (i =Ae ik(x 2 -x 1 -l*R-VI2<,) > (35) making, as before, C = A, if there be no interruption. Also, when <ii = a.i absolutely, A' = ~^~to ' (36) indicating a change of phase of 90, and an intensity referred to that of the incident waves equal to 382 PROPAGATION OF SOUND IN NARROW TUBES OF VARIABLE SECTION [402 As an example let us take the case of a tube of revolution for which y, being equal to y over the regular part, becomes y + Sy between x l and # 2 . We have , Also 1 2* + -<^ the terms of the first order in Sy disappearing. Thus in the exponent of (35) , ...(39) of which the right-hand member, taken with the positive sign, expresses the retardation of the transmitted wave due to the departure from regularity. * Theory of Sound, 308. 403. ON THE ELECTRICAL CAPACITY OF APPROXIMATE SPHERES AND CYLINDERS. [Philosophical Magazine, Vol. xxxi. pp. 177 186, March 1916.] MANY years ago I had occasion to calculate these capacities* so far as to include the squares of small quantities, but only the results were recorded. Recently, in endeavouring to extend them, I had a little difficulty in retracing the steps, especially in the case of the cylinder. The present communi- cation gives the argument from the beginning. It may be well to remark at the outset that there is an important difference between the two cases. The capacity of a sphere situated in the open is finite, being equal to the radius. But when we come to the cylinder, supposed to be entirely isolated, we have to recognize that the capacity reckoned per unit length is infinitely small. If a be the radius of the cylinder and b that of a coaxal enveloping case at potential zero, the capacity of a length I isf _Jl log (6/a)' which diminishes without limit as b is increased. For clearness it may be well to retain the enveloping case in the first instance. In the intervening space we may take for the potential in terms of the usual polar coordinates <f> = H log (r/b) + H,r- 1 cos (0 - ei ) + A> cos (6 - e/) + . . . + H n r~ n cos (n6 e,,) + K n r n cos (n6 - e n '). Since < = when r = b, e n ' = e n , K n = -H n b-\ and </> = # log (r/6) + ^(i -^cos(0-e l )+H 2 (-fycos(W-e*)+.... (1) * "On the Equilibrium of Liquid Conducting Masses charged with Electricity," Phil. Mag. Vol. xiv. p. 184 (1882) ; Scientific Papers, Vol. n. p. 130. t Maxwell's Electricity, 126. 384 ON THE ELECTRICAL CAPACITY OF [403 At this stage we may suppose b infinite in connexion with H^ H a , &c., so that the positive powers of r disappear. For brevity we write cos (nd e n ) = F n , and we replace r" 1 by u. Thus H^F a _ + .................. (2) We have now to make </> = fa at the surface of the approximate cylinder, where ^ is constant and u = u + Bu = (1 + C^ + (7 2 (r a +...) Herein G n = cos (nd e n ), and the (7s are small constants. So far as has been proved, e n might differ from e n , but the approximate identity may be anticipated, and at any rate we may assume for trial that it exists and consider G n to be the same as F n , making u = u + 8u = u (l + C 1 F l + C 2 F 2 + ...) ................ (3) On the cylinder we have and in this Su/u = C 1 F J + C,Ft + C 3 F 3 + ...................... (5) The electric charge Q, reckoned per unit length of the cylinder, is readily found from (2). We have, integrating round an enveloping cylinder of radius r, . and Q/<f>! is the capacity. We now introduce the value of 8w/w from (5) into (4) and make successive approximations. The value of H n is found by multiplication of (4) by F n , where n = 1 , 2, 3, &c., and integration with respect to 6 between and 2?r, when products such as F t Fy t F Z F 3 , &c., disappear. For the first step, where O 2 is neglected, we have M, ..................... (7) or H n u.-H.C n ....................... ' .......... (8) Direct integration of (4) gives also * = - H, log (&) + /^ ^ {H,v.F, + ZHvfF, .} + VI.:' .......... (9) 1916] APPROXIMATE SPHERES AND CYLINDERS 385 cubes of G being neglected at this stage. On introduction of the value of H n from (8) and of Su from (5), & = -H \og(uJ>) + lH 9 {W l *+5Cf + 1Cf + ...} ......... (10) Thus <MQ = 21o g (<u &)-{3C 1 2 + 5C' 2 2 + 7C'3 2 =...} ........... (11) In the application to an electrified liquid considered in my former paper, it must be remembered that U Q is not constant during the deformation. If the liquid is incompressible, it is the volume, or in the present case the sectional area (cr), which remains constant. Now so that if a denote the radius of the circle whose area is <r, iC = a~ 2 {l +f(C?+<7 2 2 + a, 2 +...} ................ (12) Accordingly, log w 2 = - 2 loga + f (Cf + <7 2 2 + (7 S 2 + ...), and (11) becomes hlQ = 2\og(bla)-Cf-2C t *-...-(p-l)a p ; ......... (13) the term in d disappearing, as was to be expected. The potential energy of the charge is |^Q. If the change of potential energy due to the deformation be called P', we have P' = -l<f{Cf + 2C t '+...+(p-I)Cf} t ............ (U) in agreement with my former results. There are so few forms of surface for which the electric capacity can be calculated that it seems worth while to pursue the approximation beyond that attained in (11), supposing, however, that all the e's vanish, everything being symmetrical about the line = 0. Thus from (4), as an extension of (7) with inclusion of C' 2 , F n (CM + C 2 F 2 +...) (H^F, + 2H.UJF, + 3H 3 u *F 3 4- . . . ) F n (C 1 F 1 + C 2 F 2 + C*F s +...?> ........................... (15) or with use of (8) = G n - F n (C 1 F 1 + C,F 2 ...), ............ (16) R. VI. 25 386 ON THE ELECTRICAL CAPACITY OF [403 by which H n is determined by means of definite integrals of the form i 2 ' F n F p F q dS ............................... (17) .'0 n, p, q being positive integers. It will be convenient to denote the integral on the right of (16) by /, / being of the second order in the (7s. Again, by direct integration of (4) with retention of C 3 , 2 + S F S + . . . )' {H 2 u *F 2 In the last integral we may substitute the first approximate value of H p from (8). Thus in extension of (11) ^ (C.F, + 0,F, + C 3 F 3 + . . .)' {C 2 F, + 3C 3 F 3 + . . . + $P(P-1)0 P F P } ............. (18) The additional integrals required in (18) are of the same form (17) as those needed for /. As regards the integral (17), it may be written rddcosndcospdcosqO. . Now four times the latter integral is equal to the sum of integrals of cosines of (n - p q) 8, (n-p + q) 6, (n+p q) 6, and (n + p + q) 6, of which the last vanishes in all cases. We infer that (1?) vanishes unless one of the three quantities n, p, q is equal to \he sum of the other two. In the excepted cases (17) = *7T .................................. (19) If p and q arc equal, (17) vanishes unless n = 2p; also whenever n, p, q are all odd. We may consider especially the case in which only C p occurs, so that W = tt (1 +(^008^0) ......................... (20) In (16) / = (2p + 1) C p ' F n F p \ 1916] APPROXIMATE SPHERES AND CYLINDERS 387 so that / vanishes unless n = 2p. But I v disappears in (18), presenting itself only in association with C^,, which we are supposing not to occur. Also the last integral in (18) makes no contribution, reducing to which vanishes. Thus the same as in the former approximation, as indeed might have been antici- pated, since a change in the sign of C p amounts only to a shift in the direction from which 6 is measured. The*corresponding problem for the approximate sphere, to which we now proceed, is simpler in some respects, though not in others. In the general case M, or r~ l , is a function of the two angular polar coordinates 6, &>, and the expansion of Bu is in Laplace's functions. When there is symmetry about the axis, a> disappears and the expansion involves merely the Legendre functions P n (/u), in which /* = cos 0. Then u = U Q + Bu = u {l + C l P l Oi) + C,P,00 + ...}, (22) where C lt (7 2 ,... are to be regarded as small. We will assume Bu to be of this form, though the restriction to symmetry makes no practical difference in the solution so far as the second order of small quantities. For the form of the potential (<) outside the surface, we have <}> = H u + H l u*P 1 (ri + H,u 3 P 2 (ri + ...; (23) and on the surface fa = ff ollo + Bu {H + (8u)* {H.P, +3w # 2 P., + ... + $p(p+l)uf-*H p P p }, ...(24) in which we are to substitute the values of S, (Buy from (22). In this equation fa is constant, and H 1} H^, ... are small in comparison with H . The procedure corresponds closely with that already adopted for the cylinder. We multiply (24) by P n , where n is a positive integer, and inte- grate with respect to fj, over angular space, i.e, between - 1 and + 1. Thus, omitting the terms of the second order, we get ufH n = -H.C n (25) as a first approximation to the value of H n . 252 388 ON THE ELECTRICAL CAPACITY OF [403 Direct integration of (24) gives fc [d/- JET.M. [<*/* + a,, [{(7,^ + C 8 P a + ...} {2tt.fr, = flX | dp + M, f {2ti.fr, (W + 3uSH s C 9 Pf or on substitution for fT n from (25) ....... (26) inasmuchas J + ' P p a (/*) dp = g 2 + x ......................... (27) As appears from (23), H is identical with the electric charge upon the sphere, which we may denote by Q, and Q/fa is the electrostatic capacity, so that to this order of approximation Capacity = t,.-' jl + f + . . . + j\ C,j . . . .(28) Here, again, we must remember that w -1 differs from the radius of the true sphere whose volume is equal to that of the approximate sphere under consideration. If that radius be called a 2C? 2CV 2<7 P 2 3 -' ......... and Capacity = ajl+y + ... 4- ~~ \CA , (30) in which (J l does not appear. The potential energy of the charge is ^Q 2 -=- Capacity. Reckoned from the initial configuration (C = 0), it is P' ^ 2 1 2 *j. a. P.Z-L.r'sl /QI\ J ~ o^: 1~E" "" " + o^ , i P ( (9 L ) It has already been remarked that to this order of approximation the restriction to symmetry makes little difference. If we take &u/u 9 = F l + F t +...+F p (32) where the Fs are Laplace's functions, y- Fp" dfidco corresponds to p . This substitution suffices to generalize (30), (31), and the result is in harmony with that formerly given. The expression for the capacity (30) may be tested on the case of the planetary ellipsoid of revolution for which the solution is known*. Here * Maxwell's Electricity, 151. 19] 6] APPROXIMATE SPHERES AND CYLINDERS 389 C 2 = Je 2 , e being the eccentricity. It must be remembered that a in (30) is not the semi-axis major, but the spherical radius of equal volume. In terms of the semi-axis major (a), the accurate value of the capacity is ae/sin" 1 e. We may now proceed to include the terms of the next order in C. The extension of (25) is u n H n jH Q = -C n + t (2n + 1) J* 1 dp P n {CiA + . . . + G P P P ] (2C 1 P 1 + ... + (^ + l)(7 3 P 9 } ) (33) where in the small term the approximate value of H n from (25) has been substituted. We set dn P n [C l P l + . . . + C p P p ] [20^ + ... + (q + l)C q P q } = J n ,.. .(34) where J n is of order C 2 and depends upon definite integrals of the form J* 1 PnPpP.dp, (35) n, p, q being positive integers. In like manner the extension of (26) is V + i {20^ + 3^/2 + 4(73/3+ -..} p P p }. (36) Here, again, the definite integrals required are of the form (35). These definite integrals have been evaluated by Ferrers* and Adams f. In Adams' notation n -f p + q = 2s, and ... 1.3.5 ...(2r?,-l) where 4n = In order that the integral may be finite, no one of the quantities n, p, q must be greater than the sum of the other two, and n+ p + q must be an even integer. The condition in order that the integral may be finite is less severe than we found before in the two dimensional problem, and this, in general, entails a greater complication. But the case of a single term in 8u, say C P P P (/i), remains simple. In (36) J n occurs only when multiplied by C n , so that only J p appears, and (39) * Spherical Harmonics, London, 1877, p. 156. t Proc. Roy. Soc. Vol. xxvn. p. 63 (1878). [Following Adams, A (o) must be taken as equal to unity. W. F. S.] 390 ON THE ELECTRICAL CAPACITY OF [403 Thus (36) becomes When p is odd, the integral vanishes, and we fall back upon the former result; when p is even, by (37), (38), For example, if p = 2, and Again, if two terms with coefficients C p> C q occur in SM, we have to deal only with J p , J q . The integrals to be evaluated are limited to Ifp be odd, the first and third of these vanish, and if q be odd the second and fourth. If p and q are both odd, the terms, of the third order in G disappear altogether. As appears at once from (34), (36), the last statement may be generalized. However numerous the components may be, if only odd suffixes occur, the terms of the third order disappear and (36) reduces to (26). [1917. Cow/. Cisotti, R. 1st. Lombardo Rend. Vol. XLIX. May, 1916. In his Kelvin lecture (Journ. Inst. El. Eng. Vol. xxxv. Dec. 1916), Dr A. Russell quotes K. Aichi as pointing out that the capacity of an ellipsoidal conductor is given very approximately by (8/4nr) , where S is the surface of the ellipsoid, and he further shows that this expression gives approximate values for the capacity in a variety of other calculable cases. As applied to an ellipsoid of revolution, his equation (6) gives Capacity - ^ . , (43) where e is the eccentricity of the generating ellipse, the plus sign relating to the prolatum and the minus to the oblatum. It may thus be of interest to obtain the formula by which u in (28) is expressed in terms of S rather than, as in (29), (30), by the volume of the conductor. For a reason which will presently appear it is desirable to include the cube of the particular coefficient C* 2 . 1916] APPROXIMATE SPHERES AND CYLINDERS 391 In terms of u, equal to l/r, the general formula for 8 is By ( 22 ) -> h = sin 2 0( 1 P; + aP 2 ' + ...) 2 (l-2C' 2 P 2 ), w Vaay and hence with regard to well-known properties of Legendre's functions we find - <V P <2p {(1 - /*) P 2 P 2 "+ 2P 2 ')J. By (41) and by use of the particular form of P 2 we readily find -^)P 2 P/ 2 = 12/35. -i Accordingly _ fW . ..,46) If we omit C 2 3 and combine (45) with (28), we get the terms in d and C t disappearing. When the cubes of the C's are neg- lected, the capacity is less than \/(S/4nr), the radius of the sphere of equal surface. If the surface be symmetrical with respect to the equatorial plane, as in the case of ellipsoids, the C's of odd order do not occur, so that the earliest in (46) is G 4 . For a prolatum of minor axis 26 arid eccentricity e, whence u = u (1 - e 2 P 2 4- terms in e 4 ), so that C 2 = J e 2 , C t is of order e 4 , &c. In like manner for an oblatum C 2 = + \e*, C 4 is of order e 4 , &c. In both cases the corrections according to (46) would be of order e 8 , but we obtain a term in e 6 when we retain (7 2 8 . 392 CAPACITY OF APPROXIMATE SPHERES AND CYLINDERS [403 By (40), (41) we obtain as an extension of (28), Capacity = ur l {l +W + W + ... + -j^CS -ftC^ ,... (47) and by comparison with (43) g In the case of the ellipsoid C. 2 = + ^ , and as far as e 6 inclusive we get 8 as given by Russell in (43).] 404. ON LEGENDRE'S FUNCTION P n (0), WHEN n IS GREAT AND HAS ANY VALUE*. [Proceedings of the Royal Society, A, Vol. xcn. pp. 433437, 1916.] As is well known, an approximate formula for Legendre's function P n (d), when n is very large, was given by Laplace. The subject has been- treated with great generality by Hobsonf, who has developed the complete series proceeding by descending powers of n, not only for P n but also for the "associated functions." The generality aimed at by Hobson requires the use of advanced mathematical methods. I have thought that a simpler derivation, sufficient for practical purposes and more within the reach of physicists with a smaller mathematical equipment, may be useful. It had, indeed, been worked out independently. The series, of which Laplace's expression constitutes the first term, is arithmetically useful only when n0 is at least moderately large. On the other hand, when 6 is small, P n tends to identify itself with the Bessel's function J (n0), as was first remarked by Mehler. A further development of this approximation is here proposed. Finally, a comparison of the results of the two methods of approximation with the numbers calculated by A. Lodge for n = 20 j is exhibited. The differential equation satisfied by Legendre's function P n is If we assume u v (sin 6) ~ , and write m for n + |, we have * [1917. It would be more correct to say P n (cos 0), where cos 9 lies between 1.] t " On a Type of Spherical Harmonics of Unrestricted Degree, Order, and Argument," Phil. Trans. A, Vol. CLXXXVII. (1896). J " On the Acoustic Shadow of a Sphere," Phil. Trans. A, Vol. ccin. (1904) ; Scientific Papers, Vol. v. p. 163. 394 ON LEGENDRE'S FUNCTION P n (0), [404 If we take out a further factor, e?**, writing w = vsin~*0= we im$ am~* 0, (3) of which ultimately only the real part is to be retained, we find . dw w We next change the independent variable to z, equal to cot 6, thus obtaining < 5 > From this equation we can approximate to the desired solution, treating m as a large quantity and supposing that w = 1 when z = 0, or Q = \ir. The second approximation gives dw i iz -j- = - 5 , whence w = 1 - ^ . dz 8m 8m After two more steps we find . / 1 9 \ -"fe~128^) Thus in realized form a solution of (1) is 9cot0 75 cot 3 6) . and this may be identified with P n provided that the constants C, 7, can be so chosen that u and du/d0 have the correct values when 6 = ^ir. For this value of we must have P n (^7r) = Ccos(|m7r + 7 ), ........................ (8) (9) We may express (dP n /d0). by means of P n+l (^TT). In general = (n + 1} (C S e ' Pn ~ PM+I) ' so that when 0= \ir, dP n /d0 = -dP n /dcos0 = (n + l)P n+ ,. ............... (10) When n is even,(dP n ld0), vanishes, and, C being, still undetermined, we may take to satisfy (9), 7 = \ir ; and then from (8) 1916] WHEN n IS GREAT AND 6 HAS ANF VALUE 395 so that 1.3.5...(n-l) ~2.4.6... rT~ Here n is even, say 2r, and it is supposed to be great. Thus . -l) 2r (2r) ! 2 2 .4 2 .6 2 ............ (2r) 2 2 2r (r!) 2 ' and when r is great, r ! = _ 128r 2 1024r When n is even and with this value of C, 'When w, is odd, the same value of 7, viz. \TT, secures the required evanescence in (8), and we may conjecture that the same value of C will also serve. Laplace* indeed was content to determine 7 from the case of n odd and G from the case of n even. I suppose it was this procedure that Todhunterf regarded as unsatisfactory. At any rate there is no difficulty in verifying that (9) is satisfied by the same value of C. From that equation and (10), and 1.3.5 2.4.6...(n+l) 2 ) f, 1 Here, as throughout, m = n +, and when we expand these expressions in descending powers of n we recover (11). Equations (11) and (12) are thus applicable to odd as well as to even values of n. * M6c. Cel. Supplement au V e volume. t Functions of Laplace, etc. p. 71. 396 ON LEGENDRE'S FUNCTION P n (0), [404 But whether n be even or odd, (12) fails when 6 is so small that nd is not moderately large. For this case our original equation (1) takes approxi- mately the form S+JS+-*-* ........................... < 13 > where a 2 is written for n (n + 1) ; and of this the solution is M = J (a0) ............................... (14) It is evident that the Bessel's function of the second kind, infinite when = 0, does not enter, and that no constant multiplier is required, since u is to be unity when 6 = 0. For a second approximation we replace (13) by d*u 1 du du (\ cos B\ 6 du a6 T or, if aB = z, In order to solve (15) we assume as usual u = v.J (z) ............................... (16) This substitution gives d*v dv /2J ' 1\ z J' a linear equation of the first order in dv/dz. In this sothat s-jj? Here TtJi'di = 4 ^V - fjfrdz = i z n -J * - i z"- (Jo 2 + Jo' 2 ) = - $ ^ a J ' 2 . s-A-c? ............................ <"> which has now to be integrated again. regard being paid to the differential equation satisfied by J . 1916] Thus and WHEN n IS GREAT AND 9 HAS ANY VALUE 397 .(19) .(20) For the present purpose A = 0, B = 1 ; so that for P n , identified with u, we get PW-J t (*) + {*J t (*) + 2*J 9 '( g )}, (21) in which z = ad, a- = n (n + 1). The functions J , J ' = J 1} are thoroughly tabulated*. The Table annexed shows in the second column P w calculated from (21) for values of 6 ranging from to 35. The third column gives the results from (11), (12), beginning with = 10. In the fourth column are the values of P& calculated directly by A. Lodge. It will be seen that for 6 = 15 and 20 the discrepancies are small in the fifth place of decimals. For smaller values of 0, the formula involving the Bessel's functions gives the best results, and for larger values of d the extended form of Laplace's expression. When 6 exceeds about 35 the latter formula gives P w correct to six places. For n greater than 20 the combined use. of the two methods would of course allow a still closer approximation. Table for P 20 . e Formula (21) From (11) and (12) ! Calculated by Lodge | 1-000000 1-000000 5 0-346521 0-346521 10 -0-390581 - 0-390420 -0-390588 15 -0-052776 -0-052753 -0-052772 20 + 0-300174 + 0-300191 + 0-300203 25 -0-078051 -0-078085 -0-078085 30 -0-216914 -0-216997 -0-216999 35 + 0-155472 + 0-155635 + 0-155636 40 +0-127328 +0-127328 45 -0-193065 -0-193065 * See Gray and Mathew'a BesseVs Functions. 405. MEMORANDUM ON FOG SIGNALS. [Report to Trinity House, May 1916.] PROLONGED experience seems to show that, no matter how much power may be employed in the production of sound-in-air signals, their audibility cannot be relied upon much beyond a mile. At a less distance than two miles the most powerful signals may be lost in certain directions when the atmospheric conditions are unfavourable. There is every reason to surmise that in these circumstances the sound goes over the head of the observer, but, so far as I know, there is little direct confirmation of this. It would clear up the question very much could it be proved that when a signal is prematurely lost at the surface of the sea it could still be heard by an observer at a con- siderable elevation. In these days of airships it might be possible to get a decision. But for practical purposes the not infrequent failure of sound-in-air signals must be admitted to be without remedy, and the question arises what alter- natives are open. I am not well informed as to the success or otherwise of submarine signals, viz. of sounds propagated through water, over long distances. What I wish at present to draw attention to is the probable advantage of so- called "wireless" signals. The waves constituting these signals are indeed for the most part propagated through air, but they are far more nearly independent of atmospheric conditions temperature and wind than are ordinary sound waves. With very moderate appliances they can be sent and observed with certainty at distances such as 10 or 20 miles. As to how they should be employed, it may be remarked that the mere reception of a signal is in itself of no use. The signal must give information as to the distance, or bearing, or both, of the sending station. The estimation of distance would depend upon the intensity of the signals received and would probably present difficulties if any sort of precision was aimed at On the other hand the bearing of the sending station can be determined at the receiving station with fair accuracy, that is to within two or three degrees. The special apparatus required is not complicated, but it is rather cumbrous since coils of large area have to be capable of rotation. I assume that this 1916] MEMORANDUM ON FOG SIGNALS 399 part of the work would be done at the Shore Station. A ship arriving near the land and desirous of ascertaining her position would make wireless signals at regular short intervals. The operator on land would determine the bearing of the Ship from which the signals came and communicate this bearing to the Ship. In many cases this might suffice; otherwise the Ship could proceed upon her course for a mile or two and then receive another intimation of her bearing from the Shore Station. The two bearings, with the speed and course of the Ship, would fix her position completely. I do not suppose that much can be done at the present time towards testing this proposal, but I would suggest that it be borne in mind when considering any change in the Shore Stations concerned. I feel some con- fidence that the requirements of liners making the land will ultimately be met in some such way and that they cannot be met with certainty and under unfavourable conditions in any other. [1918. Reference may be made to Phil. Mag. Vol. xxxvi, p. 1 (1918), where Prof. Joly discusses lucidly and fully the method of " Synchronous signals." In this method it is distance which is found in the first instance. It depends upon the use of signals propagated at different speeds and it in- volves the audibility of sounds reaching the observer through air, or through water, or through both media.] 406. LAMB'S HYDRODYNAMICS. [Nature, Vol. xcvu. p. 318, 1916.] THAT this work should have already reached a fourth edition speaks well tor the study of mathematical physics. By far the greater part of it is entirely beyond the range of the books available a generation ago. And the improvement in the style is as conspicuous as the extension of the matter. My thoughts naturally go back to the books in current use at Cambridge in the early sixties. With rare exceptions, such as the notable one of Salmon's Conic Sections and one or two of Boole's books, they were arid in the extreme, with scarcely a reference to the history of the subject treated, or an indication to the reader of how he might pursue his study of it. At the present time we have excellent books in English on most branches of mathematical physics and certainly on many relating to pure mathematics. The progressive development of his subject is often an embarrassment to the writer of a text-book. Prof. Lamb remarks that his " work has less pre- tensions than ever to be regarded as a complete account of the science with which it deals. The subject has of late attracted increased attention in various countries, and it has become correspondingly difficult to do justice to the growing literature. Some memoirs deal chiefly with questions of mathe- matical method and so fall outside the scope of this book ; others though physically important hardly admit of a condensed analysis ; others, again, owing to the multiplicity of publications, may unfortunately have been over- looked. And there is, I am afraid, the inevitable personal equation of the author, which leads him to take a greater interest in some branches of the subject than in others." Most readers will be of opinion that the author has held the balance fairly. Formal proofs of " existence theorems " are excluded. Some of these, though demanded by the upholders of mathematical rigour, tell us only what we knew before, as Kelvin used to say. Take, for example, the existence of a possible stationary temperature within a solid when the temperature at the surface is arbitrarily given. A physicist feels that nothing can make this any clearer or more certain. What is strange is that there' should be so wide a gap between his intuition and the lines of argument necessary to satisfy the pure mathematician. Apart from this question it may be said that every- where the mathematical foundation is well and truly laid, and that in not a few cases the author's formulations will be found the most convenient starting 1916] LAMB'S HYDRODYNAMICS 401 point for investigations in other subjects as well as in hydrodynamics. To almost all parts of his subject he has made entirely original contributions; and, even when this could not be claimed, his exposition of the work of others is often so much simplified and improved as to be of not inferior value. As examples may be mentioned the account of Cauchy and Poisson's theory of the waves produced in deep water by a local disturbance of the surface ( 238) the first satisfactory treatment of what is called in Optics a dispersive medium and of Sommerfeld's investigation of the diffraction of plane waves of sound at the edge of a semi-infinite screen ( 308). Naturally a good deal of space is devoted to the motion of a liquid devoid of rotation and to the reaction upon immersed solids. When the solids are " fair " shaped, this theory gives a reasonable approximation to what actually occurs ; but when a real liquid flows past projecting angles the motion is entirely different, and unfortunately this is the case of greatest practical importance. The author, following Helmholtz, lays stress upon the negative pressure demanded at sharp corners in order to maintain what may be called the electric character of flow. This explanation may be adequate in some cases ; but it is now well known that liquids are capable of sustaining negative pressures of several atmospheres. How too does the explanation apply to gases, which form jets under quite low pressure differences?* It seems probable that viscosity must be appealed to. This is a matter which much needs further elucidation. It is one on which Kelvin and Stokes held strongly divergent views. The later chapters deal with Vortex Motion, Tidal Waves, Surface Waves, Waves of Expansion (Sound), Viscosity, and Equilibrium of Rotating Masses. On all these subjects the reader will find expositions which could hardly be improved, together with references to original writings of the author and others where further developments may be followed. It would not have accorded with the author's scheme to go into detail upon experimental matters, but one feels that there is room fora supplementary volume which should have regard more especially to the practical side of the subject. Perhaps the time for this has not yet come. During the last few years much work has been done in connexion with artificial flight. We may hope that before long this may be coordinated and brought into closer relation with theoretical hydrodynamics. In the meantime one can hardly deny that much of the latter science is out of touch with reality. * The fact that liquids do not break under moderate negative pressure was known to T. Young. "The magnitude of the cohesion between liquids and solids, as well as of the particles of fluid with each other, is more directly shewn by an experiment on the continuance of a column of mercury, in the tube of a barometer, at a height considerably greater than that at which it usually stands, on account of the pressure of the atmosphere. If the mercury has been well boiled in the tube, it may be made to remain in contact with the closed end, at the height of 70 inches or more " (Young's Lectures, p. 626, 1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated. R. vi. 26 407. ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE. [Philosophical Magazine, Vol. xxxii. pp. 16, 1916.] IT is well known that according to classical Hydrodynamics a steady stream of frictionless incompressible fluid exercises no resultant force upon an obstacle, such as a rigid sphere, immersed in it. The development of a " resistance " is usually attributed to viscosity, or when there is a sharp edge to the negative pressure which may accompany it (Helmholtz). In either case it would seem that resistance involves something of the nature of a wake, extending behind the obstacle to an infinite distance. When the system of disturbed velocities, although it may mathematically extend to infinity, remains as it were attached to the obstacle, there can be no resistance. The absence of resistance is asserted for an incompressible fluid ; but it can hardly be supposed that a small degree of compressibility, as in water, would affect the conclusion. On the other hand, high relative velocities, exceeding that of sound in the fluid, must entirely alter the conditions. It seems worth while to examine this question more closely, especially as the first effects of compressibility are amenable to mathematical treatment. The equation of continuity for a compressible fluid in steady motion is in the usual notation dp dp dp fdu dv d U ^ + V J+ W J+P[-J- + J- dx dy dz r \dx dy or, if there be a velocity-potential <f>, d<f> dlogp d<f> dlogp d<f> dlogp _ dx dx dy dy dz dz In most cases we may regard the pressure p as a given function of the density p, dependent upon the nature of the fluid. The simplest is that of Boyle's law where p = a z p, a being the velocity of sound. The general equation rdn .(3) 1916] ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE 403 where q is the resultant velocity, so that (4) reduces in this case to or a 2 log (p/ po ) = - %q\ .............................. (5) if p correspond to q = 0. From (2) and (5) we get dy dy^ dz dz\ (6) When q 2 is small in comparison with a 2 , this equation may be employed to estimate the effects of compressibility. Taking a known solution for an incompressible fluid, we calculate the value of the right-hand member and by integration obtain a second approximation to the solution in the actual case. The operation may be repeated, and if the integrations can be effected, we obtain a solution in series proceeding by descending powers of a 2 . It may be presumed that this series will be convergent so long as q 2 is less than a 2 . There is no difficulty in the first steps for obstacles in the form of spheres or cylinders, and I will detail especially the treatment in the latter case. If U, parallel to = 0, denote the uniform velocity of the stream at a distance, the velocity-potential for the motion of incompressible fluid is known to be the origin of polar coordinates (r, 0) being at the centre of the cylinder. At the surface of the cylinder r = c, dtfr/dr = 0, for all values of 0. On the right hand of (6) dx dx dy dy dr dr r 2 d8 dO ' and from (7) &~v{(f>'+*$fi~ 1 +$-7* e - (9) 1 d<$> ( c 2 \ a 1 d<f> 1 dq 2 4C 4 4c 2 1 dq 2 4c 2 . _ -f- = + -cos 20, ==- -^ = sin 20. U 2 dr r 5 r 3 U 2 rd& r 3 Accordingly The terms on the right of (10) are all of the form rPcosnff, so that for the present purpose we have to solve (11) r dr r* 00* 262 404 ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE [407 If we assume that <j> varies as r m cosn0, we see that m = p + 2, and thai the complete solution is (12) A and B being arbitrary constants. In (10) we have to deal with n = 1 associated with p = 5 and - 7, and with n = 3 associated with p = - 3. The complete solution as regards terms in cos 6 and cos 30 is accordingly < = (Ar + Br- 1 ) cos + (CV 3 + Dr~*) cos 30 20V [ Q ( c 2 c 4 \ cos30~| + - |*(-- +s - ; )-._J ....... (13) The conditions to be satisfied at infinity require that, as in (7), A = U, and that (7=0. We have also to make dfyjdr vanish when r = c. This leads to Thus satisfies all the conditions and is the value of </> complete to the second approximation. That the motion determined by (15) gives rise to no resultant force in the direction of the stream is easily verified. The pressure at any point is a function of q-, and on the surface of the cylinder q* c~* (d<f>/d0)*. Now (rf</(/0) 2 involves in the forms sin 2 0, sin 2 30, sin sin 30, and none of these are changed by the substitution of TT for ; the pressures on the cylinder accordingly constitute a balancing system. There is no particular difficulty in pursuing the approximation so as to include terms involving the square and higher powers of U*la*. The right- hand member of (6) will continue to include only terms in the cosines of odd multiples of with coefficients which are simple powers of r, so that the integration can be effected as in (11), (12). And the general conclusion that there is no resultant force upon the cylinder remains undisturbed. The corresponding problem for the spftere is a little more complicated, but it may be treated upon the same lines with use of Legendre's functions P n (cos0) in place of cosines of multiples of 0. In terms of the usual polar coordinates (r, 0, &>), the last of which does not appear, the first approxima- tion, as for an incompressible fluid, is u (16) 1916] OX THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE 405 c denoting the radius of the sphere. As in (8), -d+df ddg ld<f>dq*_ (f 36c 9c\ dx dx dr dr + r 2 d6 dd ~ \ 5r* + 2r">) r * on substitution from (16) of the values of <f> and (f. This gives us the right- hand member of (6).- In the present problem while P n satisfies so that V 2 < = r*>P n ................................. (20) reducesto ^ + 2^_( + l) dr* r dr r 2 The solution, corresponding to the various terms of (17), is thus r p+zp * = (;> + 2)<p + 3)-n(n + l) ................... (22) With use of (22), (6) gives U* J &P, c 9 P, 8*P. 3*P, 3c 9 P 3 ) a 2 ( Sr 5 + 24r 10r 2 lOr 6 I76r) + ^IrP! + .Br- 2 ^ + C^Ps + Dr- 4 P 3 , ............... (23) A, B, C, D being arbitrary constants. The conditions at infinity require A= U, (7 = 0. The conditions at the surface of the sphere give and thus </> is completely determined to the second approximation. The P's which occur in (23) are of odd order, and are polynomials in p (= cos 6) of odd degree. Thus d<f>ldr is odd (in fi) and d<f>/d0 = sin 6 x even function of /z. Further, (f = even function + sin 2 x even function = even function, d<ffdr = even function, dq 2 /dO = sin 6 x odd function. Accordingly and can be resolved into a series of P's of odd order. Thus not only is there no resultant force discovered in the second approximation, but this character 406 ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE [407 is preserved however far we may continue the approximations. And since the coefficients of the various P's are simple polynomials in 1/r, the integra- tions present no difficulty in principle. Thus far we have limited ourselves to Boyle's law, but it may be of interest to make extension to the general adiabatic law, of which Boyle's is a particular case. We have now to suppose .............................. (25) if a denote the velocity of sound corresponding to p . Then by (3) If we suppose that /o corresponds to q = 0, C = a?/(y 1), and The use of this in (2) now gives 1 td+df d+df + + 2a -(7-1)9" dx dx dy Ty Tz from which we can fall back upon (6) by supposing 7 = !. So far as the first and second approximations, the substitution of (30) for (6) makes no difference at all. As regards the general question it would appear that so long as the series are convergent there can be no resistance and no wake as the result of com- pressibility. But when the velocity U of the stream exceeds that of sound, the system of velocities in front of the obstacle expressed by our equations cannot be maintained, as they would be at once swept away down stream. It may be presumed that the passage from the one state of affairs to the other synchronizes with a failure of convergency. For a discussion of what happens when the velocity of sound is exceeded, reference may be made to a former paper*. * Proc. Roy. Soc. A, Vol. LKXIV. p. 247 (1910) ; Scientific Papert, Vol. v. p. 608. [1917. See P. 8. to Art. 411 for a reference to the work of Prof. Cisotti.] 408. ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES. [Philosophical Magazine, Vol. xxxii. pp. 177187, 1916.] THE problem of the passage of gas through a small aperture or nozzle from one vessel to another in which there is a much lower pressure has had a curious history. It was treated theoretically and experimentally a long while ago by Saint- Venant and Wantzel* in a remarkable memoir, where they point out the absurd result which follows from the usual formula, when we introduce the supposition that the pressure in the escaping jet is the same as that which prevails generally in the recipient vessel. In Lamb's notationf, if the gas be subject to the adiabatic law (p oc pf), P'^J^M! /P^l 2 ) f p 7-1 Pol W ) 7-1 where q is the velocity corresponding to pressure p ; p , p the pressure and density in the discharging vessel where q = 0; c the velocity of sound in the gas when at pressure p and density p; c that corresponding to p , p . According to (1) the velocity increases as p diminishes, but only up to a maximum, equal to c \/{2/(y - 1)}, when p = 0. If 7 = 1-408, this limiting velocity is 2'214c . It is to be observed, however, that in considering the rate of discharge we are concerned with what the authors cited call the " reduced velocity," that is the result of multiplying q by the corresponding density p. Now p diminishes indefinitely, with p, so that the reduced velocity corresponding to an evanescent p is zero. Hence if we identify^ with the pressure p^ in the recipient vessel, we arrive at the impossible con- clusion that the rate of discharge into a vacuum is zero. From this our authors infer that the identification cannot be made ; and their experiments showed that from p t = upwards to p l = '4<p the rate of discharge is sensibly constant. As p^ still further increases, the discharge falls off, slowly at first, * "M^moire et experiences sur 1'ecoulement de 1'air, determine' par des differences de pressions considerables," Journ. de VEcole Polyt. t. xvi. p. 85 (1839). t Hydrodynamics, 23, 25 (1916). 408 ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES [408 afterwards with greater rapidity, until it vanishes when the pressures be- come equal. The work of Saint- Venant and Wantzel was fully discussed by Stokes in his Report on Hydrodynamics*. He remarks "These experiments show that when the difference of pressure in the first and second spaces is considerable, we can by no means suppose that the mean pressure at the orifice is equal to the pressure at a distance in the second space, nor even that there exists a contracted vein, at which we may suppose the pressure to be the same as at a distance." But notwithstanding this the work of the French writers seems to have remained very little known. It must have been unknown to O. Reynolds when in 1885 he traversed much the same ground f, adding, however, the important observation that the maximum reduced velocity occurs when the actual velocity coincides with that of sound under the conditions then prevailing. When the actual velocity at the orifice reaches this value, a further reduction of pressure in the recipient vessel does not influence the rate of discharge, as its effect cannot be propagated backwards against the stream. If 7 = 1*408, this argument suggests that the discharge reaches a maximum when the pressure in the recipient vessel falls to '527 p , and then remains constant. In the somewhat later work of HugoniotJ on the same subject there is indeed a complimentary reference to Saint- Venant and Wantzel, but the reader would hardly gather that they had insisted upon the difference between the pressure in the jet at the orifice and in the recipient vessel as the explanation of the impossible conclusion deducible from the contrary supposition. In the writings thus far alluded to there seems to be an omission to consider what becomes of the jet after full penetration into the receiver. The idea appears to have been that the jet gradually widens in section as it leaves the orifice and that in the absence of friction it would ultimately attain the velocity corresponding to the entire fall of pressure. The first to deal with this question seem to have been Mach and Salcher, but the most elaborate examination is that of R. Emden||, who reproduces interesting pictures of the effluent jet obtained by the simple shadow method of Dvorak * . Light from the sun or from an electric spark, diverging from a small aperture as source, falls perpendicularly upon the jet and in virtue of differences of refraction depicts various features upon a screen held at some distance behind. A permanent record can be obtained by photography. Eraden thus describes some of his results. When a jet of air, or better of carbonic B.A. Report for 1846; Math, and Phys. Papers, Vol. I. p. 176. t Phil. Ma<t. Vol. xxi. p. 185 (1886). * Ann. de Chim. t. ix. p. 383 (1886). Wied. Ann. Bd. XLI. p. 144 (1890). || Wied. Ann. Bd. LXIX. pp. 264, 426 (1899). IF Wied. Ann. Bd. ix. p. 502 (1879). 1916] ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES 409 acid or coal-gas, issues from the nozzle into the open under a pressure of a few millimetres, it is seen to rise as a slender column of the same diameter to a height of perhaps 30 or 40 cm. Sometimes the column disappears without visible disturbance of the air ; more often it ends in a small vortex column. When the pressure is raised, the column shortens until finally the funnel-shaped vortex attaches itself to the nozzle. At a pressure of about one-fifth of an atmosphere there appears again a jet 2 or 3 cm. long. As* the pressure rises still further, the jet becomes longer and more distinct and suddenly exhibits thin, bright, and fairly equidistant disks to the number of perhaps 10 or 12, crossing the jet perpendicularly. The first disks have exactly the diameter of the nozzle, but they diminish as the jet attenuates. Under still higher pressures the interval between the disks increases, and at the same time the jet is seen to swell out between them. These swellings further increase and oblique markings develop which hardly admit of merely verbal description. Attributing these periodic features to stationary sound waves in the jet, Emden set himself to determine the wave-length (X), that is the distance between consecutive disks, and especially the pressure at which the waves begin to develop. He employed a variety of nozzles, and thus sums up his principal results : 1. When air, carbonic acid, and hydrogen escape from equal sufficiently high pressures, the length of the sound waves in the jet is the same for the same nozzle and the same pressure. 2. The pressure at which the stationary sound waves begin to develop is the same in air, carbonic acid, and hydrogen, and is equal to '9 atmosphere. This is the pressure-excess behind the nozzle, so that the whole pressure there is T9 atmosphere. The environment of the jet is at one atmosphere pressure. Emdeu, comparing his observations with the theory of Saint- Venant and Wantzel, then enunciates the following conclusion: The critical pressure, in escaping from which into the atmosphere the gas at the nozzle's mouth . moves with the velocity of sound, is equal to the pressure at which stationary . sound waves begin to form in the jet. So far, I think, Emden makes out his case ; but he appears to over-shoot the mark when he goes on to maintain that after the critical pressure-ratio is exceeded, the escaping jet moves everywhere with the same velocity, viz. the sound- velocity ; and that every- where within it the free atmospheric pressure prevails. He argues from what happens when the motion is strictly in one dimension. It is true that then a wave can be stationary in space only when the stream moves with the velocity of sound ; but here the motion is not limited to one dimension, as is shown by the swellings between the disks. Indeed the propagation of any wave at all is inconsistent with uniformity of pressure within the jet. 410 ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES [408 At the surface of the jet, but not within it, the condition is imposed that the pressure must be that of the surrounding atmosphere. The problem of a jet in which the motion is completely steady in the hydrodynamical sense and approximately uniform was taken up by Prandtl*, both for the case of symmetry round the axis (of z) and in two dimensions. In the former, which is the more practical, the velocity component w is supposed to be nearly constant, say W, while u and v are small. We may employ the usual Eulerian equations. Of these the third, dw dw dw dw 1 d aw aw aw aw _ l ap dt dx dy dz p dz ' dy dz p reducesto W~ = ---, P -, (2) dz p dz' when we introduce the supposition of steady motion and neglect the terms of the second order. In like manner the other equations become w du I dp w dv_ I dp rr -j = j , rr -j- - ~j~ \&j dz p dx dz p dy Further, the usual equation of continuity, viz. d(pu) + d(pv) d(pw) = Q (4 , dx dy dz here reduces to -ffi+J+S+^- ft < 5 > If we introduce a velocity-potential <, we have with use of (2) V<6-_ *?- = d Q (6) where a, = V (dp/dp), is the velocity of sound in the jet. In the case we are now considering, where there is symmetry round the axis, this becomes ^7^ I, ' \ * a I .I... v > and a similar equation holds for w, since w = d<f>/dz. If the periodic part of w is proportional to cos j3z, we have for this part r dr \ a a / and we may take as the solution w= W+Hcos/3z. J o y(W*-a*)./3r/a], (9) since the Bessel's function of the second kind, infinite when r = 0, cannot here appear. The condition to be satisfied at the boundary (r R) is that Phys. Zeitschrift, 5 Jahrgang, p. 599 (1904). 1916] ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES 411 the pressure be constant, equal to that of the surrounding quiescent air, and this requires that the variable part of w vanish, since the pressure varies with the total velocity. Accordingly J o y(W*-a*).j3R/a} = 0, ..................... (10) which can be satisfied only when W > a, that is when the mean velocity of the jet exceeds that of sound. The wave-length (X) of the periodic features along the jet is given by \ = Zir//3. The most important solution corresponds to the first root of (10), viz. 2-405. In this case 2-405 The problem for the two-dimensional jet is even simpler. If b be the width of the jet, the principal wave-length is given by \=2&v / (W ra /a s -l) ............................ (12) The above is substantially the investigation of Prandtl, who finds a sufficient agreement between (11) and Emden's measurements*. It may be observed that the problem can equally well be treated as one of the small vibrations of a stationary column of gas as developed in Theory of Sound, 268, 340 (1878). If the velocity-potential, symmetrical about the axis of z, be also proportional to e i(kat+ft!!} , where k is such that the wave- length of plane waves of the same period is 27T/&, the equation is 340 (3) and if k > ft &).r} ..................... (14) The condition of constant pressure when r = R gives as before for the principal vibration VX& 2 - /8 s ). R = 2-405 ......................... (15) The velocity of propagation of the waves is ka//3. If we equate this to W and suppose that a velocity W is superposed upon the vibrations, the motion becomes steady. When we substitute in (15) the value of k, viz. W/3/a, we recover (11). It should perhaps be noticed that it is only after the vibrations have been made stationary that the effect of the surrounding air can be properly represented by the condition of uniformity of pressure. To assume it generally would be tantamount to neglecting the inertia of the outside air. The above calculation of X takes account only of the principal vibration. Other vibrations are possible corresponding to higher roots of (10), and if * When JF<a, /3 must be imaginary. The jet no longer oscillates, but settles rapidly down into complete uniformity. This is of course the usual case of gas escaping from small pressures. 412 ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES [408 these occur appreciably, strict periodicity is lost. Further, if we abandon the restriction to symmetry, a new term, r~*d?<f>ld6 % , enters in (13) and the solution involves a new factor cos(?20 + e) in conjunction with the Bessel's function / in place of / The particular form of the differential equation exhibited in (13) is appropriate only when the section of the stream is circular. In general we have the same equation as governs the vibrations of a stretched membrane (Theory of Sound, 194). For example, in the case of a square section of side b, we have </> = cos . cos .e f <*<+**>, ..................... (17) vanishing when x = + 6 and when y = 6. This represents the principal vibration, corresponding to the gravest tone of a membrane. The differential equation is satisfied provided - & = 27T 2 /& 2 , ........................... (18) the equation which replaces (15). It is shown in Theory of Sound that provided the deviation from the circular form is not great the question is mainly one of the area of the section. Thus the difference between (15) and (18) is but moderate when we suppose TrR 2 equal to 6 2 . It may be worth remarking that when V the wave-velocity exceeds a, the group- velocity U falls short of a. Thus in (15), (18) ka JT d(0V) dk /3a > -~~ a ~~' so that UV=a? .................................. (19) Returning to the jet of circular section, we may establish the connexion between the variable pressure along the axis and the amount of the swellings observed to take place between the disks. From (9) <f> = wdz = Wz + H/3-* sin 0e.J { V( W'fa 9 1) . /3r}, and ( = H V( TT'/a'-l). sin /3z. J ' (2-405) ............. (20) The latter equation gives the radial velocity at the boundary. If oR denote the variable part of the radius of the jet, 1916] ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES 413 Again, if Bp be the variable part of the pressure at the axis (r = 0), & = C - $q* = C' - $w* = - Wbw, where p is the average density in the jet and 8w the variable part of the component velocity parallel to z. Accordingly ^ = - WHcos/Sz; ........................... (22) ................... < In (23) we may substitute for /8 its value, viz. 2'405a and for Jp' (2-405) we have from the tables of Bessel's functions -0'5191, so that - 0-2158 (a- 2 -TF- 2 ) ...................... (24) As was to be expected, the greatest swelling is to be found where the pressure at the axis is least. A complete theory of the effects observed by Mach and Emden would involve a calculation of the optical retardation along every ray which traverses the jet. For the jet of circular section this seems scarcely practicable ; but for the jet in two dimensions the conditions are simpler and it may be worth while briefly to consider this case. As before, we may denote the general thickness of the two-dimensional jet by 6, and take b + ij to represent the actual thickness at the place (z) where the retardation is to be deter- mined. The retardation is then sufficiently represented by A, where fi(&+iJ) /"*(&+>) A= (p- pl )dy = pdy-^ Pl (b + r,), ......... (25) Jo Jo p being the density in the jet and p^ that of the surrounding gas. The total stream rk(b+ri) ri(6+i) rift = p(W + 8w)dy = Wl pdy + p\ Swdy; Jo Jo Jo and this is constant along the jet. Thus & = C-1sp l T)-fl*Swdy, ..................... (26) C being a constant, and squares of small quantities being omitted. In analogy with (9), we may here take -l, ............... (27) 414 ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES [408 and for the principal vibration the argument of the cosine is to become ^TT when y \b. Hence ...................... (28) Also <f>=lwdz=Wz + ft~ l Hsin @z . cos {/3y V( W*/a? - 1)1, * - 1} . sin 0*. Thus it; = 4. HT ) rf* JP J Uy/j Accordingly _ , __ ; ...... (29) so that the retardation is greatest at the places where ij is least, that is where the jet is narrowest. This is in agreement with observation, since the places of maximum retardation act after the manner of a convex lens. Although a complete theory of the optical effects in the case of a symmetrical jet is lacking, there seems no reason to question Emden's opinion that they are natural consequences of the constitution of the jet. But although many features are more or less perfectly explained, we are far from anything like a complete mathematical theory of the jet escaping from high pressure, even in the simplest case. A preliminary question is are we justified at all in assuming the adiabatic law as approximately governing the expansions throughout ? Is there anything like the " bore " which forms in front of a bullet advancing with a velocity exceeding that of sound ?* It seems that the latter question may be answered in the negative, since here the passage of air is always from a greater to a less pressure, so that the application of the adiabatic law is justified. The conditions appear to be simplest if we suppose the nozzle to end in a parallel part within which the motion may be uniform and the velocity that of sound. But even then there seems to be no reason to suppose that this state of things terminates exactly at the plane of the mouth. As the issuing gas becomes free from the constraining influence of the nozzle walls, it must begin to expand, the pressure at the boundary suddenly falling to that of the environment. Subsequently vibrations must set in ; but the circum- stances are not precisely those of Prandtl's calculation, inasmuch as the variable part of the velocity is not small in comparison with the difference between the mean velocity and that of sound. It is scarcely necessary to call attention to the violence of the assumption that viscosity may be neg- lected when a jet moves with high velocity through quiescent air. * Proc. Roy. Soc. A, Vol. LIXXIV. p. 247 (1910); Scientific Paper*, Vol. v. Art. 346, p. 608. 1916] ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES 415 On the experimental side it would be of importance to examine, with more accuracy than has hitherto been attained, whether the asserted inde- pendence of the discharge of the pressure in the receiving vessel (supposed to be less than a certain fraction of that in the discharging vessel) is absolute, and if not to ascertain the precise law of departure. To this end it would seem necessary to abandon the method followed by more recent workers in which compressed gas discharges into the open, and to fall back upon the method of Saint- Venant and Wantzel where the discharge is from atmospheric pressure to a lower pressure. The question is whether any alteration of discharge is caused by a reduction of this lower pressure beyond a certain point. To carry out the investigation on a sufficient scale would need a powerful air-pump capable of absorbing the discharge, but otherwise the necessary apparatus is simple. In order to measure the discharge, or at any rate to determine whether it varies or not, the passage of atmospheric air to the nozzle might be somewhat choked. The accompanying diagram will explain the idea. A is the nozzle* which would be varied in different series of experiments ; B the recipient, partially exhausted, vessel ; G the passage to the air-pump. Above the nozzle is provided a closed chamber E D into which the external air has access through a metal gauze D, and where consequently the pressure is a little below atmospheric. F represents (dia- grammatically) a pressure-gauge, or micromanometer, whose reading would be constant as long as the discharge remains so. Possibly an aneroid barometer would suffice ; in any case there is no difficulty in securing the necessary delicacy*. Another manometer of longer range, but only ordinary sensitiveness, would register the low pressure in B. In this way there should be no difficulty in attaining satisfactory results. If F remains unaffected, notwithstanding large alterations of pressure in B, there are no complications to confuse the interpretation. * See for example Phil. Trans, cxcvi. A, p. 205 (1901) ; Scientific Papers, Vol. iv. p. 510. [1918. The experiments here proposed have been skilfully carried into effect by Hartshorn, working in my son's laboratory, Proc. Roy. Soc. A, Vol. xciv. p. 155, 1917.] 409. ON THE ENERGY ACQUIRED BY SMALL RESONATORS FROM INCIDENT WAVES OF LIKE PERIOD. [Philosophical Magazine, Vol. xxxn. pp. 188-190, 1916.] IN discussions on photo-electricity it is often assumed that a resonator can operate only upon so much of the radiation incident upon it as corresponds to its own cross-section*. As a general proposition this is certainly not true and may indeed differ from the truth very widely. Since 1878 f it has been known that an ideal infinitely small acoustical resonator may disperse energy corresponding to an area of wave-front of the primary waves equal to \ a /Tr, an efficiency exceeding to any extent the limit fixed by the above mentioned rule. The questions of how much energy can be absorbed into the resonator itself and how long the absorption may take are a little different, but they can be treated without difficulty by the method explained in a recent paper *. The equation (4U) there found for the free vibration of a small symmetrical resonator was (1) in which p denotes the radial displacement of the spherical surface from its equilibrium value r, M the mass, /* the coefficient of restitution, a the density of the surrounding gas, and k = 2?r -f- wave-length (X) of vibrations in the gas. The first of the two terms containing a operates merely as an addition to M. If we write M' = M + 47TOT 3 , .............................. (2) (1) becomes .O ...................... (3) * See for example Millikan's important paper on a direct determination of Planck's constant "; Physical Review, Vol vii. March 1916, p. 385. | Theory of Sound, 319 : X = wave-length. J Phil. Mag. Vol. xxix. Feb. 1915, p. 210. [This volume, p. 289.] 