> un beth rere. anes i " i + fa . a he at ow - P ie + 7 7) : oo . H nv ww -** dy or, er. Eee UU eee _ 554 ENTIFIC LIBRARY j A Sc ra) MAMMA Sacamacsacsa g “ (hala Cid oe ATES PATENT 0 D ST UNITE MAaAMMasAscAww%w GOVERNMENT PREVTING OFFICE * at Et ie nee a ‘ y ji 148 70% ‘ ener bh THE s LONDON, EDINBURGH, ayn DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.R.S. SIR JOSEPH JOHN THOMSON, O.M., M.A., 8c.D., LL.D., F.R.S. JOHN JOLY, M.A., D.S8c., F.R.S., F.G.S. AND WILLIAM FRANCOIS, F.L.S8. “Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex alienis libamus ut apes.”’ Just. Lres. Polit. lib. i. cap. 1. Not. VOL. XXXVIII.—SIXTH SERIES. J ULY—DECEMBER 1919. S26 a LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. SOLD BY SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO. DUBLIN,— AND VRUVE J. BOYVEAU, PARIS, “Meditationis est perscrutari occulta; contemplationis est admirari perspicua .... Admiratio generat questionem, questio investigationem, investigatio inventionem.”—Hugo de S. Victore. “ Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina celo, Quo micet igne Iris, superos quis conciat orbes Tam vario motu.” J. B. Pinelli ad Mazonium. CONTENTS OF VOL. XXXVIII. (SIXTH SERLES). NUMBER CCXXIII.—JULY 1919. Prof. W. M. Hicks on the Series System in the Spectrum et Grell fs Seine base? ile GARIN ser OAR cP a ne Pe elt) Mr. W. B. Hardy and Flight Lieutenant J. K. Hardy on Static Friction and on the Lubricating Proper ties of certain DireniUCals SUVSEATICCS We Ath talks a eeials | ue wise lle Shee ee Mr. W. B. Hardy on the Spreading of Fluids on Glass ... Prof. ¥. Allen on the Discovery ot Four Transition Points in the Spectrum and the Primary Colour Sensations Prof. F. Allen on the Persistence of Vision of Colours of Beier simoplemtlenStbyici 28 sai avis Bice ithe ius secoe occas ale Dr. Balth. van der Pol, Jun., on the Production and Mea- surement of Short Continuous Electromagnetic Wayes Prof. R. W. Wood on Optical Properties of Homogeneous and Granular Films of Sodium and Potassium. (Plate I.) Dr. Harold Jeffreys on the Force-Function in the Theory of Oronmary: elaiipatiy ois... csaieds svelte eel alse vale nS Dr. L. Silberstein: Further Contributions to Non-Metrical Recor Ce bray ye tl maw aia Sie ieee te ee vie wiwle cies Dr. T. J. ’a. Bromwich on Electromagnetic Waves........ Prof. Barton and Miss Browning on the Resonance Theory of Audition subjected to Experiments. (Plates II. & III.) Dr. F. A. Lindemann on the Vapour Pressure and Affinity of NSO WOE SHEEN ean toe TS WS lialbarticaw ess ha eM Wate a dle € Dr. S. Chapman on the possibility of separating Isotopes Prof. A. W. Porter and Dr. R. E. Slade on the Fundamental Law for the True Photographic Rendering of Contrast .... 1V CONTENTS OF VOL. X\XXVIII.—SIXTH SERIES. Page Notices respecting New Books :— Prof. W.C. McC. Lewis’s A System of Physical Chemistry, 197 A. Findlay’s Osmotic Pressure... . .1: 2). 1s eee 198 G. H. Livens’s The Theory of Electricity ..........-. 199 Annuaire du Bureau des Longitudes pour l’an 1919.... 200 Intelligence and Miscellaneous Articles :— Problems in Conduction of Heat, by H. 8S. Carslaw .... 200 NUMBER CCXXIV.—AUGUST. Prof. A. Gray on Electric and Magnetic Field Constants and their Expression in terms of Bessel Functions and Hlliptie Inteonralsaeee Ae. oe See ee 20a Dr. Norman Campbell on Time-Lag in the Spark Discharge. 214 Dr. T. J. Pa. Bromwich on approximations in the Theory ot Probabiliticste weet fo. Sa a ee eee 231 Mr. R. F. Gwyther on the Equations for Material Stresses, and their formal soliton. 20... 04. 6. he. keer eee 235 Mr. D. L. Hammick on Latent Heat and Surface Energy.— Parh eos ee bei ed Se 240 Mr. A. T. Mukerjee on a Method of Measuring the Capacity of Gold-leat Hlectroscopes n..5 2.2.5. 4. cs. 245 Mr. L. C. Jackson: Mathematical Investigation of the Sta- bility of Dr Al We Stewart's Atom:..!..:.) 2.0252. ee 256 Proceedings of the Geological Society :— Mr. Rk. Hansford Worth on the Geology of the Meldon Valleys, near Okehampton, on the Northern Verge of Darbinoorme ies 7 tea a che ctl ve ee eh i eee 267 NUMBER CCOXXV.—SEPTEM BER. Prof. L. Natanson on the Molecular Theory of Refraction, Reflexion, andi Ebr time momen pay uel. «0. say. eee ee 269 Mr. 8. K. Mitra on the Large-Angle Diffraction by Apertures with Curvilinear Boundaries. (Plate IV.).............. 289 Prof. W. M. Hicks on the Value of the Silver Oun ........ 301 Prof. W. E. Adeney and Mr. H. G. Becker on the Determi- nation of the Rate of Solution of Atmospheric Nitrogen and Oxygen by Water. 7Part.I, (Plate! V.) .......qeuue 317 CONTENTS OF VOL, XXXVIII.—SIXTH SERIES. Vv Page Prof. Barton and Miss Browning: A Syntonic Hypothesis of | Colour Vision with Mechanical Illustrations. (Plate VI.) 3388 Dr. Harold Jeffreys on the Derivation of the Lorentz- Einstein PicaMs OLMAbIOM. > 54). eeewes ENe Bie sig ASS See ER COAG Dr. B. van der Pol, Jun., on a Method of Measuring without Electrodes the Conductivity at various points along a Glow recharee and in Wlamesi ein q's i.» «caeile tlh seth wees 352 Dr. B. van der Pol, Jun., on the Propagation ‘of Electro- magnetic Waves POU MURO MEET. MEMO Ge acy 4 365 Dr. G. W. Todd on a Simple Theory of the Knudsen Vacuum SALTS eee ceca Gan, Wo ta, sy AUB AH Oech aoa eS 0 reg . 381 Dr. L. Silberstein on a Time-Scale independent of Space Measurement. Parabolic and Hyperbolic Kinematics.... 382 Mr. A. B. Eason on Critical Speeds of Machinery placed on Upper Floors of Buildings, as related to Vibration ...... 395 Dr. R.A. Houstoun ; A Theory of Colour Vision ........ 402 Prof. G. N. Antonoff on Surface Tension and Chemical PSEA VON sh i's vc 2 AMP Gs tebe ay co's 64 i! SPOR rN 417 The late Lord Ray leigh on the Travelling Cyclone ........ 420 Proceedings of the Geological Society:— _ . Dr. C. Taylor Trechmann on a Bed of (nterglacial Loess and some Pre-Glacial Freshwater Clays on the Hrs vera COasb ive melee eiaeeees wy? a5 4o Uma ale. aa 425 ecw a OLO MN iy NS TS he ae as. yo a TS eee 427 NUMBER CCXXVI—OCTOBER. .. Lt.-Col. A. R. Richardson on Stream-line Flow from a AD ISGUIG A CCUPANEC AN, FiGe aly ace «0 ome SIRO or ee cael acyarclelaiee ahs . 4383 Capt. J. Hellingworth on a New Form of Catenary........ 452 Dr. T. KR. Merton on an Experiment relating to Atomic Oitemtabio ny Meme waved oss vai occu aieeeest als ole yo Ae epee wiledats| ake gates 463 Mr. N. R. Sen on the Potentials of Uniform and Heterogeneous Elliptic Cylinders at an External Point ................ 465 Mr. A. Everett on Proofs of Elementary Theorems of Oblique IVEMEAGLLON, .,) ho Pstehshanmerain cic momma © Halve. aie sing gle islsole 480 Prof. F. Soddy on the Relation between Uranium and Radium. SURAT RR VAL LG, chty GNP yehee Ser HOA OL US WR a ahh RUINS eg 483 Dr. D. M. Y. Sommerville on the ‘ Slip-Curves ” of an Amsler ATTAINS CEN ca haunt iatahekaciehsnecrersy oss act g ROE eEaIN eS wleey 489 Major W. T. David on the Origin of Radiation in a Gaseous EDU UOS TOM aa enka ABMs 8) acbralle Wards w enaeh eens Inga NG Ab eine 402 vi CONTENTS OF VOL, XXXVIII.—SIXTH SERIES. Notices respecting New Books :— J. Byrnie Shaw’s Lectures on the Philosophy of Mathe- MACs. jie Vics Se OUT se es Proceedings of the Geological Society ;— Prof. P. Fry Kendall on ‘‘ Wash-outs” in Coal-Seams and the Effects of Contemporary Earthquakes ...... Dr. A. Gilligan on Sandstone Dykes or Rock-Riders in the Cumberland @ealfield: .......225%. .. 93 NUMBER CCXXVII.—NOVEMBER. Sir George Greenhill on the Bessel-Clifford Function, and its applications “Soeeg eee cee... aes sae ee Dr. A. O. Rankine and Dr. F. B. Young on the Magnetic Effects of Vibration i iron’ Rods .......... +... 3s eon Dr. L. B. Loeb on the Recoil of Alpha Particles from Light AGOMS Jn. Sek ere se hte. ee ee ea as sen Messrs. A. Bursill and H. Bedson on a New Magnet-testing Instrument, “(Plate wal) .. oes. s oe ie eee Prof. A. R. McLeod on Terrestrial Refraction............. Prof. C. V. Raman on the Scattering of Light in the Re- tractive Media of the Miye “v.00. -.0.6..2.5...4 ee Prof. C. V. Raman on the Partial Tones of Bowed Stringed Tnsiruments \..\.celeaw oe kos Me dee er Prof. Kia-Lok Yen on an Absolute Determination of the Coefficients of Viscosity of Hydrogen, Nitrogen, and Oxygen. (Plate VII) -o2.85 02. 6. ting oe Prof. W. M. Hicks on the Mass carried forward by a Vortex. Prof. W. M. Thornton on the Ignition of Gases by Hot Wates oo ARP ing oid oy tere eT 6a ee Mr. F. F. Renwick on the Fundamental Law for the true photographic rendering of Contrast .................. Prof. A. W. Porter and Dr. R. E. Slade on the Funda- mental Law for the true photographic rendering of Contrast? “2 ee hers oS: sees. ah. fs Dr. Manne Siegbahn on Precisiun-measurements in the X-Ray Spectra: Parisi 22. trees. ish es. os... Dr. Manne Siegbahn and Mr. A. B. Leide on Precision- ineasurements in the X-Ray Spectra.—Part III]. -....... Dr. Norman Campbell on the Measurement of Time and other Macwitudes") 2. Geese Sey... o. | Prof. G. W. Todd and Mr. 8. P. Owen on a Vapour Pressure Hguation ©. is.25 (0s sep e e Ss ee Prof, A. R. McLeod on a New Form of Catenary ........ Notices respecting New Books :— J. W. French’s translation of Steinheil and Voit’s Applied Optics, “Valo Wh, ci... ee eee Page 495 496 499 501 528 533 O42 546 568 573 582 597 613 633 637 639 647 CONTENTS OF VOL. XXXVIII.—-SIXTH SERIES. Proceedings of the Geologicai Society :— Mr. CO. J. Gilbert on the Occurrence of Extensive De- posits of High-Level Sands and Gravels resting upon the Chalk of Little Heath, near Berkhamsted ...... Mr. G. Barrow on the Correlation of the Deposits described in Mr. C. Gilbert’s paper with the High- Level Gravels of the South of England (or the London J BST) Ue fo is a5 34 Arie Re Ah ots PCA ee Dr. A. Logie Du Toit on the Geology of the Marble 1D SIRES GINEN 22 BYR tes hest cA hSDN arc OR Sr SR Lieut. EZ. Hall Pascoe on the Early History of the Indus, Bralimap ibra, and Games ee ices <« -. sabaeyd eee ah») Mr. W. Whitaker on the Section at Worms Heath (Surrey), with Remarks on Tertiary Pebble-Beds andkon Clay-with=Mlintsien oO ee oe eee eas. ee ee NUMBER CCXXVIII.— DECEMBER. Prof. F. A. Lindemann on the Theory of Magnetic Storms .. Prof. Horace Lamb on the Kinematics of the Hye ........ Mr. 8. C. Bradford on the Molecular Theory of Solution.... Prof, W. M. Thornton on the Thermal Conductivity of Solid Piaten sp CLO SheMet oi N ras as Pa sich cltige ies oi + Sucac aat oh SP OUR ae al epee Dr. F. W. Aston on a Positive Ray Spectrograph. (Plate IX.) Miss Dorothy Wrinch and Dr. Harold Jeffreys on Some Eepeers OF the lheory of Probability... 2.02. .62 205. Prof. H. C. Plummer on the Form of the Trailing Aerial Sir Oliver Lodge on the Connexion between Light and Pr GACLOUMME ene ka dein Nu eie)s <-cw RNa a eo Meibalee sehen ies The late Lord Rayleigh on Periodic Precipitates .......... Proceedings of the Geological Society :— Major R. W. Brock on the Geology of Palestine ...... Mr. C. I. Gardiner on the Silurian Rocks of May Hill.. Dr. A. Gilligan on the Petrography of the Millstone Grit DeLies OlMOT KSI!) 2)... dutwere ewe eet se auataln'ls\s Mr. Frank Debenham on a New Theory of Transportation by Ice: the Raised Marine Muds of South Victoria Land (Ge SOUB CIO)" 3G Pe Ate REM 5 (Sie AER Rt a ANE Meee Mr. A. E. Kitson on the Geology of Southern Nigeria (British West Africa) with especial reference to the Renviary Wemosttsea meen ii gies Sao . Cy . 2e Spectrum and the Primary Colour Sensations. 63 | . SECONDS, a.) Pie eh » ee Ee WAVE LENGTHS. : .60 64 Prof. F. Allen on Four Transition Points in the The persistency curve obtained when the eye was fatigued with the wave-length \°620 is here reproduced in fig. 7 from a previous paper to which reference has already been made. This curve is of the double red and green elevation type, similar to the curves of the two figures immediately preceding. Fig. 9. eS a O06 ; \ Fatigue colour -560 \ NA me = SSS Fatigue ———— \ Second, “404 ‘50 -60 -70 Wave Length. Further consideration led to the conclusion that a second transition point ought to exist between curves of the double red and green elevation type and those of the single green elevation type. Since the curve for the wave-length °520 p, fig. 13, was known from a previous investigation, persistency curves were successively determined with the fatiguing colours ) 560 4 and A°540 w. The data for these are given Spectrum and the Primary Colour Sensations. 65 in Table V. and Table VI., and are plotted in fig. 9 and fig. 10 respectively. Both curves, it will be noticed, are characterized by a single elevation in the green. Fatigue colour 540 Normal - -—--— Wave Length. TABLE V. x Normal __ Eye fatigued \ Normal Hye fatigued ; Persistence. with A °560 pu. a Persistence. with A*560 pu. . Joo % ‘(Oded sec, 0323 sec. || ‘522 ¢ ‘(0164 sec. ‘0189 sec. } 690-0221 0224 495 0209 0214 ‘647 ‘0167 0164 | *458 0360 0380 093 0137 ‘O144 || *423 ‘0538 ‘0500 D038 ‘O141 ‘0164 || “410 0620 . Purple-red ; just out- O05 p. “470 p. side spectrum. ESSN sh Red complementary to "508 pe. “475 py. 494 w. (About °66 p.) SAUNE sao bc 25. Between end of spec- ‘570 uto ‘470m Violet, between trum and ‘66 p. ‘420 w and end. led. Green. Yellow. Blue. MMERING ...... Purple-red ; "495 p. “OT45 p. “471 p. non-spectral, SEAMED ec. 3 Purple-red ; “490 pe. 570 wu. “460 p. non-spectral. There is no conflict necessarily arising between my own fundamental hues for red and green and those of Helmholtz, Konig, and Exner; there is divergence in the case of violet. But if it is at all justifiable for Helmholtz and the others to differ in their green hues by nearly ‘06 w, no exception need be taken to my violet determination because it differs from the others by an equal amount. In endeavouring to connect the transition points with other colour-phenomena, | have found specially suggestive the colour-sensation curves obtained by Abney and Watson, fig. 23, and by Konig and Dieterici, fig. 24. On each of these figures I have indicated the transition points by short vertical lines with wave-lengtlis attached. In the curves of Abney and Watson the transition points 660 and A470 are practically at the ends of the sensation curve for green ; and the remaining transition points atrX°420 wand X°570 w are near the ends of the sensation curve for blue. . While the green and red sensation curves extend to the wave-length 1°43 «, yet their influence compared with blue seems quite negligible on the more refrangible side of 470. The transition point at 1°570 wu is also very close * J. W. Baird, “The Colour Sensitivity: of the Peripheral Retina,” Carnegie Institution, 1905. 80 On Four Transition Points in the Spectrum. to the intersection of the green andred sensation curves. The transition point at 4°420y is shown by the curves to be practically outside the influence of the red and green sen- sations, and hence may be regarded as a region of a single | i | pT 4/00 4500 5000 5500 ae 6500. 7000 ! | | | A.U. "4244 "FAT {L SITLL “6644 Normal Sensation Curves. (Abney and Watson.) (From Parsons’s ‘ Colour Vision,’ p. 244.) CNA alo ce sates a/een\ees's BEES ARES ANSse 720 660 480 420 pure-colour sensation. Similarly, the spectral region from the transition point “660 % to the end of the spectrum is pure red free from the influence of the other two sensations. The sensation curves of Konig and Dieterici, fig. 24, show some differences. The ends of the ereen sensation curve are the transition points °660 p and 2 "420 yw, while the transition point °470 wu is still the more refrangible end of the sensation curve for red. The transition point 970 po Persistence of Vision of Colours of Varying Intensity. 81 marks one end of the sensitive curve for violet. It will be noticed that the regions’ between the transition points \°660 wu and X°420 and the respective ends of the spectrum are, as before, characterized by purity and simplicity of colour- sensation. The transition point at X°570w is again at the intersection of the red and green sensation curves, that is, where the two sensations balance the influence of each other, without sufficient of the third sensation to make itself perceptible. In both figures one can see reasons why the regions between X °420 w and X°470 w, and between 2°570 @ and 2 °660 w, should be characterized by compound sensations. But it is difficult to understand why the exceedingly complex region between X°470 w and A*570m should exhibit, as it does, persistency curves with only one elevation in the green. The occurrence of the four transition points at the boun- daries of the sensation curves cannot be in any sense accidental, and is certainly not intentional, since the transition points were discovered long before any examination of the sensation curves was made in this conrexion. The cumulative effect of all the experimental evidence presented in this paper seems to warrant most strongly the conclusions already expressed, that there are but three colour sensations, which correspond to red, green, and violet, and that these sensations are primarily excited by hues which lie between the limits set by the four transition points. V. The Persistence of Vision of Colours of Varying In- tensity. By Frank Auten, PALD., F.RS.C., Professor of Physics, University of Manitoba, Canada”. [* a former communication to the Philosophical Magazinet a method of measuring the luminosity of the spectrum was discussed, which may also be used for the determination of the persistence of vision of colours of varying intensity. The arrangement of apparatus employed in the present in- vestigation is very similar to that described in the paper referred to, and is shown diagrammatically in fig. 1. Light from the acetylene flame (A), after concentration by a lens (B), passed through an open sector (90°) of the disk (C) which was rotated by an electric motor, then through two Nicol prisms (D and E) arranged with their principal * Communicated by the Author. + “A New Method of Measuring the Luminosity of the Spectrum,” May 1911. Phil. Mag. 8. 6. Vol. 38. No. 223. July 1919. GC 82 Prof. F. Allen on the Persistence of sections horizontal, thence through the spectrometer (F), and was finally viewed in an eyepiece in which all the light of the spectrum, except a narrow central band of any desired colour, was cut off by means of adjustable shutters. Fig. 1. A B C ! EST [7 aa [ee In making the measurements, the eye, which before each reading was always restedin ordinary daylight, observed any selected patch of colour which was caused to flicker by the rotation of the sectored disk. The speed of the disk was increased until the critical frequency of flicker was reached, and then electrically recorded on a chronograph. At first the principal sections of the nicols were parallel, so that a persistency curve might be obtained for the greatest intensity of light possible with the apparatus. It may be pointed out that any error arising from the use of Nicol prisms in colour measurements does not affect these results, as they are purely comparative. On subsequent days similar curves were obtained for spectra whose intensities were diminished in determinate amounts by rotating the polarizer through suitable angles. In all, nine curves were determined for a series of luminosities varying from the fullest brightness of the spectrum down to the point where very little ight was perceptible. In Table I. the first column shows the wave-lengths for which determinations of the duration of the luminous im- pression at the critical frequency of flicker were made, viz. 725 pw, X°665 pw, A°590 w, X°920 w, A °460 w, and r°430 ws the other wave-lengths in the table, and the corresponding data, were obtained by inspection from the curves formed by the first six experimental values. The non-experimental data are placed in the table to indicate the lowest points on the respective persistency curves, and hence denote the brightest points in the corresponding spectra. The numbers, 0°, 20°, 40°, &e., at the heads of the pairs of columns in the table, are the angles between the principal sections of the nicols from which the relative brightnesses of the corre- sponding spectra are computed. Vision of Colours of Varying Intensity. 83 TaBLeE I. Persistence of Vision and Luminosity. O°: 20°. 2s) 60°. SS SSS SS = SS aN a a ty EEE eA | rN D L D L D L D L “725 pp 0286 sec. 11°7 ‘0290 sec. 11°3 0312 sec. 9°5 -0366 sec. 6:8 “665 0184 AGH OUST 43:3 -0198 35:1 -0224 23°2 590 0153 100°6 :0155 945 -0162 EO AO rhs: 52°77 "520 0169 646 ‘0172 60°3 0182 48:0 -0202 32°8 “460 0254 16:0 -0261 149 -0279 12-4 -0317 9:2 “430 0448 4:9 -0461 aet Os Or 4:0 -0622 3'1 Oz: 80°. Soa Som Sir Mews Ema i eT Game ON RGITIEN © r D L D L D L D “725 ‘0486sec. 51 ‘0608sec. 3:3 665 0251 166 -0324 87 ‘0454sec. 48 -0625sec. 3:0 690 0196 36°5 0238 12 O03) oy 0366 6°8 520 "0224 23°2 -0268 13:9 -0340 79 -0415 5d ‘460 0359 72 “0476 44 -0694 2°38 -0795 2°45 "430 0805 2°5 0980 Zale 0936 DE ee : “585 0194 37°9 ~— ies oe “583 we ae "0232 20°7 ne Be 5) ce Het a O00 L040. 2 ae 572 eee Aa uN we hos OOO Wl 88° 127. 88° 30’. MELEE Fe Os LN X D L D L > (eS be aA a 665 "0820 sec. 2°4 tls ee -590 0397 59 °0484sec. 51 “520 0467 4:6 ve ee 370 ‘0391 61 In the columns under D are the values of the persistence of vision, or the duration of the visual impressions. In the columns under lL are the luminosities corresponding to the values of D. These are obtained by assuming a luminosity of 100 for the smallest value of D, 0°0153 sec., in the whole table, and then applying the Ferry-Porter law in its simple form 1 Der k log Li for the other values of D in all the columns. In fig. 2 the values of D and the wave-lengths are used as coordinates, which give a series of nine pers sistency curves for intensities varying ‘almost to invisibility. The curve for the brightest spectrum corresponding to the angle 0° between the sections of the nicols is at the bottom of the figure, and that for the dimmest spectrum at the top. In the “curve” for 88° 30’ only one point was obtained. It will be noticed G 2 oe —-— 84 Prof. F. Allen on the Persistence of that the brightest points of the spectra, represented by the- lowest values in the cur ves, gradually shift from the yellow towards the blue as the spectral intensities grow smaller. Fig, 2. \ | gy -08 th 03 06 cS iS So QO % 5 05 & 7 L- 00068 LHL -04 Le 250 /} 587 ‘03 \ ZF ae. es “45 nt [55 “60 cae 70 Wave Length. \ The displacement of the brightest point here indicated is not so great as other observers have obtained by different methods. In fig. 3 the coordinates are the values of the luminosity, L, and the wave-lengths in Table I. These give a corr esponding series of luminosity curves for spectra of diminishing in- tensities, the brightest spectrum being represented by the highest curve. ‘These are of the usual type of luminosity curves, and also show the displacement of the brightest point of the spectrum towards the blue when the intensities Vision of Colours of Varying Intensity. 85 ‘are diminished. Theré is a marked flattening of the curves as they approach the lowest, which is that for the dimmest spectrum. Luminosity. LEE : ee : 9) 40u -45 “50 45)5) “60 “65 70 Wave Length. A somewhat similar group of three luminosity curves was obtained by Haycraft* by the flicker method, who, however, employed colour disks at three different intensities of illumi- nation instead of spectral colours. His curves are somewhat the same as those in fig. 3, but are noticeable for a much greater displacement towards the blue of the brightest point than is indicated in my own curves. In Table IT. the first column shows the angles, «, between the principal sections of the nicols, the second the values of cos?a to which the luminosities are proportional, and the remaining columns in pairs under the wave-lengths, iN*725 pr, &c., show the values of the persistence of vision, D, and their reciprocals, * Journ, of, Physiol, vol. xxi. (1897). 86 Angle between A= Or iaagy [Le N=-6609 H. A='590 HB. Pol. and 2=cos? a. ———* ; Ss A~A— ~ Anal. 1 Gat cos seteataay pa Do me D Og 1000 ‘0286 sec. 34°96 ‘0184sec. 54:35 -0153sec. 65°36; 20° 0'8838 ‘0290 34:48 -0187 03°48 :0155 64:52 40° D87 "0312 32°05 -0198 30°50 -0162 61°73 60° 250 ‘0366 27°32 “0224 44:64 ‘0178 56°18: 70° laley, "0436 22-93" . Of the 15 per cent. which enters the film, about one half is removed by absorption in the blue, and somewhat more in tbe red, consequently the film is fairly opaque, and bluish green in colour. In the case of incidence on the inner surface, however, the reflexion falls to 6 per cent. for X=5400 (dotted curve Ref.). Since this wave-length does not appear in excess in the transmitted light, we must assign a value of over 85 per cent. for the absorption in this cases This absorption, I believe, results from the action of the spaces between the crystals as resonators or selective radiation traps. The absorption which occurs in the case of incidence on the outer surface may be of a ditferent type, though both types probably oecur simultaneously to a greater or less degree. We will now examine in detail the phenomena which occur as the thickness of the film is gradually increased. In the case of exceedingly thin deposits, i.e. consisting of — ‘operties of al P XG Prot. R. W. Wood on Opt 108 UL U9. 0S..07 OLy Usen0SS0r- 02> 09 US207 se OL Ou OSU SUL. Os) Oa Ol 2 UA eae re OG NOY "9 SALMA 'G SUTUAS “p SALW ‘G SHLAA = BM eee es ‘| Saruag Films of Sodium and Potassium. 