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Full text of "Scientific papers"

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 relatively moderate distance outside the boundary. 
At the boundary itself the pressure is zero, so long as the motion is free. 
Mr Cook considers the pressure here developed when the fluid strikes an 
absolutely rigid sphere of radius R. If the supposition of incompressibility 
is still maintained, an infinite pressure momentarily results; but if at this 
stage we admit compressibility, the instantaneous pressure P' is finite, and 
is given by the equation 



' being the coefficient of compressibility. P, P', $' may all be expressed in 
atmospheres. Taking (as for water) & = 20,000, P = 1, and R = ^R , Cook 
finds 

P' = 10,300 atmospheres = 68 tons per sq. inch, 

and it would seem that this conclusion is not greatly affected by the neglect 
of compressibility before impact. 

The subsequent course of events might be traced as in Theory of Sowid, 
279, but it would seem that for a satisfactory theory compressibility would 
have to be taken into account at an earlier stage. 



424. 

ON THE COLOURS DIFFUSELY REFLECTED FROM SOME 
COLLODION FILMS SPREAD ON METAL SURFACES. 

[Philosophical Magazine, Vol. xxxiv. pp. 423428, 1917.] 

IT is known that " when a thin transparent film is backed by a perfect 
reflector, no colours should be visible, all the light being ultimately reflected, 
whatever the wave-length may be. The experiment may be tried with a thin 
layer of gelatine on a polished silver plate*." An apparent exception has 
been described by R. W. Woodf: "A thin film of collodion deposited on a 
bright surface of silver shows brilliant colours in reflected light. It, more- 
over, scatters light of a colour complementary to the colour of the directly 
reflected light. This is apparently due to the fact that the collodion film 
" frills," the mesh, however, being so small that it can be detected only with 
the highest powers of the microscope. Commercial ether and collodion should 
be used. If chemically pure ether obtained by distillation is used, the film 
does not frill, and no trace of colour is exhibited. Still more remarkable is 
the fact that if sunlight be thrown down upon the plate at normal incidence, 
brilliant colours are seen at grazing emergence, if a Nicol prism is held before 
the eye. These colours change to the complementary tints if the Nicol is 
rotated through 90, i.e. in the scattered light, one half of the spectrum is 
polarized in one plane, and the remainder in a plane perpendicular to it." 

I have lately come across an entirely forgotten letter from Rowland in 
which he describes a similar observation. Writing to me in March 1893, he 
says : " While one of my students was working with light reflected from a 
metal, it occurred to me to try a thin collodion film on the metal. This not 
only had a remarkable effect on the polarization and the phase but I was 
astonished to find that it gave remarkably bright colours, both by direct 
reflexion and by diffused light, the two being complementary to each other. 
I have not gone into the theory but it looks like the phenomenon of 
thick plates as described by Newton in a different form. The curious point is 

* " Wave Theory of Light," Enc. Brit. 1888; Scientific Papers, Vol. in. p. 67. 
t Physical Optics, Macmillan, 1914, p. 172. 



1917] COLOURS DIFFUSELY REFLECTED FROM SOME COLLODION FILMS 509 

that I cannot get the effect by making the film on glass and then pressing it 
down hard upon speculum metal or mercury although I think the contact is 
very good in the case of the speculum metal. Possibly, however, it is not. 
Gelatine films on metal give good colours by direct reflexion but not by diffused 
light: only faint ones. It would seern that the collodion film must be of 
variable density or full of fine particles. However, I leave it to you. I send 
by express two of the plates used." Probably it was preoccupation with other 
work (weighing of gases) that prevented my giving attention to the matter at 
the time. 

Wishing to repeat the observation of the diffusely scattered colours, I made 
some trials, but at first without success. On application to Prof. Wood, I was 
kindly supplied with further advice and with a specimen of a suitably coated 
plate of speculum metal. Acting on this advice, I have since obtained good 
results, using very dilute collodion poured upon a slightly warmed silvered 
plate (plated copper) warmed again as soon as the collodion was set. That 
'the film is no longer a thin homogeneous plate seems certain. Wood speaks 
of " frilling," a word which rather suggests a wrinkling in parallel lines, but 
the suggestion seems negatived by the subsequent use of " mesh." I should 
suppose the disintegration to be like that sometimes seen on varnished paint, 
where under exposure to sunshine the varnish gathers itself into small detached 
heaps. At any rate there is no apparent change when the plate is turned 
round in its own plane, showing that the structure is effectively symmetrical 
with respect to the normal of the plate. 