1916] ENERGY ACQUIRED BY SMALL RESONATORS FROM INCIDENT WAVES 417 Thus, if in free vibration p is proportional to e int , where n is complex, the equation for n is n 2 (-M' + i. 4ircrfcr) + /x = ...................... (4) The free vibrations are assumed to have considerable persistence, and the co- efficient of decay is e~ qt , where q = ZTTffki* V(/V^' 3 ) = Z-rrapk^/M', .................. (5) We now suppose that the resonator is exposed to primary waves whose velocity-potential is there 4> = ae i P t .................................. (6) The effect is to introduce on the right hand of (3) the term 47rr 2 cra . ipe ipt ; and since the resonance is supposed to be accurately adjusted, p 2 = /*/J/'. Under the same conditions id 2 p/dt- in the third term on the left of (3) may be replaced by pdp/dt, whether we are dealing with the permanent forced vibration or with free vibrations of nearly the same period which gradually die away. Thus our equation becomes on rejection of the imaginary part (7) which is of the usual form for vibrations of systems of one degree of freedom. For the permanent forced vibration M'd 2 pjdt 2 + pp = absolutely, and dp _ asinpt ~dt~ kr* The energy located in the resonator is then Ma 2 .(9) and it may become very great when M is large and r small. But when M is large, it may take a considerable time to establish the permanent regime after the resonator starts from rest. The approximate solution of (7), applicable in that case, is q being regarded as small in comparison with p ; and the energy located in the resonator at time t We may now inquire what time is required for the accumulation of energy equal (say) to one quarter of the limiting value. This occurs when e~* = J, or by (5) when Iog2_ log 2. JIT ( q -p.kr.2Tr<n*' R. vi. 27 418 ENERGY ACQUIRED BY SMALL RESONATORS FROM INCIDENT WAVES [409 The energy propagated in time t across the area 8 of primary wave-front is (Theory of Sound, 245) (13) where a is the velocity of propagation, so that p = ak. If we equate (13) to one quarter of (9) and identify t with the value given by (12), neglecting the distinction between M and M' , we get The resonator is thus able to capture an amount of energy equal to that passing in the same time through an area of primary wave-front comparable with \ z lir, an area which may exceed any number of times the cross-section of the resonator itself. log 2 = 410. ON THE ATTENUATION OF SOUND IN THE ATMOSPHERE. [Advisory Committee for Aeronautics. August, 1916.] IN T. 749, Major Taylor presents some calculations which " shew that the chief cause of the dissipation of sound during its transmission through the lower atmosphere must be sought for in the eddying motion which is known to exist there. The amount of dissipation which these calculations would lead us to expect from our knowledge of the structure of the lower atmosphere agrees, as well as the rough nature of the observations permit, with the amount of dissipation given by Mr Lindemann." The problem discussed is one of importance and it is attended with con- siderable difficulties. There can be no doubt that on many occasions, perhaps one might say normally, the attenuation is much more rapid than according to the law of inverse squares. Some 20 years ago (Scientific Papers, Vol. IV. p. 298) I calculated that according to this law the sound of a Trinity House syren, absorbing 60 horse-power, should be audible to 2700 kilometres ! A failure to propagate, so far as it is uniform on all occasions, would naturally be attributed to dissipative action. I am here using the word in the usual and narrower technical sense, implying a degradation of energy from the mechanical form into heat, or a passage of heat from a higher to a lower temperature. Although there must certainly be dissipation consequent upon radiation and conduction of heat, it does not appear that these causes are adequate to explain the attenuation of sound sometimes observed, even at moderate distances. This question is discussed in Phil. Mag. XLVII. p. 308, 1899 (Scientific Papers, Vol. iv. p. 376) in connexion with some observations of Wilrner Duff. If we put dissipation out of account, the energy of a sound wave, advancing on a broad front, remains mechanical, and we have to consider what becomes of it. Part of the sound may be reflected, and there is no doubt at all that, whatever may be the mechanism, reflection does really occur, even when no obstacles are visible. At St Catherine's Point in 1901, 1 heard strong echoes 272 420 ON THE ATTENUATION OF SOUND IN THE ATMOSPHERE [410 from over the sea for at least 12 seconds after the syren had ceased sounding. The sky was clear and there were no waves to speak of. Reflection in the narrower sense (which does not include so called total reflection !) requires irregularities in the medium whose outlines are somewhat sharply defined, the linear standard being the wave-length of the vibration ; but this require- ment is probably satisfied by ascending streams of heated air. In considering the effect of eddies on maintained sounds of given pitch, Major Taylor does not include either dissipation (in the narrower sense) or reflection. I do not understand how, under such conditions, there can be any general attenuation of plane waves. What is lost in one position in front of the phase-disturbing obstacles, must be gained at another. The circumstances are perhaps more familiar in Optics. Consider the passage of light of given wave-length through a grating devoid of absorbing and reflecting power. The whole of the incident light is then to be found distributed between the central image and the lateral spectra. At a sufficient distance behind the grating, supposed to be of limited width, the spectra are separated, and as I under- stand it the calculation refers to what would be found in the beam going to form the central image. But close behind the grating, or at any distance behind if the width be unlimited, there is no separation, and the average intensity is the same as before incidence. The latter appears to be the case with which we are now concerned. The problem of the grating is treated in Theory of Sound, 2nd edition, 272 a. Of course, the more important anomalies, such as the usual failure of sound up wind, are to be explained after Stokes and Reynolds by a refraction which is approximately regular. In connexion with eddies it may be worth while to mention the simple case afforded by a vortex in two dimensions whose axis is parallel to the plane of the sound waves. The circumferential velocity at any point is proportional to 1/r, where r is the distance from the axis. By integration, or more imme- diately by considering what Kelvin called the "circulation," it is easy to prove that the whole of the wave which passes on one side of the axis is uniformly advanced by a certain amount and the whole on the other side retarded by an equal amount. A fault is thus introduced into the otherwise plane character of the wave. [1918. Major Taylor sends me the following observations: NOTE ON THE DISPERSION OF SOUND. Observations have shown that sound is apparently dissipated at a much greater rate than the inverse square law both up and down wind. The effect of turbulence on a plane wave front is to cause it to deviate locally from its 1916] ON THE ATTENUATION OF SOUND IN THE ATMOSPHERE 421 plane form. The wave train cannot then be propagated forward without further change, but it may be regarded as being composed of a plane wave train of smaller amplitude, together with waves which are dispersed in all directions, and are due to the effect of the turbulence of the original train. If d is the diameter of an eddy, X is the wave length of the sound, U is the velocity of the air due to the eddy, and V is the velocity of sound, the amount of sound energy dispersed from unit volumes of the main wave is where E is the energy of the sound per unit volume. If the turbulence is uniformly distributed round the source of sound then, as Lord Rayleigh points out, the sound energy will be uniformly distributed because the energy dispersed from one part of the wave front will be replaced by energy dispersed from other parts ; but if the turbulence is a maximum in any particular direc- tion then more sound energy will be dispersed from the wave fronts as they proceed in that direction than will be received from the less turbulent regions. Regions of maximum turbulence should, therefore, be regions of minimum sound. The turbulence is usually a maximum near the ground. The intensity of sound should, therefore, fall off near the ground at a greater rate than the inverse square law, even although there is no solid obstacle between the source of sound and the listener.] 411. ON VIBRATIONS AND DEFLEXIONS OF MEMBRANES, BARS, AND PLATES. [Philosophical Magazine, Vol. xxxn. pp. 353364, 1916.] IN Theory of Sound, 211, it was shown that "any contraction of the fixed boundary of a vibrating membrane must cause an elevation of pitch, because the new state of things may be conceived to differ from the old merely by the introduction of an additional constraint. Springs, without inertia, are supposed to urge the line of the proposed boundary towards its equilibrium position, and gradually to become stiffer. At each step the vibrations become more rapid, until they approach a limit corresponding to infinite stiffness of the springs and absolute fixity of their points of application. It is not necessary that the part cut off should have the same density as the rest, or even any density at all." From this principle we may infer that the gravest mode of vibration for a membrane of any shape and of any variable density is devoid of internal nodal lines. For suppose that ACDB (fig. 1) vibrating in its longest period (T) has an internal nodal line GB. This requires that a membrane with the fixed boundary ACS shall also be capable of vibration in period T. The im- possibility is easily seen. As ACDB gradually contracts through ACD'B to ACB, the longest period diminishes, so that the longest period of ACB is less than T. No period possible to ACB can be equal to T. 1916] VIBRATIONS AND DEFLEXIONS OF MEMBRANES, BARS, AND PLATES 423 If we replace the reactions against acceleration by external forces, we may obtain the solution of a statical problem. When a membrane of any shape is submitted to transverse forces, all in one direction, the displacement is everywhere in the direction of the forces. Similar conclusions may be formulated for the conduction of heat in two dimensions, which depends upon the same fundamental differential equation. Here the boundary is maintained at a constant temperature taken as zero, and " persistences " replace the periods of vibration. Any closing in of the boundary reduces the principal persistence. In this mode there can be no internal place of zero temperature. In the steady state under positive sources of heat, however distributed, the temperature is above zero everywhere. In the application to the theory of heat, extension may evidently be made to three dimensions. Arguments of a like nature may be used when we consider a bar vibrating transversely in virtue of rigidity, instead of a stretched membrane. In Theory of Sound, 184, it is shown that whatever may be the constitution of the bar in respect of stiffness and mass, a curtailment at either end is associated with a rise of pitch, and this whether the end in question be free, clamped, or merely " supported." In the statical problem of the deflexion of a bar by a transverse force locally applied, the question may be raised whether the linear deflexion must everywhere be in the same direction as the force. It can be shown that the answer is in the affirmative. The equation governing the deflexion (w) is where Zdx is the transverse force applied at dx, and B is a coefficient of stiffness. In the case of a uniform bar B is constant and w may be found by simple integration. It suffices to suppose that Z is localized at one point, say at x = b; and the solution shows that whether the ends be clamped or supported, or if one end be clamped and the other free or supported, w is everywhere of the same sign as Z. The conclusion may evidently be extended to a force variable in any manner along the length of the bar, provided that it be of the same sign throughout. But there is no need to lay stress upon the case of a uniform bar, since the proposition is of more general application. The first integration of (1) gives and fZdx = from x = at one end to x = 6, and takes another constant value (Zj from x = b to the other end at x = I. A second integration now shows 424 ON VIBRATIONS AND DEFLEXIONS OF [411 that Bcfrwlda? is a linear function of x between and 6, and again a linear function between 6 and I, the two linear functions assuming the same value at x = b. Since B is everywhere positive, it follows that the curvature cannot vanish more than twice in the whole range from to I, ends included, unless indeed it vanish everywhere over one of the parts. If one end be supported, the curvature vanishes there. If the other end also be supported, the curva- ture is of one sign throughout, and the curve of deflexion can nowhere cross the axis. If the second end be clamped, there is but one internal point of inflexion, and again the axis cannot be crossed. If both ends are clamped, the two points of inflexion are internal, but the axis cannot be crossed, since a crossing would involve three points of inflexion. If one end be free, the curvature vanishes there, and not only the curvature but also the rate of change of curvature. The part of the rod from this end up to the point of application of the force remains unbent and one of the linear functions spoken of is zero throughout. Thus the curvature never changes sign, and the axis cannot be crossed. In this case equilibrium requires that the other end be clamped. We conclude that in no case can there be a deflexion anywhere of opposite sign to that of the force applied at x = b, and the conclusion may be extended to a force, however distributed, provided that it be one-signed throughout. Leaving the problems presented by the membrane and the bar, we may pass on to consider whether similar propositions are applicable in the case of a flat plate, whose stiffness and density may be variable from point to point. An argument similar to that employed for the membrane shows that when the boundary is clamped any contraction of it is attended by a rise of pitch. But (Theory of Sound, 230) the statement does not hold good when the boundary is free. When a localized transverse force acts upon the plate, we may inquire whether the displacement is at all points in the same direction as the force. This question was considered in a former paper* in con- nexion with a hydrodynamical analogue, and it may be convenient to repeat the argument. Suppose that the plate (fig. 2), clamped at a distant boundary, is almost divided into two independent parts by a straight partition CD extending across, but perforated by a narrow aperture AB\ and that the force is applied at a distance from CD on the left. If the partition were complete, w and dwjdn would be zero over the whole (in virtue of the clamping), and the displacement in the neighbourhood on the left would be simple one-dimensional bend- ing, with w positive throughout. On the right w would vanish. In order to maintain this condition of things a certain couple acts upon Fi 2 the plate in virtue of the supposed constraints along CD. * Phil. Mag. Vol. xxxvi. p. 354 (1893); Scientific Papert, Vol. rv. p. 88. 1916] MEMBRANES, BARS, AND PLATES 425 Along the perforated portion AB the couple required to produce the one- dimensional bending fails. The actual deformation accordingly differs from the one-dimensional bending by the deformation that would be produced by a couple over AB acting upon the plate, as clamped along CA, BD, but other- wise free from force. This deformation is evidently symmetrical with change of sign upon the two sides of CD, w being positive on the left, negative on the right, and vanishing on AB itself. Thus upon the whole a downward force acting on the left gives rise to an upward motion on the right, in opposition to the general rule proposed for examination. If we suppose a load attached at the place where the force acts, but that otherwise the plate is devoid of mass, we see that a clamped plate vibrating freely in its gravest mode may have internal nodes in the sense that w is there evanescent, but of course not in the full sense of places which behave as if they were clamped. In the case of a plate whose boundary is merely supported, i.e. acted upon by a force (without couple) constraining w to remain zero*, it is still easier to recognize that a part of the plate may move in the direction opposite to that of an applied force. We may contemplate the arrangement of fig. 2, where, however, the partition CD is now merely supported and not clamped. Along the unperforated parts CA, BD the plate must be supposed cut through so that no couple is transmitted. And in the same way we infer that internal nodes are possible when a supported plate vibrates freely in its gravest mode. But although a movement opposite to that of the impressed force may be possible in a plate whose boundary is clamped or supported, it would seem that this occurs only in rather extreme cases when the boundary is strongly re-entrant. One may suspect that such a contrary movement is excluded when the boundary forms an oval curve, i.e. a curve whose curvature never changes sign. A rectangular plate comes under this description ; but according to M. Mesnagerf, "M. J. Resal a montr6 qu'en applicant une charge an centre d'une plaque rectangulaire de proportions convenables, on produit tres probable- ment le soulevement de certaines regions de la plaque." I understand that -the boundary is supposed to be " supported " and that suitable proportions are attained when one side of the rectangle is relatively long. It seems therefore desirable to inquire more closely into this question. The general differential equation for the equilibrium of a uniform elastic plate under an impressed transverse force proportional to Z isj =Z. ..................... (3) * It may be remarked that the substitution of a supported for a clamped boundary js equiva- lent to the abolition of a constraint, and is in consequence attended by a fall in the frequency of free vibrations. t C. E. t. CLXII. p. 826 (1916). J Theory of Sound, 215, 225 ; Love's Mathematical Theory of Elasticity, Chapter xxn. 426 ON VIBRATIONS AND DEFLEXIONS OF [411 We will apply this equation to the plate bounded by the lines y = 0, y = IT, and extending to infinity in both directions along x, and we suppose that external transverse forces act only along the line x 0. Under the operation of these forces the plate deflects symmetrically, so that w is the same on both sides of x = and along this line dw/dx = 0. Having formulated this condition, we may now confine our attention to the positive side, regarding the plate as bounded at x = 0. The conditions for a supported edge parallel to x are Q; ........................... (4) and they are satisfied at y = and y = TT if we assume that w as a function of y is proportional to sin ny, n being an integer. The same assumption intro- duced into (3) with Z= gives of which the general solution is w={(A + Bx)e- nx +(C + Dx)e nx }s\nny, ............... (6) where A, B, C, D, are constants. Since w when x = + ao , C and D must here vanish ; and by the condition to be satisfied when x = 0, B = nA. The solution applicable for the present purpose is thus w = A sin ny . (1 + nx) e* ......................... (7) The force acting at the edge x = necessary to maintain this displacement is proportional to . d 2 dw in virtue of the condition there imposed. Introducing the value of w from (7), we find that d s w/da*=2n*A sinny, ........................... (9) which represents the force in question. When n = 1, w = A sin y. (l+x)er*\ ........................ (10) and it is evident that w retains the same sign over the whole plate from x = to x =00. On the negative side (10) is not applicable as it stands, but we know that w has identical values at x. The solution expressed in (10) suggests strongly that Resal's expectation is not fulfilled, but two objections may perhaps be taken. In the first place the force expressed in (9) with n=l, though preponderant at the centre y = ^?r r is not entirely concentrated there. And secondly, it may be noticed that we have introduced no special boundary condition at x = oo . It might be argued that although w tends to vanish when x is very great, the manner of its evanescence may not exclude a reversal of sign. 1916] MEMBRANES, BAPS, AND PLATES 427 We proceed then to examine the solution for a plate definitely terminated at distances I, and there supported. For this purpose we resume the general solution (6), w = sinny{(A + Bx) e~ + (C + Dx) e}, ............... (11) which already satisfies the conditions of a supported edge at y = 0, y = TT. At x = 0, the condition is as before dw/dx = 0. At x=l the conditions for a supported edge give first w = 0, and therefore dhu/dy 2 = 0. The second con- dition then reduces to d 2 w/dx* = 0. Applying these conditions to (11) we find D = Be-**, C=-e~ Znl (A+2lB) ................ (12) It remains to introduce the condition to be satisfied at x = 0. In general and since this is to vanish when x = 0, -nA + B+nC + D = ......................... (14) By means of (12), (14) A,C,D may be expressed in terms of B, and we find + W - *> e ~ 2nl 1 + -**- I- < In (15) the square bracket is negative for any value of a; between and I, for it may be written in the form - xe~ (1 - e-2<a-*) } - (21 - x)e~ Znl {e nx - e-} .......... (16) When x = it vanishes, and when x = I it becomes - 2le~ 2nl (e nl - e-"0- It appears then that for any fixed value of y there is no change in the sign of dw/dx over the whole range from x=Q to x = l. And when n = l, this sign does not alter with y. As to the sign of w when x = 0, we have then from (11) g2nJ_g-2nZ w = sin ny(A + C} = B sin ny -- so that dwjdx in (15) has throughout the opposite sign to that of the initial value of w. And since w = when x = I, it follows that for every value of y the sign of w remains unchanged from x to x = I. Further, if n = 1, this sign is the same whatever be the value of y. Every point in the plate is deflected in the same direction. Let us now suppose that the plate is clamped at x = I, instead of merely supported. The conditions are of course w = 0, dw/dx = Q. They give (17) (18) The condition at x = is that already expressed in (14). * [The factor e" 1 has been omitted from the denominator; with l = <x> the corrected result agrees with (7) when x = 0, if B = nA. W. F. S.] 428 ON VIBRATIONS AND DEFLEXIONS OF [411 As before, A, C, D may be expressed in terms of B. For shortness we may set B = 1, and write H = I+e-(2nl-l) ................... . ..... (19) We find D = (2nJ + 1 - Thus -j? = sin ny [-"* (- nA + I - nx) + e (nC + D + nDx)] = H- 1 sin ny . e' [InWe' 1 - nx (1 + <r*"' (2nl - 1)}] f H~ l sin ny . e n(x ~^ [ 2nH" + nx {2nl + 1 vanishing when x = 0, and when x = I. This may be put into the form d^ju -r- = H- 1 sin ny [2n*l (I - x) e~ 2nl (e 1lx < )~\ ................ (20) in which the square bracket is positive from x = to x = I. It is easy to see that ^Talso is positive. When nl is small, (19) is positive, and it cannot vanish, since It remains to show that the sign of w follows that of sin ny when x = 0. In this case w = (A + C)smny; ........................... (21) and n(A+C)H=l- e~ Znl (2 + 4w 2 / 2 ) + e~ 4nl Znl 2nl ~ Znl - 2 - 4n 2 / 2 ) ................ (22)* The bracket on the right of (22) is positive, since We see then that for any value of y, the sign of dwfdx over the whole range from x = to x = I is the opposite of the sign of w when # = Of ; and since w = when a; = I, it follows that it cannot vanish anywhere between. When n = 1, w retains the same sign at x = whatever be the value of y, and therefore also at every point of the whole plate. No more in this case than when the edges at x = I are merely supported, can there be anywhere a deflexion in the reverse direction. In both the cases just discussed the force operative at x = to which the deflexion is due is, as in (8), proportional simply to d'w/da?, and therefore to * [Some corrections have been made in this equation. W. F. 8.] t This follows at once if we start from x- I where tr = 0. 1916] MEMBRANES, BARS, AND PLATES 429 sin ny, and is of course in the same direction as the displacement along the same line. When n = l, both forces and displacements are in a fixed direction. It will be of interest to examine what happens when the force is concentrated at a single point on the line a; = 0, instead of being distributed over the whole of it between y = and y = ir. But for this purpose it may be well to simplify the problem by supposing I infinite. On the analogy of (7) we take w = 2A n (l +nx)e- nx sin ny, (23) making, when x = 0, d'w/dx 3 = 2^n 3 A n sin ny (24) If, then, Z represent the force operative upon dy, analysable by Fourier's theorem into Z = Z l sin y + Z 2 sin 2y + Z 3 sin 3y -f . . ., (25) we have 2 /""' Z n = - Z sin ny dy = - Z- sin rnj, (26) 7T./0 7T if the force is concentrated at y = rj. Hence by (24) that (y i}) cos n(y n where n = 1, 2, 3, etc. It will be understood that a constant factor, depending upon the elastic constants and the thickness of the plate, but not upon n, has been omitted. The series in (28) becomes more tractable when differentiated. We have dw = xZ 1 ,^cosn(y-r ) )-cosn(y + 'r)) c _ nx , ^ dx 2ir n and the summations to be considered are of the form S^cosnySe-"* (30) This may be considered as the real part of 2w~ J e- n <*-*>, (31) that is, of - log (1-e- <*-#>) (32) Thus, if we take 2n- l e- n <*-*>=-X' + iT, (33) e -x-iv = l _ e -(x-iv ) an d e~ x+iY = 1 - -<*+*>, so that e- 2jr =l + e- 2 *-2e- a; cosy8 (34) 430 ON VIBRATIONS AND DEFLEXIONS OF [411 Accordingly 71-' cos n$ e-* = - log (1+g- 2 *- 2<r*cos); ......... (35) and dw _ x Z, . l+e-**-2e-* C os(y-r,) ~dx~ 47T g l+6- to -2e-*cos-Hi7 ' From the above it appears that W= a; log {1 + g- 2 * - 2e-* cos (y + 77)} = a; log h must satisfy V*TF = 0. This may readily be verified by means of VlogA = 0, and V 2 W = x V s log h + 2d log h/dx. We have now to consider the sign of the logarithm in (36), or, as it may be written, --- ( 3 Since the cosines are less than unity, both numerator and denominator are positive. Also the numerator is less than the denominator, for cos (y r))- cos (y + 77) = 2 sin y sin 77 = + , so that cos (y fj) > cos (y + 77). The logarithm is therefore negative, and dw/dx has everywhere the opposite sign to that of Z n . If this be supposed positive, iv on every line y= const, increases as we pass inwards from x = oo where w = Q to x = Q. Over the whole plate the displacement is positive, and this whatever the point of application (?) of the force. Obviously extension may be made to any distributed one-signed force. It may be remarked that since the logarithm in (37) is unaltered by a reversal of x, (36) is applicable on the negative as well as on the positive side of x = 0. If y = i), x = 0, the logarithm becomes infinite, but dw/dx is still zero in virtue of the factor x. I suppose that w cannot be expressed in finite terms by integration of (36), but there would be no difficulty in dealing arithmetically with particular cases by direct use of the series (28). If, for example, r\ = ^TT, so that the force is applied at the centre, we have to consider 2n- 8 sin \mr . sin ny . -"(! 4- nx) ................... (38) and only odd values of n enter. Further, (38) is symmetrical on the two sides of y = ^TT. Two special cases present themselves when x = and when y = TT. In the former w is proportional to sin r/-g- 3 sin 3y+ -sin5y-..., .................. (39) and in the latter to .......... (40) August 2, 1916. 1916] MEMBRANES, BARS, AND PLATES 431 Added August 21. The accompanying tables show the form of the curves of deflexion denned by (39), (40). y (39) y (39) oooo 50 7416 10 1594 60 8574 20 3162 70 9530 30 4675 80 1-0217 40 6104 90 1-0518 X (40) X (40) o-o 1-0518 3-0 1992 0'5 9333 4-0 0916 1-0 7435 5-0 0404 2-0 4066 10-0 0005 In a second communication * Mesnager returns to the question and shows by very simple reasoning that all points of a rectangular plate supported at the boundary move in the direction of the applied transverse forces. If z denote V 2 w, then V 2 ^, = V 4 w, is positive over the plate if the applied forces are everywhere positive. At a straight portion of the boundary of a supported plate z = 0, and this is regarded as applicable to the whole boundary of the rectangular plate, though perhaps the corners may require further con- sideration. But if V 2 2 is everywhere positive within a coutour and z vanish on the contour itself, z must be negative over the interior, as is physically obvious in the theory of the conduction of heat. Again, since V 2 w is negative throughout the interior, and w vanishes at the boundary, it follows in like manner that w is positive throughout the interior. It does not appear that an argument on these lines can be applied to a rectangular plate whose boundary is clamped, or to a supported plate whose boundary is in part curved. P.S. In connexion with a recent paper on the "Flow of Compressible Fluid past an Obstacle" (Phil. Mag. July 1916)f, I have become aware that the subject had been treated with considerable generality by Prof. Cisotti of Milan, under the title " Sul Paradosso di D'Alembert " (Atti R. Istituto Veneto, t. Ixv. 1906). There was, however, no reference to the limitation necessary when the velocity exceeds that of sound in the medium. I understand that this matter is now engaging Prof. Cisotti's attention. * C. R. July 24, 1916, p. 84. t [This volume, p. 402.] 412. ON CONVECTION CURRENTS IN A HORIZONTAL LAYER OF FLUID, WHEN THE HIGHER TEMPERATURE IS ON THE UNDER SIDE. [Philosophical Magazine, Vol. XXXII. pp. 529546, 1916.] THE present is an attempt to examine how far the interesting results obtained by Bnard* in his careful and skilful experiments can be explained theoretically. Benard worked with very thin layers, only about 1 mm. deep, standing on a levelled metallic plate which was maintained at a uniform temperature. The upper surface was usually free, and being in contact with the air was at a lower temperature. Various liquids were employed some, indeed, which would be solids under ordinary conditions. The layer rapidly resolves itself into a number of cells, the motion being an ascension in the middle of a cell and a descension at the common boundary between a cell and its neighbours. Two phases are distinguished, of unequal duration, the first being relatively very short. The limit of the first phase is described as the " semi-regular cellular regime " ; in this state all the cells have already acquired surfaces nearly identical, their forms being nearly regular convex polygons of, in general, 4 to 7 sides. The boundaries are vertical, and the circulation in each cell approximates to that already indicated. This phase is brief (1 or 2 seconds) for the less viscous liquids (alcohol, benzine, etc.) at ordinary temperatures. Even for paraffin or spermaceti, melted at 100 C., 10 seconds suffice; but in the case of very viscous liquids (oils, etc.), if the flux of heat is small, the deformations are extremely slow and the first phase may last several minutes or more. The second phase has for its limit a permanent regime of regular hexa- gons. During this period the cells become equal and regular and align Revue generate des Science*, Vol. xn. pp. 1261, 1309 .