109 exceedingly minute crystals separated by considerable dis- tances, a faint light of a distinctly reddish colour is scattered, when sun or arc light is concentrated on the outer surface of the bulb. This deposit is too attenuated to affect the colour of the regularly reflected or transmitted light to any marked degree. If the deposition is continued, we obtain a film. which is pale blue by transmission, and which reflects light of a straw (pale yellow) colour, both for external and internal incidence. The curves for this film are shown in series 1, fig. 3. Light of a deep violet colour is scattered in this case, though the total amount is very small. If the. thickness is still further increased, the external reflexion becomes whiter, and the yellow colour of the internal reflexion more marked. (Series 2 and 3.) The kinks in the absorption curves are of no significance, in my opinion, resulting from errors resulting from the use of ray filters for obtaining monochromatic light. Further increnients in thickness give us films of different shades of purple. (Series 4, 5, and 6.) These films with aoa. S50) 529) S4. Sa i58 60 '6e 64 66 68° 70 72 internal incidence scatter green light. The reflexion for ex- ternal incidence is white. Curves for the golden and purple films made with the Hilger spectroscope are reproduced in fig. 4, and may be considered as representing with some ‘to move up into the red, givin 110 Prof. R. W. Wood on Optical Properties of accuracy the dependence of reflecting power on wave-length. ‘The other curves are more or less qualitative in character, as a result of the use of ray filters having rather broad bands -of transmission. Further increments in thickness cause ice absor ption Wave g@ a blue reflexion ; but now we begin to have a diffusion of white light, and if the -thickness is still further increased, the amount of white diffuse reflexion increases, the film resembling a matt surface of silver, such as is obtained by electrolytic deposition. We now haw optical evidence of the crystalline structure of thie film, a circular “rainbow” halo appearing around the regularly reflected image of the source of light. Films showing ‘ea. effect are best mace by gradually ‘driving the deposit around to the further side of he bulb by playing the small flame continually against the ed ‘deposit, keeping the front of the bulb clean by occasional ge of the brushing with the flame, and examining the nature of the reflexion every few seconds. A deposit made in this way will be thickest along the edge, and if properly made will show a rainbow of great intensity mixed with very little white light. The rainbow is best seen with the light incident normally on the centre of the metailic patch, which can be accom- plished by standing with the back to the light. It will be found that the rainbow is ver y sensitive to. changes in the -angle of incidence, and can be made to disappear by a slight cana of the bulb. From considerations of the angle of incidence at which the rays meet the oblique surface along the curve of the coloured halo, it seems probable that we are dealing with minute octahedral crystals of sodium. Moderately thick deposits show also a colour by trans- mitted Jight which is very sensitive to sma]l changes in the -angle of incidence. The colour sequence for transmission, o as the thickness incr eases, 1s blue, blue-green, green, and pale red. This latter colour is very sensitive to ‘changes of the incidence angle, a rotation of the bulb of a few degrees causing its disappearance. One feels as if the structure concerned must be of the nature of vertical spicules or needles somewhat as shown at the right-hand part of fic, a or it may even be possible that a dendritic type of er vstalli- zation occurs, as is frequently the case with metals. I have never sueceeded in producing these colours except ‘with the alkali metals, though an analogous phenomenon can Films of Sodium and Potassium. | LLE ‘be produced with selenium. Deposited at room temperature in an exhausted bulb, in a manner similar to that employed with sodium, the deposit i is homogeneous and yellow or orange in colour. If, however, the wall on which the ‘deposition takes ‘place is heated howe 60°C. (the critical temperature for ‘selenium deposition) the colour by transmitted light is sky- blue, and. the film scatters very powerfully light of a ver- milion-red colour. There is very little specular reflexion, ‘in. which Sosa the selenium ‘deposit differs from The sodium films. The selenium deposits are permanent in air, -and the bulbs can be broken up for further examination. I have made casts in celluloid of the blue deposit, and find that they scatter blue light, indicating a Tens structure. Doubtless much light could be thrown upon the whole ‘subject if some method could be devised of making an -accurate cast or replica of the sodium surfaces in some oiler metal. Hxperiments along this line are in progress. Ex- ‘tremely complicated colour effects are produced if a trace of -some volatile hydrocarbon is present in the bulb. Some of these effects were described in my earlier papers. In a recent experiment [introduced a trace of a heavy lubr icating oil into the bulb, and distilled it along with the sodian against a surface cooled to liquid air temperature. The film was of a deep blood-red colour which disappeared at room temperature, owing to its breaking up into granules. It is unnecessary to go ante these det rls at present, for until we have some plausible hypothesis to apply to the cases where pure sodium alone is concerned, speculations as to the action -of liquid films in contact with the crystalline deposits are hopeless. It is, perhaps, of interest to draw attention to a case ‘mentioned once by Lord Rayleigh in connexion with the hypothesis that the cavities between the metallic crystals -act as selective absorbing resonators for the light-waves. Consider a surface vents in section in fig 5 (a grating for Fig’ o. SE StF LL FM aa -example) in which the depth of the groove is one quarter of a certain wave-length X. As is w vell know n, X% will be absent by interference in the central image, and appear In 112 Optical Properties of Films of Sodium and Potassium. excess In the spectra (see Wood’s ‘ Physical Optics,” Laminary grating). If, now, the grating space be made sufficiently small, the lateral spectra disappear and A must appear again in the central. image, notwithstanding the haif-wave retardation. Lord Rayleigh expresses the opinion that in this ae the narrow grooves may, perhaps, act as resonators. So far as I know this experiment has never been tried, and it is my plan to. attempt it with the 110 « waves obtained by focal isolation with quartz lenses. It appears to me possible that the cavities may be found. to act as absorbing resonators, » still remaining absent in the reflected light, in which case we should have something quite analogous to the behaviour of the sodium films, though I am very doubtful about this, however. Whether the phenomena exhibited by sodium depend upon the form of the cavities alone or whether the optical pro- perties of the metal must be taken into consideration remains to be seen. A quantitive investigation of the transmission and reflexion of films of the alkali metals is now in progress in our laboratory under the direction of Dr. Pfund, which will furnish some of the required data. It seems to me very important to establish a dynamical theory based on a theoretical investigation of the behaviour of surfaces covered by cavities small in comparison with , with respect to ether waves. Up to the present time no such investigation has been made, so far as I am aware. My feeling at the present time is that the cavities trap the radiation in some way and cause almost complete absorption for certain values of 4. An inspection of the curves shows that a change of X from 70 to 50 may cause the reflecting power to drop from 75 per cent. to 4 per cent. This enormous change, combined with the fact that the reflexion remains high in the remainder ot the spectrum (2. e. in the ultra-violet), makes it appear certain that we are dealing with a phenomenon which cannot be explained by any of the simple laws of interference and diffraction. peer en| VIII. The Force-Function in the Theory of Ordinary Relatiwity. By Haroup Jerrreys, M/.A., D.Sc.” bs iG a recent number of the ‘ Proceedings of the Physical Society’ f Dr. W. Wilson shows that on the ordinary theory of relativity the equations of motion of an isolated particle can be reduced to the canonical form of Hamilton ; also that a similar reduction is possible on Hinstein’s theory of 1915 for a particle in a gravitational field. This result is capable of extension ; for if the coordinates of a system are q,--.q, and the corresponding momenta are p;...p,, while L is any function of the p’s and q’s, and s an independent parameter, the condition that 6| Jids be zero for small 0) variations in the path of integration is that Onl dga ol. dp rr Coe Ch Oy ae (1) where H= Xp a —L. ds Jonversely if the form (1) holds, 6\L ds is zero. Now the essential assumption of the Hinstein theory and of the ordinary theory of motion under no forces is that S\ds=0, where ds” is a quadratic function of the differentials of the space-time coordinates. Hence the equations of motion, by the above theorem, must be reducible to the canonical form ; and the same will be true if 8\@ ds=0, where ® may be any function of the coordinates and velocities. This is true of the form of general relativity obtained by Dr. L. Silberstein f, and therefore the equations of motion in this also can be put into the canonical form. 2. Let (x,y, z,«) be the measured space-time coordinates of a particle, w being cet where ¢ is the time. Put st Od tee en Ate ce ey (CQ) DEGEROS ee dada A) iO sae tO) POR oe Geant, . In ordinary relativity the 4-vector qe bls Yo = u) is zero * Communicated by the Author. Tt Vol. xxxi. pp. 69-78, Feb. 1919. { Phil. Mag. vol. xxxvi. pp. 94-128 (1918). Pint. Mag. Ser. 6. Vol. 38. No. 223. July 1919. I —— 114 Force- Function in the Theory of Ordinary Relativity. for a particle under no forces. If forces are acting on it, put gas ra ae ie pmol Ys ~ U)=( fr Ilse Is fas Sen ae (4) where w is an invariant corresponding to the mass. The momentum is d (Pr, Py Pz» Pu) = BT (2s % UW) + + + ©) If the canonical form of the equations is possible, we have eee dpe uy ” py + ee os dt dr P?™ dr dp. dpy dpz dj u —(& dat 78 dy+ 7 de+ "tu IL D) D) 2 : A nes , , 3 = Du Ap, + py +p; +p) —(fide+fydy +fdz+fudu). (6) The first term in this is d(4uc?)=U. Hence the condition that H may exist is that bles ity shall be a complete differential. Hence (j:, fy, fz, fu) 18 the gradient of a scalar, U, say ; and U must be independent of the. momenta. iivotier wor ds, the canonical form of the equations of motion implies that a force-function exists. da ai Se Or Next, fe TES Tle ae ‘he =u OW ae (7) 2 yee i f da dy 2 dz Z where y ?7=1— ai(S) ac ee ee This is the equation of conservation of energy. in three- dimensional motion. If it be multiplied by any function of y, say R(y), it shows that the rate of increase of (pe? yRdy is equal to the scalar product of R(f;, fy, fz) and dx, dy, dz. Hence if we use the terms “kinetic energy” to denote \uc’yRdy, and “force” for R( fz, fy, f2) we shall have the relation Increase of kinetic energy = work done . . (9) whatever be the form of R. There is no way of deciding between these various definitions of force and energy, the fori Further Contributions to Non-Metrical Vector Algebra. 115 of R being a pure convention. If, however, the equations of motion can be put in the canonical form, there is a definite reason of convenience for taking R=1; for this is the only value that makes the “force” the oradient of a-scalar potential not involving the velocities explicitly. Thus the force is defined as (fas Jus fz) and the energy ~ as tyc’ry’. The “mass” similarly becomes py’. 4. In accordance with the theorem first quoted we have Olinda On vain. co UR Tea pnt (10) / ‘ Oe Ma U ia Ae... ote where L=pr 7 + Py Fue ae Pe se = bet coucuamiee rs lh: ira) ee mene ud II)) Thus the only possible form of L is that obtained by Silberstein, namely a constant + a scalar independent of the velocities, which is itself the foree-function from which the force is derived. IX. Further Contributions to Non-Metrical Vector Algebra. By L. Streersrery, Ph.D., Lecturer in Mathematical Physics at the University of Romet. 1. ie a recently published little book t I have en- deavoured to build up a simple vector algebra independent of the parallel-axiom and of the axioms of con- gruence, and therefore of any idea of measurement by means of “rigid” bodies. This algebra consisted in essence only of addition and equality of vectors. Notwithstanding this poverty of means the said algebra was shown to be sufficient for the treatment of any purely geometric (?. e. projective) problems; for any such problem is ultimately concerned with crosses of straights, joins of points, with concurrency, collinearity, and similar questions, and all of these can be covered by the sole operation of addition, with its inverse and its iterations. No urgent need, therefore, was felt for the introduction of a “ multiplication ” _of vectors by vectors Also, at the time of writing and up to the last moment ot * This possibility of this value of the energy is indicated by Dr. Wilson (loc. crt.}. + Communicated by Prof. A. N. Whitehead, F.R.S. { ‘Projective Vector Algebra,’ an algebra of vectors independent ot the axioms of congruence and of parallels. Pp. 78. Bell & Sons London, 1919 (May). I 2 116 Dr. L. Silberstein: Further Contributions publishing of that book, no reasonably simple multiplicative operation, independent of congruence and of parallels, pre— sented itself to the writer. During the reading of the proof- sheets, however, a not too-complicated non-metrical generali- zation of the scalar product, and then also of the vector product, of two vectors spontaneously suggested itself. And since, for technical reasons, this theory of vector multiplication came too late to be included in the said volume, it seemed worth while to publish it separately in the present paper which, to facilitate its perusal, will be kept uniform with the treatment (in- cluding nomenclature and symbolism) adopted in ‘ Projective Vector Algebra.’ In order to avoid covering unnecessarily many pages of this Journal, the definitions and theorems concerning vector-equality and addition with their immediate consequences will not be repeated here. As often as the need arises the reader will be referred to the said book, which he will be assumed to have read or at least perused.. It will be shortly referred to as P.V.A., with the section number whenever necessary. Here but one more remark to justify this paper. As was said a moment ago vector addition is sufficient by itself to cover the field of projective geometry, 7. e. to treat its problems algebraically. Thus, strictly speaking, vector “multiplication” would be a superfluous operation. So, in fact, it is in a certain sense. But “multiplication ” enables us to treat many of those problems in a considerably simpler: manner. The multiplication of vectors by vectors, especially if it is so. defined as to be distributive, is a powerful instrument. of aloebraical investigation, and therefore a very desirable supplement of addition. 2. Standard or Unit Conic.—It will be well to treat first the relations in a plane. The passage to three-dimensional. space (section 6) will offer no difficulties. In P.V.A. the concept of equality was applied to such pairs of vectors only as have different origins, while vectors emanating from a common origin were * complete strangers to one another, no relation between them having been de- fined, unless one counts their distinctness. This distinctness consisted in their being on distinct lines (straights), and therefore in having different “‘termini”’ or 7 points, their- crosses with the conventionally fixed straight, the 7-line. In more familiar language, our co-initial vectors differ in * direction,” and since their “lengths” or ‘‘ sizes” as tested by ‘rigid transferers”’ are entirely foreign to our cirele of | ideas, all these vectors have among themselves no feature in: common. * With the only exception of a vector OA and its negative OA'=. OA). to Non-Metrical Vector Algebra. iy Such being the case we are free to fix the standard or wit vectors, say a, b, ¢, d, etc. upon the rays emanating from their common origin O quite at our pleasure, choosing them conventionally ain independently of one another... There is but one limitation to this freedom, viz. if the ray I is the continuation of the ray J, andif a= ve has been chosen as the unit vector for /, the unit vector a!= (a’ for I’ is to be such that, with reference to the fixed 7-line, ‘This means that, if 7, be the terminus of the straight (/’Ol) in question, a’, OU, a, 7, should be a harmonic range, with a, @ as conjugates. Keeping this only condition in mind let us draw round O some closed continuous curve in order to standardize at once the whole pencil of our vectors. Sucha curve could be drawn ina variety of ways. It will be remembered, however, that its only office is to enable us to describe numerically the geometrical properties of the planein as convenient a manner as possible. Now, with this aim in view, it will be found particularly convenient to choose as shel standard curve a conic which, as is well-known, can be generated by purely pr ojective processes. Let, therefore, O, the origin of all our vectors, be sur- rounded by some conventionally fixed closed * conic (ellipse) which we will call the standard or the wnit conic, and which will be denoted by « Suppose the lime has not yet been selected. Then by the very act of surrounding O with the conic « and declaring it to be our unit conic (2. e. such that all its radii vectores Ol, Ob, etc. are unit vectors) the 7-line is co-determined. Tn fact, any secant «Oa drawn through O, which will hereafter be called a diameter of x, meets the required 7-line in a point 7, which is the fourth harmonic to a’, O, a, conjugate to 0. In short, the T-line to match the sbamelanel conic is the polar of. O ‘with respect to x. Thus, having drawn x, draw through O (which will be called the centre of K) any two diameters aa’ and 6b’ (fig. 1); let ad and 6'a’ cross in M, and b’a and a/b in NV; then the join MZ will be the required 7-line, supplementing our reference system. The origin O being within the conic, its polar, the 7-line, will lie outside «. We shall denote all unit vectors, such as Oa or Ob, by ‘small clarendons, a, b, etc., using capitals, A, B, ete. for any * J. e. such that no tangents can be drawn from O to the conic. 118 Dr. L. Silberstein : Further Contributions vectors on the lines of a. b, ete. Any vector OA=A can be: nie A=oa, where o is a scalar number which can always e made positiv - choosing opri nade positive (by choosing appropriately between a andi Heioced, its negative a’). With this understanding we will denote & by | A|, so that ; | ? ) == Ala, and we will call A| the tensor of A, always of course with respect to the chosen « and 7. Sometimes A (where it cannot be confounded with ‘the point A”) will be used as a short for A}. With this notation the equation of the standard conic will be Yr =, ° ne «| ogee (x). where r is any vector drawn from ( to the conic «. Having thus fixed upon every line through the origin the points 0, 1 and © (T’-point), we can construct upon it in the well-known way (P.V.A., 6, 7) the points 2, 3, ete., 4,4 : = ° 5 E ; Pe > ete., in fact, the whole projective scale. ‘The locus of the end-points of all vectors such as 2a, 2b, etc. will again be a. conic, Ky say, whose equation wil! be 1 =2, and so on, in general, == 7 = COMMS. |Y¥ to Non-Metrical Vector Algebra. 119 will again be aconic, «,, homologous with the standard conic. For 0<7r<1, all these conics will be closed and contained within «. For +>1, however, these conics will not continue for ever to be closed. When * reaches a certain value, say in) Kp will be a parabola, and for r>F,, the conic will either at once split into two branches, a new branch appearing “beyond” the 7-line, or else we shall continue to have one- branched curves up to a certain value 7, >7,,, and only when 7, is ex- ceeded, two-branched conies (hyperbole). The former will be the case for the euclidean, and the latter for a lobatchevskyan space. (This, behaviour is closely connected with the peculiarities of the construction of the negative of a vector, cf. P.V.A. 9.) But details of this kind being irrelevant for our purposes, need not detain us here. For r->so the conic x, will tend to a pair of straight lines falling, so to speak, into the T-line from both sides. (If space is elliptic, the T-line, as every straight, has but one “side”; in such cases, as in P.V.A., let us con- template a “restricted region” broad enough to embrace all the contem- plated figures.) Let w, y be the (non-homogeneous) projective coordinates of a point of « with respect to O as origin and a, b as axes, so that r=va+yb, which reads: the end-point of r is reached from O by « projective unit steps along the a-line, followed by y steps. along b, or vice versa (addition being commutative). Then, remembering that |a/ is to be the unit of 2, and | b/ that of y, the equation of our standard conic with respect to any axes a, b whatever, can be written vty? + 2¢%y=1. The value of the coefticient c), will depend upon the choice of the particular pair a, b as axes. It will be characteristic for the pair of rays represented by a, b or, if we prefer to put it so, for the “angle a,b.” To bring this into evidence let us better introduce, instead of ¢,, the symbol (ab) or (ba), indiscriminately, since the réles of a and of b in making up the scalar number cj, are manifestly the same. Both (ab) and (ba) are hitherto mere synonyms of ¢j9, the coefficient of 2zy, and nothing else. After this warning we can write, for the standard conic, pete Mo(ab cy 1, . see 6d with respect to any (distinct) a, b as.axes. How to unravel 120 Dr. L. Silberstein: Further Contributions the properties of (ab) from this somewhat implicit definition, will be shown presently. Meanwhile, in order to be able to treat vectors with any origin other than QO, let us recall from P.V.A. (section 8) the definition according to which all equal vectors are, in the first place, co-terminal, 2. e. concurrent with one another ina 7J-point. Thus, it 7, be the terminus of a= Oa, and it is required to construct, with any given O’' as origin (not on T), the vector equal to a, we have first of all to draw O'T, ; next, to obtain its end-point, we have to draw OO! crossing the 7-line in T’, say ; then the straight a7” will cross O'T, in a’, the end-point of the required vector. Having thus obtained, on the line O'7,, the unit, we can in the well- known way, construct upon it the vectors equal to 2a, 3a, and so on, in general O/A'’=A’=|A' a. This enables us to speak in a perfectly definite sense of the tensor of a vector drawn anywhere on the contemplated plane, as of the number of equal projective (unit) steps contained in it. Similarly we can construct, from 0’, vectors equal to b= Ob.) c= Oc. vetes | and thus reproduce (not to say ‘transfer’ ) the original standard « at any spot where it is desired *. Notice that if «’ be such a reproduction of the standard round O!, the original 7-line will again be the polar of O' with respect tox’. We thus see that our original 0 is by no means a privileged point of the contemplated two-space. Any 0’ is as good a point. The 7-line, how- ever, does play a privileged réle, being the common. polar of the “centres” of all these standard conics with respect to each of them. It is also the polar of O’ with respect to any x,, the homologue of x’. But it will be remembered that even the 7-line, although fixed for the duration of an investigation, is but an arbitrary reference line, and can, in different instances, be chosen in a manner most suitable for the investigation of the given figure or figures drawn on the plane, the only geometrical entities. How the choice of this reference line influences the position of the “‘ centres” of the standard conics «’ we already know ; each of them is the pole of J with respect to x’. *-The physicist will notice the difference between the standard « and, say, the metre standard preserved at Paris. It is not enough to make a copy of the latter at the spot, in Paris, it has to be carried about; the former, however, is “reproduced” or constructed at the place where it is just required. ‘This is a capital difference. Moreover, it is enough to construct a few isolated points of x’. For, as is well known, five points of x’ would (even in the absence of 7) suffice to construct, without further appeal to «, any number of other points of x’. to Non-Metrical Vector Algebra. eae 3. Geometrical meaning of (ab). Definition of Orthogo- nality.—In order to perceive clearly the intrinsic nature of the ordinary number (ab) or (ba) as a property of the vector pair, a, b, let us construct graphically some vector expression into which (ab) will enter alongside with things already known to us. For this purpose let us return to the equation (1) of the standard conic «. From this we find easily, as the equation of the tangent to « at any of its points @, Y, [w+ (ab)y]a+ [y+ (ab) y=1, aw, y being the coordinates of any point of the tangent line. In particular, the tangents at the end-points a(#=1, y=0) and b(y=1, x=0) of the vectors a, b used as axes will have the equations 1 tr (( 0109 eet eet Dima a) cade) (HD) te eee nC Let P be the cross of the two tangents ¢,, t;, or the pole of the chord ab as polar. Then its coordinates, the solutions of the iast two equations, will be 1 oy any and the vector P= OP, which is Ea+ynb, will become S : eh ee ee 2) It will be remembered that S, the end-point of §, is the cross ona lpwith 67, (P.V.A., 4). This is the “required expression, or equation, since it contains, besides (ab), only P and §S, well determined and ‘easily constructible things. Hither to (ab) was defined only as the coefficient of 2ay in the coordinate equation of « with a, b as axes. Henceforth we can drop that equation and consider (ab) as defined by the vector equation ); which shows at once the intrinsic nature of (ab). Such being the case we can, on the other hand, expand (ab) using, for instance, the components of a and b along some auxiliary AXES ; on we shall then know that the value thus obtained is independent of these auxiliaries. We may profit from this possibility later on. Meanwhile let us concentrate our attention upon the last equation. It will teach us many interesting things. First of all, 1+(ab) being a mere number, it shows us that S is 122 Dr. L. Silberstein : Further Contributions collinear with O and P (fig. 2). In other words, the vectors. P,S differ only through their tensor. Such being the case: Fig. 2. there is no objection against writing (2) in the form 1+(ab) =8/P, or also S—P OS—OP (ab) = Pp — OP : oe) ae (2a). In the next place we see from this equation (and from the- drawing) that when b=a, 7. e. when a, b coalesce, and therefore S=2a, while P=a, we have (aa) = 1, for any unit vector a. Similarly if b=a’=-—a, the vector S to Non-Metrical Vector Algebra. 