As regards Rowland's suggestion as to the origin of the colours, it does 
not seem that they can be assimilated to those of <- thick plates." The latter 
require a highly localized source of light and are situated near the light or its 
image, whereas the colours now under consideration are seen when the plate 
is held near a large window backed by an overcast sky, and are localized on 
the plate itself, the passage from one colour to another depending presumably 
upon an altered scale in the structure of the film. The formation of well- 
developed colour at the various parts of the plate requires that the structure 
be, in a certain sense, uniform locally. The case is similar to that of coronas, 
as in experiments with lycopodium, only that here the grains must be very 
much smaller. 

When examined by polarized light the behaviour of different plates is 
found to vary a good deal. We may take the case where sunlight is incident 
normally and the diffuse reflexion observed is nearly grazing. In the case of 
the specimen (on speculum metal) sent me by Prof. Wood, the light is practi- 
cally extinguished in one position (a) of the nicol, that namely required to 
darken the reflexion from glass. In the perpendicular position () of the 
nicol good colours are seen, and also of course when the nicol is removed from 
the eye. At angles of scattering less nearly grazing there is some light in 



510 ON THE COLOURS DIFFUSELY [424 

both positions of the nicol, the fainter light in (a) showing much the same 
colour as in (). 

It will be noticed that this behaviour differs from that observed by Wood 
(on another plate) and already quoted. On the other hand, one of the (silvered) 
plates prepared by me shows a better agreement, more light than before being 
scattered at a grazing angle when the nicol is in the (a) position, while the 
colours in the (a) and (/9) positions of the nicol are roughly complementary. 

No more than Rowland have I succeeded in getting diffusely reflected 
colours from collodion films on glass or, 1 may add, quartz, either with or 
without the treatment with the breath suggested by Wood. The latter 
observer describes an experiment (p. 174) in which a film, deposited on the 
face of a prism, frilled under the action of the breath and then afforded a 
nearly three-fold reflexion. But, as I understand it, this augmented reflexion 
was specular. The only thing that I have seen at all resembling this was 
when I treated a coated glass with dilute hydrofluoric acid with the intention 
of loosening the film. Even when dry, the film remained out of optical contact 
with the glass, except I suppose at detached points, and gave an augmented 
specular reflexion, as was to be expected, inasmuch as three surfaces were 
operative. 

Two views are possible with regard to the different behaviour of films on 
metal and on glass. One is to suppose that the actual structure is different 
in the two cases ; the other, apparently favoured by Wood, refers the differ- 
ence to the copious reflexion of light from metallic surfaces. The first view 
would seem the more probable a priori and is to a certain extent supported 
by Rowland's experiment. I have not succeeded in carrying out any decisive 
test. On either view we may expect the result to be modified by the metallic 
reflexion. 

As to the explanation of the colours, anything more than a rough outline 
can hardly be expected. We do not know with any precision the constitution 
of the film as modified by frilling. And, even i we did, a rigorous calculation 
of the consequences would probably be impracticable. But some idea may be 
gained from considering the action of an obstacle, e.g. a sphere, of material 
slightly differing optically from its environment and situated in the neigh- 
bourhood of a perfectly reflecting plane surface upon which the light is incident 
perpendicularly. Under this condition the reflected light may still be sup- 
posed to consist of plane waves undisturbed by the previous passage through 
and past the obstacle. 

The calculation, applying in the absence of a reflector but without limita- 
tion to the spherical form of obstacle, was given in an early paper*. In 
Maxwell's notation, /, g, h are the electric displacements. The magnetic 

* "On the Electro-maguetic Theory of Light," Phil. Mag. Vol. XH. p. 81 (1881) ; Scientific 
Paper*, Vol. i. p. 518. 



1917] REFLECTED FROM SOME COLLODION FILMS 511 

susceptibility is supposed to be uniform throughout ; the specific inductive 
capacity to be K, altered within the obstacle to K + A/f. The suffixes and 
1 refer respectively to the primary and scattered waves. The direction of 
propagation being supposed parallel to x and that of vibration parallel to z, 
we have f = g = 0, and 



e int b e i n g the time factor for simple progressive waves. For the scattered 
vibration at the point (a, /3, 7) distant r from the element of volume (dxdydz) 
of the obstacle, we have 



(2) 



where P = -- h.e-^dxdydz, ..................... (3) 

and the integration is over the volume of the obstacle. If the obstacle is 
very small in comparison with the wave-length (X) of the vibrations, h e~ ikr 
may be removed from under the integral sign and 



T denoting the volume of the obstacle. In the direction of primary vibration 
a = ft = 0, so that in this direction there is no scattered vibration. It will be 
understood that our suppositions correspond to primary light already polarized. 
If, as usually in experiment, the primary light is unpolarized, the light scat- 
tered perpendicularly to the incident rays is plane polarized and can be 
extinguished with a nicol. 