(1900); Ann. d. Chimie et de Phytique, t. xxiu. p. 62 (1901). M. Hi'- mini does not appear to be acquainted with James Thomson's paper "On a Changing Tesselated Structure in certain Liquids" (Proc. Glatgow Phil. Soc. 18812), where is described a like structure in much thicker layers of soapy water cooling from the surface. 1916] ON CONVECTION CURRENTS IN A HORIZONTAL LAYER OF FLUID 433 themselves. It is extremely protracted, if the limit is regarded as the complete attainment of regular hexagons. And, indeed, such perfection is barely attainable even with the most careful arrangements. The tendency, however, seems sufficiently established. The theoretical consideration of the problem here arising is of interest for more than one reason. In general, when a system falls away from unstable equilibrium it may do so in several principal modes, in each of which the departure at time t is proportional to the small displacement or velocity supposed to be present initially, and to an exponential factor e?', where q is positive. If the initial disturbances are small enough, that mode (or modes) of falling away will become predominant for which q is a maxi- mum. The simplest example for which the number of degrees of freedom is infinite is presented by a cylindrical rod of elastic material under a longitudinal compression sufficient to overbalance its stiffness. But perhaps the most interesting hitherto treated is that of a cylinder of fluid disinte- grating under the operation of capillary force as in the beautiful experiments of Savart and Plateau upon jets. In this case the surface remains one of revolution about the original axis, but it becomes varicose, and the question is to compare the effects of different wave-lengths of varicosity, for upon this depends the number of detached masses into which the column is eventually resolved. It was proved by Plateau that there is no instability if the wave- length be less than the circumference of the column. For all wave-lengths greater than this there is instability, and the corresponding modes of dis- integration may establish themselves if the initial disturbances are suitable. But if the general disturbance is very small, those components only will have opportunity to develop themselves for which the wave-length lies near to that of maximum instability. It has been shown* that the wave-length of maximum instability is 4-508 times the diameter of the jet, exceeding the wave-length at which instability first enters in the ratio of about 3 : 2. Accordingly this is the sort of disintegration to be expected when the jet is shielded as far as possible from external disturbance. It will be observed that there is nothing in this theory which could fix the phase of the predominant disturbance, or the particular particles of the fluid which will ultimately form the centres of the detached drops. There remains a certain indeterminateness, and this is connected with the circum- stance that absolute regularity is not to be expected. In addition to the wave-length of maximum instability we must include all those which lie sufficiently near to it, and the superposition of the corresponding modes will allow of a slow variation of phase as we pass along the column. The phase * Proc. Lond. Math. Soc. Vol. x. p. 4 (1879) ; Scientific Papers, Vol. i. p. 361. Also Theory of Sound, 2nd ed. 357, &c. 11 VT 28 434 ON CONVECTION CURRENTS IN A [412 in any particular region depends upon the initial circumstances in and near that region, and these are supposed to be matters of chance*. The super- position of infinite trains of waves whose wave-lengths cluster round a given value raises the same questions as we are concerned with in considering the clumicter of approximately homogeneous light. In the present problem the case is much more complicated, unless we arbitrarily limit it to two dimensions. The cells of Benard are then reduced to infinitely long strips, arid when there is instability we may ask for what wave-length (width of strip) the instability is greatest. The answer can be given under certain restrictions, and the manner in which equilibrium breaks down is then approximately determined. So long as the two-dimensional character is retained, there seems to be no reason to expect the wave-length to alter afterwards. But even if we assume a natural disposition to a two- dimensional motion, the direction of the length of the cells as well as the phase could only be determined by initial circumstances, and could not be expected to be uniform over the whole of the infinite plane. According to the observations of Be*nard, something of this sort actually occurs when the layer of liquid has a general motion in its own plane at the moment when instability commences, the length of the cellular strips being parallel to the general velocity. But a little later, when the general motion has decayed, division-lines running in the perpendicular direction present themselves. In general, it is easy to recognize that the question is much more complex. By Fourier's theorem the motion in its earlier stages may be analysed into components, each of which corresponds to rectangular cells whose sides are parallel to fixed axes arbitrarily chosen. The solution for maximum instability yields one relation between the sides of the rectangle, but no indication of their ratio. It covers the two-dimensional case of infinitely long rectangles already referred to, and the contrasted case of squares for which the length of the side is thus determined. I do not see that any plausible hypothesis as to the origin of the initial disturbances leads us to expect one particular ratio of sides in preference to another. On a more general view it appears that the function expressing the dis- turbance which develops most rapidly may be assimilated to that which represents the free vibration of an infinite stretched membrane vibrating with given frequency. The calculations which follow are based upon equations given by Bous- sinesq, who has applied them to one or two particular problems. The special limitation which characterizes them is the neglect of variations of density, * When a jet of liquid is acted on by an external vibrator, the reiolution into drops may be regularized in a much higher degree. 1916] HORIZONTAL LAYER OF FLUID 435 except in so far as they modify the action of gravity. Of course, such neglect can be justified only under certain conditions, which Boussinesq has dis- cussed. They are not so restrictive as to exclude the approximate treatment of many problems of interest. When the fluid is inviscid and the higher temperature is below, all modes of disturbance are instable, even when we include the conduction of heat during the disturbance. But there is one class of disturbances for which the instability is a maximum. When viscosity is included as well as conduction, the problem is more complicated, and we have to consider boundary conditions. Those have been chosen which are simplest from the mathematical point of view, and they deviate from those obtaining in Benard's experiments, where, indeed, the conditions are different at the two boundaries. It appears, a little un- expectedly, that the equilibrium may be thoroughly stable (with higher temperature below), if the coefficients of conductivity and viscosity are not too small. As the temperature gradient increases, instability enters, and at first only for a particular kind of disturbance. The second phase of Benard, where a tendency reveals itself for a slow transformation into regular hexagons, is not touched. It would seem to demand the inclusion of the squares of quantities here treated as small. But the size of the hexagons (under the boundary conditions postulated) is determinate, at any rate when they assert themselves early enough. A.n appendix deals with a related analytical problem having various physical interpretations, such as the symmetrical vibration in two dimensions of a layer of air enclosed by a nearly circular wall. The general Eulerian equations of fluid motion are in the usual nota- tion : Du I dp Dv v I dp Dw _ 1 dp m = -~' Dt~ ~~d' Dt~ -dz" D whore and X, Y, Z are the components of extraneous force reckoned per unit of mass. If, neglecting viscosity, we suppose that gravity is the only impressed force, X = 0, F=0, Z=-g, ..................... (3) z being measured upwards. In equations (1) p is variable in consequence of variable temperature and variable pressure. But, as Boussinesq* has shown, in the class of problems under consideration the influence of pressure is * Thlorie Analytique de la Chaleiir, t. n. p. 172 (1903). 282 436 ON CONVECTION CURRENTS IN A [412 unimportant and even the variation with temperature may be disregarded except in so far as it modifies the operation of gravity. If we write p = p + &p, we have 9P = 9Po U + &P/PO) = ffPo - 9Po&0, where , 0/is the temperature reckoned from the point where p = p and o is the coefficient of expansion. We may now identify p in (1) with p , and our equations become Du__\dP Z>w__ldP &^__\dP * Dt~ pdx' Dt~ p dy' Dt ~ p dz + 7 ' where p is a constant, 7 is written for go., and P for p + gpz. Also, since the fluid is now treated as incompressible, + * + *_a.. ...(5) dx dy dz The equation for the conduction of heat is in which K is the diffusibility for temperature. These are the equations employed by Boussinesq. In the particular problems to which we proceed the fluid is supposed to be bounded by two infinite fixed planes at z = 0and z = %, where also the temperatures are maintained constant. In the equilibrium condition u, v, w vanish and 9 being a function of z only is subject to d^d jdz* = 0, or d6jdz = ft, where ft is a constant representing the temperature gradient. If the equi- librium is stable, ft is positive ; and if unstable with the higher temperature below, ft is negative. It will be convenient, however, to reckon as the departure from the equilibrium temperature . The only change required in equations (4) is to write is for P, where dz (7) In equation (6) DO/Dt is to be replaced by D0/Dt + wj3. The question with which we are principally concerned is the effect of a small departure from the condition of equilibrium, whether stable or un- stable. For this purpose it suffices to suppose u, v, w, and 6 to be small. When we neglect the squares of the small quantities, D/Dt identifies itself with d/dt and we get du Icfer dv I dvr dw I dvr 1916] HORIZONTAL LAYER OF FLUID 437 which with (5) and the initial and boundary conditions suffice for the solution of the problem. The boundaiy conditions are that w 0, 6 = 0, when z = or We now assume in the usual manner that the small quantities are proportional to W*vtf*, ................................. (10) so that (8), (5), (9) become iltff iirns 1 d-sr n nu = -- , nv = -- , nw = --- ; t-fyo .......... (11) p p p dz Q, ........................ (12) m 2 )0, .................. (13) from which by elimination of u, v, nr, we derive n d 2 w Having regard to the boundary conditions to be satisfied by w and 0, we now assume that these quantities are proportional to sinsz, where s = q7r/^, and q is an integer. Hence 0=0, (15) ! )0 = 0, (16) and the equation determining n is the quadratic n 2 (I 2 + m 2 + s 2 ) + UK (I 2 + m 2 + s 2 ) 2 + 7 (I 2 + m 2 ) = (17) When K = 0, there is no conduction, so that each element of the fluid retains its temperature and density. If /3 be positive, the equilibrium is stable, and *- /rv.r. =P. ( is > indicating vibrations about the condition of equilibrium. If, on the other hand, /3 be negative, say /3', _ ' V!* 2 + m* + * 2 } When n has the positive value, the corresponding disturbance increases exponentially with the time. For a given value of l 2 + m 2 , the numerical values of n diminish without limit as s increases that is, the more subdivisions there are along z. The greatest value corresponds with q = 1 or s = 7r/'. On the other hand, if s be given, j n \ increases from zero as I 2 + m 2 increases from zero (great wave- lengths along x and y} up to a finite limit when I 2 + m 2 is large (small wave- lengths along a; and y). This case of no conductivity falls within the scope 438 ON CONVECTION CURRENTS IN A [412 of a former investigation where the fluid was supposed from the beginning to be incompressible but of variable density *. Returning to the consideration of a finite conductivity, we have again to distinguish the cases where /? is positive and negative. When ft is negative (higher temperature below) both values of n in (17) are real and one is positive. The equilibrium is unstable for all values of I* + m? and of s. If ft be positive, n may be real or complex. In either case the real part of n is negative, so that the equilibrium is stable whatever I- + m 2 and s may be. When ft is negative ( ft), it is important to inquire for what values of I 2 + m- the instability is greatest, for these are the modes which more and more assert themselves as time elapses, even though initially they may be quite subordinate. That the positive value of n must have a maximum appears when we observe it tends to vanish both when I* + m 2 is small and also when I 2 + m 2 is large. Setting for shortness I 2 + m 2 + s 2 = a, we may write (17) 7i 2 o- + w*a 2 -/3 / 7 (<r-s 2 ) = 0, (20) and the question is to find the value of a for which n is greatest, s being supposed given. Making dn/da = 0, we get on differentiation tt 2 +2rtK<r-/8'7 = 0; (21) and on elimination of ?i 2 between (20), (21) Using this value of n in (21), we find as the equation for <r When K is relatively great, 0- = 2s 2 , or Z 2 + 7H 2 = S 2 (24) A second approximation gives p + w a = ^ + |L2. (25) The corresponding value of n is Q'~ ( Q'~. \ (26) The modes of greatest instability are those for which s is smallest, that is equal to TT/, and * Proc. Lond. Math. Soc. Vol. nv. p. 170 (1883) ; Scientific Papers, Vol. n. p. 200. 1916] HORIZONTAL LAYER OF FLUID 439 For a two-dimensional disturbance we may make ra = and where X is the wave-length along a;. The X of maximum instability is thus approximately X=2f ................................. (28) Again, if I = m = 2ir/\, as for square cells, X=2x/2. ................................. (29) greater than before in the ratio V2 : 1. We have considered especially the cases where K is relatively small and relatively large. Intermediate cases would need to be dealt with by a numerical solution of (23). When w is known in the form w= We ilx e im ysinsz.e nt , ........................... (30) n being now a known function of I, m, s, u and v are at once derived by means of (11) and (12). Thus il dw im dw M = P+^-dJ v = ^?Tz ................... (31 The connexion between w and 6 is given by (15) or (16). When fi is negative and n positive, 6 and w are of the same sign. As an example in two dimensions of (30), (31), we might have in real form u W cos x . sin z . e nt ......................... (32) u = ~ Wsinz.cosz.e nt , v=0 ................... (33) Hitherto we have supposed the fluid to be destitute of viscosity. When we include viscosity, we must add v (V 2 zi, V 2 v, V*w) on the right of equations (1), (8), and (11), v being the kinematic coefficient. Equations (12) and (13) remain unaffected. And in (11) V 2 = d*/dz* - I 2 - m 2 ............. < ............... (34) We have also to reconsider the boundary conditions at z = and z = We may still suppose B = and w = ; but for a further condition we should probably prefer dw/dz = Q, corresponding to a fixed solid wall*. But this entails much complication, and we may content ourselves with the sup- position d*w/dz* = Q, which (with w = 0) is satisfied by taking as before w proportional to sin sz with s = q^l^. This is equivalent to the annulment of lateral forces at the wall. For (Lamb's Hydrodynamics, 323, 326) these forces are expressed in general by dw du dw dv * [It would appear that the immobility and solidity of the walls are sufficiently provided for by the condition w = 0, and that for " a fixed solid wall " there should be substituted " no slipping at the walls." W. F. S.] 440 ON CONVECTION CURRENTS IN A [412 \ while here / = at the boundaries requires also dwldx = 0, dw/dy=0. Hence, at the boundaries, d?u/dxdz, cPv/dydz vanish, and therefore by (5), d^w/dz*. Equation (15) remains unaltered : /3w + {n + tc(l* + m* + f?)}0 = 0, (15) and (16) becomes {n + v(l t + m* + 8*)}(l* + m? + s?)w-y(l* + m*)e = (36) Writing as before <r t* + ra j + s 2 , we get the equation in n (N + /c(r)(H + i>o-)<7 + 7(- > + w a ) = 0, (37) which takes the place of (17). If 7 = (no expansion with heat), the equations degrade and we have two simple alternatives. In the first n + K<T = with w = 0, signifying con- duction of heat with no motion. In the second n + vcr = 0, when the relation between w and 6 becomes /3w + <r(tc-v)e = Q (38) In both cases, since n is real and negative, the disturbance is stable. If we neglect K, in (37), the equation takes the same form (20) as that already considered when i/ = 0. Hence the results expressed in (22), (23), (24), (25), (26), (27) are applicable with simple substitution of v for K. In the general equation (37) if ft be positive, as 7 is supposed always to be, the values of n may be real or complex. If real they are both negative, and if complex the real part is negative. In either case the disturbance dies down. As was to be expected, when the temperature is higher above, the equilibrium is stable. In the contrary case when ft is negative ( ft') the roots of the quadratic are always real, and one at least is negative. There is a positive root only when 7 (/ 2 + w 2 ) > KVO* (39) If K. or v, vanish there is instability ; but if K and v are finite and large enough, the equilibrium for this disturbance is stable, although the higher temperature is underneath. Inequality (39) gives the condition of instability for the particular dis- turbance (I, m, s). It is of interest to inquire at what point the equilibrium becomes unstable when there is no restriction upon the value of I* + m*. In the equation '7 (fi + w 2 ) - KVO* = 7 (a - s 2 ) - KVO* = 0, (40) we see that the left-hand member is negative when I 3 + m? is small and also when it is large. When the conditions are such that the equation can only just be satisfied with some value of I* + ?n 2 , or <r, the derived equation (41) 1916] HORIZONTAL LAYER OF FLUID 441 must also hold good, so that F + w a = s 2 , ..................... (42) and #7 = 27*1^/4 ............................... (43) Unless ft'y exceeds the value given in (43) there is no instability, however I and m are chosen. But the equation still contains s, which may be as large as we please. The smallest value of s is w/f, The condition of instability when I, m, and s are all unrestricted is accordingly If $'7 falls below this amount, the equilibrium is altogether stable. I am not aware that the possibility of complete stability under such circumstances has been contemplated. To interpret (44) more conveniently, we may replace /3' by ( 2 i and 7 by g (p 2 - p,)/p, (0 2 - @0*> so that /?7 = , ........................... (45) PI where @ 2 , i> pz> and p are the extreme temperatures and densities in equilibrium. Thus (44) becomes Pi In the case of air at atmospheric conditions we may take in C.G.S. measure v = '14, and K = \ v (Maxwell's Theory). Also g = 980, and thus For example, if " = 1 cm., instability requires that the density at the top exceed that at the bottom by one-thirtieth part, corresponding to about 9 C. of temperature. We should not forget that our method postulates a small value of (pz-p^/p^ Thus if icv be given, the application of (46) may cease to be legitimate unless be large enough. It may be remarked that the influence of viscosity would be increased were we to suppose the horizontal velocities (instead of the horizontal forces) to be annulled at the boundaries. The problem of determining for what value of I 2 + m\ or a, the instability, when finite, is a maximum is more complicated. The differentiation of (37) with respect to a- gives ri* + 2n<r(tc + v) + 3i/<r 2 - '7 = 0, .................. (48) /3V 2 ~ whence n= , ........................... (49) (7* (K 4- v) * [If pj is taken to correspond to 0j , and p., to 9 2 , "ft -ft." must be substituted for "pj- throughout this page. W. F. S.] 442 ON CONVECTION CURRENTS IN A [412 expressing n in terms of a. To find <r we have to eliminate n between (48) and (49). The result is <rKi> (K - v? + <r 4 0'y (tc + i>)* - a 3 . 2/S V ( K * + "') ~ ^V* 4 = . -( 50 ) from which, in particular cases, a- could be found by numerical computation. From (50) we fall back on (23) by supposing i>=0, and again on a similar equation if we suppose K = 0. But the case of a nearly evanescent n is probably the more practical. In an experiment the temperature gradient could not be established all at once and we may suppose the progress to be very slow. In the earlier stages the equilibrium would be stable, so that no disturbance of importance would occur until n passed through zero to the positive side, corresponding to (44) or (46). The breakdown thus occurs for s = irj and by (42) I* + w 8 = Tr 8 / 2*. And since the evanescence of n is equivalent to the omission of djdt in the original equations, the motion thus determined has the character of a steady motion. The constant multiplier is, however, arbitrary ; and there is nothing to determine it so long as the squares of u, v, w, ft are neglected. In a particular solution where w as a function of x and y has the simplest form, say w = 2 cos x . cos y, (51) the particular coefficients of x and y which enter have relation to the par- ticular axes of reference employed. If we rotate these axes through an angle <f>, we have w = 2 cos \x cos <j> y' sin $} . cos {x' sin <f> + y' cos <} = cos \x' (cos < sin <f>)} . cos \y' (cos <f> + sin <)} + sin {x' (cos <j> - sin <)} . sin {y' (cos </> + sin <f>)} + cos \x (cos <j> 4- sin <)} . cos {y' (cos $ - sin $)} - sin \x' (cos < + sin </>)} . sin \y (cos <f> sin <)} (52) For example, if = |TT, (52) becomes w = cos(yV2) + cos(a?V2) (53) It is to be observed that with the general value of <f>, if we call the coefficients of x', y 1 ', I and m respectively, we have in every part I* + m 2 = 2, unaltered from the original value in (51). The character of w, under the condition that all the elementary terms of which it is composed are subject to I 2 + wt 2 = constant (& 2 ), is the same as for the transverse displacement of an infinite stretched membrane, vibrating with one definite frequency. The limitation upon w is, in fact, merely that it satisfies (d*/da* + d*ldy*+k*)w = (54) The character of w in particular solutions of the membrane problem is naturally associated with the nodal system (w = 0), where the membrane may be regarded as held fast ; and we may suppose the nodal system to divide 1916] HORIZONTAL LAYER OF FLUID 443 the plane into similar parts or cells, such as squares, equilateral triangles, or regular hexagons. But in the present problem it is perhaps more appropriate to consider divisions of the plane with respect to which w is symmetrical, so that dw/dn is zero on the straight lines forming the divisions of the cells. The more natural analogy is then with the two-dimensional vibration of air, where w represents velocity-potential and the divisions may be regarded as fixed walls. The simplest case is, of course, that in which the cells are squares. If the sides of the squares be 2?r, we may take with axes parallel to the sides and origin at centre w = cos x + cos y, ........................... (55) being thus composed by superposition of two parts for each of which A?= 1. This makes dw/dx = sin#, vanishing when a; = IT. Similarly, dw/dy vanishes when y = + TT, so that the sides of the square behave as fixed walls. To find the places where w changes sign, we write it in the form os, ..................... (56) Fig l giving x + y = Tr,x-y=-jr, lines which constitute the inscribed square (fig. 1). Within this square w has one sign (say +) and in the four right- angled triangles left over the sign. When the whole plane is considered, there is no want of symmetry between the + and the regions. The principle is the same when the elemen- tary cells are equilateral triangles or hexagons; but I am not aware that an analytical solution has been obtained for these cases. An experi- mental determination of & 2 might be made by observing the time of vibration under gravity of water contained in a trough with vertical sides and of corresponding section, which depends upon the same differential equation and boundary conditions*. The particular vibration in question is not the slowest possible, but that where there is a simultaneous rise at the centre and fall at the walls all round, with but one curve of zero elevation between. In the case of the hexagon, we may regard it as deviating comparatively little from the circular form' and employ the approximate methods then applicable. By an argument analogous to that formerly developed! for the boundary condition w = 0, we may convince ourselves that the value of k* for the hexagon cannot differ much from that appropriate to a circle of the same area. Thus if a be the radius of this circle, k is given by JJ (ka) = 0, * See Phil. Mag. Vol. i. p. 257 (1876) ; Scientific Papers, Vol. i. pp. 265, 271. t Theory of Sound, 209 ; compare also 317. See Appendix. 444 ON CONVECTION CURRENTS IN A [412 / being the Bessel's function of zero order, or ka = 3'832. If b be the side of the hexagon, a a = 3 V3 . b*/2ir. APPENDIX. On the nearly symmetrical solution for a nearly circular area, when w satisfies (d*/da? + d*/dy* + k*) w = and makes div/dn = on the boundary. Starting with the true circle of radius a, we have w a function of r (the radius vector) only, and the solution is w = J (kr) with the condition J ' (ka) = 0, yielding ka = 3'832, which determines & if a be given, or a if k be given. In the problem proposed the boundary is only approximately circular, so that we write r = a + p, where a is the mean value and p = a, cos 6 + fii sin 6 + . . . + a n cos nd + @ n sinn0 .......... (57) In (57) 6 is the vectorial angle and t etc. are quantities small relatively to a. The general solution of the differential equation being w = A J (kr) + J l (kr) {A l cos 6 + B 1 sin 0} + ...+J n (kr) {A n cos n0 + B n sin n0], . . .(58) we are to suppose now that A lt etc., are small relatively to A . It remains to consider the boundary condition. If <f> denote the small angle between r and the normal dn measured outwards, dw dw dw . sin*. ..................... (59) and ten0 = = = (-a n sinw0 + /8 n cosn0) ......... (60) with sufficient approximation, only the general term being written. In formulating the boundary condition dwldn=Q correct to the second order of small quantities, we require dw/dr to the second order, but dw/dB to the first order only. We have i d ~ = ^ (J ' (ka) + kpJ " (ka) + PW" (ka)} + [J n f (ka) + kpj n " (ka)} [A n cos nd + B n sin n0|, ~30 = I J n (ka) {- A n sin nd + B n cos nd} and for the boundary condition, setting ka = z and omitting the argument in the Bessel's functions, A (Jo' . cos <f> + kp Jo" + + {J n ' + kpj n "} [A n cos nB + B n sin nB] - /{- A n sin nB + B n cos nB} {- a n sin nB + /3 n cos nB\ = 0. (61) 1916] HORIZONTAL LAYER OF FLUID 445 If for the moment we omit the terms of the second order, we have A J ' + kA J " [a n cos n0 + @ n sin n6] + J n ' {A n cos nO -I- B n sin n0} = ; (62) so that JQ (z) 0, and kA J".ctn + Jn .A n = 0, kA J " . n + </'. B n = ....... (63) To this order of approximation z, = ka, has the same value as when p 0; that is to say, the equivalent radius is equal to the mean radius, or (as we may also express it) k may be regarded as dependent upon the area only. Equations (63) determine A n> B n in terms of the known quantities o n , n . Since </' is a small quantity, cos $ in (61) may now be omitted. To obtain a corrected evaluation of z, it suffices to take the mean of (61) for all values of 6. Thus A. {2 ^ + P 2 /,,"' K 2 + /3 2 )j + [kJ n " - rfJnlaz] K A n + $ n B n ] = 0, or on substitution of the approximate values of A n , B n from (63), J.' = VfW + A.") jg (/."-=) - ~f\ ............. (64) This expression may, however, be much simplified. In virtue of the general equation for J n , and since here J ' = approximately, J " = _ J , J"' = - 2-1 J " = z-i J . Thus / / (^) = P 2 / .S(an 2 +^ n 2 ) / + ~> ............... (65) the sign of summation with respect to n being introduced. Let us now suppose that a + da is the equivalent radius, so that 7 ' (ka + kda) = 0, that is the radius of the exact circle which corresponds to the value of k appropriate to the approximate circle. Then and Again, if a + da' be the radius of the true circle which has the same area as the approximate circle da' = ^ 2 (a M 2 + '), ........................ (67) and daj '- da = -l!^^ ........ (68) za J n (z) where z is the first root (after zero) of /,' (z) = 0, viz. 3'832. 