123. vanishes, and (aap These values correspond to what may be called the “zero angle,” and the “‘straight angle,” respectively,—using these names only as. synonyms of the pairs (a, a) and (a,a’). For hitherto we have not introduced the general concept of “ angle” as a magnitude. But the most important thing for our present purpose 1s: to note that when P=S=a+b, that is to say, when the pole of ab coincides with the cross of aT, and bT,, our equation gives 3 (CD) URE Ose pisnin oc Naa tO) I propose to call such, manifestly remarkable, pairs a, b normal or orthogonal, or perpendicular vectors. Their definition can be conveniently written P.,=atb, ° 5 . e ° . Bie (4), the double suffix reminding us that P is the pole of the chord ab. It will be noticed that, when used as coordinate: axes, any such pair a, b reduces the standard-conic equation to Hig. 3: Any number of such orthogonal pairs can at once be con-- structed. In fact, from the form (4) of their definition we see that both 7b and 7ja are tangents to «. Thus, if any a= Oa (fig. 3) be given, prolong it up to its terminus 7%, EE OK bh Sai) | ail Oe ps | 124 Dr. L. Silberstein: Further Contributions and draw from 7, a tangent to x. Then the contact point d will be the end-point of the required unit vector b normal toa. The proof that 7;a will then also be a tangent may be left to the reader. Again, the other tangent from 7; will touch « in b', which will be found to be collinear with b, O, so that b’=—b. Similarly, the other tangent from 7, will touch « in a’ the end-point of —a. ‘Thus, together with Oa, Ob also Oa, Ob' and Oa', Ob wiil become auto- matically pairs of orthogonal rays. In short, the straights aa’ and bb' will be orthogonal. ‘Since 7, is ihe pole of 60’, and 7, that of aa', we see that our orthogonal lines are, in usual nomenclature, pairs of conjugate lines with respect tox. At the same time we see that, in technical language, OT, T, will be a selj-polar triangle with respect to x. We thus know how to draw. to any straight passing through O a perpendicular in O. This limitation to the origin can be easily removed. In fact, let us generalize our 5 original definition by postulating that if (ab) =0, and if v be any vector equal to b, then also (av) =0 more generally, if a’ =a and b’ ie then (a'b’)= (ab). Now, all vectors equal to a are coter minal witha. Thus, returning to fig. 3, all lines of the pencil centred at 7; will be ortho- gonal to all those of the pencil centred at 7,. In particular, the tangent to « at any of its points (as Ta) will be normal to the corresponding diameter (a'a) * This shows us also that the proper representatives of the orthogonality thus defined are the pairs of corresponding pencil-centres themselves, each pair being so correlated as ee a ities tes 3. (These pairs of points are, In common nomenclature, ‘conjugate with respect to the conic fe.) We may have an opportunity to return to similar questions. Meanwhile let us proceed with our chief subject. 4. Scalar Product defined. Distributivity. Let A= A a and B=|B/b be any two vectors. Let us denote by (AB), and eall the scalar product of A into B, the number (AB) =|A|.|Bib).:. == Since the tensors are ordinary numbers and (ab) =(ba), we have also (AB) =(BA), the commutative property. / * Notice in passing that the 7-line itself will be orthogonal to every other straight line whatever. \§ ROB to Non-Metrical Vector Algebra. 125: If m, n be any scalars, then |mA|=m|A|, |nB|=7 | B\, so. that, by the definition (6), the product of mA into nB is equal to mn times (AB). In view of this property we can henceforth drop the brackets and write the scalar product simply AB and, instead of (6), | ABER A) Blab.) (Ga) If A,B are orthogonal, i.e. if ab=0, we have AB=0- unless, of course, A ov B are infinite or T-vectors. We have already seen that aa=1, for any unit vector a; instead of this we shall write a?=1. Thus, for any vector A, ee Ae ee C7 In words, the scalar autoproduct of any vector is equal to the square of its tensor, exactly as in ordinary, euclidean, vector algebra. But what makes our generalized scalar multiplication of vectors a powerful operation is the fact that it shares with the ordinary one the property of distributivty. This can be proved in a variety of ways. The shortest of these seems to. be the following one. Fig. 4. By (6) or (6a) the definition of AB is reduced to that of ab, given originally by (1). In the preceding Section we have already given a geometrical representation of this number. Another, still simpler representation is obtained by drawing from } the perpendicular to a. Let n (tig. £) be SSS ee a & SS ae 126 Dr. L. Silberstein: Further Contributions normal to a, and let the terminus of n be 7» (this is also obtained by simply drawing the tangent ata). Draw 7,6 crossing the a-line in N. Then the a-coordinate of V will be ab or, vectorially, ON=(abja. . °. 2 2a 0 aaeitan To see this, take the orthogonal pair a, n as axes of a, y, -so that the equation of « will be ge +y=1. Now, if a, v are the coordinates of } with these axes, we ‘have b=aat+m, ¢?+v?=1. Thus, if 2’, y' be the coordinates of any point (2, y) with ‘respect to a, b as axes, we have vat yn=ev'at y b= (u' +ay')atvy'n, whence pS oe 8 v0 -and the conic equation with a, b as axes becomes vw? ty" + 2Qaa'y'=1. Therefore, by the original definition of ab, as given in (1), ab=a, which is the a-coordinate of b, 7. e. that of V. This proves the statement made in (8). Thus ab is obtained by the orthogonal projection of b upon a or, similarly, by that of a upon b, exactly as in ordinary vector algebra, notwithstanding the different, more general meaning of our concepts. Now, such being the case, the distributivity of the scalar produet follows at once. fn fact, if A, B, C be any vectors and B+C=8§, we have, first of all, A(B+C)= A}. aS. Now, whatever B,C, we can always reduce them to a chain, so that S will be the vector drawn from the origin of B to the end-point of C. But the orthogonal projection of S upon a is instantly seen to be equal to the algebraic ‘sum of the orthogonal projections of Band of C. Cf. fig. 5, in which 7), is the 7-point of the vector n normal to a. Notice that the intercepts of the three perpendiculars will to Non-Metrical Vector Algebra. 127 not re-cross one another; for they already crossed one -another in 7%. ‘Thus we have the announced distributive property, A(B+C) =AB+ AC. And since the commutative law holds as well, we have also (A+B)(C+D)=AC+AD+BC+BD, and so on. : In fine, all ordinary rules hold for the generalized scalar product of vectors, as defined ahove by means of the standard conte. The far-reaching geometrical consequences of this simple property are obvious. Hvery formula known from euclidean geometry (or “ trigonometry”) will continue to hold, with the only difference that tensor or “size” of a vector will not stand for the number of centimetres or inches contained in it, but for the number of equal projective steps with reference to the conventional unit-conic « and 7-line, and that right angles (orthogonality), equal angles, ete., have in our case a different, more general meaning. 128 Dr. L. Silberstein: Further Contributions Thus for instance, if A is normal to B, and if C=A—B, we have, by the distributive law, C?= A’ + B? or CP= | A SB lee the generalization of Pythagoras’ theorem, which reads: the squared number of projective unit steps contained in the hypotenuse is equal to the sum of those contained in the sides. The reader will find it a useful recreation to verify this graphically in a convenient case, such as| A,;=3,{/B =4, drawing any ellipse as x round O, constructing its polar it ‘and so on. If he draws the rather numerous lines, necessary for the whole construction, with some eare he will find the number of unit steps contained in C to be hardly distin- euishable from 5. The writer has actually constructed this case with sood success, his naive aim being originally to verify thus the analytic result. More generally, for any triangle with A, Band C=A—B as vector sides, \C =A | B/?—2) Al. (Blab. een Many other examples of generalized theorems could at once be given, but those quoted above will suffice for the present. 5. Angles, their arc-measure and trigonometric functions.— We have incidentally called the pair a,b an “angle,” in particular a, aa zero angle and a, —aa straight angle. We may now, without fear of a misunderstanding, call a,b, i. e. aOb or bOa, in the case when ab=0, a right angle. In general, for any a, b, or their equals constructed anywhere, we can speak of the angle aVb. Still, it remains true that we do not yet know w hat “ equal angles” and “‘ multiples of an angle”? mean, simply because we have not yet fixed the meaning of these words. There is nothing to prevent us from denoting the angle ab by a single letter, say 6, provided we keep in mind that hitherto this 6 is not figelioa magnitude. We shall define it presently as such. * Meanwhile we can speak of the number ab, already familiar to us,as @ certain “indirect measure” or cence ie of the angle @=a, b. As such let us call it the cosine of 0, writing: ab= cos 0= cos (a, b). We know from Section 4 that this number is represented by the orthogonal projection V of b upon a, that is to say, that Cosid = cos (Gn b) ==: | OV ee one to Non-Metrical Vector Algebra. 129 according as WV falls on Oa or Oa’. We may, therefore, consider the latter equation as the definition of cosine. Similarly, let us define the sine of @ by the tensor of the normal b/V, writing SIN] — tenet) =) FOU ee enn (EL) a, 6 being points on the unit conic «, as before. (The sine and the cosine of an angle /O'm constructed anywhere can at once be reduced to the above by joining the 7-points of O'l, O'm with O, and so on.) Using these familiar symbols, however, we must not yet confound them with the familiar trigonometrical functions, until we have learnt that they are identical with them. Such, in fact, they will be shown to be presently; and their analytical identity will become complete after an appropriate measure has been introduced for @, still to be defined as a magnitude. At any rate we know already that if 6, 6,, 0, stand for the zero angle, the straight, and the right angle respectively, we have cose p— by casio, —— I cos 6, —— Oh 2a) although we do not know, for example, what the numerical value of @, is. Again, since ODN is a right-angled triangle and Ob is a unit vector, we have, by the generalized Pythagorean theorem, for any 9, SIMO Cos? Oana cin. ne mic lZ0) Moreover, again in virtue of the distributive property, we ean deduce for our sin and cos the familiar additivity Fig. 0 a. y 6) theorems. For this purpose it is enough to declare that if a=bOi and B=aQ be co-vertical angles haying the side Oz in common, aOb is the sum of these angles ; and similarly for the difference. In fact, if 7, a, b (fig. 5a) be the end- Phil. Mag. 8. 6. Vol. 38. No. 223. July 1919. K 130 Dr. L. Silberstein: Further Contributions points of the unit vectors i, a, b, and if the unit vector j be normal to i, we have, by what has been explained previously, a=icos a +jsin «, b=icos 8 —jsin B, whence, by the distributive law of multiplication, ab= cos («+8)=cosa.cos@—sine.sinB, (126) whence also, by (120), after easy transformations, sin (a+ 8)=sina.cos8+cose.sinB, . (12c¢} as in ordinary trigonometry. ) Fig. 6, Leaving other details to the reader, let us still notice that tan 6= tan (a, b) can be defined either by ian @= sin O/cos0, = . = (eae or graphically (fig. 6) by | fang === 129), to Non-Metrical Vector Algebru. 131 where S is the cross of the b-line with the conic tangent at a, the upper or the lower sign being used according as S falls on the segment a7, or outside it (7. e. below a or beyond 7;,). It is not difficult to see that these two de- finitions are equivalent. In fact, if n be normal to a, and therefore co-terminal with aS, we have S=a+An=pb=p(a cos 6 +nsin 8), whence pcos6=1, wsind=A, .. X= sin O/cos O, and aS=S—a=a~n, -so that proving the statement. Notice that the jump of tan 9 from o to —x when @ passes through -a right angle (é,) obtains the following elegant representation: When b (fig. 6) approaches n, the point S approaches 7',, the value of tan 6 thus ‘Increasing to --oo, and when b exceeds n, the point S passes across the Y-line thus indicating the jump of tan é to —o. It will be remembered ‘from the construction of the projective scale (P.V.A.) that, on every straight, to one side of TJ’ corresponds # ,and to the other side —o. The discussion of the obvious geometrical correlate of the fact that cos 6, for instance, does not exceed the values +1 may be left to the reader. It will be enough for this purpose to note that the pencil of all lines 7,6 Aleading to our previous JV) is limited by the two tangents 7a, T,a'. Let us now pass to compare angles with one another, so cas to be able to treat @ as an algebraical magnitude. And to begin with let us give a definition of angle equality. This can, in the first place, be based upon ab= cos @ itself. Limiting ourselves to concave angles we may define two angles a and 8 as equal if their cosines are equal, no matter where their vertices are situated *. In symbols, if AD ln wae. ce as (CLS) we will say that angle a, b = angle 1, m. The simplest sub-case of such a relation occurs when 1 and a, m and b are equal to one another but situated in different places. This gives, as a corollary of the above * Notice that every angle having its vertex on the 7-line (being of the form 1,1) is a zero angle, unless one of its sides is the T-line, when the angle is aright angle. Of. supra. K 2 132 Dr. L. Silberstein : Further Contributions definition, the equality of all angles (such as@ and @ in fig. 7) whose sides are coterminal. Such was the only case of angle comparison treated in P.V.A., Appendix A, where it was- shown to be sufficient to prove that the non-metrical sum of angles in every triangle (not traversed by the 7-line nor- having it as one of its sides) is equal to a straight angle or, as we now can say, to two right angles. We may return to- this triangle property later on. Meanwhile let us proceed with the application of the definition (13) of angle equality to eases in which 1, m are vectoria ly different from a, b. Without loss to generality we can assume all these four vectors to be co-initial in O, so that a, b, 1, m are on the conic x. Let us put the question in the form of the problem: Given a, b, the sides of the angle a, and given I, one side of the angle , to construct m, so that a=B, 2. e. so that Im=ab. From } draw the normal to Oa meeting it in NV ; through: NV imagine drawn the conic homologous to «; let this eut Ol in N'.. Through V’ draw the normal to 1 cutting « in two points; either of these will be the end-point of the- required m, giving @=«. It is, for our purpose, by no. means necessary to execute this construction in detail.. (And a more speedy construction will be given in Section 9.) It is enough to contemplate it in order to see that, what-- ever 1, an angle equal to « can be constructed on either side- of 1, in a perfectly determined manner. In particular, if / coincides with 6, when the problem: consists in the doubling of a, the actual construction becomes avery simple process. For, involuntarily, we have already to Non-Metrical Vector Algebra. 133 -solved the inverse preblem, viz. to bisect a given angle aO0m. In fact, if P be the pole of the chord am we have, by (2), -and replacing b by m _ atm ~ |+am" Multiply this equation scalarly by a or by m. The result will, in both eases, be 1. Thus Pa=Pm, and therefore, anole a, P = angle P, m. But P and S, the end-point of a+ m, are collinear with O. ‘Thus also aOS=SOni, giving the required bisection of the angleaOm. Remembering ithat S is the cross of a7, with mT',, this construction is obtained at once, as shown in fig. 8. (Notice in passing Ate '@) Tig, 8. that a S= Om, Oa=msS, and that all four are unit vectors. ‘Thus OaSm isa generalized rhombus. In particular, if am= 0, it will be a generalized square.) Thus far the bisection of an angle. The solution of the inverse problem, originally required, follows at once by remembering that P, the pole of the required chord am, is collinear with O, S.. Thus to «4.0... aes in case of orthogonal axes. This subject does not call for any further explanations. Nor does the proof that the are of any conic «,, 2. e. Y?=7?=const. (homologous with «) subtending the angle @ is equal to Or, offer any difficulties. Thus also the whole circumference of a conic «, (or of its reproduction obtained anywhere by the transformation r'=r+const.) will be equal to 2rr staudtians, no matter whether «, is still an ellipse or has already become, say, a parabola (whose length would in the ordinary treatment be infinite). Hyperbolee may offer some peculiar- ities whose discussion will be left to the reader. 7 Polar coordinates 7, 8 can at once be introduced, with v=recos0, y=rsind as their relations to the previous orthogonal z,y. ‘The use of such coordinates amounts to writing, for the position vector of any point of the plane, r=ricos#+jsin@), - - 2. sane where i,j is a pair of orthogonal unit vectors. 6. Extension to Three-Space.—In order te extend the preceding investigation to three-dimensional space let us take, instead of «, a non-ruled closed quadric (ellipsoid), surrounding the origin O, as the standard or unit surface. We will denote it shortly by Q. The 7T-plane will be the polar plane of O with respect to Q. The equation of Q with any non-coplanar triad a, b, ¢ as axes will be vty? + 27+ 2do3y2 + 2d3124 + 2ayry=1. to Non-Metrical Vector Algebra, 137 The plane section z=0, and therefore any diametral section * of Q, will be a conie whose equation will be of the form a +y?+ 2ayovy=1. This will play, in the said plane, the part of our previous «. Thus, defining ab as the corresponding coeficient of 2vy and reasoning as before, we shall obtain the scalar product AB with all its properties, provided that A, B are in a diametral plane. But if A, B are not in a diametral plane we ean construct vectors equal to them in an appropriate plane, viz. that passing through O and the straight 7,47. Moreover we can draw these, and all other vectors, a O as origin. It is enough to declare, as in the case Oe ate dimensions, that A/B’'=AB whenever A’=A and B’=B, no matter where A’, B’ are drawn All fundamental questions are thus reduced to those treated previously, and need not detain us here. It remains only to show how ¢riads of mutually orthogonal vectors, reducing the Q-equation to w?+y?4+<7=1, can be constructed. And, by what has just been said, we can without loss to generality, confine our attention to vectors drawn from O as origin. Let, therefore, a= Ow be any given unit vec tor, and 7, its terminus, i. e. the cross of the a-line with the 7 /-plane, The second vector b normal to a being required to be ina given plane, lay that plane through O and draw from 7, a tangent to Kaw, the section of Q. The contact point ) will be the end-point of the required b. (The other tangent will give .b'=—hb, as before.) It remains to construct a “third vector ¢ Suenethat ca—0 and ch=—0. Lay through 7,7; a plane tangent to Q. ‘Then its contact point wil! be the end-point ¢ of the required unit vector ¢ To verify this statement notice that if ¢ is to be normal to a, b, which are already normal to one another, the equation of Q with respect to a, b,c as axes will be x+y" fal, and therefore the equation of the plane touching it at any point &, n, ¢ will be easily found to be Ev-+ny+ c=, so that the tangent plane at c (E=1, 7=¢=0) should simply be x=}. Now, this is precisely the plane through Gd 1. The second tangent plane through 7),7% will touch Qin ¢, the end-point of the vector —c. Thus any number of orthogonal triads a, b, ¢ can be * I. e. by a plane passing through O. = = tT. 3 138 Dr. L. Silberstein: Further Contributions constructed *. We shall henceforth denote any such triad of normal unit vectors by i,j, k, assuming this to be their order in a right-handed system. It may be interesting to notice that the end-point P of the vector P=i+j-+K .. ....«\> sey is the cross of the three tangent planes to Q at the end- points 2,7, & of these normal vectors. More generally, if a, b, c are any three unit vectors “equally inclined” to one another, 7. e. such that ab=be=ca=X, the planes touching Q at a, b, ¢ cross in the end-point of the vector pet Bake Bee 2 Ny The proof, similar to that of equation (2), offers no difficulty. All other details concerning the passage from a plane to- three-space may be left to the care of the reader. He will easily convince himself that the distributive property, A(B+C) =AB+ AC, SB). with its consequences, will remain valid, whether A, B, C are (or can be made) coplanar or not. One of the consequences will be the generalized Pytha- gorean theorem for three dimensions, IP =D?=A?+B°+C, where A, B, Care any normal vectors and D their sum. Thus. also the line-element, 7. e. the tensor ds of the vector dy =ida+jdy+kdz, will be given by ds*=da? + dy? + dz’, i) re (17): the extension of (14) to space. The introduction of polar coordinates and _ allied questions do not call for any. explanations. 7. Vector Product of two Vectors.——QOnce in possession of the concept of orthogonality we can define the vector product of two vectors in much the same way as in common vector algebra. Let A= Aa and B= bb, and let @ be the concave angle * In technical language, O7,7,7., will be a self-polar tetrahedron. with respect to x. to Non-Metrical Vector Algebra. 139° (2. e. not exceeding +) made by a,b. Then we call vector product of A into B and denote by C= VAB, a third vector whose tensor is C= ABsin 0, drawn perpen- dicularly to a,b and so that a right-handed rotation about C carries the vectoratob. — From this definition it follows at once that VAB=— VBA= 4A Vab. Further, if A, B are coterminal vectors, VAB=O (corre-- sponding to the common case of ‘ parallel”? vectors). In particular, for our normal, right-handed triad of unit vectors i, J, k Vii=0, ete. Vij=k, Vik=i, Vkisj. The distributivity of this vector product can be proved in a variety of ways. The shortest and easiest seems to be the: following proof. Let A= Ajit Agj + Ask, and similarly for Band C. Then, C being normal to A, B,, by definition, we have CA = CA, == CyAs + C,A., = 0, CB= OB, + CB. - CB; = 0, whence, solving these two equations for the ratios = we find at once C,=A(A,B;— A;B,), C,=)(A;B,—A,B;), C;=2(A,B,—A.B,), ° . ° ° (18) where X is a scalar. Remembering that, by definition, (7=C,74+ C,?+ C3’ = A*B? sin? 8, we have ?=1, thatisXA=+1. To decide the sign take, for imsuance: | A—ieb—j. Dheny by Ydetmition, C=k. 2. e¢. oO, —0.0,—- andesinee:4,— 6,— 1, while A. 6B), ete. all vanish, we have, by (18'), X=1. Thus the expanded form of the vector product (defined intrinsically at the outset) is VAB=i(A,B,— A;B,) +j(A3B,— A,B;) + k(A,B,—A,B,), (18) familiar from ordinary vector aloebra. Now, if M be any vector, the components of B+M along i, j, k are (by the 140 Dr. L. Silberstein : Further Contributions ‘distributive property of scalar multiplication of vectors) 5B, +M;, ete. Thus VA(B+M)=i| A,(6,+ 1/;) — A;(6,+ M,)] +ete., -and this is the same thing as VA(B+M)=VAB+ VAM, ose eS ithe distributive property of the vector product. This being valid, all other properties of such products and of mixed products, AVBC, etc., known from ordinary vector algebra, will continue to hold. It is therefore needless to -dwell upon them any further. It will be enough to notice that the so-called ‘* parallelepipedal ” property, AVBC]BVCA=—CVAB, . ue enieen which is commonly proved by saying that each of these three expressions represents the “ volume” of the parallele- pipedon A, B, C (right-handed), will in our case be better proved without the aid of the concept of “ volume,’ since we have not defined any such concept. The vector VBC being expanded as in (18), we see at once that its scalar product into A is A,(.6,C;— BC) + ete., or Beh A um rAs AVBC= B, By Be eC and this is, by the well-known property of determinants, the same thing as BVCA or CVAB. ().E.D. The vector multiplication of two veetor polynomials will be handled as the scalar multiplication, the only difference being that the order of factors must be preserved Otek inverted, the sign of the pirtial product must be changed, the well-known rule of ordinary vector algebra. 9 8. General remarks.