The formation of colour depends upon other factors. When the obstacle 
is very small, P is constant, and the secondary vibration varies as Ar*, so that 
the intensity is as the inverse fourth power of the wave-length, as in the 
theory of the blue of the sky. In this case it is immaterial whether the 
obstacles are of the same size or not, but for larger sizes when the colour 
depends mainly upon the variation of P, strongly marked effects require an 
approximate uniformity. If the distribution be at random, the colours due 
to a large number may then be inferred from the calculation relating to a 
single obstacle ; but if the distribution were in regular patterns, complications 
would ensue from the necessity for taking phases into account, as in the 
theory of gratings. For the present purpose it suffices to consider a random 
distribution, although we may suppose that the centres, or more generally 
corresponding points, of the obstacles lie in a plane perpendicular to the 
direction of the primary light. 

When the obstacle is a sphere, the integral in (3) can be evaluated*. The 
centre of the sphere, of radius R, is taken as the origin of coordinates. It is 
* Proc. Roy. Soc. A, Vol. xc. p. 219 (1914). [This volume, p. 220.] 



512 COLOURS DIFFUSELY REFLECTED FROM SOME COLLODION FILMS [424 

evident that, so far as the secondary ray is concerned, P depends only on the 
angle ( \) which this ray makes with the primary ray. We suppose that % = 
in the direction backwards along the primary ray, and that % = TT along the 
primary ray continued. Then with introduction of the value of h from (1), 
we find 



p . cos m 

~~ -~~ > ' 



where m = '2kRcosx ............................... ( 6 ) 

The secondary disturbance vanishes with P, viz. when tan m = m, and on 
these lines the formation of colour may be understood. Some further par- 
ticulars are given in the paper just referred to. 

The solution here expressed may be applied to illustrate the scattering of 
light by a series of equal spheres distributed at random over a plane perpen- 
dicular to the direction of primary propagation. The effect of a reflector 
will be represented by taking, instead of (1), 



tf expressing the distance between the plane of the reflector and that con- 
taining the centres of the spheres. The only difference is that 

m~ 3 sin m m~ 2 cos m 
is now replaced by 

sin m cos m /sin m cos m' 

~ +e l ~ 



\ri* 

where m is as before, and m' = 2kR sin i %. In the special case where, while 
the incidence is perpendicular, the scattered light is nearly grazing, x = TT, 
sin $% = cos $x ~ VV2, and m = m = V2 . kR ; so that (8) becomes 



This vanishes if cos 2kx * = 1 ; otherwise the reflector merely introduces a 
constant factor, not affecting the character of the scattering. At other angles 
the reflector causes more complication on account of the different values of 
m and m. 

[* The results given in the original text have been corrected by the substitution of - 2x for XQ . 
It is assumed, as apparently in the original, that no change of phase occurs at the reflector. 
W. F. 8.] 



425. 

MEMORANDUM ON SYNCHRONOUS SIGNALLING. 
[Report to Trinity House, 1917.] 

I HAVE been impressed for some time with the unsatisfactory character of 
the present fog signals. We must recognize that powerful siren signals are 
sometimes inaudible at distances but little exceeding a mile. It is true that 
these worst cases of inaudibility may not recur during fogs as to this there 
seems to be insufficient evidence. But even when a sound-in-air signal is 
audible, the information conveyed is far from precise. The bearing of the 
source cannot be told with much accuracy, indeed some say that it cannot be 
told at all. The distance is still more uncertain. I should say that no system 
is satisfactory which does not give either the one or the other element, 
bearing or distance. 

The system of synchronous signalling explained by Prof. Joly claims to 
give the distance with sufficient precision, and the American and Russian 
trials show that the claim is justified, as might indeed have been expected 
with some confidence, provided both signals themselves are well defined in 
time. The wireless electric signals are easily made sharp. Submarine signals 
from a bell, or explosive, would also be sharp enough. So probably would be 
explosive signals in air.. The case of siren signals is more doubtful. Possibly 
the end might be sharp enough. Even so, the objection of the uncertain 
carrying of air signals remains. 