446 ON CONVECTION CURRENTS IN A HORIZONTAL LAYER OF FLUID [412 The question with which we are mainly concerned is the sign of da' - da for the various values of n. When n = 1, Jj (z) = J ' (z) = 0, so that da = da', a result which was to be expected, since the terms in cti,j3i represent approxi- mately a displacement merely of the circle, without alteration of size or shape. We will now examine the sign of /,//' when n = 2, and 3. For this purpose we may employ the sequence equations 2n "n-H = ~ J ~ J 1> Jn = jJn-i ~~ jJ+i> which allow J n and J n ' to be expressed in terms of J^ and J 0) of which the former is here zero. We find J. 2 =-J , J 3 = - 4s- 1 J 0) J 4 = (1 - 24*-') / ; J/ = J , J 2 ' = 2*-J , J t ' = (12z~*-l)J a . Th,, Jl - J * z J * 4z 7/-' J7 = ~2' J7 = *rri2' whence on introduction of the actual value of z, viz. 3'832, we see that J 2 /J 2 ' is negative, and that J 3 /J 3 ' is positive. When n > z> it is a general proposition that J n (z) and J n ' (z) are both positive*. Hence for ?i = 4 and onwards, J,,/J U ' is positive when = 3*832. We thus arrive at the curious conclusion that when n = 2, da > da, as happens for all values of n (exceeding unity) when the boundary condition is tv = 0, but that when n > 2, da' < da. The existence of the exceptional case n = 2 precludes a completely general statement of the effect of a de- parture from the truly circular form ; but if the terms for which n = 2 are absent, as they would be in the case of any regular polygon with an even number of sides, regarded as a deformed circle, we may say that da' < da. In the physical problems the effect of a departure from the circular form is then to depress the pitch when the area is maintained constant (da' = 0). But for an elliptic deformation the reverse is the case. At first sight it may appear strange that an elliptic deformation should be capable of raising the pitch. But we must remember that we are here dealing with a vibration such that the phase at both ends of the minor axis is the opposite of that at the centre. A parallel case which admits of com- plete calculation is that of the rectangle regarded as a deformed square, and vibrating in the gravest symmetrical modef. It is easily shown that a de- parture from the square form raises the pitch. Of course, the one-dimensional vibration parallel to the longer side has its pitch depressed. [1918. This problem had already been treated by Aichi (Proc. Tokio Math.-Phys. Soc. 1907).] * See, for example, Theory of Sound, 210. t Theory of Sound, 267 (p = g = 2). 413. ON THE DYNAMICS OF REVOLVING FLUIDS. [Proceedings of the Royal Society, A, Vol. xcm. pp. 148154, 1916.] So much of meteorology depends ultimately upon the dynamics of revolving fluid that it is desirable to formulate as clearly as possible such simple con- clusions as are within our reach, in the hope that they may assist our judgment when an exact analysis seems impracticable. An important contribution to this subject is that recently published by Dr Aitken*. It formed the starting point of part of the investigation which follows, but I ought perhaps to add that I do not share Dr Aitken's views in all respects. His paper should be studied by all interested in these questions. As regards the present contribution to the theory it may be well to premise that the limitation to symmetry round an axis is imposed throughout. The motion of an inviscid fluid is governed by equations of which the first expressed by rectangular coordinates may be written du , du' , du' , du' dP -JT + U'-J- +v -j- + w' -j- =--j- , (1) dt dx dy dz dx where -jdp/p+V, (2) and V is the potential of extraneous forces. In (2) the density p is either a constant, as for an incompressible fluid, or at any rate a known function of the pressure p. Referred to cylindrical coordinates r, 6, z, with velocities u, v, iv, reckoned respectively in the directions of r, 6, z increasing, these equations become f du du f du v\ du dP dv dv i dv u\ dv dP X + *X + '(f3 + -r) +W airii dw dw dw dw dP * "The Dynamics of Cyclones and Anticyclones. Part 3," Boy. Soc. Edin. Proc. Vol. xxxvi. p. 174 (1916). t Compare Basset's Hydrodynamics, 19. 448 ON THE DYNAMICS OF REVOLVING FLUIDS [413 For the present purpose we assume symmetry with respect to the axis of z, so that u, v, w, and P (assumed to be single-valued) are independent of 6. So simplified, the equations become du du v* du dP -j7 + u- r *-- + w- r = --j-, (6) dt dr r dz dr dv dv uv dv a + . jj , + _ + . a ..O (7) dw dw . dw dP of which the second may be written /d d d\, ( :n + M "j" + w-r~ I (rv)= 0, (9) \dt dr dz) signifying that (n>) may be considered to move with the fluid, in accordance with Kelvin's general theorem respecting "circulation." If r , v , be the initial values of r, v, for any particle of the fluid, the value of v at any future time when the particle is at a distance r from the axis is given by rv = r v . Respecting the motion expressed by v, w, we see that it is the same as might take place with v = 0, that is when the whole motion is in planes passing through the axis, provided that we introduce a force along r equal to v*/r. We have here the familiar idea of " centrifugal force," and the conclusion might have been arrived at immediately, at any rate in the case where there is no (u, w) motion. It will be well to consider this case (u = 0, w = 0) more in detail. The third equation (8) shows that P is then independent of z, that is a function of r (and t) only. It follows from the first equation (6) that v also is a function of r only, and P = Iv^dr/r. Accordingly by (2) (10) If V, the potential of impressed forces, is independent of z, so also will be p and p, but not otherwise. For example, if gravity (g) act parallel to z (measured downwards), (11) gravity and centrifugal force contributing independently. In (11) p will be constant if the fluid is incompressible. For gases following Boyle's law a'(logp, or log p) = C + gz+jv*dr/r ................ (12) 1916] ON THE DYNAMICS OF REVOLVING FLUIDS 449 At a constant level the pressure diminishes as we pass inwards. But the corresponding rarefaction experienced by a compressible fluid does not cause such fluid to ascend. The heavier part outside is prevented from coming in below to take its place by the centrifugal force*. The condition for equilibrium, taken by itself, still leaves v an arbitrary function of r, but it does not follow that the equilibrium is stable. In like manner an incompressible liquid of variable density is in equilibrium under gravity when arranged in horizontal strata of constant density, but stability requires that the density of the strata everywhere increase as we pass down- wards. This analogy is, indeed, very helpful for our present purpose. As the fluid moves (u and iv finite) in accordance with equations (6), (7), (8), (vr) remains constant (k) for a ring consisting always of the same matter, and v*/r = fr/r 3 , so that the centrifugal force acting upon a given portion of the fluid is inversely as r 3 , and thus a known function of position. The only difference between this case and that of an incompressible fluid of variable density, moving under extraneous forces derived from a potential, is that here the inertia concerned in the (u, w) motion is uniform, whereas in a variably dense fluid moving under gravity, or similar forces, the inertia and the weight are proportional. As regards the question of stability, the difference is immaterial, and we may conclude that the equilibrium of fluid revolving one way round in cylindrical layers a*nd included between coaxial cylindrical walls is stable only under the condition that the circulation (k) always in- creases with r. In any portion where k is constant, so that the motion is there " irrotational," the equilibrium is neutral. An important particular case is that of fluid moving between an inner cylinder (r = a) revolving with angular velocity &> and an outer fixed cylinder (r = b). In the absence of viscosity the rotation of the cylinder is without effect. But if the fluid were viscous, equilibrium would require f k = vr = a?u (b n - - r 2 )/(6 2 - a 2 ), expressing that the circulation diminishes outwards. Accordingly a fluid without viscosity cannot stably move in this manner. On the other hand, if "it be the outer cylinder that rotates while the inner is at rest, k = vr = 6 2 w (r 2 - a 2 )/(6 2 - a 2 ), and the motion of an inviscid fluid according to this law would be stable. We may also found our argument upon a direct consideration of the kinetic energy (T) of the motion. For T is proportional to \v*rdr, or * When the fluid is viscous, the loss of circulation near the bottom of the containing vessel modifies this conclusion, as explained by James Thomson. t Lamb's Hydrodynamics, 333. R. VI. 29 450 ON THE DYNAMICS OF REVOLVING FLUIDS [413 Suppose now that two rings of fluid, one with k = k v and ?- = ?'i and the other with k = k t and r = r 2 , where ?- 2 > i\, and of equal areas rfr^ or dr, are inter- changed. The corresponding increment in T is represented by (rfr, = dr*) {*,/', + kf/rf - h'/rf - k a */r t '\ and is positive if k. 2 *>ki*', so that a circulation always increasing outwards makes T a minimum and thus ensures stability. The conclusion above arrived at may appear to conflict with that of Kelvin*, who finds as the condition of minimum energy that the vorticity, proportional to r~ l dk/dr, must increase outwards. Suppose, for instance, that k = r*, increasing outwards, while r^dk/dr decreases. But it would seem that the variations contemplated differ. As an example, Kelvin gives for maximum energy v = r from r to r = b, v = b*/r from r = b to r = a ; and for minimum energy v = from r = to r = v / (a 2 -6 2 ), v = r (a 2 b z )/r from r = ^(a z b' 2 ) to r = a. In the first case l*m*dr = 1 bl (2a 2 - b"), Jo and in the second case I vr*dr=b<; Jo so that the moment of momentum differs in the two cases. In fact Kelvin supposes operations upon the boundary which alter the moment of momentum. On the other hand, he maintains the strictly two-dimensional character of the admissible variations. In the problem that I have considered, symmetry round the axis is maintained and there can be no alteration in the moment of momentum, since the cylindrical walls are fixed. But the variations by which the passage from one two-dimensional condition to another may be effected are not themselves two-dimensional. The above reasoning suffices to fix the criterion for stable equilibrium ; but, of course, there can be no actual transition from a configuration of unstable equilibrium to that of permanent stable equilibrium without dissipative forces, any more than there could be in the case of a heterogeneous liquid under gravity. The difference is that in the latter case dissipative forces exist in any real fluid, so that the fluid ultimately settles down into stable equilibrium, it may be after many oscillations. In the present problem ordinary viscosity does not meet the requirements, as it would interfere with the constancy of the circulation of given rings of fluid on which our reasoning depends. But Nature, Vol. xxm. October, 1880 ; Collected Papers, Vol. iv. p. 175. 1916] ON THE DYNAMICS OF REVOLVING FLUIDS 451 for purely theoretical purposes there is no inconsistency in supposing the (u, w) motion resisted while the v motion is unresisted. The next supposition to u = 0, w = in order of simplicity is that u is a function of r and t only, and that w = 0, or at most a finite constant. It follows from (8) that P is independent of z, while (6) becomes du du v* dP ~T- + u-, --- = --r- , ........................ (13) dt dr r dr ' determining the pressure. In the case of an incompressible fluid u as a function of r is determined by the equation of continuity ur = C, where C is a function of t only ; and when u and the initial circumstances are known, v follows. As the motion is now two-dimensional, it may conveniently be ex- pressed by means of the vorticity which moves with the fluid, and the stream-function -ty-, connected with by the equation The solution, appropriate to our purpose, is gr-B0, .................. (15) where A and B are arbitrary constants of integration. Accordingly d B dr 2 r , A T*-, -T- ............. (16) rdd r' dr rr r In general, A and B are functions of the time, and is a function of the time as well as of r. A simple particular case is when f is initially, and therefore permanently, uniform throughout the fluid. Then -> '......(17)* Let us further suppose that initially the motion is one of pure rotation, as of a solid body, so that initially A=0, and that then the outer wall closes in. If the outer radius be initially R and at time t equal to R, then at time t R 2 , ........................... (18) since vr remains unchanged for a given ring of the fluid ; and correspondingly, v = Z{r + (R Q *-R*)r-*} ......................... (19) Thus, in addition to the motion as of a solid body, the fluid acquires that of a simple vortex of intensity increasing as R diminishes. * It may be remarked that (17) is still applicable under appropriate boundary conditions even when the fluid is viscous. 29-2 452 ON THE DYNAMICS OF REVOLVING FLUIDS [41 S If at any stage the u motion ceases, (6) gives dp/dr = ptf/r, (20) and thus P/P = P II * + 2 W - -R 2 ) log r - W - #) r-*} + const. . . .(21) Since, as a function of r, v 2 continually increases as R diminishes, the same is true for the difference of pressures at two given values of r, say r*i and r a , where r 2 > r,. Hence, if the pressure be supposed constant at r,, it must continually increase at r a . If the fluid be supposed to be contained between two coaxial cylindrical walls, both walls must move inwards together, and the process comes to an end when the inner wall reaches the axis. But we are not obliged to imagine an inner wall, or, indeed, any wall. The fluid passing inwards at r = r, may be supposed to be removed. And it remains true that, if it there pass at a constant pressure, the pressure at r = i\ must continually increase. If thia pressure has a limit, the inwards flow must cease. It would be of interest to calculate some case in which the (u, w) motion is less simple, for instance, when fluid is removed at a point instead of uniformly along an axis, or inner cylindrical boundary. But this seems hardly practicable. The condition by which v is determined requires the expression of the motion of individual particles, as in the so-called Lagrangian method, and this usually presents great difficulties. We may, however, formulate certain conclusions of a general character. When the (u, w) motion is slow relatively to the v motion, a kind of " equilibrium theory " approximately meets the case, much as when the slow motion under gravity of a variably dense liquid retains as far as possible the horizontal stratification. Thus oil standing over water is drawn off by a syphon without much disturbing the water underneath. When the density varies continuously the situation is more delicate, but the tendency is for the syphon to draw from the horizontal stratum at which it opens. Or if the liquid escapes slowly through an aperture in the bottom of the containing vessel,, only the lower strata are disturbed. In like manner when revolving fluid is drawn off in the neighbourhood of a point situated on the axis of rotation,, there is a tendency for the surfaces of constant circulation to remain cylindrical and the tendency is the more decided the greater the rapidity of rotation. The escaping liquid is drawn always from along the axis and not symmetrically in all directions, as when there is no rotation. The above is, in substance, the reasoning of Dr Aitken, who has also described a simple experiment in illus- tration. P.S. It may have been observed that according to what has been said above the stability of fluid motion in cylindrical strata requires only that the- square of the circulation increase outwards. If the circulation be in both 1916] ON THE DYNAMICS OF REVOLVING FLUIDS 453 directions, this disposition involves discontinuities, and the stability exists only under the condition that symmetry with respect to the axis is rigorously maintained. If this limitation be dispensed with, the motion is certainly unstable, and thus the stability of motion in cylindrical layers really requires that the circulation be one-signed. . On the general question of the two- dimensional motion of liquids between fixed coaxial cylindrical walls reference may be made to a former paper*. The motion in cylindrical strata is stable provided that the " rotation either continually increase or continually decrease in passing outwards from the axis." The demonstration is on the same lines as there set out for plane strata. * Proc. Lond. Math. Soc. Vol. xi. p. 57 (1880) ; Scientific Papers, Vol. i. p. 487. See last paragraph. 414. PROPAGATION OF SOUND IN WATER. [Not hitherto published.] FROM the theoretical point of view there is little to distinguish propagation of sound in an unlimited mass of water from the corresponding case of air; of course the velocity is greater (about four times). It is probable that at a great depth the velocity increases, the effect of diminishing compressibility out-weighing increased density. As regards absorption, it would appear that it is likely to be less in water than in air. The viscosity (measured kinematically) is less in water. But the practical questions are largely influenced by the presence of a free surface, which must act as a nearly perfect reflector. So far the case is analogous to that of a fixed wall reflecting sound waves in air ; but there is an important difference. In order to imitate the wall in air, we must suppose the image of the source of sound to be exactly similar to the original ; but the image of the source of sound reflected from the free surface of water must be taken negatively, viz., in the case of a pure tone with phase altered by 180. In practice the case of interest is when both source and place of observation are somewhat near the reflecting surface. We must expect phenomena of interference vaiying with the precise depth below the surface. The analogy is with Lloyd's interference bands in Optics. If we suppose the distance to be travelled very great, the paths of the direct and reflected sounds will be nearly equal. Here the distinction of the two problems comes in. For air and wall the phases of the direct and reflected waves on arrival would be the same, and the effect a' maximum. But for the free surface of water the phases would be opposite and the effect approximately zero. This is what happens close to the surface. By going lower down the sound would be recovered. It is impossible to arrive at quantitative results unless all the circumstances are specified distance, depths, and wave-length. If there are waves upon the surface of the water there is further complication; but in any case the surface acts as a nearly perfect reflector. The analogy is with a rough wall in air. There is also the bottom to be considered. This, too, must act as a reflector in greater or less degree. With a rocky bottom and nearly grazing incidence, the reflection would be nearly perfect. Presumably a muddy or sandy bottom would reflect less. But I imagine that at grazing incidence as when the distance between source and place of observation is a large multiple of the depth the reflection would be good. This makes another complication. 415. ON METHODS FOR DETECTING SMALL OPTICAL RETARDA- TIONS, AND ON THE THEORY OF FOUCAULT'S TEST. [Philosophical Magazine, Vol. xxxin. pp. 161178, 1917.] As was, I think, first emphasized by Foucault, the standard of accuracy necessary in optical surfaces is a certain fraction of the wave-length (X) of the light employed. For glass surfaces refracting at nearly perpendicular incidence the error of linear retardation is about the half of that of the surface ; but in the case of perpendicular reflexion the error of retardation is the double of that of the surface. The admissible error of retardation varies according to circumstances. In the case of lenses and mirrors affected with "spherical aberration," an error of |X begins to influence the illumi- nation at the geometrical focus, and so to deteriorate the image. For many purposes an error less than this is without importance. The subject is dis- cussed in former papers*. But for other purposes, especially when measurements are in question, a higher standard must be insisted on. It is well known that the parts of the surfaces actually utilized in interferometers, such as those of Michelson and of Fabry and Perot, should be accurate to T ^\ to ^X, and that a still higher degree of accuracy would be advantageous. Even under difficult conditions interference-bands may be displayed in which a local departure from ideal straightness amounting to ^ of the band period can be detected on simple inspection. I may instance some recent observations in which the rays passing a fine vertical slit backed by a common paraffin-flame fell upon the object-glass of a 3-inch telescope placed some 20 feet away at the further end of a dark room. No collimator was needed. The object-glass was pro- vided with a cardboard cap, pierced by two vertical slits, each ^ inch wide, and so placed that the distance between the inner edges was T % inch. . The parallelism of the three slits could be tested with a plumb-line. To observe the bands formed at the focus of the object-glass, a high magnifying- power * Phil. M,ig. Vol. vm. pp. 403, 477 (1879) ; Scientific Papers, Vol. i. p. 415, 3, 4. . 456 ON METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, [415 is required. This was afforded by a small cylinder lens, acting as sole eye- piece, whose axis is best adjusted by trial to the required parallelism with the slits. Fairly good results were obtained with a glass tube of external diameter equal to about 3 mm., charged with water or preferably nitro- benzol. Latterly, I have used with advantage a solid cylinder lens of about the same diameter kindly placed at my disposal by Messrs Hilger. With this arrangement a wire stretched horizontally across the object-glass in front of the slits is seen in fair focus. When the adjustment is good, the bands are wide and the blacknesses well developed, so that a local retardation of ^jy\ or less is evident if suitably presented. The bands are much disturbed by heated air rising from the hand held below the path of the light. The necessity for a high magnifying-power is connected with the rather wide separation of the interfering pencils as they fall upon the object-glass. The conditions are most favourable for the observation of very small retar- dations when the interfering pencils travel along precisely the same path, as may happen in the interference of polarized light, whether the polarization be rectilinear, as in ordinary double refraction, or circular, as along the axis of quartz. In some experiments directed" to test whether " motion through the aether causes double refraction*," it appeared that a relative retardation of the two polarized components could be detected when it amounted to only X/12000, and, if I remember rightly, Brace was able to achieve a still higher sensibility. The sensibility would increase with the intensity of the light employed and with the transparency of the optical parts (nicols, &c.), and it can scarcely be said that there is any theoretical limit. Another method by which moderately small retardations can be made evident is that introduced by Foucaultt for the figuring of optical surfaces. According to geometrical optics rays issuing from a point can be focussed at another point, if the optical appliances are perfect. An eye situated just behind the focus observes an even field of illumination ; but if a screen with a sharp edge is gradually advanced in the focal plane, all light is gradually cut off, and the entire field becomes dark simultaneously. At this moment any irregularity in the optical surfaces, by which rays are diverted from their proper course so as to escape the screening, becomes luminous ; and Foucault explained how the appearances are to be interpreted and information gained as to the kind of correction necessary. He does not appear to have employed the method to observe irregularities arising otherwise than in optical surfaces, but H. Draper, in his memoir of 1864 on the Construction of a Spherical Glass TelescopeJ, gives a picture of the disturbances due to the heating action of the hand held near the telescope mirror. Topler's work dates from Phil. Mag. Vol. iv. p. 678 (1902); Scientific Payers, Vol. v. p. 66. + Ann. de VObterv. de Paris, t. v. ; Collected Memoirs, Paris, 1878. * Smithsonian Contribution to Knowledge, Jan. 1864. 1917] AND ON THE THEORY OF FOUCAULT's TEST 457 the same year, and in subsequent publications* he made many interesting applications, such as to sonorous waves in air originating in electric sparks, and further developed the technique. His most important improvements were perhaps the introduction of a larger source of light bounded by a straight edge parallel to that of the screen at the observing end, and of a small telescope to assist the eye. Worthy of notice is a recent application by R. Cheshire f to determine with considerable precision for practical purposes the refractive index of irregular glass fragments. When the fragment is surrounded by liquid* of slightly different index contained in a suitable tank, it appears luminous as an irregularity, but by adjusting the composition of the liquid it may be made to disappear. The indices are then equal, and that of the liquid may be determined by more usual methods. We have seen that according to geometrical optics (\ = 0) the regular light from an infinitely fine slit may be cut off suddenly, and that an irregularity will become apparent in full brightness however little (in the right direction) it may deflect the proper course of the rays. In considering the limits of sensibility we must remember that with a finite A, the image of the slit cannot be infinitely narrow, but constitutes a diffraction pattern of finite size. If we suppose the aperture bounding the field of view to be rect- angular, we may take the problem to be in two dimensions, and the image consists of a central band of varying brightness bounded by dark edges and accompanied laterally by successions of bands of diminishing brightness. A screen whose edge is at the geometrical focus can cut off only half the light and, even if the lateral bands could be neglected altogether, it must be further advanced through half the width of the central band before the field can become dark. The width of the central band depends upon the horizontal aperture a (measured perpendicularly to the slit supposed vertical), the distance f between the lens and the screen, and the wave-length \. By elementary diffraction theory the first darkness occurs when the difference of retardations of the various secondary rays issuing from the aperture ranges over one complete wave-length, i.e. when the projection of the aperture on the central secondary ray is equal to \. The half-width () of the central band is therefore expressed by =/X/a. If a prism of relative index /u,, and of small angle t, be interposed near the lens, the geometrical focus of rays passing through the prism will be displaced through a distance (/i 1) if. If we identify this with as expressed above, we have (/*-l)i = X/a, (1) * Pogg. Ann. Bd. cxxvm. p. 126 (1866); Bd. cxxxi. pp. 33, 180 (1867). t Phil. Mag. Vol. xxxn. p. 409 (1916). J The liquid employed was a solution of mercuric iodide, and is spoken of as Thoulet's solution. Liveing (Camb. Phil. Proc. Vol. in. p. 258, 1879), who made determinations of the dispersive power, refers to Sonstadt (Chem. News, Vol. xxix. p. 128, 1874). I do not know the date of Thoulet's use of the solution, but suspect that it was subsequent to Sonstadt's. 458 ON METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, [415 as the condition that the half maximum brightness of the prism shall coincide with approximate extinction of the remainder of the field of view. If the linear aperture of the prism be b, supposed to be small in comparison with a, the maximum retardation due to it is X.6/o; .......... ....... ............. (2) and we recognize that easy visibility of the prism on the darkened field is consistent with a maximum retardation which is a small fraction of X. In Cheshire's application of Foucault's method (for I think it should be named after him) the prism had an angle i of 10, and the aperture a was 8 cms., although it would appear from the sketch that the whole of it was not used. Thus in (1) \/ia would be about 5 x 10~ 5 ; and the accuracy with which fj. was determined (about -00002) is of the order that might be expected. It is of interest to trace further and more generally what the wave theory has to tell us, still supposing that the source of light is from an infinitely narrow slit (or, what comes to the same, a slit of finite width at an infinite distance), and that the apertures are rectangular. The problem may then be supposed to be in two dimensions*, although in strictness this requires that the elementary sources distributed uniformly along the length of the slit should be all in one phase. The calculation makes the usual assumption, which cannot be strictly true, that the effect of a screen is merely to stop those parts of the wave which impinge upon it, without influencing the neighbouring parts. In fig. 1, A represents the lens with its rectangular Fig. l. aperture, which brings parallel rays to a focus. In the focal plane B are two adjustable screens with vertical edges, and immediately behind is the eye or objective of a small telescope. The rays from the various points Q of the second aperture, which unite at a point in the focal plane of the telescope, or of the retina, may be regarded as a parallel pencil inclined to the axis at Compare " Wave Theory," Encyc. Brit. 1888 ; Scientific Papers, Vol. in. p. 84. 1917] AND ON THE THEORY OF FOUCAULT'S TEST 459 a small angle <. P is a point in the first aperture, AP = x, BQ = , AB =/. Any additional linear retardation operative at P may be denoted by R, a function of x. Thus if V be the velocity of propagation and K = 27T/X, the vibration at the point of the second aperture will be represented by or, if //= 0, by (3) the limits for 6 corresponding to the angular aperture of the lens A. For shortness we shall omit **, which can always be restored on considering " dimensions," and shall further suppose that R is at most a linear function of 6, say p 4- <r6, or, at any fate, that the whole aperture can be divided into parts for each of which -R is a linear function. In the former case the con- stant part p may be associated with Vt /, and if T be written for Vt -f- p, (3) becomes a)0 ............. (4) Since the same values of p, a apply over the whole aperture, the range of integration is between + 6, where 6 denotes the angular semi-aperture, and then the second term, involving cos T, disappears, while the effect of & is represented by a shift in the origin of , as was to be expected. There is now no real loss of generality in omitting R altogether, so that (4) becomes simply 28inT^, .............. : ...... .. ..... ..