—hus, having defined vector addition and vector multiplication with reference to a conventional 7-line and a standard conic « (or 7-plane and quadrie Q), we have seen that the resulting Vector Algebra and differ- ential analysis are formally identical with ordinary, euclidean Vector Algebra and Analysis (the only difference being that inches, etc. are replaced by staudtians, and that the angle concept is modified and generalized). Does this mean that we have involuntarily lapsed into euclidean space? By ne means. We have only set up, in the general projective space, a euclidean or a parabolic system of measurement. In fact, that our vector algebra is entirely equivalent to para- bolic metrics in the sense of Cayley-Klein, will be seen at to Non-Metrical Vector Algebra. eae once by noticing that the 7-points or termini of our pairs of” orthogonal tities (such as Z;, 7) form on the 7-line an inv olution, with imaginary doubie points *. Now, this with the réle played throughout by the 7-line is the well-knowth characteristic of the parabolic system of measurement (“absolute involution,” “circular points”). Only the way in which it was here set up is, if 1 may judge, a much more. natural one and easier to follow than that based on the usual application of Caylev’s ideas. Such, however, being the nature of the measurement system equivalent to the proposed Vector Algebra, it is important to notice that it can be set up and used as well in HKuelid’s as in Lobatchevsky’s or Riemann’s (elliptic) space, i.e. no matter whether Huclid’s parallel-axiom is valid or whether there is a whole pencil of non-intersecting lines, whether there are none at all. (In order to avoid cen it would be well to speak in this connexion of euclidean, lobatchevskyan, and riemannian spaces and to reserve the names parabolic, hyperbolic, and elliptic to the metrical systems themselves.) To make these remarks more plain let us take the case of — two dimensions, which will suffice, and let us imagine an ordinary spherical surface in or dinary euclidean space. In order to avoid antipodal complications take an appropriately limited portion o of this surface and consider only such figures which, together with all their auxiliaries, do not surpass the limits of c. Let the geodesics of (creat circles) be substituted for the straight lines of our fundamental definitions (P.V.A.). Thus an arrowed segment OY of such 2 Jine willbe calleda vector, X. One of these geodesics being chosen as the 7-line, the sum X+Y will be defined as the vector whose origin is O and whose end-point is the cross of the geodesics XT, and YT. Thus vector addition will be commutative. And since Desargues’ theorem is obviously valid for spherical tri iangles i in perspective +, the addition of * The easiest way to see this is to draw from O, the centre of x, a pencil of lines 7 and to cut it by any transversal s. Let A be the foot of © the normal ? drawn from O to s, and let a line Zand its corresponding (orthogonal) ZU’ cut sin Z and L’ , respectiv ely. Then, writing 2’ and —2 for the coordinates of ZL’, L with respect to A, we have wx '= —h?, by the generalized Pythagorean theorem. Whence also the correspondence l,l in the pencil of lines is seen to be an involution of the said kind. + As far as I can gather this property is not mentioned in text-books, but it is, none the less, a most immediate consequence of the well-known thecrem on perspective trihedra, The usual theorem of Desargues is obtained by taking of the space figure a plane section. In order to obtain the above theorem, it is enough to intersect it with our spherical surface. 142) Further Contributions to Non-Metrical Vector Algebra. these vectors will also be associative (P.V.A., p. 8), that is to say, (K+Y)+Z=X+4(¥+4+Z). ‘Thus all addition rules will continue to hold, together with their consequences. among which is the construction of the staudtian scale, and so on. Again, “conics” can be generated on o, as loci of crosses of correlated lines of two projective pencils, say by the process R=([X+Y |i, as explained in P.V.A., Section 15. One such closed conic « can be drawn round O and used as a standard conic, and the whole line of reasoning given above can be literally repeated, leading to the definition of orthogonal lines and thence to the concept of the scalar product of two vectors. This product will again obey the distributive law so that, for instance, the generalized theorem of Pythagoras, C?=A?+ B?, will continue to hold for “ right-angled”’ spherical triangles, and so on. In short, we shall have an ordinary vector -algebra, or a parabolic measurement system, set up upon the contemplated portion of the spherical surface. On the same surface we can also set up a hyperbolic system (by taking one of the said real ‘‘ conics” as Cayley’s ‘‘ absolute” and adopting his definition of distance and of angle) or an elliptic system of measurement, which latter will simply be the usual spherical trigonometry. In the last two cases we are deprived of the facilities of an associative and distributive vector algebra, while in the first case we have a vector algebra consisting of addition and scalar multiplication formally identical with ordinary vector algebra. The same remarks and explanations apply, mutatis mutandis, to the case of three-dimensional space, where also the vector product of vectors comes to its rights. 9. Supplementary Constructions to Section 5.—It has seemed advisable to add a few words about the constructions involved in what was treated in Section 5, and in particular on p. 132. A. (riven an angle «= (a,b) =a0b, say with the centre of « as vertex, to construct an angle a' =a with any given O' as vertex and | as one of its sides, coplanar with a, b. Let 7; be the terminus of 1. Draw the join Tia cutting « -again in c. Double the angle « in the explained way, making aOd=2«. Join cd cutting the J-line in 7’. Then O'T" will be the required second side of a’ =a. In fact, the angle acd is 2 (3) =a; the termini of the sides of this angle -are 7,7’: hence, 7,0! T'=a' =a. Klectromagnetic Waves. 143 An explanatory figure is scar recessary. roo A planatory figure is scarcely ne ry. The f just given is based upon the lemma: All angles 8 subtended by the are ad of the “central” angle aQd and having their vertices on « are equal to 4aQd. The proof of this lemma, a generalization of the familiar theorem on peripheral angles, -offers no difficulties, and may be left to the reader as a simple exercise in vector algebra. The generalization of the above construction to non-coplanar «’,a is again recommended to the reader. B. Any vector R=OR being given, construct upon a given ray OT, the vector S= OS, whose tensor rs equal to that of R, in symbols, ‘$|=(|R/, or in (generalized) familiar language, construct a segment OS congruent to OR. The vector r= Or being the unit of R, and s=Os the unit of the required §, draw the join s7 up to its terminus '’. Then T7’R will cut O7,in the end-point S of the required vector. Thus also we can easily draw a conic cp, homologous to x, through any given point /. In fact, all rays centred at the point just called T’ will cut the r-line and the s-line in points MZ, NV, such that | ON| =| OJZ), or in other words, in points ‘, N lying on the same conic homologous with x. The extension to non-coinciding origins or centres offers no difficulty. London, March 12th, 1919, Research Dept., Adam Hilger, Ltd. X. Llectromagnetic Waves. By T. J. Va. Bromwicu, Sc.D., F_R.S* Introductory Summary. | eae following pages contain a general solution of the electromagnetic equations of wave-propagation : in §1 these formule are expressed in terms of general orthogonal coordinates, and in § 2 it is proved that these for mule are the most general possible—at any rate for spherical polar coordinates. It naturally follows that the solution of § 1 should include all known types of solution ; and this fact is confirmed by detailed examination. In § 3 the spherical polar solution * Communicated by the Author. 144 Dr. T. J. Va. Bromwich on is derived from $1; and it is shown that this solutiom coincides with a Cartesian form given by Prof. A. E. H. Love*. In particular in §4 the field of an oscillating electron is. determined and the rate of radiation of energy :. it is found that the analysis is much simpler and easier to grasp in terms of spheric al polar coordinates than in the usual Cartesian forms. A further restriction is introduced in $5, by supposing the waves to be simple harmonic with respect to the time ; the solutions so found (in spherical polar coordinates) are substantially the same as those given by Profs. J. W. Nicholson ¢ and G. Mie Tf. It is shown in § 6 Gan these spherical polar solutions are actually equivalent to the Cartesian solutions originally given by Prof. H. liamb §. Finally in § 7, it is proved that the solution of § 1 also. includes certain general solutions given (for problems with an axis of symmetry) by Hertz |] and FitzGerald 4; and further that it leads to Lord Rayleigh’s solution ** for waves. travelling along two long parallel conducting wires. The solution of §3 (in spherical polar coordinates) was. originally worked out in 1899 ; and has been found also by Prof. H! M. Macdonald. It was published as a question in Part Il. of the Mathematical Tripos, 1910; and has formed the basis of a paper on the scattering of plane waves by spheres, communicated to the Royal Society in 1916. Tt may be convenient to explain here the system of numbering the formule adopted in this paper: the principal point to be borne in mind is that the decimal notation is. followed. The figure before the decimal point indicates the section of the paper; and those following the point ir supposed to be arranged in order of magnitude as. decimal fractions. or instance, DAS OLS eae Had wily all refer to the third section (§ 3) and follow in the order indicated. * Phil trans As wo. 197, p. 10 (1901). + Phil. Mag. ser. 6, vol. xin. p. 259 (1907). } Annalen der Physik, ser. 4, vol. xxv. p. 382 (1908). § Proc. Lond. Math. Soe. ser. 1, vol. xiii. p. 51 (1881). | ‘ Electric Waves’ (English edition), p. 140. | ‘Scientific Writings,’ p. 122. ** Phil. Mag. ser. 5, vol. xliv. p. 199 (1897) ; ‘Scientific Papers,” vol. iv. p. 327. Electromagnetic Waves. 145 § 1. A general solution of the electromagnetic equations. Let &, 7, € denote a set of orthogonal curvilinear co- ordinates, arranged so that their directions of increase form a right-handed set of axes when taken in this order (or in any cyclical permutation): further let the element of length in terms of these coordinates be expressed in the form ds? = A?d#? + Bedn? + (2d&, where A, B, C are supposed positive, and are in general functions of 2 n, €. To specify the electric and magnetic forces we shall use electromagnetic units; further we write X, Y, Z for the components of the Slceme force, resolved along the directions of &, », € respectively, while «, 6B, y will denote the corre- sponding components of magnetic force. Then the fundamental equations of electromagnetism, in a medium of dielectric constant K, and of magnetic permeability , can be written as follows : Ora BC = 2 (oy -2 (88), (2) | Oa SF = 5 (As) 2G), ive au a eee ae iG AB FY, = Tone, and 0% _ f —BC (M) ae. 2 (A He (CZ), | Ot ce 0 | TABySt = 5p BY)— . GAEX.)) where c is the velocity of radiation. For these equations see for instance Macdonald’s ‘ Electric Waves,’ ch. vi. § 36. Or we may proceed directly as follows :— Apply Ampére’s circuital relation to the curvilinear rectangle on a surface §=const. bounded by the curves n=, n=n9, = Gis a Go. Phil. Mag. 8. 6. Vol. 38. No. 223. July 1919. L 146 Dr. T. J. Va. Bromwich on The line-integral of the magnetic force is readily seen to be Ns Ns GBs Piven [Cy], a (BB},,d— i (Crue: UBT Now 1» 3 [Cy], [Crh =| 2 Cran 2 Wh On and 27 a) , (BBl. (BB, = | (BB), , 8 Thus the line-integral becomes “Me 20 9 3 : , { Jin ~ ae Bp) f ana Using Ampére’s relation erga the line-integral of magnetic force with the total electric current through the rect- angle, and Maxwell’s hypothesis for displacement currents, it is clear that the last integral must be equal to Ne 2 i ( ee = —< BOdndé. And so the first of equations (E) is proved; the second and third follow by symmetry. Similarly equations (M) are deduced from Faraday’s circuital relation applied to the same rectangle. It will be proved in §2 below that, under certain con- ditions, the most general solutions of the equations (H) and (M) can be derived by superposition of two special solutions, which can be obtained by writing first «=0, and secondly X = 0. We proceed to find the first of these solutions, corre- sponding to a=0,; then the first equation in (M) gives Mere) oP ‘ BY=<, OZ=9,.--.- @y where P is an arbitrary function. J£ we substitute from equations (1°1) in the second and third equations in (I) we see that they become CA Oy'K OB Vigo AB io) or) eee B onl aar) = ae lm “eae agr) = +gz@8> es In order to proceed further with the solution we shall now restrict the coordinates by assuming that A is a function Electromagnetic Waves. 147 of € only ; and that B/C is a function of 7, € only. These restrictions are satisfied in all the applications which have been made hitherto. It may be noticed further that when A depends only on &, we may reduce our work by making a change of variable to & where d&’'=Adé; and so we may effectively put A=1, without real loss of generality. then, on making use of the facts that A=1 and that B/C is independent of & the last pair of equations (1°2) give eset B a O . 5 me es = —< (By) oe) da oe Sol a : ; : @lre2it) 2 a 2 acy amelye eile) ae! a2 | It is therefore possible to write mele C3) mee 0, KOU ere Beas ar) Hee e o> | am where P= ou | . OE j and U is still left arbitrary. If we substitute for Y, Z and 8, y from equations (1:1) and (1°3), the second id third of equations (M) become (making use of the fact that A= 1), 6 VOI Ps) oP 3 - 3, (48 37) = ge(8-32) > ar “28 | (1:31) eek oy . oor oO O*| Tan oe ie anloe ae on aaa Oe ) The equations (1°31) lead to the relation On Un kee) ie Den oil aon 4) Substitute next for 8, y from (1°3) in the first of equations (Hj), and we obtain the result: : oh Gola © (Bou BCX = Beh | a2 Oli Teh ce, a aa If we substitute for X from (1°4) in (1°41) we obtain a differential Sone for U: pK o'U _ OU, =e (5 ar 7 OFT BO Lon\B (1-41) +a Sey he (5) , L 2 148 Dr. T. J. Pa. Bromwich on We can now write out the complete solution of the first type in the form Oa cal) hi | an oe ee Oe oe EGS olor ace c a) ie pa tor Ss | Z beso, Kou | = 6 9£0¢ |" Bone" oye where U is an arbitrary solution of equation (1°5). Similarly, by starting from X=0, we obtain the complete solution of the second kind : x= 0; | ca= — '— 7 | y) [oe eee ee) | 1 a°v ie —Gaele xo) : o — Bocor i. OO Pia orl ae Boel eC O80 J where V is another arbitrary solution of equation (1°5). § 2. Discussion of the solution of § 1. In order to determine conditions under which we can assert that the most general solution of equations (E) and (M) can be deduced by superposing the two solutions (1:6) and (1°7), it will be necessary to show that all. the other components of force must vanish when X=0 and a=Q0. For then it is evident that all the components are uniquely determined by means of the two X and a. The proof given below refers specially to spherical polar coordinates, and to problems in which the whole of angular space is used. ‘The earlier part of the proof is arranged so that it can be applied to other types of orthogonal coordinates; but the details of the final reasoning need modification, and must be adapted in other cases so as to. suit the problem in hand*. . If we begin by writing X=0, a=0, it will be evident that th prelirninary analysis used in §1~is not affected: and we can express our conclusions in the same form (1°6), except that now 1 + * Tt may be useful to remind readers that the conclusion that all the other components of force must vanish when X=O, and a=0 is not necessarily true for al/ types of orthogonal coordinates; an example ee contrary is given by Lord Rayleigh’s solution found in §7 elow. Electromagnetic Waves. 149 U is to satisfy the two equations found by writing X=0 in (1:4) and (1°41), namely, | oe hy ene Te ON OEs kGuOe and ; * (55) e G57 )= 2 Namen 2) (22) On o6\C of Now consider the integral | i a) Low \2 2) = ee ln d ee tyben CHS) — (ee{( fe +(¢ S| {dn de (2°3) ap which | 1S taken over the rectangle on the surface §=const. bounded by the curves n=), yn. and (=¢), 9. Using equation (2-2) it will be seen that the integral (2: 3) can. be written Ber S eset) ae os | -{" Lee ane a nt eel \ (Lo @az].- Ler a . a. 5 281) It is now convenient to introduce the special features of spherical polar coordinates, where we have : == io n = 6, (= ~, so that * A\ == Jb, Bei 0) eae sin (2), We shall take the area of integration for the integral I to be the whole of a sphere r=const. Thus the limits for ¢ (or @) will be €,=0 and ¢,=27; and since U and its differential coefficients must be single-valued functions of position in space, it is evident that now [gu all =[5u$ oy Thus (2°31) becomes 20 : aU aU ne { | sin 0 U! a —| sin ou il, fa, | (2°32) Now, in order to include the whole surface of the sphere, we must make m->9 and y.->7; and both U and its differential coeflicients minsi) remain wanitermat’ these limits, otherwise our Sr ~ $$ * It may be noted that these alunite do aie the hypotheses used in (1:21) that A=1, and that B/C is independent of € (=r, here). 150 Dr. T. J. Va. Bromwieh on solutions would not be valid in ali angular space. Thus, on making 7,—>0 and 7 .—>z7, it is evident that both terms in equation (2°32) will tend to zero: and accordingly the value of I as zero. Returnivg now to the form (2°3), we see that since B and C are positive =—_ = 6) — ) ° a . . . st 2°4 On 9 ( ) at all points: and accordingly U is wdependent of n, € (or of 9, ¢). It now follows from equations (1:6) that Y=0, Z=0, B=0, y=9: and accordingly all the components of foree are zero, as already stated. It may be noted that if the solutions do not refer to all angular space (so that the arguments used above may no longer apply), we can often draw the same conclusion, by using the boundary-conditions laid down in the special problem, to prove that (2°31) is zero. § 3. Special form of solution in terms of spherical polar coordinates. Taking ci wee 0,» == . n(n+1) | z =a { z es =e (3°51) where , is a solid harmonic of order x; while the radial magnetic force is zero. Thus, on comparison with (3°5) we see that Love’s radial components of force agree aa those found above ; and hence, by the general theorem proved in § 2, it follows that all the other components of the fields must also agree. A direct verification of the agreement is not difficult ; but the details of the work are somewhat tedious. Sone special examples will be found in §§ 4, 6 below. §4. The field of an electron oscillating in a known manner along the axis of z, with the origin as its mean position. We take this problem as an example of the general formule of §3; then we write n= eG = cos0, ee and also jo Ue iso that esa Thus equation (3°49) gives i) OB ieaent) \ ae 9 Fy = <4 , (ie oe fe . (4°2) where the solution is restricted to divergent waves. On substituting from (4°1) and (4:2) in (3°5) we find 9 >) X= te) cos 0, nec — 0) | ve 2 (f+0f'+ rf") sin 8, bee =, > (4:3) i; en L =. | Oy = +1f") sin @. On considering the character of (X, Y, Z) in (4:3) for small values of 7 the field is readily recognized as due to an electric doublet of strength /(ct); and this may be interpreted as due to an oscillating electron in the usual way. * Phil. Trans. Roy. Soc. A, vol. 197. p. 10 (1901); or Proc, Lond. Math. Soe. ser. 2, vol. ii. p. 92 (a9 04). Electromagnetic Waves. 155 To compare the results found here with those previously given by Prof. Love * it will be convenient to resolve first the components (X, Y, Z) perpendicular to and along the axis 82=(: we obtain the values | si oy Xsin6+ Y cos @ = i ee i OW ia ve | ‘ * Se, es 8G (4-4) X cos9#—Y sin 6 = — 3 Ciao + ty )+ os (f+ rf’), | Finally, to obtain the Cartesian components, multiply the first line of (44) by cos ¢ and sing; then writing #—rsin@cos¢?, y=rsindsingd, z= rcosé, we obtain the three components of electric force as : Bae ! 2 ee) 5 (37+ 3rf" +rf"’), = (3ft+ 3rf! + pape). | — (a? +7’) a, 2 ee) LO ap Bef +P 4 FET, | es J agreeing with Prof. Love’s formule. The Cartesian components of the magnetic force are found by multiplying the last formula of (4°3) by (—sin@, cos, 0) ; thus they are given by ee ee ee esc) As afurther example, we can readily calculate the radiation of ‘energy in the field (4:3) ; for the Poynting vector has the radial component ipl, isk | Se on Ant VOLCTINEr! " ig (1-28) Tee hh see MUR air Ds Integrating over a sphere of radius 7 we obtain for the whole rate of radiation 2 ! TES 7 lai Peas time ts Oye setts ) Zea: iL 5 nI\s = Pt ge SU tO ED which agrees with known formule. * Phil. Trans. Roy. Soc. A, vol. 197. p. 12 (1901), see formule (28) ; other equivalent formule were given by Drude, Physik des Aethers, 1894, p.415 (102). Drude’s solution is found by writing U=f(ct—r)/r, in Hertz’s general solution given in (7:1) below. 156 Dr, T. J. I’a. Bromwich on It will be recognized at once that the calculation of (4°7) from the equivalent Cartesian formule (4°5) and (4°6) would be much longer, although it must lead to the same formula finally. It is also more troublesome to form a mental picture of the field from the Cartesian formule than from those given in (4°3) in terms of spherical polar coordinates. $5. Special case of simple harmonic waves. If we assume that the waves are simple harmonic with a wave-length equal to 27/« in free space, we can suppose that ¢ occurs only in a time-factor e ; as usual we suppose that in the final results the real (or the imaginary) parts of the formule are retained. Then the functions f, g in (3°49) above will be exponentials of the types si = eei(cit—11) , a erxulcitt) t R- where KiGl— KG, OL. el == Kn (py f ee Thus, if we now suppress the time-factor for brevity, we see that the functions I, are expressed by the formule j= al — : 2) (— ) divergent waves, : \ (5:2) ee cae ange e ) | an fa) internal problems. J r Or ; In place of using the general formule (5°2) it is often more convenient to write F, = Eyer) or 8,07), . ... eee where the functions E,, S, are defined by * ad NieF | EE, (2) = aoe as ee | ae I ah Ny Sih B= 2'( 2 jae Some of the properties of these juncticns will be given below, as well as their connexion with the functions used in other solutions of these problems. (5-22) * Tables of the functions S,(z), Cx(z), En(z)=Ca(z)rSn(z) have been calculated trom z=] to z=10 by Mr. A. T. Doodson, and will be found in the British Association Report for 1914. Electromagnetic Waves. 157 The formule (3°5) then become opal ae 1) S.(«yr) Y All (0, d), Ca = 0, ah LDS. OY, Se ON, | a ae Or ° 06’ co) — i Se ae Oo’ ( 3) Z ») 1 OSn il OY» | yee eK, OY, } Macon) sind Oo, Ue Lea. where the time-factor e“* is suppressed, and Y,(0,) is a surface-harmonic of order x. The solution for divergent waves is obtained by writing H,(«,7) in place of Sn(«,r). The correlated solution, found by using V instead of U in the solutions of § 3, is X = 0, cas ae” S.Cer) Yn, 6), } Meee eck OYn ih et ON: ee sin bod. | OS) a Ree 00’ 7 Os) Bis UK LL a OYn att 1 OSn Ik ONG, 6 ee 00’ | oY > Or “sin @ 0d As explained in § 2, the most general solution containing ext as a time-factor is obtained by superposing solutions of the types (5°3) and (5°4). Solutions equivalent to (5:3) and (54) but differing somewhat in notation and arrangement have been given by Profs. J. W. Nicholson * and G. “Mie t. Summary of properties of the functions S,(z), E (Z). By term-by-term differentiation we see that (5:22) gives i Husa sin z Oe at i) ae 2 ot = T5506 OaeD) |! Het) + FA BMT) Hence we can write also aay aes \ » + (85) Sa) = ye (F) re Be et fess in terms of Bessel’s functions f. * Phil. Mag. ser. 6, vol. xiii. p. 259 (1907). + Annalen “der Physik, ser. 4, vol. xxv. p. 3882 (1908). { It is therefore easy to see ahead S,(z) is the function denoted by x in Prof. H. M. Macdonald’s paper (Phil. Trans. A, vol. 210, 1910, p. 115). 158 Dr. T. J. Va. Bromwich on The function u,(z), used by Lamb*, is connected with §,(z) by the equation \ Shei 2° (2), >. a oe ee as may be seen by comparing the series for W,(z) with (5:5). Similarly we have ie E@) =e (-; ss ‘ VG a) = Cz ere . = = : - = (5:6) where ee Se (Se PAY) == 2n+1 ee ty tS \ Gee aise: C,,(z) ( = PE - oe ) Ny i 8. Cn 2 ye i zl ~ L280 2a yi 2. 4(1 —2n)(3 —2n) = } sea In terms of the modified Bessel function K,, we find thatt E,(2) = a/ (=) ent emK aa a(1Z). ine (5°62) For divergent waves Lamb uses a function f,{2) which is connected with E,,(z) by the equation BZ) ]2"1f,(2). ... . . 2 Gia He writes further Inl2) = Vn(z)— hrlZ), C, (eS 24 Iw (Z). 2 3) eee so that Difference relations amongst the functions Sn, Cn, En. These relations are the same for all three functions, but will be stated for the S,-function only. From (5:5) we have Sati2) = ng 1 ao re se Sn(z)—-5, (Zz). (8°7) gn+l Again, it will be seen that (Ltr = eater and if this result is applied to the series in (5’5) we find that l ones ae) Sn(Z) = Sn—1(2) or Siz) = : S,(z)+8,'@)s « . - Gaal * ¢ Hydrodynamics,’ 1906, Art. 287. + And thus E,(z)=v—vz in the notation of Macdonald’s paper quoted above, so that s=Cp(z). . Electromagnetic Waves. 159 Combining (5°7) and Cr 71) we find the results : 2n+ 1 3. :@)eseue)— “21 S,)p (n4-1)8y-1(2)—Sn4i(z) = (2n+1)8,'(2). The functions S,, C,, En are solutions of the differential equation (5 sie 2) Su(e )+8,(z) = 0 az 2 aes found by writing n—1 for m in equation (5°7), and then eliminating S,_; from (5°71). This equation may be written in the form cree 1 ee Si 00 G73) dz z which can be deduced directly from (3-42). Since H,, is a second solution of (5°73), we see that E,(z)Sn(z) — E, (2)S2(Z) = const., and supposing z to be small, we may replace S,,(z), E,(z) by the first terms of the formule (5°5) and (5°61). Thus the value of the constant is seen to be unity, so that EGS Ean) La eee (Os) Again, by considering the form of (5-6), it is easy to see that we can write Ba) =e" (142. sp otto +25) and by substituting the last expression in the differential equation (5°73) it will be found that he ae), My Se ree eS a | one These lead to the values 1) (n—1).. (nh 2) (n—2)...(n+3) 2 i a oe ea SD 2 2.4 a eG oes 1 ae Semel 247.0 Vern ; (575)* and it easy to confirm the value of A, by direct differentiation of (5°6 * The values (5°75) agree with those given by Sir George Stokes, quoted in (3:492) above. 160 Dr. T. J. ’'a. Bromwich on § 6. Comparison of the formule of § 5 with those given by Lamb. Prof. H. Lamb’s solutions * are given in Cartesian form, and have formed the starting-point for a large number of investigations on electromagnetic waves, in spaces bounded by spheres. In view of the general proof of § 2 that the solutions (31), (3°2) include all solutions suitable for spherical problems, it is evident that Lamb’s solutions must be included as well as others. Buta direct proof is not without interest, as the complete verification is less obvious than might perhaps be expected ; the work depends on the difference-relations, (5°7)-(5°72), obtained above for S,(z). Lamb’s solutions are divided into two classes, corresponding to the two functions U, V employed in equations (3°1) and (3:2). All these solutions contain the simple harmonic time-factor e'*, so that they can be compared directly with those given in (5:3) and (5:4). For simplicity of statement, let us take K=1, p=1, so that «=k. Lamb’s solution of the “ first class” has a Cartesian component (parallel to «), Sih OX NL OW 5 een dalor) (y Kt eX) a = =F =) ; (6:1) where W = rdaler)xn- Here yn is a solid harmonic of order n, and so is capable of being regarded as a multiple of 7”Y,(0. @): thus we may write W = AS, (er) ¥.