I do not know whether there is already sufficient experience of submarine 
signals. If it be true that they can be depended upon up to distances of at 
least 4 or 5 miles, the case is strong for a combination of them with electric 
signals. 

In some respects the system described in my former memorandum of 
1916* has its advantages. It would give the bearing with electric signals 
only, but requires further experimenting, which if desired could be arranged 
for at the National Physical Laboratory but perhaps not during the war. 

I am strongly of opinion that whatever is possible at the present time 
should be done to prepare the way for a better system. 

* This volume, p. 398. 
R. vi. 33 



426. 



A SIMPLE PROBLEM IN FORCED LUBRICATION. 
[Engineering, Dec. 14, 28, 1917*.] 

THE important case of a shaft or journal running in bearings has been 
successfully treated by Reynolds, Sommerfeld and others. As Tower showed, 
the combination acts as a pump, and of itself maintains the layer of lubricant 
between the opposed solid surfaces f. There are other cases, and some of them 
are of practical importance, where the layer can be maintained only with the 
aid of special devices, such as Michell bearings, or by the forcible introduction 
of fluid from outside, in order to compensate inevitable escapes. Thus, Fig. 1, 



Fig.1 




when a shaft E with a flat end bears against a flat surface AB, the included 
oil tends to escape from the pressure, whether at G when the flat surface is 
continued, or at D when it is surmounted by a cylindrical cup. The perman- 
ence of the layer requires a continuous forcible feed, which may be through 
an axial perforation at F\ for here, in contradistinction to the case of the 
journal, the rotation of the shaft does not avail. It is proposed to consider 
the problem thus presented, supposing in the first instance, that there is no 
cup. The small distance between the flat surfaces, i.e., the thickness of the 
oil layer, is denoted by h, and the angular velocity of the shaft by <w. The 
motion is referred to cylindrical coordinates r, 6, z, where z is measured parallel 

* In the original statement there was an error, pointed out by Mr W. Pettingill. 

t As was noticed at an early date by the present writer (President's address to British 
Association in 1884), this requires that the layer be thicker on the ingoing than on the outgoing 
side. [This collection, Vol. n. p. 344.] 



1917] 



A SIMPLE PROBLEM IN FORCED LUBRICATION 



515 



to the axis of symmetry, and r is the distance of any point from that axis, 
Fig. 2. The velocities in the three directions are respectively u, v, w, and in 
virtue of the symmetry, they are all independent of 6. The motion is supposed 
to be steady, that is, the same at all times, and the inertia of the fluid is 
neglected. Under these conditions it is easy to recognize that w may be 
supposed to vanish throughout, and that v is given by 

v = <ozr/h, (1) 

where z is measured from the fixed surface, so that v there vanishes. 



Fig.2. 



ft 



In like manner the boundary conditions at z = and z = h, as well as the 
equation of continuity, are satisfied by 



where C is a constant. The total flow U, representing the volume of lubricant 
fed in unit time, which flows past every cylindrical surface of radius r, is 

......................... (3) 

When the inertia terms are neglected, and attention is paid to the 
symmetry, the formal equations in cylindrical coordinates* are 

.............................. < 4 > 

(5) 

(6) 
p denoting the pressure and /z the viscosity, where 



Basset's Hydrodynamics, Vol. u. p. 244, 1888. 



332 



516 A SIMPLE PROBLEM IN FORCED LUBRICATION ' [426 

Of these (5) is satisfied by v in (1), and (6) is satisfied when w = and p is 
independent of z. Also, with use of (2), (4) becomes 



dr~ r ' 7rh*r' ........................... ^ 

so that 

6/u/\ ri 
p-ft-^log- ............................ (9) 

where p l is the pressure at the outer radius r t . If the layer is open at r,, 
and we reckon only pressures above atmosphere, p^ may be omitted. 

The whole force sustained by the layer of fluid between the radii r and r, 
is independent of o>, being given by 



(10) 



If we suppose r = 0, so that the supply takes place on the axis itself, this 
becomes simply 

i s , .............................. (11) 



but we have then to face an infinite pressure at the axis. In practice r 
would have to be finite though small, and would correspond to the radius of 
the perforation in the lower fixed plate, not much disturbing (11). In fact, 
if p be the pressure of the feed corresponding to r , 



The moment of the forces due to viscosity, by which the rotation is resisted, 
has the expression 

-fV9 ................ (13) 



It may be worth remarking that if geometric similarity is preserved, so 
that r lt r , h are in constant ratios, a consideration of "dimensions" suffices 
to show that P is proportional to /tf/rr 1 , at least when we assume inde- 
pendence of the rotation (o>) which does not influence u. A deficiency of 
viscosity may thus always be compensated by an increase of supply. 