(5) as in the usual theory. The borders of the central band correspond to f 6, or rather /c0, = + TT, or = + X, which agrees with the formula used above, since 26 = a/f. When we proceed to inquire what is to be observed at angle < we have to consider the integral (6) sin (T + <) g! = sin T f<L4> ? + si <" ~ *) 1 It will be observed that, whatever may be the limits for , the first integral is an even and the second an odd function of 0, so that the intensity (/), represented by the sum of the squares of the integrals, is an even function. The field of view is thus symmetrical with respect to the axis. * Equivalent to supposing X = 2ir. 460 ON METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, [415 The integrals in (6) may be at once expressed in terms of the so-called sine-integral and cosine-integral denned by ~. , f*sin# , ~. . pcosa; , Si (#) = I - dx, Ci (x) = - dx. If the limits of be ft and ft we get sin T[Si ((6 + </>) ft} - Si {(0 + *) ft} + Si ((0 - 0) ft} - Si {(0 - </>) ft}] + cos T[Ci {(0 - $) ft} - Ci {(ff - <#>) ft} - Ci {(0 + </>) ft} + Ci {(0 + *) ft}]. ......... (7) If ft = ft = ft so that the second aperture is symmetrical with respect to the axis, the Ci's, being even functions, disappear, and we have simply 2 sin T [Si {(0 + <)}+ Si 1(0 -</>)}] ................ (8) If the aperture of the telescope be not purposely limited, the value of ft or rather of /eft is very great, and for most purposes the error will be small in supposing it infinite. Now Si( oc )= + |TT, so that if < is numerically less than 0, I = 4nr 2 , but if <f> is numerically greater than 0, 1 = 0. The angular field of view 20 is thus uniformly illuminated and the transition to darkness at angles is sudden that is, the edges are seen with infinite sharpness. Of course, cannot really be infinite, nor consequently the resolving power of the telescope ; but we may say that the edges are defined with full sharpness. The question here is the same as that formerly raised under the title "An Optical Paradox*," the paradox consisting in the full definition of the edges of the first aperture, although nearly the whole of the light at the second aperture, is concentrated in a very narrow band, which might appear to preclude more than a very feeble resolving power. It may be well at this stage to examine more closely what is actually the distribution of light between the central and lateral bands in the diffraction pattern formed at the plane of the second aperture. By (5) the intensity of light at is proportional to ~ 2 sin 2 0g or, if we write 77 for #ft to rj~- sin* r). The whole light between and 17 is thus represented by J can be expressed by means of the Si-function. As may be verified by differentiation, t /=Si(27/)-7/- 1 sin 2 7; ......................... (10) vanishing when 17 = 0. The places of zero illumination are defined by rj = tnr, when n = 1, 2, 3, &c. ; and, if ij assume one of these values, we have simply (11) Phil. Mag. Vol. ix. p. 779 (1905); Scientific Papert, Vol. v. p. 254. 1917] AND ON THE THEORY OF FOUCAULT'S TEST 461 Thus, setting n = 1, we find for half the light in the central band J = Si 27r = 7r- -15264. On the same scale half the whole light is Si (x ), or |TT, so that the fraction of the whole light to be found in the central band is or more than nine-tenths. About half the remainder is accounted for by the light in the two lateral bands immediately adjacent (on the two sides) to the central band. We are now in a position to calculate the appearance of the field when the second aperture is actually limited by screens, so as to allow only the passage of the central band of the diffraction pattern. For this purpose we have merely to suppose in (8) that $=TT. The intensity at angle $ is thus .(13) The further calculation requires a knowledge of the function Si, and a little later we shall need the second function Ci. In ascending series + | 1 - 2 ~4-...; ......... (15) 7 is Euler's constant '5772157, and the logarithm is to base e These series are always convergent and are practically available when x is moderate. When x is great, we may use the semi-convergent series 1.2 1.2.3.4 1.2...0 1 1 1.2.3 1.2.3.4.5 -^ + ^ 1 1.2 1.2.3.4 -- + - - 1 1.2.3 1.2.3.4.5 ) Tables of the functions have been calculated by Glaisher*. For our present purpose it would have been more convenient had the argument been TT.C, rather than x. Between x= 5 and x= 15, the values of Si (x) are given for integers only, and interpolation is not effective. For this reason some * Phil. Tram. Vol. CLX. p. 367 (1870). 462 OX METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, '[415 values of <f>/8 are chosen which make (I 4- <J>/B)TT integral. The calculations recorded in Table I refer in the first instance to the values of TABLE I. */ (18) (18)2 o-oooo 3-704 13-72 0-2732 3-475 12-08 0-5000 2-979 8-87 0-5915 2-721 7-40 0-9099 1-707 2-91 1-0000 .1-418 2-01 1-2282 0-758 0-57 1-5465 0-115 o-oi 20000 -0-177 - 0-03 It will be seen that, in spite of the fact that nine-tenths of the whole light passes, the definition of what should be the edge of the field at < = 6 is very bad. Also that the illumination at (f> = is greater than what it would be (7T 2 ) if the second screening were abolished altogether (+ = oo ). So far we have dealt only with cases where the second aperture is sym- metrically situated with respect to the geometrical focus. This restriction we will now dispense with, considering first the case where i = and .( = f) is positive and of arbitrary value. The coefficient of sin T in (7) becomes simply Si {(0 + </>)} + Si 1(0 -<)} ...................... (19) In the coefficient of cos T, Ci {(0+ <)}, Oi {(0 -</>)} assume infinite values, but by (15) we see that Ci.{ ( + ^^-Ci{(0-^)^=lo g !||| .......... (-20) so that the coefficient of cos T is .......... (21) The intensity I at angle < is represented by the sum of the squares of (19) and (21). When < = at the centre of the field of view, / = 4 (Si &!)*, but at the edges for which it suffices to suppose = + 6, a modification is called for, since Ci {(6 <J>) } must then be replaced by 7 + log j (6 <) |. Under these circumstances the coefficient of cos T becomes and / = {Si + ( 7 + log (20) - Ci (20f )} 2 ............. (22) 1917] AND ON THE THEORY OF FOUCAULT's TEST 463 If in (22) be supposed to increase without limit, we find 7=iir + {log0} > (23) becoming logarithmically infinite. Since in practice f, or rather KJ~, is large, the edges of the field may be expected to appear very bright. As may be anticipated, this conclusion does not depend upon our sup- position that & = 0. Reverting to (7) and supposing <f> = 6, we have sin T [Si (20&) - Si (20fc)] + cos T[Ci (20fc) - Ci (20f 2 ) + log (ft/ft)], (24) and 7 = oo, when 2=00. If & vanishes in (24), we have only to replace Ci (20) by 7 + log (20) in order to recover (22). We may perhaps better understand the abnormal increase of illumination at the edges of the field by a comparison with the familiar action of a grating in forming diffraction spectra. Referring to (5) we see that if positive values of be alone regarded, the vibration in the plane of the second aperture, represented by -1 sin (#), is the same in respect of phase as would be due to a theoretically simple grating receiving a parallel beam perpendicularly, and the directions </> = + tf are those of the resulting lateral spectra of the first order. On account, however, of the factor g~ l , the case differs somewhat from that of the simple grating, but not enough to prevent the illumination becoming logarithmically infinite with infinite aperture. But the approxi- mate resemblance to a simple grating fails when we include negative as well as positive values of , since there is then a reversal of phase in passing zero. Compare fig. 2, where positive values are represented by full lines and Fig. 2. negative by dotted lines. If the aperture is symmetrically bounded, the parts at a distance from the centre tend to compensate one another, and the intensity at </> = does not become infinite with the aperture. We now proceed to consider the actual calculation of 7 = (19) 2 + (21) 2 for various values of <f>/6, which we may suppose to be always positive, since 7 is independent of the sign of <j>. When j-0 is very great and <f>/0 is not nearly equal to unity, Si {(0 + <) } in (19) may be replaced by TT and Si {(0 -<) by ^7r, according as <f>/0 is less or greater than unity. Under the same conditions the Ci's in (21) may be omitted, so that 7='7T 2 (1, or 0) + 464 ON METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, [415 But if we wish to avoid the infinity when $ = 6, we must make some supposition as to the actual value of 6g, or rather of 2ir61~l\. In some obser- vations to be described later a = 1 inch, = \ inch, 1/X = 40,000, and 6 = \a\f, Also / was about 10 feet = 120 inches. For simplicity we may suppose /= 40-7T, so that 2-7r0f/X = 500, or in our usual notation 6% = 500. Thus (19) = Si {500(1 + $10)} + Si {500 (1 - $/0)}, (26) and (21) = Ci {500 (1 - $/0)} - Ci {500 (1 + <t>/8)} + log(l+4>/0)-log|l-4>/0! (27) For the purposes of a somewhat rough estimate we may neglect the second Ci in (27) and identify the first Si in (26) with TT for all (positive) values of $10. Thus when $ = 0, / = 7r 2 ; and when $ = x , 7 = 0. When $JO = 1, we take (26) = \TT = 1-571, (26) 2 = 2-467. In (27) Ci {500 (1 - $16)} = 7 + log 500 + log (1 - $/0), so that (27) = 7 + log 1000 = 7-485, (27) 2 = 56'03; and . 7 = 58-50. For the values of $jO in the neighbourhood of unity we may make similar calculations with the aid of Glaisher's Tables. For example, if $J6 = 1 T '02, we have 500(1 -</#)= 10. From the Tables Si ( 10) = '+ 1-6583, Ci ( 10) = - -0455, and thence 7(-98) = 31-13, 7(1'02) = 20-89. As regards values of the argument outside these units, we may remark that when x exceeds 10, Si(#) \TT and Ci (x) are approximately periodic in period 2?r and of order ar l . It is hardly worth while to include these fluctuations, which would manifest themselves as rather feeble- and narrow bands, superposed upon the general ground, and we may thus content our- selves with (25). If we apply this to . 10, we get / (-98) = 30-98, 7(1-02) = 21-30; and the smoothed values differ but little from those calculated for 10 more precisely. The Table (II) annexed shows the values of 7 for various values of $fd. Those in the 2nd and 8th columns are smoothed values as explained v and they would remain undisturbed if the value of 0% were increased. It will be seen that the maximum illumination near the edges is some 6 times that at the centre. 1917] AND ON THE THEORY OF FOUCAULT S TEST TABLE II. 465 m I tie I <t>/6 I 0/0 I o-ooo 9-87 0-980 31-13 1-001 56-28 1-05 13-76 0-250 10-13 0-990 35-78 1-002 52-89 MO 9-24 0-500 11-08 0-992 39-98 1-004 44-09 1-20 5-76 0-800 14-71 , 0-994 46-81 1-006 35-27 1-50 2-59 0-900 18-51 0-996 54-13 1-008 29-03 2-00 1-21 0-950 23-27 0-998 58-81 1-010 26-14 oc 0-999 59-36 1-020 20-89 1-000 58-50 TABLE III. K0& = IT, K0& = 500. */9 I 0/4 I o-oo 0-32 1-01 8-98 0-50 0-48 1-02 6-57 0-91 2-46 1-23 0-58 0-98 7-55 1-55 0-13 0-99 9-90 1-86 0-05 1-00 25-51 00 o-oo In the practical use of Foucault's method the general field would be darkened much more than has been supposed above where half the whole light passes. We may suppose that the screening just cuts off tihe central band, as well as all on one side of it, so that 0^ = IT. In this case (7) becomes sin T [Si (0 + 0) + Si(0- 0) - Si(l + 0/0) TT - Si (1 -0/0)7r] + cos T[Ci (0 - 0) f -Ci(0 + 0) f + Ci (1 + 0/0) TT- Ci (1 - 0/0) TT]. ......... (28) We will apply it to the case already considered, where 0% = 500, as before omitting Ci (0 + 0) and equating Si (0 + 0) to \ -rr. Thus / = [TT + Si 500 (1 - <f)/0) - Si (1 + <f>/0) TT - Si (1 - 0/0) -rrj + [Ci 500 (1 - <f>/0) + Ci (1 + </>/#) TT - Ci (1 - </0) ir]\ ......... (29) When < = oc , 7 = 0. When 0=0, When / = [TT - Si (27T)] 2 + [log (500/Tr) + Ci (2-Tr)] 2 = 25'51 ; R. vi. 30 466 ON METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, [415 so that the brightness of the edges is now about 80 times that at the centre of the field. The remaining values of / in Table III have been calculated as before with omission of the terms representing minor periodic fluctuations. Hitherto we have treated various kinds of screening, but without additional retardation at the plane of the first aperture. The introduction of such retardation is, of course, a complication, but in principle it gives rise to no difficulty, provided the retardation be linear in 6 over the various parts of the aperture. The final illumination as a function of < can always be expressed by means of the Si- and Ci-functions. As the simplest case which presents something essentially novel, we may suppose that an otherwise constant retardation (R) changes sign when 0=0, is equal (say) to + p when is positive and to' p when 6 is negative. Then (3) becomes sin (T + p + 0)d0+ I sm(T ...(30) reducing to (5) when p = 0. This gives the vibration at the point of the second aperture. If f=0, (30) becomes 20 cos p sin T, and vanishes when cos p = ; for instance, when the whole difference of retardation 2p = TT, or (reckoned in wave-lengths) \. The vibration in direction </> behind the second aperture is to be obtained by writing T+<f>i- for T in (30) and integrating with respect to This gives 2 sin TJd cos tf jcos p **g* + sin p ^ + 2 cos T^sin # coep + rin, , ... (31) and the illumination (/) is independent of the sign of <f>, which we may hence- forward suppose to be positive. If the second aperture be symmetrically placed, we may take the limits to be expressed as f, and (31) becomes 28in If we apply this to = x to find what occurs when there is no screening, we fall upon ambiguities, for (32) becomes 2 sin T cos p {\-rr %ir] + 2cosrsinp {2 Si(<)-$7r ITT}, 1917] AND ON THE THEORY OF FOUCAULT'S TEST 467 the alternatives following the sign of 6 </>, with exclusion of the case <j> = 6. If <f> is finite, 2 Si (<f ) may be equated to TT, and we get / = 47r 2 (l orO), according as < is positive or negative. But if <f> = absolutely, Si (</>) disappears, however great may be ; and when < is small, / = 4?r 2 cos 2 p + 4 sin 2 p [2 Si (<f)} 2 , in which the value of the second term is uncertain, unless indeed sinp = 0. It would seem that the difficulty depends upon the assumed discontinuity of R when 6 = 0. If the limits for 9 be a (up to the present written as + 0), what we have to consider is d9 sin T- \ > in which hitherto we have taken first the integration with respect to 9. We propose now to take first the integration with respect to , introducing the factor e ^ to ensure convergency. We get 2 sin (T - R) e-* cos (0 + 0) g . d = ~ . .-(33) There remains the integration with respect to 6, of which R is supposed to be a continuous function. As fj, tends to vanish, the only values of 6 which contribute are confined more and more to the neighbourhood of <, so that ultimately we may suppose 6 to have this value in R. And /: +a AI dd _j < + a _! ~ ~ which is TT, if <f> lies between + a, and if </> lies outside these limits, when /* is made vanishing small. The intensity in any direction is thus independent of R altogether. This procedure would fail if R were discontinuous for any values of 6. Resuming the suppositions of equation (31), let us now further suppose that the aperture extends from to | 2 , where both and | 2 are positive and 2 > 1 O ur expression for the vibration in direction < becomes sin T [cos p {Si (0 + </>) f + Si (0 - $) fj + sin p (2 Ci (0|) - Ci (0 + 0) - Ci (0 - + cos T[cos p {Ci (0 - </>) f - Ci (0 4- 0) (} ......... (34) We will apply this to the case already considered where ,0 = 500, = TT ; and since we are now concerned mainly with what occurs in the neighbourhood of ^ = 0, we may confine < to lie between the limits and 0. Under these circumstances, and putting minor rapid fluctuations out of account, we may 302 468 ON METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS, [415 neglect Ci (6 <f>) & and equate Si (6 </>) , to TT. A similar simplification is admissible for Si ($9), Ci (<>)> unless <f>/0 is very small. When = 0, (34) gives sin T [cos p {TT - 2 Si (ir)j + sin p (2 log (500/7r) + 2 Ci (TT)}], in which TT - 2 Si (TT) = - "5623, Ci (TT)= '0738, log (500/7r) = 5'0699. Thus for the intensity / (0) = [--3623 cos p + 10-2874 sin pj (35) If p = 0, we fall back upon a former result (-3162). If p = \ TT, / (0) = 47 3. Interest attaches mainly to small values of p, and we see that the effect depends upon the sign of p. A positive p means that the retardation at the first aperture takes place on the side opposite to that covered by the screen at the second aperture. As regards magnitude, we must remember that p stands for an angular retardation icp, or 2?r/3/X ; so that, for example, p = \ir above represents a linear retardation A./8, and a total relative retardation between the two halves of the first aperture equal to \/4. The second column of Table IV gives the general expression for the vibration in terms of p for various values of <p/0, followed by the values of the intensity (/) for sin p = 1/10 and sin p = 1/V2. TABLE IV. *0f , = 7T, K0& = 500. 1 I e Formula for Vibration ship sin p + 1 -1 + 1/V/2 -iWi sin T { - -56 cos p + 10-29 sin p} 22 2-53 47-3 58-9 ool sin T 7 !- -56 cos/) + 10- 16 sin/)} + co87 T x-99sinp 22 2-50 46-6 68-0 010 . sin T { - -56 cos p + 5-53 sin p} + cos T x 3'10 sin p 10 1-34 17-2 23-4 050 sin T { --55 cos p + 2*71 sin/)} 11 83 6-0 9-6 100 sin T { - '53 cos p + 1 '37 sin p} -1- cos T { - -20 cos p + 2-52 sin p} 16 66 3-0 6-5 250 sin T{- '37cosp - '17 sin p} + cos T { - -46 cos p + 1 -66 sin p} 23 52 86 2-3 500 sin T{ + -16 cos p - -67 sin p} + cos T { - -67 cos p + -64 sin p} 38 59 13 1-2 1917] AND ON THE THEORY OF FOUCAULT'S TEST 469 It will be seen that the direction of the discontinuity (<j> = 0) is strongly marked by excess of brightness, and that especially when p is small there is a large variation with the sign of p. Perhaps the next case in order of simplicity of a variable R is to suppose R = from 6 = - 6 to 6 = 0, and R = <r0 from 6 = to 6 = + 6, corresponding to the introduction of a prism of small angle, whose edge divides equally the field of view. For the vibration in the focal plane we get sin T M + ffi=5il + cos T P --(I-<^ _ ! IL I I I"* J I -<r (36) In order to find what would be seen in direction <f>, we should have next to write (T+<) for T and integrate again with respect to between the appropriate limits. As to this there is no difficulty, but the expressions are rather long. It may suffice to notice that whatever the limits may be, no infinity enters at </> = 0, in which case we have merely to integrate (36) as it stands. For although the denominators become zero when = or <7, the four fractions themselves always remain finite. The line of transition between the two halves of the field is not so marked as when there was an actual dis- continuity in the retardation itself. In connection with these calculations I have made for my own satisfaction a few observations, mainly to examine the enhanced brightness at the edges of the field of view. The luminous border is shown in Draper's drawing, and is described by Topler as due to diffraction. The slit and focussing lens were those of an ordinary spectroscope, the slit being drawn back from the " colli- mating " lens. The telescope was from the same instrument, now mounted independently at a distance so as to receive an image of the slit, and itself focussed upon the first lens. The rectangular aperture at the first lens was originally cut out of the black card. The principal dimensions have already been given. A flat paraffin-flame afforded sufficient illumination. The screens used in front of the telescope were razor-blades (Gillettes), and were adjusted in position with the aid of an eyepiece, the telescope being temporarily removed. It is not pretended that the arrangements used corresponded fully to the suppositions of theory. The brightness of the vertical edge of the field of view is very conspicuous when the light is partly cut off by the advancing screen. A question may arise as to how much of it may be due to light ordinarily reflected at the edges of the first aperture. With the aperture cut in cardboard, I think this part was appreciable, but the substitution of a razor-edge at the first aperture made no important difference. The strongly illuminated border must often have been seen in repetitions of Foucault's experiment, but I am not aware that it has been explained. 470 OX METHODS FOR DETECTING SMALL OPTICAL RETARDATIONS [415 To examine the sudden transition from one uniform retardation to another, I used a piece of plate glass which had been etched in alternate strips with hydrofluoric acid to a depth of about JX*. When this was set up in front of the first aperture with strips vertical, the division-lines shone out brightly, when the intervening areas were uniformly dark or nearly so. No marked difference was seen between the alternate division-lines corresponding to opposite signs of p. Perhaps this could hardly be expected. The whole relative retardation, reckoned as a distance, is \\, and is thus intermediate between the values specified in Table IV. It would be of interest to make a similar experiment with a shallower etching. [1919. For further developments reference may be made to Banerji, Phil. Mag. Vol. xxxvn. p. 112, 1919.] * Compare Nature, Vol. LXIV. p. 385 (1901); Scientific Papers, Vol. iv. pp. 546, 547. 416. TALBOT'S OBSERVATIONS ON FUSED NITRE. [Nature, Vol. xcvni. p. 428, 1917.] AMONG the little remembered writings of that remarkable man H. F. Talbot, there is an optical note in which he describes the behaviour of fused nitre (nitrate of potash) as observed under the polarizing microscope. The experi- ments are interesting and easily repeated by any one who has access to a suitable instrument, by preference one in which the nicols can be made to revolve together so as to maintain a dark field in the absence of any interposed crystal. "Put a drop of a solution of nitre on a small plate of glass, and evaporate it to dryness over a spirit-lamp ; then invert the glass, and hold it with the salt downwards and in contact with the flame. By this means the nitre may be brought into a state of fusion, and it will spread itself in a thin transparent film over the surface of the glass. "Removed from the lamp it immediately solidifies, and the film in cooling cracks irregularly. As soon as the glass is cool enough, let it be placed beneath the microscope (the polariness being crossed, and the field of view consequently dark)." I have found it better to use several drops spread over a part of the glass. And instead of inverting the plate in order to melt the nitre, I prefer to employ the flame from a mouth blow-pipe, caused to play upon the already heated salt. The blow-pipe may also be used to clean the glass in the first instance, after a preliminary heating over the flame to diminish the risk of fracture. Further security is afforded by keeping down the width of the strip, for which half an inch suffices. Talbot describes how under the microscope there appeared crystalline plates of irregular shape, often fitted together like a tesselated pavement, each plate forming a single crystal. If one plate is darkened by rotation of the nicols, the others remain visible in varying degrees of brightness. If the plates are thin, the light is white; but with more salt they display colour, and the 472 TALBOT'S OBSERVATIONS ON FUSED NITRE [416 colour is not always uniform over the whole plate, indicating a variable thickness. But this condition of things is not permanent. After perhaps a quarter of an hour, the plates break up in a surprising fashion and the general appearance is totally changed. Moreover the transformation may be accelerated. " Let a film of fused nitre be obtained in the manner already mentioned, and let it be allowed to cool during three or four minutes. The plate of glass should be turned round upon the stage of the microscope until the crystalline film is darkened as accurately as possible. Things being thus adjusted, let the observer touch the film with the point of a needle, while he is observing it in the microscope. He will perceive that the touch immediately produces a luminous spot on the dark surface, and this spot will slowly expand itself in all directions like a luminous wave. This is a very curious object, but difficult to describe." And further on " If however we touch it prematurely, as, for instance, during the first minute after it has become solid, this change does not take place." I have made a few trials to ascertain whether the life of the plates can be prolonged. Protection from atmospheric moisture did little good. Another plate kept for five hours at a temperature not much short of that of boiling water was found to have undergone transformation. But, as might be expected, a higher temperature over a diminutive gas flame acted as a safe- guard, and the plate after removal behaved like one newly formed. I have found that nitre may be replaced by chlorate of potash, with the advantage that the plates will keep (at any rate in an artificially warmed room) for weeks and perhaps indefinitely. The appearances are similar but less beautiful, as colour is not so often developed. The chlorate is more fusible than nitre, and the heat should not be pushed beyond what is needed for fusion. Other salts, for example silver nitrate, which fuse in the anhydrous state without decomposition may also be employed, as is probably known to those who prepare objects for the microscope. But Talbot's early observations on nitre are rather special and deserve recall as they seem to be but little known. 417. CUTTING AND CHIPPING OF GLASS. [Engineering, Feb. 2, 1917, p. 111.] WITH all its advantages, the division of labour, so much accentuated in modern times, tends to carry with it a regrettable division of information. Much that is familiar to theorists and experimenters in laboratories percolates slowly into the workshop, and, what is more to my present purpose, much practical knowledge gained in the workshop fails to find its way into print. At the moment I am desirous of further information on two matters relating to the working of glass in which I happen to be interested, and I am writing in the hope that some of your readers may be able to assist. Almost the only discussion that I have seen of the cutting of glass by the diamond is a century old, by the celebrated W. H. Wollaston (Phil. Trans. 1816, p. 265). Wollaston 's description is brief and so much to the point that it may be of service to reproduce it from the " Abstracts," p. 43 : "The author, having never met with a satisfactory explanation of the property which the diamond possesses of cutting glass, has endeavoured, by experiment, to determine the conditions necessary for this effect, and the mode in which it is produced. The diamonds chosen for this purpose are naturally crystallised, with curved surfaces, so that the edges are also curvilinear. In order to cut glass, a diamond of this form requires to be so placed that the surface of the glass is a tangent to a curvilinear edge, and equally inclined laterally to the two adjacent surfaces of the diamond. Under these circumstances the parts of the glass to which the diamond is applied are forced asunder, as by an obtuse wedge, to a most minute distance, without being removed ; so that a superficial and continuous crack is made from one end of the intended cut to the other. After this, any small force applied to one extremity is sufficient to extend this crack through the whole substance, and successively across the whole breadth of the glass. For since the strain at each instant in the progress of the crack is confined nearly to a mathe- matical point at the bottom of the fissure, the effort necessary for carrying it through is proportionately small. " The author found by trial that the cut caused by the mere passage of the diamond need not penetrate so much as -%fa of an inch. 474 CUTTING AND CHIPPING OF GLASS [417 " He found also that other mineral bodies recently ground into the same form are also capable of cutting glass, although they cannot long retain the power, from want of the requisite hardness." I suppose that no objection will be taken to Wollaston's general description of the action, but it may be worth while to consider it rather more closely in the light of mathematical solutions of related elastic problems by Boussinesq and Hertz ; at the same time we may do well to remember Mr W. Taylor's saying that everything calculated by theorists is concerned with what happens within the elastic limit of the material, and everything done in the workshop lies beyond that limit. A good account of these theoretical investigations will be found in Love's Elasticity, Chap. vm. It appears that when a pressure acts locally near a point on the plane surface of an elastic solid, the material situated along the axis is in a state of strain, which varies rapidly with the distance from the surface. The force transmitted across internal surfaces parallel to the external surface is a pressure all along, but the force trans- mitted in a perpendicular direction, although at first a pressure, at a very small distance below changes to a tension, which soon reaches a maximum and afterwards gradually diminishes. I suppose it is this tension which determines the crack, an action favoured by the longitudinal character of the pressure on the surface, and, once started, easily propagated as the diamond travels. Doubtless cutters of hardened steel discs, sharpened on the edge, act in a similar manner. It is possible that examination under the microscope by a skilled observer would throw light upon the matter. Among the questions which suggest themselves, one may ask whether the diamond cut necessarily involves a crushing at the surface, and what materials, besides glass, can be dealt with in this way. Would a bending force, such as is afterwards applied to separate the parts, facilitate the original formation of the crack? The other matter in which I have been interested is the preparation of what I believe is called "chipped" glass. The only mention of it that I know is a casual one in Threlfall's Laboratory Arts. In an experiment tried some yi-ars ago, a glass plate was coated thickly with a warm solution of gelatine and allowed to dry on a levelling stand. Nothing particular happened afterwards for days or weeks; but eventually parts of the gelatine film lifted, carrying up with them material torn away from the glass. The plate is still in my possession, and there is now but little of the original glass surface left. If the process is in regular use, I should much like to know the precise procedure. It seems rather mysterious that a film of gelatine, scarcely thicker than thick paper, should be able to tear out fragments of solid glass, but there is no doubt of the fact. [1919. Interesting information in response to the above will be found in Engineering for March 11 and 16, and April 27, 1917.] 