(0, 9)- Now the vector of which (6*1) is the x-component is the vector product of the two vectors TG eB ow OW OW ¢ L 9 =) and eC 37 ’ Oz } >) See (6 11) Translating to spherical polar coordinates, the two vectors (6-11) become ow Vow) I) some a Ge (ale 0: 0) and ( Or’ r 06) rsing O¢ : * Originally given in Proc. Lond. Math. Soc. vol. xiii. p. 51 (1881) ; here we shall quote results given in Art. 335 of Lamb’s ‘ Hydro- dynamics’ (1906), where other applications of the formule are referred to. It should perhaps be noted that the function denoted by #,(z) has a different numerical factor in Lamb’s earlier work; thus the formule corresponding to (63) are somewhat different in appearance in the earlier papers. Electromagnetic Waves. 161 Hence (6:1) is equivalent to the vector with spherical polar components OW) | > r+ sin @ 00’ Ts 30 J Tf we identify the vector (6-2) with electric force* we see that Lamb’s solution is the same as (5:4) if W = uk Sn(xr)Yn(0, >), which agrees with the form quoted in (6:1) above if A=«. Lamb’s solution of the “second class” has an «-component of electric force equal to . (62) (n str 1) Wn- 1(k?’) fe = Oeics (in aaa S ( oe ) ¢ (6:3) x pent where ¢, is a solid harmonic of order n. The expressions (6°3) can be transformed at once by simply : : OV LS MeOH N Cis using the spherical polar operators le AG? anne =) cs place of (< 2 2) oz’ Oy Oz) Then, since gn 18 of the form Br" Y,,(0, ¢), we have Obn ce gn fe) ( Dn ) ale _ (n+1 on (OE NG ae or ia , pond pen+2 y and so (6°3) gives the radial component of electric force - nnt+l aa —- via § p,a(«r) + (er) dn galer)t bp mn+l1) cg ' = ae : Dekh) ails Sn ii(xr) bon Gap Ae 1) Boi , (xr)? +7 by using (5°52) and (5°72) above. The formula (6°4) agrees with (5°3) if we write Coe = Ne eva om tare) (6-41) which is of the appropriate form since ¢, is a solid harmonic of order 2. Similarly the 6-component of electric force is equal to Sn(er ons More Ce Unie fe (6-4) ss (n+ 1) dna (wr) — SO — Yn +1 (Kr) ee ce 1 6 /,, \ ~ ) 1 0 a — eal ) os 1) Sp—1(er) —2S8p 41 (Kr) ¢ 30 Ww lt : ASnx Oon im (year xn eve again using (5°52) and (5'72). (6°42) * To identify with magnetic force bas merely tle effect of inter- changing the functions U, V of our general solutions (3'1), (3°2), Phil. Mag. 8. 6. Vol. 38. No. 223. July 1919. M - : -——-— 162 Dr. T. J. a. Bromwich on The form (6°42) agrees with (5°3) in consequence of the relation (6°41) already found ; this would be anticipated from the general theorem of §2, that the radial component of force suftices to determine the other components. Similarly we deal with the g-component of electric force. 5 $7. Comparison of the general formule of § 1 with other known solutions. Before concluding it may be of interest to note that the formule (1:6) and (1:7) include certain other known solutions of very general character. The first group of solutions is obtained by using cylindrical polar coordinates p, , ¢, such that ds* = dp?+ p*dd’+ dz’, taking Bye 9, (AC ie= AL We make, however, a cyclical interchange of coordinates , so that z corresponds to the & of our general formule of § 1*; then the solutions (1°6) can be written in the forms :— toe UE | EO Ese BOE. fee ee ole ot J1 2 hue. 0 (ee ) TPL ODIOE: Te Oorce =) cee _ BU pK 3°U o es Pe ee | da J where U is any solution of the equation AOU TBE) LIU 2 Pat ag Tae pop Op Oe oe? 92° . (‘4 2) Similarly the solutions (1°7) lead to the ie a peed YO. (ye ON) Ae ee ean Hie r- 1308) |=] nO ON dees Uae 3) es aaa ¢ (73) ooh om, DN pal OV | O22 re C2 PY E) J where V is also a solution of equation (7:2). It may be noted that here equation (7:2) reduces to the * It is therefore necessary to see that C=1, and that A/B is inde- pendent of z: both of these conditions are satisfied here. Electromagnetic Waves. : 163 ordinary wave-equation expressed in terms of cylindrical coordinates. In the case of symmetry about the axis of z, the solutions (7-1) reduce to a solution given by Hertz*; the function which Hertz calls II being in fact the same as the function —U. It will be remembered that U is now independent of d, and consequently that Y=0, «:=0. Similarly, still assuming symmetry about the axis of z, the ‘solutions (7°3) represent the field of FitzGerald’s magnetic oscillator + ; here V is independent of ¢ and X=0, B=0. The solutions (1°6), (1°7) include also a solution given by Lord Rayleigh { and applied by him to the propagation of electromagnetic waves in the space outside perfect con- ductors ; we present the formulee here in the form given by Macdonald §. We take & 7 as conjugate coordinates in ‘the plane of wy, and identify € with <, (making as above a -cyclical interchange of coordinates). Then we have A= B= SJ, Ci 1, ‘because ds? = J*(d& + dn’) + dz’. Lord Rayleigh’s solution is found by assuming that LO, ey), -so that the electric and magnetic forces both lie in the ~wave-fronts. Then we find the formule corresponding to (1°6) : Bek hs.) yea | feces ‘i Pade Omen oe | 7, eds i eee ete Y=5 9532 ‘O= —y seh an) | 4— 0, y = 0, 3 where U is now a solution otf the two equations : 2 2 2 2 cad HES On 9, ind StS = 0, (7°5) ae OF OF ‘Lord Rayleigh’s actual solutions are given e taking w=1, Be and Ue h (Ey). * ¢Blectric Waves’ (English edition), p. 140. + ‘Scientific Writings,’ p. 122: it will be seen that FitzGerald does not ‘give the explicit formule (7:3) in this paper, although he obtains the corresponding formula for the rate of radiation of energy which can be -derived from (7'3) in the same way as (4°7). Ina later paper (J. ¢. p. 418) the paralleiism of the fields (7°1) and (7°38) is pointed out. -4 Phil. Mag. xliv. P; 199 (1897) 5 Scientific Papers, vol. iv. p. 327. RS ‘Blectrie Waves,’ ch. vil. § 43. M 2 164 Prof. Barton and Miss Browning on the Resonance Of course there is a similar group of formule corre- sponding to (17); but it seems hardly necessary to write them out, as the formule (7:4) correspond most closely to the problems of physical importance. The formule (7:4) given above agree with Macdonald’s. on writing but it should be observed that his J is the reciprocal of ours. Attention may be directed to the fact that in the formule: (7:4), the vanishing of Z and y does not involve the vanishing of the other components of force, in contrast to what was. proved in § 2 for the case of spherical polar coordinates. XI. Lhe Resonance Theory of Audition subjected to Expe- riments. By W.H. Barton, F.R.S., and H. M. Brownine,. M.Sc.* [Plates IT. & IIL. NHEORIES of audition have been recently under consi- derable discussion}: indeed this controversial subject appears to be of perennial interest. Helmholtz long ago. advanced his hypethesis of sympathetic resonance, and supposed at first that the réle of resonator was played by each of the arches of Corti. This latter detail was afterwards. modified, the basilar membrane being then considered to act somewhat like a set of resonators or harp strings. It was. shown that this was possible owing to its fibrous nature with high lateral tension and retative slackness longitudinally. There are of course difficulties in the hypothesis in matters. of detail. But this is only to be expected in the case of so small a mechanism working at such high frequencies and with such minute displacements. Indeed, there are difficulties in any hypothesis, and that of sympathetic resonance seems: to deserve careful examination froma new standpoint. Some anatomists have felt considerable difficulty in accepting it, ° others accept it unreservedly. Some of its critics have obviously not quite grasped the meaning of the hypothesis, * Communicated by the Authors. + ‘ Analytical Mechanism of the Internal Ear’: Sir T. Wrightson and Dr. A, Keith. London, 1918. ‘‘The Internal Ear,” Nature, Aug. 8, 1918. Letters in ‘Nature,’ Oct. 17 & 31, Nov. 7 & 21, Dec. 5 &19, 1918, Jan. 9, 1919. “On Sir T. Wrightson’s Theory of Hearing,” by c W. B. Morton, Phys. Soc. Proc. xxxi. Part III. April 1919. Theory of Audition subjected to Huperiments. 165 and so unfortunately base upon their mistaken view a eriticism which falls wide of its mark. : The term “resonance” used in the present connexion is open to misunderstanding. In the minds of some, it recalls simply the familiar case of the actual resounding tor several seconds by a lamp-shade of a musical sound originally due to the voice or piano. ‘Taken in this crude sense of the actual reproduction of a sound probably no one, competent to judge, has ever believed in a resonance theory of hearing. But the essential facts of the hypothesis are present in the case just referred to. The lamp-shade has a certain period of vibration natural to it. But when practically the same note is sung, the very feeble vibrations cf the air reaching it, being of the right period and repeated hundreds of times, elicit a powerful response. If the periods had been utterly different instead of nearly alike, the response would have been unnoticeable instead of arresting. Hence the sufficiently powerful vibrational response of an elastic system to very weak forces, owing to the almost exact tuning between the period of the forces and that of the responder, is of the essence of the theory of sympathetic resonance, whether that responder makes any sound or not. Perhaps sympathetic response would have described more precisely what is intended by the commoner phrase sympa- thetic resonance. Further, it must be borne in mind that the degree of falling off in response amoung resonators, owing to their mistuning with the forces impressed upon them, depends upon the damping of their own natural vibrations. ‘Thus, theory shows that the nore highly the vibrations natural to a responder are damped, the less is the falling off of their response owing to a mistuning of the impressed forces. In other words, the response is more widely spread among a graduated set of responders when they are highly damped, but is more con- centrated when the responders are but slightly damped. This may be clearly seen by reference to the plates of previous papers”. Hypotheses of audition may be approached in a variety of ways. Perhaps the most natural basis is that of dissection and microscopic examination of the anatomical structure of the ear. These investigations have been carried out by a number of workers, among whom it may suffice to mention * See figs. 1. 2, & 3, Plate VIIL. Forced Vibrations, &c., Phil. Mag. August 1918. Plate V. Mechanical “ Resonators,’ &c., Phil. Mag. April 1919. Eee ES —_—— 166 ~=Prof. Barton and Miss Browning on the Resonance Bowman, Corti, Deiters, Hasse, Henle, Hensen, Kolliker, Kuile, Reissner, Retzius, Schultze. But it is not sufficient to regard the internal ear as a structure merely. It must be recognized that it is a working mechanism. Turther, it must not be looked upon as a mechanism capable only of slow displacements. For it is of the very essence of this mechanism that it is movable at acoustic frequencies, and highly susceptible to very feeble forces provided they alternate at any such frequency. The question may be asked here, Is it easy to imagine these- mechanisms responding to such feeble forces as are usually present except on the principle of forced vibrations? Further, the presence of a graduation in these mechanisms suggests that they are elastic systems with natural periods of vibration which form a series according to their dimensions and other conditions. Without at all prejudging the case for or against the resonance hypothesis, it is allowable to consider what are the chief facts of audition, and whether they are explicable on the resonance theory. If they appear to be so, we may further ask what number and disposition of responders are: needed. Since the mathematical theory of forced vibrations remains. essentially unchanged for a great variety in the forms of the vibrators, we may reduce the probiem to its simplest terms by arranging a set of simple pendulums of graduated periods. to represent these vibrators. Then their behaviour may be compared with the facts of audition and the agreement or conflict noted. Any conflict, if observed, would seriously discredit the resonance hypothesis. On the other hand, any agreement that may be observed will essentially support the. hypothesis in general terms, and might conceivably give some clue as to which parts of the ear could act as responders.. For the facts of audition might be reproducible only by a certain number of responders with given frequencies and dampings, and the properties requisite might be possible to- certain anatomical structures only. The anatomical method of studying the subject would: probably be best of all could it be carried out in its entirety ona living subject. But as this is impossible and the alter- native post mortems are in some respects inconclusive, we- seem justified in taking any indirect method of approach that is available. Hence the method of using a set of pendulums, though they are confessedly unlike any structure: in the ear, may throw a valuable side-light on the subject by revealing and displaying what number and arrangement of Theory of Audition subjected to Experiments. 167 responders would prove adequate to account for the known facts of the case. It would appear from the experiments thus made and their accompanying photographic records, that about twelve re- sponders to the octave or a total of about one hundred in all, if of suitable damping, would probably suffice to account for some of the chief facts of audition. And this number is only about one thirtieth of the number of Corti arches present in the human ear. Accordingly on this view the resonance theory would not make the demand for so large a number of structures with separate nerves as its adherents have usually supposed. On the contrary, it leaves a liberal allowance for the possibility of the number of nerve- fibres being much smaller than the number of Corti arches. So that if a single nerve-fibre is distributed to a number of arch segments (as found by Held), this would not necessarily invalidate the resonance theory. Fundamental Facts of Audition—We may now review some of the basai facts of audition, so as to set up a standard such that the success or failure of the resonance theory to account for these facts would afford a confirmation or disproof of the hypothesis. For those whose hearing is norma] the following may be taken as fairly representative of the fundamental facts with which we are now concerned. 1. When two different notes at a considerable interval are sounded together, we can hear both notes and estimate their interval, but do not mistake them for a single note of inter- mediate pitch. Thus C and G sounded together are recognized us forming the interval of the fifth and are not mistaken for a single note of pitch E or Ep. (This is the direct contrary of the case with colour vision in some parts of the spectrum. Hor, when beams of red and green light are converged on to tire same white screen, the impression received is that of yellow, und the unassisted eye furnishes no hint of the dual nature of tuis composite light which might be a monochromatic yellow for aught we are able to perceive.) 2. When two near notes are sounded successively the small interval between them can be perceived by a specially keen ear down to something of the order of two vibra- tions in a thousand or one twentieth of an equal-tempered semitone. 3. When two very near notes of almost equal intensities are sounded simultaneously, the difference of their frequen- cies can be recognized by anyone as the number of beats per second. And this may serve to discriminate an interval 168 Prof. Barton and Miss Browning on the Resonance of say one vibration ina thousand, or about the fortieth of an equal-tempered semitone. 4, The range of audition is limited at both ends, each limit varying iin the individual, but about eleven acres are usually audible. 5. Before either limit of audition is absolutely reached, the note is recognized to be very high or very low, but the power of distinct location of pitch is lost, only about seven octaves being musically audible. 6. A musical shake of about ten notes per second on a tone of frequency a hundred and ten per second is heard quite distinctly. Variables of Responding System.—Ilf a set of vibrating responders is provisionally postulated as existing in the ear, and if in order to test the validity of the postulate we are té make a working model on this principle, it is evident that many variables are at our disposal, and probably the success of the model in accounting for the actual facts of audition, should that prove possible, will depend somewhat upon the right choice of these variables. The chief variables in question may be stated as follows :— (a) The total range of responders. (6) The musical intervals between adjacent responders. (c) The damping natural to these responders. (d) The constancy or otherwise of the intervals and of the damping throughout the range. We must also suppose that (e) acertain order of discrimination of relative amplitudes of vibrations of different responders is possible by means of the nerves attached to them. Obviously these variables must be chosen so as to accord as far as possible with the facts of audition previously enumerated. Thus the facts under headings 1, 2, and 3 -give some elue to the smallness of the interval between adjacent responders and also show that the damping must not be too large. Otherwise the sharpness of resonance would not be great enough to facilitate location of pitch. The facts of limited range of audition and loss of exact sense of pitch near ends (headings 4 and 5) show that the range of responders should extend to about seven octaves. Biom dhe cece tach astto the vlearuccaioure shake, Helmholtz conciuded that the expiring tone is reduced to one tenth of its original amount in the fifth of a second. If therefore the ear has vibrational responders, their natural damping at this pitch must correspond to a logarithmic decrement of the order 7~7=0°:06. This shows that the damping must not be too small. Theory of Audition subjected to Experiments. 169 We have thus obtained some light on (a), (d), and (c), but not upon (d), which would require more refined examination. Further, it should be pointed out that these facts of audition do not lead us to any exact determination of the variables at our disposal in the set of responders. On the contrary, they suggest values of a certain order of magnitude, or furnish us with approximate upper and lower limits. Thus it would be sometimes possible to account for the facts of the case with certain values of the variables or to account for them equally well by changing one variable, some other being adjusted in compensation. For example, the less the damping of the responders, the sharper is the resonance and the easier would it be to locate the pitch of maximum response. Buta greater damping of the responders, leading to less sharp- ness of resonance, could be compensated by an enhanced discrimination of relative amplitudes of responding vibrations near the maximum. Experimental Arrangements. — No attempt has been made to set up the whole seven octaves of responders postulated, but only a single representative octave, which suffices for experimental tests. In order to have a definite and constant interval between adjacent responders throughout the octave, they were set at distances from one end, and adjusted to lengths, which formed a geometrical progression. And it seemed desirable to make the intervals correspond musically to those of the tempered chromatic scale. Thus the ratio of adjacent pendulum lengths was v2, the ratio of periods being accordingly 4/2. Thus the thirteen res- ponders for the one octave may be referred to by the letters ixperimental Arrangement. used for the notes of the scale with sharps and flats where required. Hight of these are indicated in the figure by C, D, E, F, G, A, B, C, the five corresponding to the sharps and flats being left unlettered. All these responding pendulums have bobs in the form of 170 ~—sOW Prof. Barton and Miss Browning on the Resonance paper cones and weighted with a ring of copper wire (like those used in Mechanical “ Resonators,” Phil. Mag. April 1919). These hang by suspensions of black thread from the stout cord HJI, to which is attached the driving pendulum JK whose true length must be reckoned from J’. Itis adjustable to various required lengths by a tightener as shown. A second cord is shown by HML, from which is suspended a second driving pendulum MN (of virtual length M’N) and whose bob N is equal in mass to the bob K. These two cords are connected by the wooden bridge T, thus the two driving pendulums are only loosely coupled to each other but each acts quite distinctly upon the set of responders. The camera lens is along the line HJ produced so that in the photographs all the responding pendulums will seem to. hang from the same point. Further, the responding bobs all lie along a straight line QH in order that each responder shall experience the same driving influence (see “ Forced Vibrations,” p. 176, Phil. Mag. Aug. 1918). Results and their Significance.—Plate IL. gives six repro- ductions of time exposures of the responders actuated hy two drivers of widely differing periods, the slower of the two being gradually increased in length from figure to figure throughout the series. In fig. 1 it is obvious that the length of the driver is about midway between those of the responders whose bobs are third and fourth from the bottom. Or, in musical terms, we might say that the pitch of the driver was about midway between Ep and D, the Ep being slightly favoured. In the second figure we may, in like manner, refer to the pitch of the driver as being slightly nearer D than Eb, since the bob third from the bottom responds better than the fourth. In the third figure, the third bob from the bottom responds much better than any other, but the fourth shows a distinctly better response than the second. Hence the pitch of the driver is recognized as. distinctly sharper than D. In the fourth figure the driver is still slightly sharper than D, whereas in the fifth it is slightly flatter than D. In the sixth figure the driver is distinctly flatter than D. Hence, in five steps we have passed over about a quarter of a tone, giving an average interval of a twentieth of a tone, or ten logarithmic cents. It is evident by inspection of these figures that pitches. midway between the adjacent ones given would be discernible as differing one from the other. Thus between figures 1 and 2 with fourth bob favoured and third bob favoured we might have had another case with neither favoured. Hence, with nervous discrimination of relative amplitudes equivalent. / Theory of Audition subjected to Experiments. Lt to our perception of: these figures, it would be possible to discriminate between successive notes differing in pitch by only the twentieth of an equal-tempered semitone or five logarithmic cents. And this is just about what a good ear can accomplish. In the case of the six figures of Plate II. it should be noted that the shorter driver is allowed to swing the whole time, and in no way interferes with the discrimination of pitch of the longer one as seen throughout. And this is known to be the case with hearing when the other note is neither too near in pitch nor too loud. Hence the six figures of this Plate corroborate the facts of audition L and 2 in our list. Indeed, fact 1 was supported also by photographs 1-18 on plates V. and VI. of the paper on Mechanical ‘ Resonators,” &e. (Phil. Mag. April 1919). Further, photographs 14 and 15 of plate VI. in that paper showed by flash exposures the presence of beats, which obviously allow of a finer discrimination of pitch between simultaneous notes. And this constitutes our third fact of audition. As for fact 4 ofaudition, any finite set of responders would obviously accord with that experience. Photographs 7-12 (PI. ITI.) are devoted to the test of fact 5,. which is the failure to recognize with precision pitches lying near either limit of audition. Thus in figures 7-9 we test the upper limit on the supposition that our single octave represents the top of the whole set of responders in the ear. In fig. 7 the responder second from the top has maximum amplitude, in fig. 8 the driving pendulum has been shortened so that the shortest pendulum responds best. In fig. 9 the driving pendulum was shorter than any responder, but that fact can scarcely be inferred from inspection of the figure, so that the exact location of pitchislost. Indeed, as soon as the pitch of the driver passes beyond either limit of the pitches of the set of responders, so that no one bob exhibits a maximum vibration with a falling off above and below, the exact location of pitch must be lost. For the lower limit of this single representative octave of responders the gradual failure to locate the pitch is illustrated hy photographs 10-12. In fig. 10 the pitch is evidently between the lowest and the second, that is between what we have called Cand C#. In fig. 11 it is about at the lowest C, and in fig. 12 itis still lower, but by an amount which cannot be precisely inferred from the figure. It may be noted that although by a set of responders, the exact pitch is not detectable for notes near either limit, it is SSS SS tc i | + | i} 2 The Resonance Theory of Audition. yet clearly recognizable whether the note in question is near the upper or lower limit of the range. And this cor- responds with one of the facts (5) of audition as already pointed out. Referring to fact (6) of audition, the test corresponding to a musical shake was carried outasfollows. The two driving pendulums were set to what we may call the notes D and E (that is the bobs third and fifth from the bottom of the series). One driver was started while the other remained at rest, and in ashort time the maximum vibration of the corre- sponding bob was elicited, the other bobs exhibiting the ordered amplitudes and phases characteristic of forced vibrations. Then the first driver was stopped and the second started. After five or ten vibrations the new pitch was clearly established, as shown by the ordered state of things with the new maximum. So there is here quite sufficient damping to make clear shakes possible. And we have previcusly seen that the damping is small enough to give fairly sharp resonance and so render possible a fine discrimination of pitch. Summary and Conclusion. 