The work which must be done in unit time to maintain the rotation is 
3/o). In addition to this, there is the work required to introduce the feed of 
lubricant, represented by p U. Thus, altogether, for the work required 



1917] A SIMPLE PROBLEM IN T FORCED LUBRICATION 517 

In practice the diminution of h calls for the utmost accuracy in fitting 
together the two opposed surfaces, which, however, need not be accurately 
plane, as well as the removal of all suspended solid matter from the lubricant. 

When this is attended to, there should be no wear of the solid surfaces, 
which should never come into contact. To attain this ideal it is evidently 
necessary .that the feed of lubricant should be established before the rotation 
commences. 

It should be observed that no property of oil beyond viscosity is involved, 
and that the investigation may be expected to remain valid until the thickness 
(h) of the layer is approaching molecular limits. 

P.S. I may perhaps mention that I have made a small model, in which 
the opposed surfaces are those of two pennies ground to a fit, and the " lubri- 
cant " is water supplied from a tap. 



427. 

ON THE SCATTERING OF LIGHT BY SPHERICAL SHELLS, AND 
BY COMPLETE SPHERES OF PERIODIC STRUCTURE, WHEN 
THE REFRACTIVITY IS SMALL. 

[Proceedings of the Royal Society, A, Vol. xciv. pp. 296300, 1918.] 

THE problem of a small sphere of uniform optical quality has been treated 
in several papers*. In general, the calculations can be carried to an arith- 
metical conclusion only when the circumference of the sphere does not exceed 
a few wave-lengths. But when the relative refractivity is small enough, this 
restriction can be dispensed with, and a general result formulated. 

In the present paper some former results are quoted, but the investigation 
is now by an improved method. It commences with the case of an infinitely 
thin spherical shell, from which the result for the complete uniform sphere 
is derived by integration. Afterwards application is made to a complete 
sphere, of which the structure is symmetrical but periodically variable along 
the radius, a problem of interest in connexion with the colours, changing 
with the angle, often met with in the organic world. 

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. 
Electric displacements being denoted by f, 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 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 = (K 1) . e int 1 1 1 e** (x ~ r) dxdydz (3) 

* Phil. Mag. Vol. XLI. pp. 107, 274, 447 (1871); Vol. xn. p. 81 (1881) ; Vol. XLVII. p. 375 
(1889); Roy. Soc. Proc. A, Vol. LXXXIV. p. 25 (1910) ; Vol. xc. p. 219 (1914); Scientific Papers, 
Vol. i. pp. 87, 104, 518; Vol. iv. p. 397; Vol. v. p. 547 ; Vol. vi. p. 220. 



1918] ON THE SCATTERING OF LIGHT BY SPHERICAL SHELLS 519 

In these equations r denotes the distance between the point (a, /9, 7), 
where the disturbance is 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 on 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 



(4) 



p denoting the distance of the point of observation from the centre of the 
sphere. In the paper of 1914 I showed that the integral in (4) can be simply 
expressed by circular functions in virtue of a theorem given by Hobson, so 
that 



where m = 2kRcos^x ............................... ( 6 ) 

In (5) the optical -quality of the sphere, expressed by (K 1), is supposed 
to be uniform throughout. In view of an application presently to be con- 
sidered, it was desired to obtain the expression for a spherical shell of 
infinitesimal thickness dR, from which could be derived the value of P for a 
complete symmetrical sphere whose optical quality varies along the radius. 
The required result is obtained at once from (5) and (6) by differentiation. 
We find 

dP = - (K - 1) . 4nrR*dR . e i(nt ~^ . sin m/m, ............ (7) 

expressing the value of P for a spherical shell of volume 4>7rR*dR. The 
simplicity of (7) suggested that the reasoning by which it had been arrived 
at is needlessly indirect, and that a better procedure would be an inverse one, 
in which (7) was established first, and the result for the complete sphere 
derived from it by integration. And this anticipation was easily confirmed. 

- Commencing then with a spherical shell of centre and radius OA equal 
to R, let xO be the direction of the primary and Op that of the secondary 
ray (Fig. 1). Draw 0% in the plane of Ox, Op, and bisecting the angle between 
these lines, and let f be a coordinate measured from in the direction Of, so 
that the plane AOA, perpendicular to