418. THE LE CHATELIER-BRAUN PRINCIPLE. [Transactions of the Chemical Society, Vol. cxi. pp. 250 252, 1917.] IN a paper with the above title, Ehrenfest (Zeitsch. physikal. Chem. 1911, 77, 2) has shown that, as usually formulated, the principle is entirely ambiguous, and that nothing definite can be stated without a discrimination among the parameters by which the condition of a system may be defined. The typical example is that of a gas, the expansions and contractions of which may be either (a) isothermal or (ft) adiabatic, and the question is a comparison of the contractions in the two cases due to an increment of pressure Bp. It is known, of course, that if Bp be given, the contraction j Bv \ is less in case (ft) than in case (a). The response of the system is said to be less in case (ft), where the temperature changes spontaneously. But we need not go far to encounter an ambiguity. For if we regard Bv as given instead of Bp, the effect Bp is now greater in (ft) than in (a). Why are we to choose the one rather than the other as the independent variable ? When we attempt to answer this question, we are led to recognise that the treatment should commence with purely mechanical systems. The equilibrium of such a system depends on the potential energy function, and the investigation of its character presents no difficulty. Afterwards we may endeavour to extend our results to systems dependent on other, for example, thermodynamic, potentials. As regards mechanical systems, the question may be defined as relating to the operation of constraints. A general treatment (Phil. Mag. 1875, [iv], Vol. XLIX. p. 218 ; Scientific Papers, Vol. I. p. 235 : also Theory of Sound, 75) shows that "the introduction of a constraint has the effect of diminishing the potential energy of deformation of a system acted on by given forces ; and the amount of the diminution is the potential energy of the difference of the deformations. "For an example take the case of a horizontal rod clamped at one end and free at the other, from which a weight may be suspended at the point Q. If a constraint is applied holding a point P of the rod in its place (for example, by a support situated under it), the potential energy of the bending 476 THE LE CHATELIER-BRAUN PRINCIPLE [418 due to the weight at Q is less than it would be without the constraint by the potential energy of the difference of the deformations. And since the potential energy in either case is proportional to the descent of the point Q, we see that the effect of the constraint is to diminish this descent." It may suffice here to sketch the demonstration for the case of two degrees of freedom, the results of which may, indeed, be interpreted so as to cover most of the ground. The potential energy of the system, slightly displaced from stable equilibrium at x = 0, y = 0, may be expressed where, in virtue of the stability, a, c, and ac - b* are positive. The forces X, Y, corresponding with the displacements x, y, and necessary to maintain these displacements, are : If only X act, that is, if F = 0, y = - bxfc, and X ~a-6 2 /c' This is the case of no constraint. On the other hand, if y is constrained to remain zero by the application of a suitable force F, the relation between the new x (say x'} and X is simply Thus X - = l--- x ac so that x', having the same sign as x, is numerically less, or the effect of the constraint is to diminish the displacement x due to the force X. An exception occurs if 6 = 0, when x = X/a, whatever y and F may be, so that the constraint has no effect. An example, mentioned by Ehrenfest, may be taken from a cylindrical rod of elastic material subject to a longitudinal pressure, X, by which the length is shortened (#). In the first case the curved wall is free, and in the second the radius is prevented from changing by the application of a suitable pressure. The theorem asserts that in the second case the shortening due to the longitudinal pressure X is less, in virtue of the constraint applied to the walls. Returning to the compressed gas, we now recognise that it is the pressure Sp which is the force and Sv the effect, corresponding respectively with X and x of the general theorem. But we may still feel a doubt as to which is the constrained condition, the isothermal or the adiabatic, and without a decision on this point no statement can be made. It is, however, evident that if the general theorem is applicable at all, the adiabatic condition must 1917] THE LE CHATELIER-BRAUN PRINCIPLE 477 be regarded as the constrained one, since the response is to be diminished by a constraint. The justification of this view does not seem difficult. The gas may be supposed to be confined in a cylinder under a piston, and the walls of the cylinder may be taken to be so massive as to control the temperature of the gas when undergoing slow alterations of volume. The necessary interchanges of heat take place of themselves, and the condition is one of freedom from constraint. We pass to the adiabatic condition by preventing this accom- modation. The How of heat may be stopped by the introduction of a non- conducting layer or in any other way, and the operation has the character of a constraint. Since the motion of heat in a conductor is due to differences of temperature, the former is assimilated to the displacement and the latter to the force of the purely mechanical problem. The same conclusion follows from a consideration of the thermodynamic potential. Instead of a gas we may take a vapour in contact with liquid, say steam in contact with water. The pressure is now a function of temperature only, so that if the pressure is increased while the temperature remains unchanged, the whole of the steam is condensed, and the volume is greatly reduced. If by a constraint the outward passage of heat is prevented, the temperature rises and the reduction of volume soon ceases, in accordance with the principle. Or again, we may suppose that the temperature is raised by a given amount, in the first case under constant pressure (no constraint), or, secondly, under constant volume, namely, with constraint. The passage of heat is less in the second case. Electrostatic problems, governed by a potential energy function, are seen to be included under those of ordinary mechanics. Imagine two conductors, near enough to influence one another, of which the first can be connected with a battery (the other pole of which is earthed), whilst the second can be connected directly to earth. In the first case (of no constraint), the second conductor is earthed, and a certain charge enters the first conductor as the result of the battery contact. In the second case, the earth connexion of the second conductor is broken before battery contact is made. The breaking of this contact introduces a constraint, and the charge on the first conductor is reduced. In all such problems potential corresponds with force and charge corresponds with displacement. In problems relating to steady electric currents maintained against re- sistance, the dissipation function takes the place of the energy function. If an electromotive force act on any branch of a network of conductors, it generates less current, and accordingly does less work, when an interruption occurs, as by breaking a contact in any part of the system. 419. ON PERIODIC IRROTATIONAL WAVES AT THE SURFACE OF DEEP WATER. [Philosophical Magazine, Vol. xxxm. pp. 381389, 1917.] THE treatment of this question by Stokes, using series proceeding by ascending powers of the height of the waves, is well known. In a paper with the above title* it has been criticised rather severely by Burnside, who concludes that " these successive approximations can not be used for purposes of numerical calculation...." Further, Burnside considers that a numerical discrepancy which he encountered may be regarded as suggesting the non- existence of permanent irrotational waves. It so happens that on this point I myself expressed scepticism in an early paper f, but afterwards I accepted the existence of such waves on the later arguments of Stokes, M c CowanJ, and of Korteweg and De Vries. In 1911 1| I showed that the method of the early paper could be extended so as to obtain all the later results of Stokes. The discrepancy that weighed with Burnside lies in the fact that the value of (see equation (1) below) found best to satisfy the conditions in the case of a = ^ differs by about 50 per cent, from that given by Stokes' formula, viz. /3 = a 4 . It seems to me that too much was expected. A series proceeding by powers of ^ need not be very convergent. One is reminded of a parallel instance in the lunar theory where the motion of the moon's apse, calculated from the first approximation, is doubled at the next step. Similarly here the next approximation largely increases the numerical value of /9. When a smaller a is chosen (-fa), series developed on Stokes' plan give satisfactory results, even though they may not converge so rapidly as might be wished. The question of the convergency of these series is distinct from that of the existence of permanent waves. Of course a strict mathematical proof of their existence is a desideratum; but I think that the reader who follows the results of the calculations here put forward is likely to be convinced that Proc. Lond. Math. Soc. Vol. xv. p. 26 (1915). t Phil. Slag. Vol. i. p. 257 (1876) ; Scientific Papers, Vol. i. p. 261. J Phil. Mag. Vol. xxxn. pp. 45, 553 (1891). Phil. Mag. Vol. xxxix. p. 422 (1895). || Phil. Mag. Vol. xxi. p. 183 (1911). [This volume, p. 11.] 1917] ON PERIODIC IRROTATIONAL WAVES 479 permanent waves of moderate height do exist. If this is so, and if Stokes' series are convergent in the mathematical sense for such heights, it appears very unlikely that the case will be altered until the wave attains the greatest admissible elevation, when, as Stokes showed, the crest comes to an edge at an angle of 120. It may be remarked that most of the authorities mentioned above express belief in the existence of permanent waves, even though the water be not deep, provided of course that the bottom be flat. A further question may be raised as to whether it is necessary that gravity be constant at different levels. In the paper first cited I showed that, under a gravity inversely as the cube of the distance from the bottom, very long waves are permanent. It may be that under a wide range of laws of gravity permanent waves exist. Following the method of my paper of 1911, we suppose for brevity that the wave-length is 2?r, the velocity of propagation unity*, and we take as the expression for the stream-function of the waves, reduced to rest, <\Jr = y ae~ y cos x f3e~- y cos 2# je~ 3y cos 3# 8e~* y cos 4# ee~ sy cos 5x, ...... ( 1 ) in which x is measured horizontally and y vertically downwards. This ex- pression evidently satisfies the differential equation to which ^ is subject, whatever may be the values of the constants a, /3, &c. And, much as before, we shall find that the surface condition can be satisfied to the order of a 7 inclusive ; /3, 7, 8, e being respectively of orders a 4 , a 5 , a 6 , a 7 . We suppose that the free surface is the stream-line ^ = 0, and the constancy of pressure there imposed requires the constancy of U 2 2gy, where U, representing the resultant velocity, is equal to ^{(d-^jdxf + (dty/dy)*}, and g is the constant acceleration of gravity now to be determined. Thus when i/r = 0, U* - 2gy = 1 + 2 (1 - g) y + oftr* -f 2/3er*v cos 2a? + 4fye-w cos 3x + 68e~' y cos 4c + See-*" cos 5# + 4a/3e-* cos x + Garyer* cos 2a? + 8aSer*v cos 3# ......... (2) correct to a 1 inclusive. On the right of (2) we have to expand the exponentials and substitute for the various powers of y expressions in terms of a?. It may be well to reproduce the process as formerly given, omitting 8 and , and carrying (2) only to the order a 5 . We have from (1) as successive approximations to y: y = ae~ y cosx = acosx; ........................... (3) * The extension to arbitrary wave-lengths and velocities may be effected at any time by attention to dimensions. 480 ON PERIODIC IRROTATIONAL WAVES [419 y=a(l y)cosa; = o l + cos# - $a*cos2#; ............... (4) a (1 + |o s ) cos x - a j cos 2x + fa 8 cos 3x, ......... (5) which is correct to a 3 inclusive, /S being of order a 4 . In calculating (2) to the approximation now intended we omit the term in ay. In association with a/3 and 7 we take e'** = 1 ; in association with /3, er*v = 1 2y ; while a? e - 3y = o 2 (1 - 2y + 2# a - fy 8 ). Thus on substitution for y* and y 8 from (5) ft 2 e -2y = a * ( i _ 2y -- a 8 - 4O 3 cos # + a 2 cos 2x - a s cos 3#}. In like manner 2/9e-^ cos 2# = 2/9 cos 2a? - 2a/S (cos a; + cos 3#). Since the terms in cos x are of the fifch order, we may replace a cos x by y, and we get U* - Igy = 1 + a 2 + a* + 2y (1 - g - a 2 - 2a 4 + ) + (a 4 + 2/9) cos 2a; + (- |a 5 + 4 7 - 2a/3) cos 3# ....... (6) The constancy of (6) requires the annulment of the coefficients of y and of cos 2x and cos 3x, so that = -K> 7 = ^ 5 , ........................... (7) and # = l-a 2 -fa 4 .................................. (8) The value of g in (8) differs from that expressed in equation (11) of my former paper. The cause is to be found in the difference of suppositions with respect to >/r. Here we have taken ^ = at the free surface, which leads to a constant term in the expression for y, as seen in (5), while formerly the constant term was made to disappear by a different choice of >/r. There is no essential difficulty in carrying the approximation to y two stages further than is attained in (5). If 8, e are of the 6th and 7th order, they do not appear. The longest part of the work is the expression of e~ y as a function of x. We get and thence from (1) a 4 125a 8 - cos 4#+ - cos5# ........................................... (10) 1917] AT THE SURFACE OF DEEP WATER 481 When we introduce the values of /? and 7, already determined in (7) with sufficient approximation, we have in agreement with equations (13), (18) of my former paper when allowance is made for the different suppositions with respect to ty, as may be effected by expressing both results in terms of a, the coefficient of cos #, instead of a. The next step is the further development of the pressure equation (2), so as to include terms of the order a 7 . Where ft, 7, etc. occur as factors, the expression for y to the third order, as in (5), suffices; but a more accurate value is required in ofe'^. Expanding the exponentials and replacing products of cosines by cosines of sums and differences, we find in the first place U*-2gy = 2(1 -g-tf}y + 1 37a 7 + cos 2a ja 4 + 2/3 + ^- - 2 + co S 3,(-^-2^ + 47- 3 ^ + cos 4# j~ + 2a 2 - 6a 7 + 6SJ- (12) From the terms in cos x we now eliminate cos x by means of a cos x = y (1 fa 2 ) + ^a 2 + a 2 cos 2arf. thus altering those terms of (12) which are constant, and which contain y and cos 2#. Thus modified, (12) becomes + cos 2x L< + 1$ + ~ [ The terms in o 3 /3(cosar, cosSar) should read +^a 3 /3cosa-, + - a 3 /3 cos 3* ; apparently the term - 4a 3 /3 cos x cos 2x had been omitted from the development of 2/3e~ 2 ' cos 2.r. t Since terms of order a 7 are retained, the term - 1 a 3 cos 3.r should be added to the expression O for a cos a;. W. F. S.] R. VI. 31 4S2 ON PERIODIC IRROTATIONAL WAVES [419 + cos 4* |^ + 2o> - 6a 7 + 6sl (13). The constant part has no significance for our purpose, and the term in y can be made to vanish by a proper choice of g. If we use only a, none of the cosines can be made to disappear, and the value of g is # = l-a 2 -2a 4 -7a 6 ............................ (14) When we include also ft, we can annul the term in .cos 2# by making ............................ < 15 > and with this value of But unless a is very small, regard to the term in cos 3# suggests a higher value of ft as the more favourable on the whole. With the further aid of 7 we can annul the terms both in cos 2# and in cos 3#. The value of ft is as before. That of 7 is given by and with this is associated , = l- a .- 5 f-^ ......................... (IS)- The inclusion of 8 and e does not alter the value of g in this order of approximation, but it allows us to annul the terms in cos 4>x and cos 5x. The appropriate values are a 6 a 7 -72' e= and the accompanying value of 7 is given by 413a' (20) while ft remains as in (15). We now proceed to consider how far these approximations are successful, for which purpose we must choose a value for a. Prof. Burnside took a = . With this value the second term of ft in (15) is nearly one-third of the first (Stokes') term, and the second term of 7 in (20) is actually larger^ than the [* With the alterations specified in the footnotes on p. 481, the terms in (13) involving a- ; i;t. and (a 7 , o 3 /3) cos 3x, become 2y . a 2 /3, and cos Sx ( - a" + - a 3 /3j. Then the highest terms in (16), (17), (18), and (20) become respectively - *g , jjj ( + ) , - ^ , and g ( + ^ a*) ; the second term in (20) being now little more than half the first when o = J. W. F. 8.] 1917] AT THE SURFACE OF DEEP WATER 483 first. If the series are to be depended upon, we must clearly take a smaller value. I have chosen a = -j^, and this makes by (15), (18), (20) = - -000,052,42, 7 = -000,000,976, g = '989,736,92 ....... (21)* The next step is the calculation of approximate values of y from (11), which now takes the form y = - -0051 + -101,165,0 cos x - -005,183,3 cos 2# + -000,399,6 cos 3x - -000,033,3 cos 4a? -I- -000,003,3 cos ox. ............... (22) For example, when x = 0,y = "091,251,3. The values of y calculated from (22) at steps of 22| (as in Burnside's work) are shown in column 2 of Table I. We have next to examine how nearly the value of y afforded by (22) really makes i|r vanish, and if necessary to calculate corrections. To this $ and e in (1) do not contribute sensibly and we find T/T = + -000,01 5,4 for x 0. In order to reduce ty to zero, we must correct the value of y. With sufficient approximation we have in general or in the present case 000,015,4 1-091 = -000,014,1, so that the corrected value of y for # = is -091,237,2. If we repeat the calculation, using the new value of y, we find i/r = 0. TABLE 14 X y from (22) y corrected f/2 - 2gy - I Corrected by 30 + 091,251,3 + 091,237,2 010,104,9 45 22* + -084,839,7 + 084,841,9 4,7 44 45 + 066,182,8 + 066,181,8 4,3 43 67^ + 036,913,1 + 036,915,1 4,1 44 90 + -000,050,0 + -000,052,4 . 4,2 46 112* - -039,782,7 - -039,780,2 4,4 47 135 - -076,316,2 -076,317,5 4,3 43 157* - -102,381,1 - -102,395,1 . . 4,7 44 180 -111,884,7 -111,907,9 010,105,1 47 [* With the corrections specified in the footnote on p. 482 we have 7 = -000,000,905, g = -989,737,42. W. F. S.] t The double use of 8 will hardly cause confusion. [J With the corrections specified in the footnotes on pp. 481, 482, and calculating direct from (2), with the inclusion of the term 65e~ tv cos 4x, I find that the first 5 figures in the value of [72 _ 20?/ - 1 are as j n the table, whilst the last 2 figures, proceeding in order from x=0 to x = 180, become 45, 45, 44, 43, 42, 42, 45, 51, 53; after making 6 modifications in "y corrected" (third column), the first 6 figures of which remain as printed, whilst the last becomes, taken in the same order, 1, 9, 9, 1, 4, 3, 6, 3, 8, these modified values of y in every case reducing \j/ to zero to 7 places of decimals. W. F. S.] 312 484 ON PERIODIC IRROTATIONAL WAVES [410 In the fourth column are recorded the values of U* 2gy l, calculated from (1) with omission of 8 and 6, and with the corrected values of y. d-ty/da;, d-ty/dy were first found separately, and then U* as the sum of the two squares. The values of 0, y, g employed are those given in (15), (18), (20). The form of -<Jr in (I) with these values of the constants vanishes when y takes the values of the third column, and the pressure at the surface is also constant to a high degree of approximation. The greatest difference is ('000,001,0), which may be compared with '4-, the latter amount representing the corresponding statical difference at the crest and trough of the wave. According to this standard the pressure at the surface is constant to 2 parts in a million*. The advantage gained by the introduction of ft and 7 will be better estimated by comparison with a similar calculation where only a (still equal to J^) and g are retained. By (2) in this case 7'-2 5 ry-l=a 2 e- 2 " + 2(l- 5 r)2/ (23) Table II shows the values of y and of a%~ 2l/ corresponding to the same values of # as before. The fourth column gives (23) when g is so determined as to make the values equal at and 180. It appears that the discrepancy in the values of U 3 Igy is reduced 200 times by the introduction of ft and 7, even when we tie ourselves to the values of ft, 7, g prescribed by approxi- mations on the lines of Stokes. TABLE II. X y a2-2 u*-*n-i + 091,276,5 008,331,4 010,207,7 22* 084,870,5 008,438,8 . . 183,4 45 066,182,4 008,760,2 . . 120,7 67$ 036,882,6 009,288,9 . 047,1 90 010,000,0 .000,0 112* - -039,823,1 010,829,0 . 010,4 135 - -076,318,5 011,649,0 .080,2 157$ - -102,344,1 012,271,4 . 167,6 180 -111,832,6 012,506,5 010,207,7 A cursory inspection of the numbers in column 4 of Table I suffices to show that an improvement can be effected by a slight alteration in the value of ft. For small corrections of this kind it is convenient to use a formula which may be derived from (2). We suppose that while a and ^ are main- tained constant, small alterations Sft, 87, Sg are incurred. Neglecting the small variations of ft, 7, g when multiplied by a 2 and higher powers of o, we get By = Bft {cos 2# fa cos a? |a cos 3a?j + Sy[cos3x- 2o cos 2# - 2a cos 4#}, (24) [* With the alterations specified in footnote % on p. 483, the greatest difference becomes 000,001,1, so that the surface pressure is constant to 2f parts in a million. W. F. 8.] 1917] AT THE SURFACE OF DEEP WATER 485 and S(U*- 2gy) = 2a (B/3 - 8#) cos x + 28/3 cos 2# 4- 2 ( 287 -a8) cos 3^-6087 cos 4# ................ (25) For the present purpose we need only to introduce 8/9, and with sufficient accuracy we may take S(U 2 -2gy) = 28j3cos2x ...................... (26) We suppose 8/8 = - '000,000,2, so that the new value of is - '000,052,6. Introducing corrections according to (26) and writing only the last two figures, we obtain column 5 of Table I, in which the greatest discrepancy is reduced from 10 to 4 almost as far as the arithmetic allows and becomes but one- millionth of the statical difference between crest and trough. This is the degree of accuracy attained when we take simply A|T = y - ae~y cos x fie~w cos 2# yer* cos 3#, ........ . . .(27) with a = -fa, g and 7 determined by Stokes' method, and /3 determined so as to give the best agreement*. [1919. Reference may be made to Wilton, Phil. Mag. Vol. 27, p. 385, 1914; also to Havelock, Roy. Soc. Proc., Vol. A 95, p. 38, 1918.] [* If we include the first 3 terms of (25), and write 5 (C/2 _ -2gy) = -000,000,2 cos x - -000,000,4 cos 2x+ -000,000,2 cosSx, corresponding to 5= --000,000,2, 8y= + -000,000,04, 8g= - -000,001,2, we find that the cor- rected values of the last two figures of U*-2gy-l, given in footnote J on p. 483, become 45, 45, 44, 45, 46, 46, 45, 46, 45, taken in the same order ; these results would not be affected by including the term in (25) involving cos 4x. Thus the greatest discrepancy is reduced from 11 to 2, becoming only half one-millionth of the statical difference. The new values of /3, 7, and g, thus determined so as to give the best agreement, are /3 = - '000,052,6, y = -000,000,94, = -989,736,2. W. F. S.] - 420. ON THE SUGGESTED ANALOGY BETWEEN THE CONDUCTION OF HEAT AND MOMENTUM DURING THE .TURBULENT MOTION OF A FLUID. [Advisory Committee for Aeronautics, T. 941, 1917.] THE idea that the passage of heat from solids to liquids moving past them is governed by the same principles as apply in virtue of viscosity to the passage of momentum, originated with Reynolds (Manchester Proc., 1874); and it has been further developed by Stanton (Phil. Trans., Vol. cxc. p. 67, 1897; Tech. Rep. Adv. Committee, 1912-13, p. 45) and Lanch ester (same Report, p. 40). Both these writers express some doubt as to the exactitude of the analogy, or at any rate of the proofs which have been given of it. The object of the present note is to show definitely that the analogy is not complete. The problem which is the simplest, and presumably the most favourable to the analogy, is that of fluid enclosed between two parallel plane solid surfaces. One of these surfaces at y = is supposed to be fixed, while the < >ther at y = 1 moves in the direction of x in its own plane with unit velocity. If the motion of the fluid is in plane strata, as would happen if the viscosity were high enough, the velocity u in permanent regime of any stratum y is represented by y simply. And by definition, if the viscosity be unity, the tangential traction per unit area on the bounding planes is also unity. Let us now suppose that the fixed surface is maintained at temperature 0, and the moving surface at temperature 1. So long as the motion is stratified, the flow of heat is the same as if the fluid were at rest, and the temperature (0) at any stratum y has the same value y as has u. If the conductivity is unity, the passage of heat per unit area and unit time is also unity. In this case, the analogy under examination is seen to be complete. The question is will it still hold when the motion becomes turbulent? It appears that the identity in the values of and u then fails. The equations for the motion of the fluid when there are no impressed forces are Du 1 d 1917] ANALOGY BETWEEN CONDUCTION OF HEAT AND MOMENTUM 487 with two similar equations, where D d d d d -m = dt + U d-x +V d-y + W dz> representing differentiation with respect to time when a particle of the fluid is followed. In like manner, the equation for the conduction of heat is -.. Although we identify the values of k and v, and impose the same boundary conditions upon u and 0, we see that the same values will not serve for both u and 6 in the interior of the fluid on account of the term in dp/dx, which is not everywhere zero. It is to be observed that turbulent motion is not steady in the hydro- dynamical sense, and that a uniform regime can be spoken of only when we contemplate averages of u and 6 for all values of x or for all values of t. It is conceivable that, although there is no equality between the passage of heat and the tangential traction at a particular time and place, yet that the average values of these quantities might still be equal. This question must for the present remain open, but the suggested equality does not seem probable. The principle of similitude may be applied in the present problem to find a general form for H, the heat transmitted per unit area and per unit time (compare Nature, Vol. xcv. p. 67, 1915)*. In the same notation as there used, let a be the distance between the planes, v the mean velocity of the stream, 6 the temperature difference between the planes, K the conductivity of the fluid, c the capacity for heat per unit volume, v the kinematic viscosity. Then K& favc cv 12 = - . - , a \ K K. or, which comes to the same, where F, F^ denote arbitrary functions of two variables. When For a given fluid cv/tc is constant and may be omitted. Dynamical similarity is attained when av is constant, so that a. complete determination of F (experimentally or otherwise) does not require the variation of both a and v. There is advantage in keeping a constant; for if a be varied, geometrical similarity demands that any roughnesses shall be in proportion. The objection that K, c, v are not constants, but functions of the tempera- ture, may be obviated by supposing that is small. [* This volume, p. 300.] 421. THE THEORY OF ANOMALOUS DISPERSION. [Philosophical Magazine, Vol. xxxin. pp. 496 499, 1917.] IN a short note* with the above title I pointed out that Maxwell as early as 1869 in a published examination paper had given the appropriate formulae, thus anticipating the work of Sellmeierf and HelmholtzJ. It will easily be understood that the German writers were unacquainted with Maxwell's formulae, which indeed seem to have been little known even in England. I have thought that it would be of more than historical interest to examine the relation between Maxwell's and Helmholtz's work. It appears that the generalization attempted by the latter is nugatory, unless we are prepared to accept a refractive index in the dispersive medium becoming infinite with the wave-length in vacuo. In the aether the equation of plane waves propagated in the direction of x is in Maxwell's notation pd*r)/dP = Ed*r)/da?, .............................. (1) where 77 is the transverse displacement at any point x and time t, p is the density and E the coefficient of elasticity. Maxwell supposes " that every part of this medium is connected with an atom of other matter by an attractive force varying as distance, and that there is also a force of resistance between the medium and the atoms varying as their relative velocity, the atoms being independent of each other"; and he shows that the equations of propagation in this compound medium are where p and v are the quantities of the medium and of the atoms respectively in unit of volume, 77 is the displacement of the medium, and tj + that of the atoms, <rp* is the attraction, and a-Rd^/dt is the resistance to the relative motion per unit of volume. * Phil. Mag. Vol. XLVIII. p. 151 (1899) ; Scientific Papert, VoL iv. p. 413. A miuprint is now corrected, see (4) below. t Pogg. Ann. CXLIII. p. 272 (1871). * Pogg. Ann. CLIV. p. 582 (1874) ; Witientchaftliche Abhandlungen, Band n. p. 213. 1917] THE THEORY OP ANOMALOUS DISPERSION 489 On the assumption that r,, = ((7, Z>)rt-,,/wn/ P >* ..... '.^'.l.* ......... (3) we get Maxwell's results* 1 1 =P + <T <rn* p*-n* v 2 l z n* E r E (p*-ri>)* + RW 2 L _<rn* Rn vhi~^ (p*-n*)* + RW ............................ ( " Here v is the velocity of propagation of phase, and I is the distance the waves must run in order that the amplitude of vibration may be reduced in the ratio e : 1. When we suppose that R = 0, and consequently that I = oo , (4) simplifies. If v be the velocity in sether (<r = 0), and v be the refractive index, For comparison with experiment, results are often conveniently expressed in terms of the wave-lengths in free sether corresponding with the frequencies in question. Thus, if X correspond with n and A with p, (6) may be written < 7 > the dispersion formula commonly named after Sellmeier. It will be observed that p, A refer to the vibrations which the atoms might freely execute when the aether is maintained at rest (77 = 0). If we suppose that n is infinitely small, or \ infinitely great, "oc 2 =l + <r/V>> ................................. (8) thus remaining finite. Helmholtz in his investigation also introduces a dissipative force, as is necessary to avoid infinities when n=p, but one differing from Maxwell's, in that it is dependent upon the absolute velocity of the atoms instead of upon the relative velocity of sether and matter. A more important difference is the introduction of an additional force of restitution (a?x), proportional to the absolute displacement of the atoms. His equations are * Thus in Maxwell's original statement. In my quotation of 1899 tRe sign of the second term in (4) was erroneously given as plus. t What was doubtless meant to be d^jdy- appears as dPydx*, bringing in x in two senses. 490 - THE THEORY OF ANOMALOUS DISPERSION [421 This notation is so different from Maxwell's, that it may be well to exhibit explicitly the correspondence of symbols. Helmholtz... I A * I y I *- ' ! * m a* I c I & Maxwell rj \ p E \ x \ I op' o- ] w ! 1/J When there is no dissipation (R = 0, y 2 = 0), these interchanges harmonize the two pairs of equations. The terms involving respectively R and 7* follow different laws. Similarly Helmholtz's results mn'-o'-ff c 2 n 2 a 1 aV M- = _^'_ i (lg) . en a 2 ft (wm 2 a 2 p 2 ) 2 -f 7 4 n 2 identify themselves with Maxwell's, when we omit R and 7* and make a 2 = 0. In order to examine the effect of a 2 , we see that when 7 = 0, (11) becomes 1 u, 8* mn* a* c 2 a 2 a 2 n 2 mn 2 -a 3 -/3 2 ' or in terms of v* (= Co'/c 8 ), ""^"f mtf-a'-V (U) If now in (14) we suppose n = 0, or X = x , we find that v = oo , unless a 2 = 0. If a 2 = 0, we get, in harmony with (6), < 15 > which is finite, unless ran 2 = yS 2 . It is singular that Helmholtz makes precisely opposite statements! : " Wenn a = 0, wird k = und 1/c = oc ; sonst werden beide Werthe endlich sein." The same conclusion may be deduced immediately from the original equations (9), (10). For if the frequency be zero and the velocity of pro- pagation in the medium finite, all the differential coefficients may be omitted ; so that (9) requires x - = and (10) then gives a 2 = 0. WullnerJ, retaining a? in Helmholtz's equation, writes (14) in the form (16) [* The result (12) is so given by Helmholtz; but the first "-" should be " + ", involving some further corrections in Helmholtz's paper. + Helmholtz, however, supposes 7*0, and on that supposition his statements appear to be correct. They cannot, however, legitimately be deduced, as appears to be assumed by Helmholtz, from the equations which in his paper immediately precede those statements, since those equations are obtained on the understanding that the ratio of the right-hand side of (12) to that of (11) is zero when n = 0, which is not the case when a absolutely = 0. W. F. S.] I Wied. Ann. xvn. p. 580; xxm. p. 306. 1917] THE THEORY OF ANOMALOUS DISPERSION 491 applicable when there is no absorption. And he finds that in many cases the facts of observation require us to suppose P = Q. This is obviously the condition that i/ 2 shall remain finite when \ = x , and it requires that a 2 in Helmholtz's equation be zero. It is true that in some cases a better agreement with observation may be obtained by allowing Q to differ slightly from P, but this circumstance is of little significance. The introduction of a new arbitrary constant into an empirical formula will naturally effect some improvement over a limited range. It remains to consider whether a priori we have grounds for the assumption that v is finite when \ = oo . On the electromagnetic theory this should certainly be the case. Moreover, an infinite refractive index must entail complete reflexion when radiation falls upon the substance, even at perpen- dicular incidence. So far as observation goes, there is no reason for thinking that dark heat is so reflected. It would seem then that the introduction of a 2 is a step in the wrong direction and that Helmholtz's formulae are no improvement upon Maxwell's*. It is scarcely necessary to add that the full development of these ideas requires the recognition of more than one resonance as admissible (Sellmeier). [* Similarly, the substitution of a dissipative force " dependent upon the absolute velocity of the atoms instead of upon the relative velocity of tether and matter " (p. 489 above) appears to be the reverse of an improvement, since Maxwell's results (4) and (5) above lead to a finite v when n = 0, but E * (cf. p. 490 and footnote t). W. F. S.] 422. ON THE REFLECTION OF LIGHT FROM A REGULARLY STRATIFIED MEDIUM. [Proceedings of the Royal Society, A, Vol. XCIIL pp. 565577, 1917.] THE remarkable coloured reflection from certain crystals of chlorate of potash described by Stokes*, the colours of old decomposed glass, and probably those of some beetles and butterflies, lend interest to the calculation of reflection from a regular stratification, in which the alternate strata, each uniform and of constant thickness, differ in refractivity. The higher the number of strata, supposed perfectly regular, the nearer is the approach to homogeneity in the light of the favoured wave-lengths. In a crystal of chlorate described by R. W. Wood, the purity observed would require some 700 alternations combined with a very high degree of regularity. A general idea of what is to be expected may be arrived at by considering the case where a single reflection is very feeble, but when the component reflections are more vigorous, or when the number of alternations is very great, a more detailed examination is required. Such is the aim of the present communi- cation. The calculation of the aggregate reflection and transmission by a single parallel plate of transparent material has long been known, but it may be convenient to recapitulate it. At each reflection or refraction the amplitude of the incident wave is supposed to be altered by a certain factor. When the light proceeds at A from the surrounding medium to the plate, the factor for reflection will be supposed to be &', and for refraction c ; the corresponding quantities when the progress at B is from the plate to the surrounding medium may be denoted by e', f. Denoting the incident vibration by unity, we have then for the first component of the reflected wave &', for the second c'e' fe~**, for the third c'e' 3 f'e~ yM , and so on, all reckoned as at the first surface A. Here B denotes the linear retardation of the second reflection as compared with the first, due to the thickness of the plate, and it is given by B = 2fjiTcosa, ............................... (1) * Roy. Soc. Proc., February, 1885. See also Rayleigh, Phil. Mag. Vol. xxiv. p. 145 (1887), Vol. xxvi. pp. 241, 256 (1888); Scientific Papers, Vol. in. pp. 1, 190, 204, 264. 1917] REFLECTION OF LIGHT FROM A REGULARLY STRATIFIED MEDIUM 493 where //, is the refractive index, T the thickness, and a the angle of refraction within the plate. Also k = 2w/X, X being the wave-length. Adding together the various reflections and summing the infinite geometric series, we find In like manner for the wave transmitted through the plate we get .................. (3) the incident and transmitted waves being reckoned as at A. The quantities b', c', e', f are not independent. The simplest way to find the relations between them is to trace the consequences of supposing 8 = in (2) and (3). For it is evident a priori that, with a plate of vanishing thickness, there must be a vanishing reflection and an undisturbed total transmission*. Accordingly, b' + e' = 0, cf = l-e'\ ........................ (4) the first of which embodies Arago's law of the equality of reflections, as well as the famous " loss of half an undulation." Using these, and substituting ij for e, we find for the reflected vibration, and for the transmitted vibration In dealing with a single plate, we are usually concerned only with inten- sities, represented by the squares of the moduli of these expressions. Thus, , Intensity of reflected light = A > o ( 1 - 7? 2 cos &S) 2 + rf sm 2 k8 1 - 2?? 2 cos k8 + q* ' Intensity of transmitted light = - ~ - rs - ; 1 2T? 2 cos k8 + r)* the sum of the two expressions being unity, as was to be expected. According to (7), not only does the reflected light vanish completely when 5 = 0, but also whenever ^k8=S7r, s being an integer; that is, whenever 6 = SX. Returning to (5) and (6), we may remark that, in supposing k real, we are postulating a transparent plate. The effect of absorption might be included by allowing k to be complex. * " Wave Theory of Light," Ency. Brit. Vol. xxiv. 1888; Scientific Papers, Vol. in. p. 64. 494 ON THE REFLECTION OF LIGHT [422 When we pass from a single plate to consider the operation of a number of plates of equal thicknesses and separated by equal intervals, the question of phase assumes importance. It is convenient to refer the vibrations to points such as 0, 0', bisecting the intervals between the plates ; see figure, where for simplicity the incidence is regarded as perpendicular. When we ^ reckon the incident and reflected waves from instead of A, we must introduce the additional factor e~* iks ', S' for the interval corresponding to 8 for the plate. Thus (5) becomes - = r. (9) I ^ *jZg CM So also if we reckon the transmitted wave at 0', instead of A, we must introduce the factor e~** <*+*'', and (6) becomes ^ _ a e -fJM =t (10) The introduction of the new exponential factors does not interfere with the moduli, so that still \r*\ + \t*\ = I (11) Further, we see that and thus (in the case of transparency) r/t is a pure imaginary. In accordance with (11) and (12) it is permitted to write r = sin0.e'> t = i cos . e', (13) in which 6 and p are real and ^.SyjjlM (M) Also from (9), (13) i ir / cj | <y \ / i e \ where s is an integer and tan v = T - - (16) 1 tj 2 cos kS The calculation for a set of equal and equidistant plates may follow the lines of Stokes' work for a pile of plates, where intensities were alone regarded*. * Roy. Soc. Proc. 1862; Math, and Phys. Papers, Vol. IT. p. 145. 1917] FROM A REGULARLY STRATIFIED MEDIUM 495 In that case there was no need to refer the vibrations to particular points, but for our purpose we refer the vibrations always to the points 0, 0', etc., bisecting the intervals between the plates. On this understanding the formal expressions are the same. <j> m denotes the reflection from ra plates, referred to the point in front of the plates ; -^r m the transmission referred to a point O m behind the last plate. " Consider a system of m + n plates, and imagine these grouped into two systems, of m and n plates respectively. The incident light being represented by unity, the light <f> m will be reflected from the first group, and i|r, n will be transmitted. Of the latter the fraction ^ n will be transmitted by the second group, and <f> n reflected. Of the latter the fraction ty m will be transmitted by the first group, and <f> m reflected, and so on. Hence we get for the light reflected by the whole system, <f> m + tm 2 <n + * and for the light transmitted which gives, by summing the two geometric series, The argument applies equally in our case, only <f> mj etc., now denote complex quantities by which the amplitudes of vibration are multiplied, instead of real positive quantities, less than unity, relating to intensities. By definition fa = r,-\fr 1 = t. Before proceeding further, we may consider the comparatively simple cases of two or three plates. Putting m = n = 1, we get from (17), (18) > ................... <> By (13), 1 - r 2 + 2 = 1 - e zi *, and thus r !-<** - (20) It appears that <f> 2 vanishes not only when r = 0, but also independently of r when cos 2/> = 1. In this case i/r 2 = - 1. When cos 2/t> = 1, r = + sin 0, t = i cos 6, so that r is real and t is a pure imaginary. From (9) we find that a real r requires that cos p (8 + 8') = 7? 2 cos &('- 8) ................... (21) or, as it may also be written, *" 7 ^ ...................... (22) 496 ON THE REFLECTION OF LIGHT [422 When V) is small we see that &(8 + 8') = (2s + !)TT, or S + S' = (2s + l)\/2. In this case only the first and second components of the aggregate reflection are sensible. If there are three plates we may suppose in (17) m = 2, n = 1. Thus ^4+Jb^J, - ...................... (23) <J> 2 and i^ 2 being given by (19). If <j> 3 = 0, <Ml-r&) + r^ = ......................... (24) In terms of p and 6 sin 0(1 -<**)<* cos^e 2 * T^in^^' "l-sin 2 ^- Using these in (24), we find that either sin 0, and therefore r, is equal to zero, or else that cos<0 + #(2-#)(l-#)cos 2 + (l-#) s = 0, ......... (26) E being written for e 2 *?. By solution of the quadratic cos 2 = - ( 1 - E) 2 /E or I - E~\ The second alternative is inadmissible, since -it makes the denominators zero in (25). The first alternative gives E = cos 2p + i sin 2p = 1 - cos 2 i cos 6 V(l - i cos 2 0), whence cos#= 2sin/> ....... . ....................... (27) When rj, and therefore r, is small, cos# = 1 nearly, and ^ in (15) may be omitted. Hence S + 8' = X(or) + sX, ........................ (28) as might have been expected. If we suppose e? = 1, </> 2 = 0, ^ 2 = t - 1 and (23) gives </> 3 = r. It is easy to recognize that for every odd number <f> m = r, and for every even number <f>,n = 0. In his solution of the functional equations (17), (18)*, Stokes regards <f> and >/r as functions of continuous variables m and n, and he obtains it with the aid of a differential equation. The following process seems simpler, and has the advantage of not introducing other than integral values of m and n. If we make m = 1 in (17), or if we write u n = r<f> n - 1, + ? = Q ...................... (30) Stirling has shown, Roy. Soc. Proe. A, Vol. xc. p. 237 (1914), that the two equations are not independent, (18) being derivable from (17). 1917] FROM A REGULARLY STRATIFIED MEDIUM 497 In this we assume u n = v n+ ^/v n , so that w n+a + (l-r a + J )v n+1 + s w n =0 .................... (31) The solution of (31) is where p + q = J 2 1, pq = t 3 , ..................... (32) and H, K are arbitrary constants. Accordingly Hp n+1 + Kq n+1 U= ~ in which there is but one constant of integration effectively. This constant may be determined from the case of n = 1, for which Ml =:r 2 -l. By means of (32) we *-ci&==JS8r (3*) and <6 ra = or since by (32) r 8 = (p + 1) (q + 1), ft _ &- , (35) where - -.(36) q In order to find -*\r m we may put n = 1 in (17); and by use of (29), with m substituted for n, we get and on reduction with use of (35), (32), By putting m = 0, we see that the upper sign is to be taken. The expressions thus obtained are those of Stokes: <f>m = ^ ...(38) &i _ -m a _ a -i ofom _ a -i fr-m The connexion between a, b and r, is established by setting m = 1. Thus * In Stokes' problem, where r, t, </>, ^ represent intensities, a and 6 are real. If there is no absorption, r + t = 1, so that a 1, 6 - 1 are vanishing quantities. In this case r t 1 6-1 a-1 a-1+6-1' R. vi. 32 498 ON THE REFLECTION OF LIGHT [422 and g = _y, / x ^ (40) mr l-r l+(ni-l)r When m tends to infinity, <, approaches unity, and i/r m approaches zero. For many purposes, equations (38), (39) may conveniently be written in another form, by making 6 = e ft , a = e a . Thus <frm ^ 1 sinh mft sinh a. sinh (a + mft) ' r t 1 .(41) sinh/3 sinha sinh(a + /3) ^ where in Stokes* problem a and ft are real, and are uniquely determined in terms of r and t by (44), (46) below*.. If we form the expression for (1 + r 3 2 )/2r by means of (42), we find that it is equal to cosh or. Also 8hih*-L lAilt L), (43) from which we see that, if ? and t are real positive quantities, such that r + 1 < 1, sinh a is real. Similarly, sinh ft, sinh (a + ft) are real. Passing now to my proper problem, where r and t are complex factors, represented (when there is no absorption) by (13), we have 1 + r- - 1* cos p cosh a = ^ = . 7; , (44) 2r sin 6 so that cosh a is real. Also f*f\a- f\ (45) sin-* If we write a = a, + iot.,, ft = /?, + ift z , where er lf cr 2 , fti, & are real, sinh^a = sinh Q-J cos a 2 4- 1 cosh aj sin 2 , cosh a = cosh ^ cos 2 4- 1 sinh a! sin a,. Since cosh a is real, either o, or sin 2 must vanish. In the first case, sinh a = i sin a a , and (45) shows that this can occur only when sin* 6 > cos 2 p. In the second case (sino 2 = 0), sinh 2 a = sinh 2 a l , which requires that sin 2 6 < cos 2 p. Similarly if we interchange r and t, so that cosh ft is real, requiring either & = 0, or sin # 2 = 0. Also Except as to sign, which is a matter of indifference. It may be remarked that hi$ equation (13) can at once be put into this form by making his o and j3 pure imaginaries. 1917] FROM A REGULARLY STRATIFIED MEDIUM 499 If ft = 0, sinh j3 = i sin ft, which can occur only when sin 2 p < cos 2 6, or, which is the same, sin 2 6 < cos 2 p. Again, if sin ft = 0, sinh 2 # = sinh 2 ft, occurring when sin 2 6 > cos 2 p. It thus appears that, of the four cases at first apparently possible, i = ft = 0, sin a z = sin ft = 0, are excluded. There are two remaining alternatives : (i) sinh 2 a = ; sin 2 6 > cos 2 p ; cti = 0, sin ft = ; (ii) sinh 2 a = + ; sin 2 6 < cos 2 p ; ft = 0, sin 2 = 0. Between these there is an important distinction in respect of what happens when m is increased. For <f) m = sinh ??i/3/sinh (a + m/3). In case (i) this becomes l/<m = cos 02 + i coth wft sin 2 , (48) and l/|< m | 2 =l+sin 2 a 2 /sinh 2 wft (48 bis) If ft be finite, sinh 2 wft tends to oo when w increases, so that | < m | 2 tends to unity, that is, the reflection tends to become complete. We see also that, whatever m may be, <f> m cannot vanish, unless ft = 0, when also r = 0. In case (ii) + l/(f> m = cosh ! i cot wft sinh a l , (49) and 1/j <f) m | 2 = 1 + sinh 2 a,/sm 2 wft, (49 bis) so that (f> m continues to fluctuate, however great m may be. Here <j> m may vanish, since there is nothing to forbid wft = sir. Of this behaviour we have already seen an example, where cos 2 /o = 1. In order to discriminate the two cases more clearly, we may calculate the value of sinh 2 a from (43), writing temporarily for brevity e liks = ^ } e * m '=' (50) Thus by (9) and (10) ( =(^1" (51) so that r + t = , . ^. ., , or (A 17) A whence ^^ A 2 I) 2 The two factors in the numerator of the fraction differ only by the sign of 17, so that the fraction itself is an even function of r). The first factor may be written {(A - 77) A' + 1 - r) A} {(A - rj) A' - (1 - T; A)} = - (1 + AA' - 7j(A + A')l }1 - AA' + 77 (A'- A)}; 322 500 ON THE REFLECTION OF LIGHT [422 and similarly the second factor may be written with change of sign of 77 - {1 + AA' + rj (A + A')} {1 - AA' - rj (A' - A)}. Accordingly .,. K1+AA7-^A + A7}{(1-AA7-17'(A-A') 2 } 2 '''-' In this, on restoring the values of A, A', + AA' i) (A + A') = 2e** ta+ *'> {cos i&(8 + 8') /cos k(S - 8% and 1 - A A' rj ( A - A') = - 2i e W+*'> {sin k (8 + 8') + rj sin k (8 - 8')}. Also 4A' 2 (A a - I) 2 = - Ue ik(S+ v sin 8 $kS, and thus _ {cos 2 ^k (8 + 8') - 77" cos4 (8 - 8Q} 7, 2 sin 2 U8 x {sin 2 p(8 + 8')-; a 8m 2 p(8-8')} ....... (55) The transition between the two cases (of opposite behaviour when w = oo ) occurs when sinh a = 0. In general, this requires either cos i A; (8 + 8') sin k (8 + 8') ^cosl^-ar or ^^inws^r ...... (56) conditions which are symmetrical with respect to B and 8', as clearly they ought to be*. In (55), (56), rj 1 is limited to values less than unity. Reverting to (43), we see that the evanescence of sinh 2 o requires that ? = + 1 T t, or, if we separate the real and imaginary parts of r and t, r= 1+^ + 0,. If, for example, we take r = 1 t, we have Also jr| 2 = l-|<| 2 ; so that jri^l + J,, j< {' = -,. In like manner by interchange of r and t, \t\*=l + r jt | r | a = _ ri) showing that in this case r,, ij are both negative. The general equation (55) shows that sinh 2 a is negative, when rj 1 lies cos 2 A; (8 + 8') sin 2 jfe (8 -t- 8') cos 2 i&(8 -8') si between This is the case (i) above defined where an increase in m leads to complete reflection. On the other hand, sinh 2 a is positive when if lies outside the * That is with reversal of the sign of 77, which makes no difference here. 1917] FROM A REGULARLY STRATIFIED MEDIUM 501 above limits, and then (ii) the reflection (and transmission) remain fluctuating however great in may be. When if is small, case (ii) usually obtains, though there are exceptions for specially related values of 8 and 8'. Particular cases, worthy of notice, occur when 8' 8 = s\, where s is an integer. If &' + 8 = s\, sinh 2 a = i7 2 cos 2 p8-l, ........................ (57) and is negative for all admissible values of 77, case (i). If 8' 8 = \, sinh a a = cos 2 pS/'; 2 -l > ........................ (58) and we have case (i) or case (ii), according as 77* is greater or less than When 77 is given, as would usually happen in calculations with an optical purpose, it may be convenient to express the limiting values of (56) in another form. We have ^ = tan i ArS . tan k8', \1 = - cot k8 . tan W. . . .(59) 1 + r) L r) When the passage is perpendicular, Young's formula, viz. 17 = (/A !)/(/& + 1), gives (!Ti7)/(li7)-/**, ........................... (60) fi being the relative refractive index. We will now consider more in detail some special cases of optical interest. We choose a value of 8 such as will give the maximum reflection from a single plate. From (5) or (9) 1 _ (I-*; 2 ) 2 . , fil , J7T J " + 2^(1- cos k&y so that | r | is greatest for a given 77 when cos k8 1. And then We may expect the greatest aggregate reflection when the components from the various plates co-operate. This occurs when e - ik(S+s<} = 1, so that in the notation of (50), A 2 = A /2 = 1. The introduction of these values into (54) yields sinh 2 a = -l, .............................. (63) coming under (i). The same result may be derived from (57), since here cos fcS = 0. In addition to o x = 0, sin & = 0, we now have by (63) sin a = 1, cos a 2 = 0, and (48) gives l^p-tanh'mft, jr| = tanh& ................ (64) We are now in a position to calculate the reflection for various values of m, since by (62) tanh ft = r^- 2 = tanh 2, 502 OX THE REFLECTION OF LIGHT [422 if 77 = tanh f , so that 2 tanh- 1 (65) Let us suppose that, as for glass and air, /A = 1'5, *) = , making & = 0-40546. The following were calculated with the aid of the Smithsonian Tables of Hyperbolic Functions. It appears that under these favourable conditions as regards B and 8', the intensity of the reflected light | < m |" approaches its limit (unity) when in reaches 4 or 5. TABLE I. M "ft tanh m/9i |0m|=tanh 8 7n/3 1 1 0-4055 0-3846 0-1479 2 0-8109 0-6701 0*4490 3 1-2164 0-8386 0-7032 4 1-6218 0-9249 0-8554 5 2-0273 0-9659 0-9330 6 2-4328 0-9847 0-9696 7 2-8382 0-9932 0-9864 10 4O55 0-9994 0-9988 oc oc 1-0000 1-0000 In the case of chlorate of potash crystals with periodic twinning 77 is very small at moderate incidences. As an example of the sort of thing to be expected, we may take & = 0'04, corresponding to 17 = 0'02. TABLE II. : taub MJ/SI |*m| i 0-0400 0-00160 2 0-0798 0-00637 4 0-1586 002517 8 0-3095 0-09579 16 0-5649 0-3191 32 0-8565 0-7336 64 0-9881 0-9763 According to (58), if &' B = sX, the same value of sinh 2 a obtains as in (63), since we are supposing cos %k& = 0, and the same consequences follow*. Retaining the same values of 8, that is those included under B = (* + ) X, we will now suppose 6' = s'X, where s' also is an integer. From (55) (1 7l2) S1Dh2g= 4,7' =sinh2of " ( 66 > But when 17 is small, a slight departure from cos$fc5 = produces very different effects in the two cases. 1917] FROM A REGULARLY STRATIFIED MEDIUM 503 since sin 2 = in this case (ii). By (49 bis) we have now, setting w = 1, J_ sinhX^l+q') 8 |r| 2 sin 2 /9 2 ~ V as we see from (62). Comparing with (66), we find sin 2 & =1, & = (* + ) TT. Thus sin 2 m0 2 is equal to 1 or 0, according as m is odd or even ; and (49 bis) shows that when m is odd \<j>\' = i*-*fl(I + iff, ........................ (67) arid that when m is even, j< m | 2 = 0. The second plate neutralizes the reflection from the first plate, the fourth plate that from the third, and so on. The simplest case under this head is when 8 = | \, 8' = X. A variation of the latter supposition leads to a verification of the general formulae worth a moment's notice. We assume, as above, &' = s'\, but leave S open. Since eW = 1, (9) and (10) become and these are of the form (39), if we suppose a = ij~ l , b = e* iks . The reflection <f) m from m plates is derived from r by merely writing b m for 6, that is, e limks f or gij^ leaving \<f> m \ equal to |rj*, as should evidently be the case, at least when 8' = 0. [* This statement does not hold in general, when S' = s'\, where s' is an integer and may be zero. We have _ ;-I + 1?) sin $kS ' sothat |T Hence * consequently, if | <f> m \ = | r \ , we must have where n is an integer, so that 8= ^. This result may be verified for m = 2 or 3 from (19), (23), and (68). It includes as a special case that dealt with in the preceding paragraph, if, when m is odd, we write n = ( + ) (mil), where 8 is an integer. When S' = the strata intervening between the plates disappear, but the theory is only applicable on the supposition that reflection and refraction continue to take place as before at each of the contiguous surfaces of the plates. W. F. S.] 423. ON THE PRESSURE DEVELOPED IN A LIQUID DURING THE COLLAPSE OF A SPHERICAL CAVITY. [Philosophical Magazine, Vol. xxxiv. pp. 9498, 1917.] WHEN reading 0. Reynold's description of the sounds emitted by water in a kettle as it comes to the boil, and their explanation as due to the partial or complete collapse of Bubbles as they rise through cooler water, I proposed to myself a further consideration of the problem thus presented ; but I had not gone far when I learned from Sir C. Parsons that he also was interested in the same question in connexion with cavitation behind screw-propellers, and that at his instigation Mr S. Cook, on the basis of an investigation by Besant, had calculated the pressure developed when the collapse is suddenly arrested by impact against a rigid concentric obstacle. During the collapse the fluid is regarded as incompressible. In the present note I have given a simpler derivation of Besant's results, and have extended the calculation to find the pressure in the interior of the fluid during the collapse. It appears that before the cavity is closed these pressures may rise very high in the fluid near the inner boundary. As formulated by Besant*, the problem is "An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated ; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant." Since the fluid is incompressible, the whole motion is determined by that of the inner boundary. If U be the velocity and R the radius of the boundary at time t, and u the simultaneous velocity at any distance r (greater than R) from the centre, then d) * Besant'a Hydrostatics and Hydrodynamics, 1859, 158. 1917] PRESSURE DEVELOPED DURING COLLAPSE OF A SPHERICAL CAVITY 505 and if p be the density, the whole kinetic energy of the motion is (2) J R Again, if P be the pressure at infinity and ^ the initial value of R, the work done is 4-n-P -~-(R Q 3 -R 3 ) (3) When we equate (2) and (3) we get expressing the velocity of the boundary in terms of the radius. Also, since U=dR/dt, //3p\ r* (R"dR) //Sp\ f* &*d& = v (*?))* w^"*V^Ho^ ...... ( } if /8 = R/R Q . The time of collapse to a given fraction of the original radius is thus proportional to R p^P~^, a result which might have been anticipated by a consideration of "dimensions." The time T of complete collapse is obtained by making = in (5). An equivalent expression is given by Besant, who refers to Cambridge Senate House Problems of 1847. Writing /3 3 = z, we have n _ t , *Jo (1- 3)4 which may be expressed by means of F functions. Thus According to (4) U increases without limit as R diminishes. This indefinite increase may be obviated if we introduce, instead of an internal pressure zero or constant, one which increases with sufficient rapidity. We may suppose such a pressure due to a permanent gas obedient to Boyle's law. Then, if the initial pressure be Q, the work of compression is 4nrQR 3 log (R Q /R), which is to be subtracted from (3). Hence and 17=0 when P(l - z) 4- Qlog* =0, ........................... (8) z denoting (as before) the ratio of volumes R'/RJ. Whatever be the (positive) value of Q, U conies again to zero before complete collapse, and if Q > P the first movement of the boundary is outwards. The boundary oscillates between two positions, of which one is the initial. 506 ON THE PRESSURE DEVELOPED DURING THE The following values of P/Q are calculated from (8) : [423 z us 1 PIQ T&V 69147 1 arbitrary iJo 4-6517 2 06931 A 2-5584 4 0-4621 i 1-8484 10 0-2558 i 1-3863 100 00465 i arbitrary 1000 0-0069 Reverting to the case where the pressure inside the cavity is zero, or at any rate constant, we may proceed to calculate the pressure at any internal point. The general equation of pressure is 1 dp _ Du_ du du pfo ~Dt~ dt U fc'" u being a function of r and t, reckoned positive in the direction of increasing r. As in (1), u = UR*/r*, and du 1 dt = ^ Tt L dU dt Also and by (4) so that dt p K* Thus, suitably determining the constant of integration, we get -1 = P 3r At the first moment after release, when R= R , we have p = P(I-R /r) (11) When r = R, that is on the boundary, p = 0, whatever R may be, in accord- ance with assumptions already made. Initially the maximum p is at infinity, but as the contraction proceeds, this ceases to be true. If we introduce z to represent Rj/R?, (10) may be written R (12) and =H V ;' -<?-4>h .(13) 1917] COLLAPSE OF A SPHERICAL CAVITY 507 The maximum value of p occurs when -r: ...04) and then ^ = 1 -j- = j_| 1 _ t ><>t (15) r 4r 4* (z 1)* So long as z, which always exceeds 1, is less than 4, the greatest value of p, viz. P, occurs at infinity ; but when z exceeds 4, the maximum p occurs at a finite distance given by (14) and is greater than P. As the cavity fills up, z becomes great, and (15) approximates to $-*--. ae) corresponding to r = 4*# = 1-587^ ............................ (17) It appears from (16) that before complete collapse the pressure near the boundary becomes very great. For example, if R = -^R , p = 1260P. This pressure occurs at a rel