1. The present position of the resonance theory of audition is reviewed. The subject is acknowledged to be controversial. But the endeavour is made to throw a side- light upon it by the trial of a graduated set of pendulums used as responders to other pendulums as drivers. In actual form these pen- dnlums make no pretensions to represent any structures to be found in the ear; but in their essential behaviour they do typify such mechanisms as are postulated for the ear by the resonance theory. ‘This typical representation may be pos- sible and useful, because we can apply to it the theory of forced vibrations in its most essential aspects. Six facts of audition are then recognized as fundamental. These include a much finer discrimination of pitch than one for each responder and the failure to locate pitches quite exactly when they lie near either limit of audition. 3. Twelve photographs of the responding )endulums in action are taken and here reproduced. 4. These experimental results nowhere conflict with the above six facts of audition. Indeed, for their explanation it suffices to have a set. of suitably damped responders and their associated nerves of . about twelve to the octave over a range of seven octaves, or _ say about a hundred in all. The supposed. necessity for a Vapour Pressure and Affinity of Isotopes. 173. much larger number of nerves and consequent conflict with some anatomical results are thus removed. 5. Any hypothesis like the resonance theory in question must be very difficult to prove to the hilt by any number of confirmatory experiments. But, if it is essentially at variance with facts, its disproof should be comparatively easy. Nottingham, March 20, 1919. XII. Note on the Vapour Pressure and Affinity of Isotopes. By F. A. Linpemann, Ph.D.* [T was shown in a recent paper tliat isotopes must be separable both by fractionation and by chemical means f.. At that time it was not possible to give any quantitative estimate of the differences in vapour pressure or affinity,. since data were lacking as to the physical properties. This omission may now be repaired, since the meiting-points of two sorts of lead of atomic weight 207:19 and 206°34 have been found to be identical within the limits of error of the experiment f. It has been shown that the melting-point may be regarded. as the temperature at which the average amplitude of oscil- lation of the atoms is equal to the distance between them §. This distance is obviously the distance between the centres of two neighbouring atoms minus the diameter of the atom which is presumably defined by some electronic orbit. Since the spectra of isotopes are practically identical, the outer elec- tronic orbits are presumably identical, so that the diameters of the atoms may be regarded as identical. oe me ounen hand, the densities are proportional to the : : From. which we may conclude that the retan nae owen ae centres of neighbouring atoms are identical. It follows from this that the amplitudes of oscillation at the melting- point are identical. As is well-known there are two alternative views which give the same formula for the atomic heat. According to the first, a linear oscillator of frequency v can only absorb or * Communicated by the Author. t+ F. A. Lindemann & F. W. Aston, Phil. Mag. xxxvu. p. 528 (1919). t T. W. Richards, ‘ Presidential Address to American Association for- the Advancement of Science ’ (1918). § F. A. Lindemann, Phys. Zeitschrift, xi. p. 609 (1910). | T. W. Richards, loc. evt. 174 Dr. F. A. Lindemann on the -emit whole quanta of energy. At temperature T its average energy is therefore hy ey cae ‘in the ordinary equation of state of a gas and N is the number of atoms in a gramme-atom. This is the view first put forward by Planck * and Hinstein fF. Planck’s modified hypothesis t allowed an oscillator to -absorb energy continuously, but restricted its emission to whole quanta. This theory, which is capable of much more consistent treatment, entails that each oscillator pos- hv sesses, on the average, half a quantum -5- over and above a! ‘the energy calculated above, 2. e. hy i ihe) x ( Byv z) nal Boy : 7 aioe = aoa e\ arse pie ta 1 2 se = N ev —] x In calculating the relative vapour pressures or affinities of ‘isotopes, these two pcints of view lead to somewhat different results. It will be convenient to consider the second hypo- thesis before the first, since the consequences are simpler. In lead, as presumably in all other elements, the melting- point, (T,, = 600°), is large compared with (@v = 95°), the characteristic temperature §. One may therefore expand V ‘ Vy ; ee — in terms of , and one finds for the energy of the 2 os an | m atomic oscillation at the melting-point an 8 17S ipa Pe (er) Boy (Tg CUT) to) ae 2 if i (iT) and the higher powers are neglected in com- parison with 1. * M. Planck, Vorieswngen tiber die Theorie der Wiérmestrahlung (1906). + A. Einstein, Ann d. Physik, (4) cexx. pp. 184 & 800 (1907). t M. Planck, Ann. d. Physik, (4) xxxi. p. 758 (1910), (4) p. 672 (1912); Sitzungsber. d. Akad. d. W. p. 723 (1911); and Verh. d. d. _Phys. Ges. xiii. p. 188 (1911). § F. A. Lindemann, Diss, Berlin (1911). Vapour Pressure and Affinity 07 Isotopes. 175 Now, if f is the restoring torce which keeps each atom e ® fL gis where / is the amplitude. At the melting-point therefore yells: ll avd ie and, since it was proved experimentally that T,, is identical and shown above that J,, must be identical, one may conclude that in isotopes the restoring force f is identical. in its position of equilibrium, the energy is obviously : A / : Since the frequency v= —— iene ap je te cece 21 m? of the atom, it follows from this that the frequency is inversely proportional to the square root of the atomic weight. Further, it is obvious that the compressibilities must be identical if the restoring force is identical, and it may be shown that it is unlikely that the coefficients of thermal expansion vary appreciably. As was shown in the paper referred to above *, it is impossible for the latent heats at the absolute zero A» to differ if the restoring force f is mathematically identical over a finite range. In the instance considered, the values of fare only equal to the second order, and this fact has not been tested over a wide range, so that the conclusion that Xo is identical is not quite binding. It may, however, be regarded as extremely probable. Now the vapour pressure p is given by UR ro ‘ ne ea) +f Rey, ees where C, is the atomic heat of the gas and c, that of the solid, whilst 2 is the chemical constant. Therefore unless the atomic heats of the gaseous isotopes differ, which is certainly not the case within a very wide range of tempera- ture, the difference of vapour pressure is given by ) T aT it r : , logh =| a. (Cp, — Cp, aT + (2; —1e)- 9a?Tv ——, where K The atomic heat at constant pressure c,=c,+ ¢y is the atomic heat at constant volume, v the atomic volume, a the expansion coefficient, and « the compressibility. v and « have been shown to be identical and & can only differ very * F. A, Lindemann & F, W. Aston, Phil. Mag. xxxvii. p. 528 (1919). 176 Dr. F. A. Lindemann on the slightly if at all in two isotopes, so that the difference of the - Gor Nae Yao? Top Ge Ge be neglected. One may therefore write correction terms must be small and may go - pMLAS ie al, (Co, — Cry) AT + (4113). It has been shown that the atomic heat may be repre- sented with remarkable accuracy by cn d 9RT "* da 2 Ges iy Goa ik? where p= ae as before, except that v,, is defined somewhat differently from v the frequency of the atom *. v,, is, however, very nearly equal to v and may almost certainly be identified with it. This equation naturally holds equally well if the hv 2? corresponding to continuous ab- sorbtion, is taken into account, for then ** Nullpunktsenergie ”’ d Haida _w\_ d 72RT (2 a mat n(- A) esa J=a ee y coth y dy Taking this value therefore ee Ho? de Ho da logs ==) ie js Jee ee) ea = | 75 (7; — te) Giadlpey keteen a v dy “1 a8 dx Sat ( at 7a ea pen lez 0) el a) ON 1 =- ae dpby — As 1) = (1; — tg) dw ( edz 9 mie Br =—9 —— ——_—— —~ _ 21 —Z25). 1 wae —1 ai (41 — He) + (4122) * P. Debye, Ann. d. Phys. xiv. p. 789 (1912}; W. Nernst & Fo AG Lindemann, Sitzungsber. d. Akad. d. W. L. 11, p. 1160. Vapour Pressure and Affinity of Isotopes. IAT At low temperatures y-is large and one may write Di ce 1 —— [= "(= oe - 2 ee ole pa? ie on n if Yew? QD \[e=F1 Taare pe aa. + 1 8 = (uae — fz) + (1 = tz), or if wis large enough 9 logh = (4H) + (AA). Since w=1 at 95° absolute the vapour pressures at these temperatures have little practical interest. At higher tem- peratures w is small and one must use another expansion. = B,, are the Bernouilli numbers as above, “oS "dp age freee \n—1 83 pent? =a" sheer —Ft 2 OU ee oF 8 ay — flo) + (% —1y) ey ee we mole =9{ fe s + 607 5040+ AR = (dH) + (iis) 1 nai =Slog + 7) (a? — #2") — go qp (Hr 2") + 4,9'0'5 + (244 - dg )- Whether this equation holds beyond the melting-point it is difficult to say, as the latent heat of fusion of the two isotopes might differ. A more serious matter is the value of the chemical constant, which has not yet been taken into account. A number of attempts * have been made within the last seven years to derive the chemical constant from the element of action h. Without entering into details a simple dimen- sional consideration enables one to predict its variation with the atomic weight, provided such a connexion exists. The dimensions of the chemical constant are those of the logarithm of a pressure. In a monatomic gas one may suppose that it only depends upon the mass m, Boltzmann’s constant k= N , Planck’s constant h, and some temperature @, . * QO. Sacker, Nernst Festschrift, p. 705 (1912); H. Tetrode, Anz. d. Phys. xxxviii. p. 787 (1912), Phil. Mag. 8. 6. Vol. 38. No. 223. July 1919. N 178 Dr. F. A. Lindemann on the probably the temperature at which, according to the quantum theory, the gas begins to deviate from the equation pu= RT. This temperature may of course depend upon the atomic dimensions, but so long as it is equal in both isotopes it does not affect the present problem. One may therefore write i= log m=2logm+ylogk+zlogh+ulog 0+ const. The dimensional ae are therefore given by (MLA (MLA: ia Sal cen (Cr 6 whence = 3/2, 0 — a2, 213, ands oy 2 ; 3/2(J.9) 5/2 1. e. e nN (= log ee +const.=const.+ log m3? i] ~ If A, and A, are the atomic weights of the two isotopes, therefore " 1) —1, = 3/2 le: Inserting this value in the equation found above, and A ‘emembering that HM (2 remembering tha em A that the first order terms cancel out and the vapour pressures Bv A,—Az, TE oe) AC Putting A;=A,(1+5), where 6 is small as it probably always is in isotopes, 1/2 ) , 1t is immediately seen only differ in the higher orders of log! = a In lead 6 is about 1 per cent. and ~=0°16 at the melting- point, so that the vapour pressures cannot be expected to differ by more than 1/50 of 1 per cent. This estimate of course only represents the order of magnitude since a number of second order terms were neglected in the course of the argument. A much larger difference, about 1 per cent., would occur if the chemical constants were identical. A very similar consideration shows that the affinities, which may be measured by the constant of the law of mass action or by the electromotive force, differ very little. In the first case po. idl (ee | log K,=— pat RE ( =(Cp—c,)dT + 22, trom which it is obvious that, the heat of combination at the Vapour Pressure and Affinity of Isotopes. 179 cabsolute zero Qo being equal, the ratio of the Guldberg- Waage constants K, is the same as that of the vapour pressures. ‘his shows that chemical separation, though possible in principle, is too small to be considered at ‘temperatures at which chemical reactions take place. The electromotive force of reversible cells containing two isotopes as electrodes is given by nF V=A, where | is Faraday’s constant 96, 540. coulombs, 2 is the valency of the ‘ion, V is the ech omanine force, and A the affinity. Now A= RT (tog Pup i a —log K,) 9 P1 "Po aa: ‘so that AC? A Rios 22? = RT log 22, KS P1 whence the difference in voltage of the two otherwise identical processes, e. g. Ph +2AGCISPbCl, + 2Ag i is Oe 8c ee Ts re al nt ak -at ordinary temperatures. Putting w==4 corresponding to Av=95° and T=285° and 6=1 per cent., one finds for the -above reaction, in which n= 2, a difference of barely 10~° volt. It is not surprising that no difference has been observed if the above theory is correct. If the chemical constants are identical, the ditference might amount to 0°18 millivolt. The whole theory takes a much more complicated form if one does not assume continuous absorption, but takes the -energy at temperature T to be a as in Planck’s earlier kT — J] publications *. As the results are quite different from those given above it may be worth while to make a rough estimate, even though this basis is probably not correct. For the sake -of simplicity, all results will only be worked out to the first powers of ou and 0. In this case putting — alle Ne MO Ae) Sy 9 ane hv = wlie- = yy Oa * M. Planck, Vorlesungen tiber die Theorie der Warmestrahlung «(1906). N 2 180 Dr. F. A. Lindemann on the one finds, since hy ae Tar m’ Ne Oe a All Ay Dn 1+5(1+ 4") Vo A, Br, Ue 1+ — ee that fi 2 6 By ae Ky a 2 se It is probable, though by no means certain, thas 5 =H | 2 2 at any rate as a first approximation. Further, at high temperatures the expansion coefficient should be proportional to the compressibility «. If one neglects the difference of : 9eT y ; the correction terms ~~——, which are small in any case * 6 and only differ by something of the order 5 . ad , one finds. 20 Th for temperatures which are great compared to @y, gh 0 ea log —(snr Pe AT, 7 +i): In lead ue » n= 600°, and Xo the most important term is approximately 50,000 cal. On the above assumptions: therefore, 2. e. taking Ay»* f and the chemical constant as derived above 1916 ee If 6 is 1 per cent. itis seen that the fepou pressure of the lighter isotope should be 5 per cent. greater than that of the heavier at the melting-point and “nearly 3 per cent. greater at 1200°. The differences in Guldberg and Waage’s constant ye be of the same order. It is, however, in tie affinity a measured by the electromotive force that the best hope ee of testing which hypothesis is right. As above RT Jat | 5 on yee V,-V,= AE », =p (vet 7 * W. Nernst & F. A. Lindemann, Zettschrift f. Eleltrochemie, xvii. p. 822 (1911). log’ P= 8(1: Yt [aia Vapour Pressure and Affinity of Isotopes. 181 Putting T=290° and-d==1 per cent., this would lead to a difference in electromotive force of almost exactly one milli- volt. This should be perceptible if due precautions were taken. : The general conclusion may be formulated as follows: Isotopes must in principle be separable both by fractionation and by chemical means. Th+ amount of separation to be expected depends upon the way the chemical constant is calculated and upon whether “ Nullpunktsenergie”’ is assumed. At temperatures large compared with @v, which are the only practicable temperatures as far as lead is concerned, the difference of the vapour pressure and of the constant of the : v law of mass action may be expanded in powers of e The Bv most important term of the type log, is cancelled by the chemical constant if this is calculated in what seems the only reasonable way. The next term in oe the “ Nullpunktsenergie” if this exists. All that remains is cancelled by v T° therefore fractionation does not appear to hold out prospects of success unless one of the above assumpticns is wrong. It the first is wrong a difference of the order of 13 per cent. should be found. If the second is wrong a difference of as much as 3 per cent. should occur at 1200° and a difference of electromotive force of one millivolt might be expected. Negative results would seem to indicate that both assumptions are right. are terms containing the higher powers of In practice @ Summary. It is shown that though isotopes cannot be identical chemically there is only a second order difference if ““ Nullpunktsenergie ” is assumed and if the chemical constant is derived from m, hk, h, and @. The difference should be measurable if there is no “ Nullpunktsenergie,” and it is suggested that experiments on the vapour pressure and affinity of isotopes would give valuable information on this important point. Sidmouth, May Ist, 1919. i aes XIII. Lhe possibility of separating [sotopes. By 8. Cuarmay, MA. D.Sc, PRS" ‘Nols, i an interesting paper bearing the above title f.. Prof. F. A. Lindemann and Dr. F. W. Aston have described various ways of separating isotopes, tent elements of slightly differing atomic weights occupying the same position in the atomic table, and inseparable by chemical means. The object of this note is to draw attention to a further method, which may possibly be of service in this. difficult task. The method depends upon thermal difeusion: a phenomenon o£ gases which was independently discovered by Dre Enskog t t and myself$ by theoretical reasoning, and sub- sequently experimentally verified by Dr. F..W. ” Dootson |. It is most simply described by stating that if free com- munication is made between like mixtures of gases in vessels maintained at different temperatures, diffusion will take place until an equilibrium condition is reached in which there is a slight excess of the heavier gas in the cold vessel, and of the lighter gas in the hot vessel. If the molecular masses are equal, but the diameters. unequal, the larger molecules will be in excess in the cold. vessel. The mathematical theory of thermal diffusion is highly complicated, but a first approximation to its results can be expressed fairly simply (Phil. Trans. A. 217, pp. 181-186). In the present note I shall confine myself to the case of a gas mixture in which the molecular masses m,, m, are nearly equal, as, for instance, in a mixture of neon and the hypo-. thetical metaneon (of molecular weights 20 and 22). This case was the one specially considered by the two authors. | first-named. The separating power of thermal diffusion depends on a constant ky whose value, when m,—my, is small (we will * Communicated by the Author. 7 F. A. Lindemann & F. W. Aston, Phil. Mag., May 1919. } D. Enskog, Phys. Zeitschrift, xii. p. 538 “(1911) ; Ann. d. Phys.. XXXvVili. p. 750 (1912) ; Dissertation, Upsala, 1917. § S. Chapman, Phil. Trans. A. 217, p- 115 (1916); Phil. Mag. xxxiv.. . 146 (1917). || S. Chapman & F. W. Dootson, Phil. Mag. xxxiii. p. 248 (1917). The possilility of separating Isotopes. 183, suppose that m,>mz.),is approximately given by * Lt my = ms AjAg fos oe my +m, 915 — 825A, A,’ where X,, A, denote the proportions by volume of each gas. in the mixture; thus A; +A,=1. The diameters have here been assumed equal, the molecules, moreover, being regarded as elastic spheres. The equation giving the variation of d, or A, corresponding to a gradient of absolute temperature T in the direction of a is ON ON sae dlogT aL on eor Oe when the steady state has been attained. If Ay, Ay do not vary much throughout the gas (and they will not do so by reason of thermal diffusion alone, which is only a weak separating agent), we have as the difference between the values of X at the two ends of a tube maintained at temperatures T, T’: YA = ay (Ay— Ay’) = hop log ay. § 2. It is of interest to compare the separating power of thermal diffusion with that of pressure diffusion. The latter is considered in the paper first cited, where an estimate is made of the difference of pressure and the resulting partial separation obtainable by centrifuging the gas. The effect in this case can be represented by the equation OM ain OX» Jonsanae Ooum tor.) ? one where NyAg (724 — My) PON Nyy Agni) Thus if the extreme pressures are p, p’ the separation is: given by Ay — Ay’ = —(Xz —),') = lp log p/p’. * A numerical error may be pointed out on p. 185 of my paper, Phil. Trans. A. 217, 1916. The first numerical constant in the denominator of the expression for kp in the oxygen-nitrogen column should be 10:1 instead of 222; the three particular values of A calculated from the expression should be 0-010, 0:013, 0:0086, giving the values 2°6, 2°5, 2°8 for D,/D, in the last column. 184 Dr. S. Chapman on the For equal values of log p/p’ and log T’/T, pressure diffusion is more powerful than thermal diffusion in the ratio 3 Mm, +mM, (9°15—8°25 A,r). ke Fa 17 Aym1 + AgMs * In the case of neon and metaneon this varies from 2°5 when the gases are equally mixed (A;=A,=4) to about 3°2 when there is a large excess of either gas. Thus, if it were equally easy to obtain a given ratio for either p or T, pressure diffusion would afford much the more advantageous method of varying the relative concentration of the two gases. It seems more convenient, however, to maintain large differences of temperature than of pressure, and it may therefore be preferable to use the feebler type of diffusion on this account. The operation can, of course, be repeated any number of times. § 3. To take a numerical example, suppose that it is desired to separate a mixture of equal proportions of neon and metaneon. Then, writing m,=22, m,=20,rA,;=A.=4, we find that Ar=0°0095. Suppose that the mixture is placed in a vessel consisting of two bulbs joined by a tube, and one bulb is maintained at 80° absolute by liquid air, while the other is heated to 800° absolute (or 527° C.). When the steady state has been attained the difference of relative concentration between the two bulbs is given by the equation Ay — Ay! = — (Ay—Ay!) =0°0095 log, 800/80, =0-0095 log. 10, = (0-022, or 2°2 per cent. ‘Thus the cold bulb would contain 48-9 per cent. metaneon to 5l1‘1 per cent. of neon, and vice versa in the hot bulb. By drawing off the contents of each bulb separately, and repeating the process with each portion of gas, the difference of relative concentrations can be much increased. But as the proportions of the two gases become more unequal, the separation effected at each operation slowly decreases. For instance, when the proportions are as 3:1, the variation at each such operation falls to 1°8 per cent., in place of 2°2 per cent. ; while if A; :”A, is 10:1, the value is 1'2 per cent. This assumes that the molecules behave like elastic spheres: if they behave like point centres of force varying as the inverse nth power of the distance, the separation is rather less; e. g., if n=9, it is just over half the above quantities. possibility of separating Isotopes. 185 § 4. Whether or not this method of separation by diffusion is of convenient practical application depends largely on the time taken for the attainment of the steady state. An estimate of this can be made, somewhat roughly. Let vy denote the number of molecules of both kinds (vy; +¥) per unit volume at the point 2 measured along a uniform cylinder, the two ends of which are maintained at different fixed temperatures. The pressure being the same throughout, vol is constant along the cylinder. If diffusion is taking place, the number of molecules m, which per unit time cross unit area of a normal section of the cylinder in the positive direction is vou, equal to the number of molecules m, crossing in the opposite direction, where heed Our 0 log T ii wDu(S- + kp VE ). Here Dy,» is the ordinary coefficient of diffusion of the mixture. In the present case, where the molecules of the two sorts are so nearly alike, we can replace Dy. by Dy, the coefficient of self-diffusion of the gas; if the molecules behave like rigid elastic spheres D,,;=1-°200 u/p, where p is the coefficient of viscosity and p the density of the gas. The equation giving the rate of variation of the relative ‘concentrations at any point is Om Loe Ot =o Yo Fy Voro i: This cannot be exactly integrated, because (on account of the temperature gradient) vy and Dj,» vary along the tube. We know, however, that initially, when the mixture of gas is uniform, 9A,/O72=0, and w= —fkrD_»0d log T/dz ; finally, wo’ tends to zero, while 2, tends to its equilibrium value C—fplog T. Clearly, the greater the value of Dy», the more rapid is the progress towards the steady state. Itseems likely that the order of magnitude of the time taken may be estimated by making the calculation from the last equation as if vy and Dj, were constant along the cylinder, giving D,, its minimum value. In this case, since /ylog T is independent of the time, we may write at The solution of this is i .— pt+or(p/D 10) W/2 At hq log i =Ay+ 2A, ¢ pe+ (p/Djy2) : where the constants p are determined by the initial and end 0 Oryx -- hep log T) = IDK os (A, + hy log i). 186 The possibility of separating Isotopes. conditions. Suppose, for instance, that the cylinder is of length /, and that the origin of x is at the middle point along the axis. A fairly uniform increase of T along the cylinder can be represented by the formula, convenient for-. the present purpose, ken log T=B,+ B im, whence we obtain the result A =Apw—B (1 —e 2 Dust] sin — 4 which is stationary at the two ends of the cylinder, as it should be ; Ayo is the value of A, at the middle of the cylinder.. When the process of diffusion has been nine-tenths com-. pleted, we have ¥ p Dist — a or Taking D,,=1°:20u/p, and assuming that jo I, ¢ as ls. pr obably nearly true, we have (Dips 208 (F) ae where wo, po are the values of w and p at fects To, say- At 0° C. and atmospheric pressure 4,=3°10° 4p, =092 tte so that at 80° absolute the value of Dy, is approximately 0°050. Hence log. 10 0:0507 if the cylinder is 10 cm. long, ¢=460 seconds, or about 8 minutes. This very short time applies to a cylinder ; for — Oe two bulbs connected by a tube (which should be vertical, to avoid convection from the lower, cold, to the upper, hot, bulb) it would be considerably greater. But unless some serious mistake has been made in the above calculation, it would seem that the time required is to be reckoned in hours,. at most. Hence thermal diffusion, many times repeated, should prove a practicable method of separating isotopes of. the kind considered. Cambridge, May 1919. a ee ee ae XIV. The Fundamental Law for ihe True Photographic: Rendering of Contrast. By AtrreD W. Porter, ).Sc., ef PeS:, University College ond’ i. House, | Se.,. British Photographic Research A ssociation *. HE modern treatment of the character of a photographic: plate is based entirely (in England at any rate) upon the methdd of examination introduced by F. Hurter and V. C. Driffield in a paper read before the Society of Chemical Industry and printed in their Journal dated May 31st, 1890.. The advance that was made by these authors was so great and the utility of their method so advantageous compared with previous methods, that the whole photographie world has joined in according ‘them the honour due to their insight into the problem. Nevertheless, a certain malaise is often felt in regard to: the logical foundation of their method. Perhaps they themselves are to blame for this, because one of the important steps in their chain of arguments is made in so cursory a fashion (in a single sentence) that any reader must simply assume it to be correct unless he himself develops the: argument in all its detail from the beginning. The whole argument is this} :—-‘‘ Since the density is the. logarithm of the opacity, and since in a theoretically pertect negative, the opacities are directly proportional to the intensities of the light which produced them, it follows that each density must be proportional to the logarithm of the light intensity which produced it.’’ They add in brackets : “More correctly the density is a linear function of the intensity of light and time of exposure.” Unfortunately in making this detailed examination one of us (A. W. P.) has recently discovered that their principal conclusion is erroneous when regardedasa general principle. We shall first of all describe such a detailed examination and then discuss in what respects difference is found from Hurter and Dritheld’s results. The problem deals with the taking of a negative followed by the printing of the negative on to another ‘plate or paper, and investigates the conditions under which the final positive: gives a true renderi ing of the contrasts in the original subject.. There e are e several parts UO) HUD * Communicated by the Authors. t+ Journ. Soc. Chem. Ind. May 30th, 1890. 188 ~=Prof. Porter and Dr. Slade on the Fundamental When the plate is exposed to the subject (exposure=E)) a certain density of deposit is produced and the negative has a certain transparency ‘TT in consequence after development. ies Ie Sl Te Negalive Faye Taking stage. In the second placea print is made by exposing the positive plate (or paper) through the negative. Let the printing light be P. Where the transparency is T, the light getting to the negative is PT,=H,; this exposes the positive pro- ducing a transparency T, after development. Kis. 2. ie Sat ee : ee i : Ne pa live oO oNo One ee SA Paka Re moo OOS Slevin en Nig SOO POS Gs . Ose 5 . e : Printing stage. Finally, this positive is viewed under an illumination V (the viewing light) and the light that issues from it is VT,=[. Fie. 3. V ek ae oe Posilive Bigs ce le | Viewing stage. Since for a true rendering of contrast it is necessary to have the same ratio between light emitted from two portions of our final picture as that which fell in the camera from the two corresponding portions of the subject, we must have I=KE where K is a constant reduction factor. Now this Law for True Photographic Rendering of Contrast. 189 relation between the ‘first and last lights concerned in the process involves a connexion between the relative transpar- encies of the positive and negative plates. Hurter and Driffield defined the transparency T as ees pee ouatted ight incident They also make use of a correlative quantity, the density: D of a plate, the connexion between these quantities being log 4 =density = D. It should be observed that since T is always less than 2G ete : i unity, its reciprocal 7 Is positive. Now [=VT, from the definition of transparency. But if also is greater than unity and its logarithm. On, we have eRe ae | T, Ke —ACOMSEANL. nt 1 one eran (1); Similarly, 3, In = 1 whence E T == ]P SS OOMR EMR Lal oy lo a CN =) The first of these equations states that the transparency of | the positive must bear a constant proportion to the light from the subject at all corresponding points. The second equation states, what is only the direct result of definition, that the light transmitted by the negative is proportional to the transparency of the negative itself. Both of these equations must be caused to be true simultaneously by a suitable choice and treatment of the two plates. To examine these equations further it is convenient to take the logarithms (to base 1U) of both sides of each and write I i= lee aut D.=7-> 1 2 whence from (1) Vv og Hi= log —D:, (3) and from (2) log E,= log P—D, (4) ‘190 =Prof. Porter and Dr. Slade on the Fundamental Let us suppose that the connexion between D, and log Ey is known. Itis represented graphically by the characteristie -curve usually obtained for a plate (tig. 4). Fig. 4. es <-- log K- } I ° Sim sass aes log | Saar = 4 — eyo 4--- 32 ) B H { ~N bss ol uf “30 lies mit oO (eve) | pie | | PENS oer i RY Aga SES NL Ne Log , e D, ie N l D | ! 1B) : A ! ! O f ——> log E, is D, <—_° The above equation will enable us to determine the values -of D, and log E, for a plate that will form a suitable positive. Such plates which suit one another as negative and positive we may advantageously refer to as conjugate plates. The former equation (3) asserts that we can obtain a suitable value of D, by taking any constant (the same for all points) and subtracting log BH, fromit. This is equivalent to shifting the origin to any point O' and measuring Dy backwards from it; thus if OM represents a particular value of log Hy, then O’M represents the suitable value of D,. On the other hand, the second equation (4) asserts that the suitable value -of log E, is obtained by taking any constant (the same for all points) and subtracting D; from it. This is equivalent to still further shifting the origin to any point O" and reckoning log E, downwards from it. Thusif O'N represents the value of D,, then O”"N represents the suitable value of log Ho. Hence the same point Q represents not only the corresponding values of D, and log Ey, but also, when measurements are made as indicated from the new origin O”, it represents the corresponding values of D, and log E,. Since this can be said tor every point Q in the characteristic curve for the negative, it follows that this curve suitably interpreted gives a ee ee a Law for True Photographic Rendering of Contrast. 191 ‘the characteristic of the conjugate plate also. We ma express the transformation most easily by saying that if the characteristic curve D, against log H, be drawn on tracing ‘paper and be then viewed from the back so that the first origin QO is at the top right hand corner, the curve, as it then appears, is the characteristic curve of the conjugate plate. ‘Such a curve is shown in fig. 5 for comparison. Fig. 6, ee log sa It should be carefully remarked that the new origin O” ‘may be any point whatever. But it must not be forgotten that the constant amounts by which it is shifted in the two directions are connected with the viewing light and printing light. In fact, OO'= log [viewing light K |= log V — log K, O’O'’= log [ printing light]= log P. To choose the point O” is equivalent to deciding upon particular values for the printing and viewing lights and the reduction factor K. The more nearly K is to unity the more nearly will the light from the positive be not only the same in gradation as that from the subject but also the same in absolute intensity. When we make a print we arrange our exposure so that the curve of the paper is in the best position with reference to the axis of log EH, to be brought by development to fit the 192 ~=Prof. Porter and Dr. Slade on the Fundamental conjugate curve; then we develop until this fit is obtained. We must not develop longer or this fitting of the curves would be spoiled. In what respects do these conclusions differ from those of Hurter and Driffeld? Their great conclusion was that it was necessary and sufficient that D should be a linear function of log E for a sensitive plate, 7. e. D=y(log H— log). This conclusion appears to have been arrived at by con-- sidering a single kind of plate only for both negative and positive, but they do not state exactly what assumptions they made. If we derive from our general results, this case- in which the two plate curves by supposition are identical, it follows that D= log E— log I for each; or in other words y can only be equal to unity.. No other curve but this can fit its conjugate. Hence the conclusion that y may have any value in such a case is erroneous. Ina later paper * onthe relation between photo- graphic negatives and their positives, they carry the question further and conclude that | ID: =y(log P —aD,), which is somewhat similar to our fourth equation ; but they only give arbitrary values to a and the value of y¥ 1s also. quite arbitrary. It would seem, then, that as the result of a somewhat quick judgment they were led to an imperfect conception of the true conditions for securing a true repre- sentation of gradation. The true result is in one case more special than theirs because for similar plates the value of y must be unity ; it is also more general than theirs because we are not constrained to keep to the linear function at all. We will examine in detail some particular cases. Case 1.—Suppose we follow the customary theoretical practice and restrict ourselves to a negative in which the straight portion of the characteristic has been utilized. So that D,=yUog E,— log z,). By plotting fig. 6 and viewing from the back it is seen at once (fig. 7) that for the positive the necessary conjugate: relation is p= : (log E,— log 2). Y * Journ. Soc. Chem. Ind. 28th February, 1891: Law for True Photographic Rendering of Contrast. 193 The constant log7,’is arbitrary as we have said before, and y= tan 6. ir Fig. 6. A low characteristic curve for the negative requires a| steep characteristic curve for the positive print. In fact) 0,+ 0,=90°, where @; and @, are their respective inclinations to the horizontal exposure axis. In real cases, only the central part of the curve is straight (figs. 4 and 5). Since the conjugate curve is obtained from the curve of the positive simply by a change of origin and axes, the length of the straight line portion in the two curves is the same, Phil. Mag. 8 6. Vol. 38. No. 223. July 1919. O 194 Prof. Porter and Dr. Slade on the Fundamental although the projections on the exposure axes are different in the two cases unless y is equal to unity. Hence, in order that the two plates may match, the straight parts utilized in each characteristic must be the same in length. Generally this is possible in lantern-slides or other transparencies. In the case of paper the straight line is very short ; it is there- fore only the gradations corresponding to a portion of the curve for the negative which will be correctly rendered. The particular part reproduced correctly depends on the printing exposure. Whether any part is correctly rendered depends on the development, for the development factors must be made reciprocals of one another (vy for the one and il. : 5 for the other). These considerations show that if the subject presents a wide range of gradations and it is treated so that these correspond to the straight part of the curve, then development of. the negative should be stopped early, so that the straixht part utilized is as short as possible (for y increases with development). The print then should have a high y which may be obtained partly by choice of plate or ‘paper and partly by prolonged development so as to make the paper curve as lone as possible. [or subjects with only a small ranve of gradation these considerations are of less moment. The three curves in fig. 6 all correspond to the same range of gradation in the subject; with increased development the straight part required increases. The corresponding portions of best positive curves which re- quire to be straight are shown in fig. 7. The lines a, 6, ¢ on fig. 6 require to be matched by the corresponding lines a, 6,¢ on fig. 7. Since the straight part on paper curves is short we require the part of the curve on the negative corresponding to extreme range of exposure to be as short as possible, 2. e. we require the negative to have a small y ; consequently the print must have a big y. This conclusion is at variance with that arrived at by Jones, Nutting, and Mees (Photographic Journal, liv. p. 342, 1914). They point out that the latitude of plates may be ~as high as 128 in exposure units, although the contrasts in most subjects is not more than 1 to 34; they conclude that the negative is easily capable of satisfying what is required of it. Buta paper has only a latitude of 5°6 exposure units (‘75 in log E,) and they say: ‘“‘The paper then will only render correctly a range from 1 to 5°6 ; hence it is obvious that in order to rendera negative having a range wider than 1 to 5°6 in transmission, it 1s necessary to utilize portions Law for True Photographic Rendering of Contrast. 195 of the characteristic curve lying outside the latitude of the paper and thus depart from direct proportional rendering.” This conclusion on their part illustrates a prevalent fallacy that the range of contrast on the exposure scales of the negative and of the positive must be alike. Consider, how- ever, a paper for which the latitude is 0°75 in log units and for which y2 is 1°5. This is shown in fig. 8, where tam@o—yo— lis. i g) Bonn? | i) De log E, b ‘ The curve of the negative which will be correctly printed by this paper under these conditions has He eel a 5 ==\00Gre VS and reproduces a range ab on the log Hy scale which is given by 0:75 tan@,=0°75 x15=1:12. This corresponds to a ratio of 1 : 13 in exposure units. Thus though the exposure latitude of this paper is only 1 : 5:6, it will correctly repro- duce a latitude in the camera exposure of 1:13 and not only the 1 : 5°6 as these authors take for granted. Case 1/.—It is not compulsory, however, to work in the straight por‘ions of the curve as Hurter and Driffield thought to be necessary. Examination of the curve B’B in fig. 9 will show that this possible extension is of limited avail- ability. No sensitive material known will yield such a curve as is shown in this figure throughout its whole length ; but this is necessary to give a true photographic reproduction. In fig. 9 we see that the curve of the paper pgr will fit 196 Photographic Rendering of Contrast. over a certain range of the conjugate curve B’/B. Thus we can reproduce correctly over a short range of contrasts in the original by using both the under exposure portion of the negative and of the positive printing process. This portion of the curve is often used in practice, and the two curves BB’ and pqr shown in fig. 9 are actually those of an Ilford Empress plate and a piece of Illingworth Slogas paper (Vigorous) respectively. Fig. 9. iw ——F Tog Ey Again, for over-exposed negatives, the shape of the con- jugate curve is A’B’ (fig. 9). We can only get the over exposure portion of the curve of a print to be of this shape. Moreover, such prints would not represent any portion of the original subject by white paper, and therefore such a print is usually of no pictorial value. When this paper had been drafted and while it was being copied out, our attention was called to the Traill-Taylor Lecture, given by Mr. Renwick in 1916, in which the same question of matching the negative and positive is discussed. In this lecture Mr. Renwick gives what he calls reciprocal curves for the positive and negative respec- tively ; but he does not describe how they have been ob- tained. In the absence of such a description it is not possible for us to test how far his work overlaps or antici- pates ours. But he certainly does not touch the validity of Hurter and Driffield’s conclusions. We have also found a paper by Lord Rayleigh “On the ~ Notices respecting New Books. 197 general problem of photographic reproduction with sug- gestions for enhancing gradation originally invisible’ Geile) Wage. xxiiip. 734, (1911)), ), reprinted (British Journal of Photography, Iviii. p. 994 (1911)). In this paper Lord Rayleigh certainly goes to the root of the matter so far as the matching of plates i is concerned, but does not leave the problem in a form suitable for practice. In particular, some of the assumptions made in the typical cases taken do not fit in with practical aims. For example, there is no object in making the gradation of the negative the same as the positive. The only condition which seems to us useful is that the gradation in the issuing light (1) shall be the same as fia) Grom! fhe original source (H,). This is the condition which we have taken as oe XV. Notices respecting New Books. A System of Physical Chemistry. By Witu1am C. McC. Lewis, Professor of Physical Chemistry, University of Liverpool. Longmans, Green & Co. In 3 volumes. (Second Edition.) HE first edition of this work is, of course, well known to all students of physical chemistry. The general arrangement of the matter in the new edition is similar to that of the first, but the large amount of new material which has been incorporated has made a third volume necessary. ‘The first volume deals with the kinetic theory, and we are introduced at once to the very recent work of Perrin, Svedberg, Wilson, and Millikan, on the physical properties of molecules, the Brownian movement of small particles suspended in liquids and gases, and their behaviour in electric and gravitational fields. Among the new additions to Volume I. may be mentioned: The study of the arrangement of the atoms in crystals by means of X rays, as developed by Laue, Bragg and others, the dissociation of the vapour of ammonium chloride in the light of recent experiments, the dual theory of homogeneous catalysis, the ‘‘ displacement effect” (a term suggested by the author for effects produced by gradually replacing one reactant by another), and Langmuir’s very recent work on hetero- geneous reaction velocity, condensation and evaporation, chemical theory of surface tension and the general mechanism of surface forces. ‘The volume contains an enormous amount of information of great interest to the general reader, and should be in the library of everyone intere sted in the recent developments of science. The subject of suspensions, emulsions and colloidal solutions is very fully treated, and is of especial interest in that numerous applications of the theories are made to matters of 198 Notices respecting New Books. everyday life, such as the rising of cream and the churning of milk, precipitation of river-sludge and sewage, sizing and colouring of paper with mixed dyes, the part played by ad- sorption in the fertility of soils, and the recently discovered and very remarkable electro-osmosis phenomena with their industrial applications. The last chapter is devoted to Langmuir’s work on evaporation and condensation, the rate of adsorption of gases by solids, the kinetics of heterogeneous reactions, and the chemical theory of surface forces, with the extremely interesting conceptions of the orientation of the molecules on the surfaces of liquids, in benzene for example the molecules arranging themselves so that the benzene rings lie flat upon the surface. In connexion with the subject of evaporation and condensation as treated in this chapter it may be of interest to draw attention to a recent paper on the subject pub- lished in this journal*, dealing with the discovery of a sort of critical temperature for each element below which the element condenses as a heterogeneous non-crystalline film, and above which it deposits in a granular crystalline form. The critical temperature is surprising low, — 90°C. for cadmium, and — 140°C. for mercury, for example. The new material in Volume II. deals with the subject of osmotic pressure and the theory of dilute solutions, and vegetable tanning is taken up in connexion with adsorption and membrane equilibria. Volume III. will treat the whole subject from the Quantum standpoint. The treatise as a whole is not only the most complete, up-to-date and best text-book on physical chemistry that I have ever read, but, in addition, one of the most interesting scientific books that has come to my notice in many years. hk. W. Woon, Paris, April 1919. Osmotic Pressure. By ALEXANDER Finpuay. Second Edition. Longmans, Green & Co. Price 6s. net. Tuts little monograph gives a concise account of the experimental work and speculations on the subject in question. With respect to the various theories of the nature and cause of osmosis it preserves an impartial tone, stating the views of the various authors without pronouncing very decidedly in favour of any particular school. This second edition gives an account of the apparatus employed by Morse for his measur-ments of large osmotie pressures which is fuller in detail than that of the first edition, and brings the subject up to date by adding a description of the new work carried out during the past six years. The * R. W. Wood, “Condensation and Refraction of Gas Molecules,” Phil. Mag. October 1916. Notices respecting New Books. Los) important discussion, at the Faraday Society in 1917 opened by Professor A. W. Porter is freely cited. Giving as it does detailed references on all the points discussed, the volume is valuable to the research worker as a guide to the literature of different branches of the subject, as well as to the student seeking only a general knowledge. The latter would probably be grateful if a little more space were devoted to the deduction of some of the formule given. The Theory of Hlectruwaty. By G. H. Livens, M.A. Cambridge University Press. Pp. vii+717. Price 30s. net. In his preface Mr. Livens acknowledges his obligations to Sir Joseph Larmor for permission to make use of notes made at his lectures, and throughout the volume the influence of this great teacher is to be traced. A consequence of this is that space is devoted to the different branches of the subject somewhat in pro- portion to the attention which they have recoived in the original work of Sir Joseph and his school, and as a result the reader who looks for an account of the new developments of electrical theory is struck by large omissions. For instance, very little space is devoted to the treatment of electric currents in gases and the problems of the motion of ions connected with them (in spite of a diagram showing the effect there is no mention of ionization by coliision, and T’ownsend’s name is only mentioned once in the book), to the electron theory of spectral series, or to the reflexion of short electromagnetic waves, while the treatment of relativity, deferred to the last five pages of the book, is very meagre indeed. In view of the success of this theory in interpreting a series of experimental results, and of the attention it has received from many of the best mathematical physicists in Hurope, it is worthy of more serious exposition or criticism. oom could have been made for adequate treatment of these and other subjects of present interest by omitting matter of secondary importance only, or matter where a reference to Maxwell’s treatise would have sufticed, such as much of the detail of Chapter II. We naturally expect to find good chapters on polarized media, on electrostatic stresses and on the electromagnetic field, and we are not disappointed. The treatment of the Maxwellian stress system is particularly satisfactory, and the development of the theory of electromagnetic waves and oscillations is very complete, especial attention being paid to the field round a Hertzian oscillator. We must also mention an excellent chapter on magnetostatics, a rather neglected subject, which includes a short account of the work of Curie, Langevin, and Weiss. In general the book appears to suffer from the lack of any very clearly defined plan. Jt is hard, for instance, to know why the reader should be assumed ignorant of the very elements of vector analysis, but well acquainted with Legendre functions. Again, after the introductory chapter on vector analysis we should expect 200 Intelligence and Miscellaneous Articles. to find an extensive use made of that method, but we are often confronted with masses of Cartesians. Round each subject is accumulated a great deal of mathematical matter, but sufficient distinction is not made between what is fundamental and what is subordinate. For instance, we cannot find anywhere the funda- mental equations of the electron theory clearly stated and con- trasted with the Maxwellian equations. It would have added greatly to the value of the book if more care had been taken to secure unity, and to make the explanatory text a logical discussion of basic principles. As itis, there is a great deal of very valuable matter in the book, and as a handbook to the writings of the famous Cambridge school of mathematical physicists it will find an honoured place on many shelves. Annuaire du Bureau des Longitudes pour Van 1919.. Gautier- Villars: Paris. Pp.-vii+680. Price 3 francs. Tus excellent little publication contains, in addition to the usual astronomical and geographical information, two valuable articles, one on the figure of equilibrium of a rotating liquid whose elements obey the Newtonian law of attraction, by M. P. Appell, and the other on the determination of the diameter of heavenly bodies by interference methods, by M. Maurice Hamy. The names of the authors vouch for the worth of these short accounts of the work done up to the present on the subject treated. We would also draw attention to the brief résumé on solar physics prepared by M. H. Deslandres. XVI. Intelligence and Miscellaneous Articles. PROBLEMS IN CONDUCTION OF HEAT. To the Editors of the Philosophical Magazine. AAO Ry The University of Sydney, Soe April 17, 1919. [ may be worth while to mention that the method of attacking problems in the Conduction of Heat, when the surface temperature varies with the time, attributed to Riemann in Mr. McLeod’s paper (Phil. Mag. ser. 6, vol. xxxvii. Jan. 1919, p. 139), is in fact, due to Duhamel. Reference may be made to the Journal del’ Ecole Polytechnique, T. xiv. Cah. 22, p. 20, 1333, and to ny book on ‘ Fourier’s Series,’ p. 208. Yours faithfully, H. S. Carsnaw. ne Woon. Phil. Mag. Ser. 6, Vol. 38. Pl. I. Refl. by e}) Quartz. Refi. by Potassium. Trans. of Potassium. Seconds Expos. #8 Refi. by 20 (irra 1 Pi Lele U Me ie | s 10 Ka ‘ ra cai Stee = Quartz. Refl, by Sodium. Fig.3. * 2 ! \ |) Trans of Sodium. NAN Barton & Brow . Ys 1 i S e \ aes . = ‘y) 2 } it i r ¢ = + i> ~~ F) a ‘ ; ‘ ‘ j i ; ' 4 r rag i mh eet f Barton & BRowNInG, Phil, Mag. Ser, 6, Vol. 38, Pl. II. rY a s = - * a = —_ i peewee 5 z ame tee nee mee Tater ae pos : esr pis eae athe : , ¢ Fi a ft Barron & Broy { ( j ti " ie — * aI re r acsseaae 3. 4 j i Ba Sg sae 6 te ae ¢ Ree On : bs aves a9 ce t-4eeeaa aia Se 4 : on < Wis, Barron & BRrowntna, AY, ore t rh cena ne an et Sem ens pra na me a eine pone merge eee ower Tecate er hoe os ge emerald one oy Pu evened a nme RN 8 eye TS een eee veere reir tny ome eh ees memes hi me mae ptr tN ag ta re oy iT nme 9 pen et meee Meas mes et wate 1 4 q i 4 4 ( et mee png etre ane a = oe we eben ns ties me gaaeines 2s VITA! er eet eect Oo corr ce ppeer oeaeniaeonanenrmentar en Pr saa ime Firma oo 0 Sn mm me Gh RS Se : as Y nim pan Wish jm poehieieonel THE LONDON, EDINBURGH, ann DUBLIN | PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. PLT: ‘ Aug. (SIXTH SERIES. ] Xe Vu ve ver 1919. EL OCRIGE 5 XVII. Notes on Electric and Magnetic Field Constants and their Expression in terms of Bessel functions and Llliptic Integrals. By Professor A. Gray, /.RS.* if YHE following notes were for the most part written out several years ago, and were suggested by points which arose in the course of a revision of part of my book on ‘Hlectrical Measurements’ on which I was engaged. I have to-day noticed that in the idea of caiculating the exhaustion of potential energy involved in the formation of a uniform disk of attracting matter, I have been anticipated by Clerk Maxwell and by Lord Rayleigh +. Some of the other processes in like manner may not be novel, but on the whole the notes may perhaps be found to be not altogether devoid of interest. Asa preliminary we take some expressions for potentials and forces in terms of Bessel Functions. Some of these, if not all, are quite well known. Case (1). A thin disk of matter of uniform surface density o and radius ais situated with its centre at O, fig. 1, and its axis along OC: it is required to find an expression in terms of Bessel functions for the potential produced at any point P. Let the axial distance of P from O be z, and its distance from the axis be y, as shown in the diagram. Consider a ring of the disk with O as centre, and of radius p and * Communicated by the Author. + “Theory of Resonance,” Rayleigh, Collected Papers, vol. i. Phil. Mag. S. 6. Vol. 38. No. 224. Aug. 1919. P 202 Prof. A. Gray on Electric breadth dp (fig. 1). Jet E be a short element of this ring, and @ be the angle which the radius OE makes with the Fig. 1. S ‘ i) ' ' ' Be ' ! + radius which lies in the plane PCO. The potential V at P is given by Vao( | piled (ae e020 ( 2 +y?+p?—2py cos )? Put R?’=7?+ p?—2ypcos¢. We get (see Gray and Mathews, ‘ Bessel Functions,’ p. 72) 2 27 a io) i a dd ( pdp| e~™ Jio(XK) dX.) eee <0: Sit aamipenes 3) 0 But by Neumann’s theorem (G. & M. p. 27) JAR) = JIo(rAp) Io Ay) +2=ZT (rp) Te(ry) cos sh. (3) 1 Hence, since the order of integration in (2) may be changed, we obtain 1 Qa ( J,(aBydebi = Bede) JoQuns 2 a J0 since the terms involving cos s@ contribute nothing to the integral. Thus (2) becomes Ve na | CA {\ pdpdo(Ap) } Jory) a. 3, She Now it is easy to prove, by the relation ody (2) +Ii(z) = 2d), or otherwise, that ( pdp Jo(Ap) => J,(ra). a (i) and Magnetic Feld Constants. 203 Hence (5) may be written in the form 10 dn We Onae { oJ, (Na) Ju(Ay) Se (6) 0 From this value of V all the results I desire to deal with at present can be derived in succession. 2. The axially-directed force Z at P due to the disk is —OV/dz, and therefore by (6) Ti rou | 7 TAMORIR OO J0 The force R at P in the direction CP is —OV/oy. Hence remembering that Jy’ (Ay) = —J, (Ay) we obtain ro 2noal OAS OM IOEN ON) ICNe hee so (Go) (j ) Case (2). A circular doublet-disk (a circular magnetic shell) of uniform strength m per unit area is placed with its centre at O, its axis along OU, and its positive side turned towards ©: it is required to find in terms of Bessel functions the potential produced at P and the component magnetic field intensities. It is required also to show that, if A be a point on the circle of centre O and radius a, the axially directed magnetic field intensity at P is equal to the radial or y-component of the electric intensity at A due to a parallel uniformly electrified circular ring (centre C). Clearly —OV/dz.dz is the potential at P due to such a doublet-disk, if m=od:. Calling this potential ©, we obtain from (7) Oz arma Cin CNG) J UN INGH ern a's) (a) xv 0 The forees Z and R for this magnetic shell are given by Li rem e~" Ji (Aa) Jo(Ay) AA, 3 J0 ’ oO (10) hi 2mina| e~* Ji (ra) Ji Ay)rAdn. 0 Obviously if this doublet-disk have its centre at C, y(=h) for its radius, and OC for its axis, and face in tha same way Ee? 204 Prof. A. Gray on Electric as the disk just considered, the magnetic force at A in’ the direction OC is by (10) Li emt ( e~ Jy(Aa) J,(rAy)rdn. : = 1G 0 But going back to (5) we see at once that the potential at A, due to a uniform ring, centre C, of electricity (or matter) of uniform linear density ~=ap, and of radius p=8, is V = 2npa| Jb) TQalan. 0 The field intensity at A in the direction of y is —QV/Ody,. or if we call it Ry R’ = 2mpa | eo Jo Ab) J(Aa) Add, 2 een 0 which, with ~=ma/b, is by (10) exactly the field intensity Z at P due to the magnetic disk centre O, but turned through iu right angle. This is an example of the theorem of reci- procity of magnetic and electric field intensities at corre- sponding points stated in my paper (Phil. Mag. May 1919). A similar result can be obtained for the other component. Case, (3). A uniform right circular cylinder of volume density p (of electricity or ordinary matter) has its axis along OC, and its right-hand end (fig. 2) at O: required the potential and the field intensities in terms. of Bessel functions. ce z wae 2] | "= = aa esos 2 '—- mee oe a = We write pdz for ¢ and integrate (6) from the end nearer to the end further from C. If we now suppose z to be the- and Magnetic Field Constants. 205 distance from C of the near end of the cylinder, and denote the length of the cylinder by J, we get © l V = 27pa i en —e™™) J, (Na) Joiry) ae | 10 y i= 2p e-*(1—e™) Ji (Aa) So (Ay) oe ie (4) 0 ~ 0 ae R = | e~*(1—e-”’) J (Aa) Jy (Ay) ‘ 0 Cass (4). A right cylindrical array of equal doublet-disks exists closely and uniformly packed from. O to O! (fig. 2) with their common axis OCU produced back- ward: to find O, Z, and R for this arrangement. Let again the length of the cylinder be /. The arrange- ment is obviously equivalent to a positive disk of surface density o at O, and a negative disk of numerically the same density at O’. Hence, if z be the distance of P from O we get at once ! : QO = 2nca | e*(L—e~”’) J (Aa) Jo(ay) ee | 0 ae | L= dren e-* (L—e™) J, (Aa) Jo(Ay) da, uy) R= 2noa | e (L-e™) J, (ra) Ji (ay) dr. J 0 3. In the four cases above specified we can find the total flux of force across a circle of radius 6 with its centre at C. Multiplying in (7) under the sign of integration by ydyd@ and integrating from y=0 to y=6, and from 0=0 to 0=2r, we find : Tor case (1), ti) 2r i B00 \ i Ly dyd@ = tnrtoa| e- Ji(ra) J (Ab) ak 0 Jo vd nr (16) Yor case (2) we find in the same way, a Tee) 3 aw) 20 in ( { Ly dy dd = 4n*mab | e* J, (Aa) J,(Xb) dr. (17) wig vo Vo | . J9 206 - Prof. A. Gray on Electric For case (3) the result is 12m | dd. ( {, Lydyd@ = 4a pal ~*7(L—e7™) J (Aa) J (Ad) x Finally, in case (4), the most important of all, b st ; oa ae se ‘ dx Ly dyd@ = An?cabt e~*(1—e-™) J, (Aa) Jy (nb) a. 0 Jo 0 In the electromagnetic analogue, in which a current sheet takes the place oP the cylindrical array of doublet-disks, we have to replace a by Ny, if the sheet be produced by N circular turns of fine wire, with a current 7 in each. 4. Comparing the R of (14) with (19), we have the remarkable result that the y-component of force at P (a point on the circumference of the circle of radius 6), due to the uniform cylindrical volume distribution of density p, is to a constant identical in value with the total magnetic induction through that circle due to the current sheet. To convert the former into the latter value it is only necessary to multiply by 27ba/p. This is a consequence of the reciprocal theorem that the radial force at a point P (on the circumference, say, of a circle of radius )) due to a uniform coaxial disk of matter of any radius 4, say, is, to a constant, equal in value to the magnetic induction through the circle of the disk produced by a current flowing in the circle of radius 6. The reader may think of two cireular coaxial cylinders, one of radius a and length / lying to the left of O, the other of radius 6 and situated to the right of C. Wither may be taken as the volume distribution and the other as the current sheet or array of magnetic shells. To fix the ideas, take that of radius a as the volume distribution, and the other as the current sheet. Then the magnetic induction through the cirele of radius a, centre O (fig. 2), due to the current sheet, is, to a constant, equal to “the value of R produced at any point of the cireumference of the circle of radius }, and centre C. Or, since the induction through the circle of radius a, due to the current sheet just specified, is equal to the induction through the circle of radius 6, due to a current sheet coinciding with the cylinder of radius a, we may suppose the volume distribution and the current sheet to occupy coincident cylinders. Then the magnetic induction through any coaxial circle due to the current sheet i is, to the (18) (19) and Magnetic Field Constants. 207 constant stated above, equal to the value of R produced by the volume distribution at anv point of that circle. ». By means of the results which have been obtained above, and various dynamical considerations, we can evaluate certain elementary Bessel function integrals, and obtain some other results which seem of interest. J am aware of course that these, and a large number of other definite integrals of greater complexity, have been evaluated analytically. Taking the value given by (7) above for the axial force F, at the point P, produced by a uniform disk of radius a, the axis of which is OC, we see from (16), if © coincide with O, so that <=0, that the surface integral of normal force over the two circular ends of a concentric disk-shaped surface of radius b, tightly enclosing the disk, is Sa2aab \ Jawa) J (Nb) we 0 But the mass (of electricity or matter, as the case may be) enclosed by the surface specified is waa”. Hence we have Sieab Ji (Aa) J y(AD) = = An eae. Hic. (20) a0 that is {100 WOU, (Oa ei) It follows that, if b=a, { “T2Qka) dry = 4 (22) Multiplying now (6) by 2aydy, and integrating with respect to y trom O to a, we obtain the surface integral of potential over the disk. It is Sf ee 29 ydy = An’ou | (Aa) ex (23) 0 «0 If, to fix the ideas, we suppose the disk to be composed of attracting matter, the exhaustion of potential energy involved in increasing o by do is the expression just found multiplied by do. Hence the total exhaustion of potential energy, involved in building the disk up from density zero to density a, is eee dr Pian o- , (Na) Sag hd es (24) By) 208 Prof. A. Gray on Electric . We can easily find another expression for this energy. We calculate first the potential at a point A on the edge of Fie. 3. ee the disk (taken first of radius 7). From the diagram (fig. 3) we see that 27 sin @ 7) Va=ol ( d0dr =4Aor, . 11 em Jo 0 The exhaustion of potential energy involved in adding a ring of breadth dr is therefore Aor .2rrordr = 81a0°r' dr. Hence the whole energy exhausted in building up the disk of radius a from matter formerly dispersed through infinite space is a. + 1s da Sa eee Equating this to the former value, found as a Bessel function integral, we get { 200) o> a Putting a=1, we get Tp, > 2 ; { Jy (A) x2 ae oye e . e . . (27 ) which is the known value of the integral. It may be possible to obtain more general forms of these results by using a more general form of potential. 6. We now eoneider the following problem. A uniform thin circular disk of attracting matter has radius 6. It is and Magnetic Field Constants. | 209 required to find the. surface integral of potential over a circle of radius a concentric with the disk, and coinciding with the face of it. It is supposed that b >a. Fig. 4, Consider a point P at distance f from the centre O, and denote the angle OPM by y. Taking the disk as of unit surface density, we get for the potential V at P V =2( rey. #4 ensue Nets AZ Os) oD The integral of this over a small area /dfdd at P is Vfdfdg. Hence for the integral over the circle of radius a we have I = 4n| { fafa, SN fhe ak caine (29) wo 2) 0 for it is clear that the potential at the point P is also the potential at every point on a circle of radius f described about the centre of the disk. We have to calculate Tt Aaa { | Jr dj dw. 0 0 210 Prof. A. Gray on Electric Keep y fixed and consider 7a { rf df. 0 6? = 7? + f?—2rf cos ¥,, By the figure which gives r= feosh+ Vb'—f? sin? w. We take the upper sign and integrate for the semicircle above OP in the diagram. Substituting, we obtain ) rf df =| f? cosw+f V¥i?—/? sin? Ww) df 0 1G ‘ay | is sete cost 2 aly Feng > ee vil 1 b3 =, @coswts i —(1-/? sin?w)?t, . (30) where £? is written for a?/b’. We have now to integrate this with respect to wy. First, it is clear that 4a* cosy contributes nothing to the definite integral between 0 and 7 as limits. Next we get ol Me aa ce, Ot ae Sl $l and this is infinite at each ., It will be found, however, to be cancelled by the same integral, arising from the term which remains to be integrated And affected by the — sign. Integrating by parts we obtain » § A dP sin? pi = — (cob thy) P sin? py sin’ wv : ie Sn —(« a Wa — k? sin? aby? bia But - (og Raine Ste one iy ss an sin? f " (1— eae sin? et » (L—F? sin? wr)? = —2)°F—-2H+42F, .. . Saige where F and E are Legendre’s chic hee alliptic integrals of the first and second kinds to modulus k=a)b. and Magnetic Field Constants. 211 Next we have Us i vr sin yr cos eee Mi 2 ig WC et sin? Wr)? [y(1—2? sin?) |, ~12( (1-2 sin? wp) dy ma) eto toe) Collecting these results we obtain T= §r{l(1+)E—0(1—22)F} —= Sm Ob? + a7) Wi 0b? — a7) Ph. oe 2 188) For surface density o we have to multiply this result ine follows that, if to the disk just considered we add a second disk of density o’ and radius a, potential energy is exhausted of amount P = 8700'b} (0? +a7)H—(b?—a?) FY. 2. BA) This is apart from the exhaustion of potential energy involved in the formation of the second disk, which is itself under the potential of its own matter. As we have seen in (26) above, this latter potential energy amounts to 87re’?a’. Now by (6) above the potential at the point P is V=2nai J (Ab) Jo( ee Nr Mae (35) and the integral of this overa ring of radius f and breadth df is nV fdf = Arab \ Jib) fIo(af) ee a Hence the total surface integral of potential over a concentric circular area of radius a is -@ 10 oO DV 2a Via nea | Jan) | Soe) rar = (36) /9 v9 v0 But we have seen that { Jef) far = ~ Ji(Aa). 212 Prof. A. Gray. on Electric Thus = dr a On 2a! { 0 Vid = 4n%ea'ba{ J,(va) Td) Jo is the potential energy exhausted in superimposing con- centrically an already constructed circular disk, of radius a and surface density o', on a previously existing disk of surface density o and radius 6 (>a). Comparing this result with (34) we obtain ie dn 2 {, Jj (Aa) J, (A) Nw = a, 1(0? + a?) Ki = (0? — a*)F}, (38) where E and F are, as before, the elliptic integrals to modulus a/b. . The whole potential energy exhausted in constructing the two disks separately, and then putting them together in the manner described, is thus DIE ie In ot ee In 277a7b? | J?(Ab) = aE 2ntata? | J (ra) = a Agaa'ab ( Ie (Aa) Ji (AD) - = §[0°b?+ 07a? + ma0'b{ (b? +a?) H— (b?—a) F} ]. (39) If c=oa' this gives ss adr apy a ine 0 e/ 9 0 - [08 +a? + wb {(b? + a2) B— (U2—a?) F}]. (40) 3 7. It is possible to express the Legendrian elliptic integrals in various ways as Bessel function integrals. Let us calculate for a uniform thin ring (of unit line density) the potential produced at a point P ona coaxial circle of radius 6, and at an axial distance ¢ (fig. 1). We find by (12) that it is V =20 \ e-°J,(ra) J o(ab) dr. 2 | Calculating directly we get easily 4a FB i 2/ ab Y= asspeap® @rbpeepl e.> oY and Magnetic Field Constants. 21 Hence Q/ab aa ee | =i7 J y(Na) To(Xb) ary, FL, f(a+b)2+er3 "| a : Le ea) ORO) a ; (42). where F is the ele: linge integral of the first kind to the modulus indicated. Again, the lateral magnetic force at the point P due toa circular magnetic shell coinciding with the cirele of radius a, at the axial distance c from P,is by (10) above (for unit strength of shell) ona, eS (Aa) IT (AD)A AX. ‘0 But it is easily found by direct integration that the magnetic force is Dane OF eye ee r) rb \a?-+-b? + c? — 2ab ‘ where the Sutpuue integrals are taken to the same modulus. 2r/ ab] ( (a+b)? + ys as before, and 7,=(a?+0?+ c+ Qab)?. Thus we coe 2¢ a+l?+¢ AN el Ne br, = a E-*) if 2a | : ae MM in From the value of F already found in (42), and this last. result, we may obtain an expression for H. The mutual induction between a circle of radius a and a coaxial helix or current sheet of radius 6 and n turns of wire per unit of the length / of the cylinder, with the positive end at distance 4/ and ane negative end at distance —3/ from the circle, is : I ee Ba (6 Inn (a+ blk E (B-F)+-° -F) |, where k& is the same modulus as before, and II is the complete elliptic integral of the third kind, that is J W = \~ Na Jo (—e? sin? y)(1—# sin? wv)” with c?=4ab/(a+b)’. | But further we have, by (19), since the integration is here supposed taken from -=3/ to -=—4/, Brel (e3’ —e-3") Ji(ha) J, (Mb) ee a) ea if il iF c Bey (a+b)?+2? Ee s 214 Dr. Norman Campbell on Thus we obtain (ato BF) +15 (0-8) | = 2ab | (e2’ —e-2!) J,(Xa) J (Ab) - (44) e 9 Hence with the aid of the preceding results a value of the third elliptic integral II can be found in terms of Bessel function integrals, | The magnetic potential w, at the point P, due to unit current in the circle of radius a and centre O, is, with OC =z, c*=4ab/(a+b)?, given by : = o( dw i 5 (1- me sin? +) (1—/? sin? yr)? c+b dy | 2 ane (? dp eee (4 5 — 2 . 2 Cee (14 : “sin? yw JU sin? y)? Jy i—#* sin |» as may be verified by caiculating » as the solid angle sub- tended at P by the circle. Thus from (15) and (19) above and the results for HE and F already acquired we obtain a Bessel function expression for this potential. A large number of alternative expressions for the elliptic integrals could be constructed, but it is doubtful whether any -of them would be found of any particular use. Mr. T. H. Havelock (Phil. Mag. xv. 1908) has evaluated in power series some of the Bessel function integrals here dealt with and applied these series to the calculation of mutual and self inductance. XVI. Time-Lag in the Spark Discharge. By Norman CampsE.., Sc.D.* Nore.—The experiments described in this paper were carried out -at the National Physical Laboratory under the direction of the Advisory Committee for Aeronautics. Their results have been communicated in a series of confidential reports to the Internal Combustion Engine Sub-Committee cf that committee; per- mission has now been given for the publication of any portions -of the work which appear of pure scientific interest. * Communicated by the Author. Time-Lag in the Spark Discharge. 215 I wish to express my gratitude to Sir Richard Glazebrook, Director of the National Physical Laboratory, and more especially to Mr. C. C. Paterson, in whose department the work was done, for many suggestions and much invaluable advice. Introduction. T is a fact familiar to all who have been concerned with spark-ignition systems that the sparking potential of a given gap in a given gas may vary with the frequency of the applied potential. In some gaps at least, sparking will occur only if the peak potential of a rapidly alternating source (such as a magneto) exceeds very greatly the steady potential which would be necessary to start a spark across the gap. In other gaps the peak potential of an alternating source required to spark the gap is very much more nearly equal to the steady potential sparking, though there is some conflict of evidence whether the power of a source to cause the spark is ever completely independent of the frequency and ever depends simply on the peak potential. It is known further that among the gaps in which the sparking potential varies greatly with the frequency are those in which the electrodes are fine points; but it is generally suspected that the dependence on frequency is not determined entirely by the shape of the electrodes, and that the nature of the material of which they are made and of the gas in which they are placed may also exert some influence. — The matter has been investigated in some detail by F. W. Peek, who described his results in his book ‘ Dielectric Phenomena in High Voltage Engineering ’ (1915). The ratio of the peak potential of an alternating source which will just spark the gap to the steady sparking potential is called by Peek the “impulse ratio” of the gap. For a “sphere gap,” or any gap in which the radius of curvature of the electrodes is large compared with the distance between them, the impulse ratio was found to be 1 for all spark potentials in air and some other gases, and for frequencies of the applied potential as high as a milliona second. For a “needle” or “point” gap, on the other hand, the impulse ratio is always greater than 1 and in- creases both with the frequency and with the spark potential, if the spark potential is varied by changing the distance between the electrodes. Values as great as 3 or 4 were found. This difference between needle and sphere gaps is associated with another difference which has been described by Peek and his colleagues of the General Electric Co. of 216 Dr. Norman Campbell on America. It is that whereas in the sphere gap the sparking potential is independent of the humidity of the air if the pressure is unchanged, the sparking potential of the needle yap varies considerably with the humidity, even when the frequency is so low (say 500 a second) that the impulse ratio: is indistinguishable from 1. Mr. Peek explains his observations on impulse ratio by supposing that there isa “time-lag ” in the passage of the spark. According to his theory, a finite time must elapse after the potential reaches the steady sparking potential before the spark will pass. If a potential greater than the steady sparking potential is imposed, the time-lag. though less than before is still finite : it diminishes with increase of the excess of the appiied potential above the spark potential. It is obvious that such a theory will explain the variation of the sparking potential with the frequency, and indeed it may seem that it is the only theory that can possibly do so. For if the spark gap can distinguish between different frequencies, there must be involved in the passage of the spark some process requiring a finite time against which the frequency can be measured ; and it is difficult to see where this process can occur except between the attainment of the requisite potential and the passage of the spark. But it remains open to inquiry by what the time-lag is determined, whether it is constant in successive discharges, and why it is so very different for a sphere gap anda needle gap. Peek supposes that the period of the finite time-iag is occupied by the development of a ‘‘corona” or brush discharge, which is a necessary forerunner of the true spark ; in the sphere gap, so long as the sparking distance is not large compared with the radius of the spheres, the corona does not develop. But he does not explain (indeed it is not his purpose to do se) why the corona is a necessary forerunner of the spark in a needle gap but not in a sphere gap. The conception of a time-lag in the spark discharge is, of course, not new. It is discussed at some length and the experimental evidence for it reviewed at the opening of Chap. XV. of J. J. Thomson’s * Conduction of Electricity through Gases.’ Thus it is well known that if the potential across any gap is raised slowly, the spark will not usually ‘start until a potential is reached considerably greater than that required to maintain the spark once started, and that a spark will sometimes pass after a steady potential has been applied for a time amounting to several minutes, although it did not pass when the potential was first applied. This. ‘lag’ in the spark is abolished by ionizing the air in the Time-Lag in the Spark-Discharge. 217 gap, or by causing the emission of electrons from the electrodes. It is generally attributed to the fact that a few casual ions must be present in the region of the electric field in order to start the ionization by collision which results finally in the passage of the spark. If this expla- nation is correct, it is to be expected that the lag should be greater in the needle gap than in the sphere gap; for in the former the volume of the region in which there is an intense electric field is much smaller than in the latter: in the former the gradient is concentrated in the immediate neighbourhood of the electrodes ; in the latter it is approximately uniform over the whole distance between the electrodes. In the needle gap therefore, the chance that there will be a casual ion at a place where the electric field is sufficiently great to cause ionization by collision is much smaller than in the sphere gap; and, if we rely on chance to produce an ion in the right place, the time that we may have to wait before that event occurs will be correspondingly longer. But if it were lag of this nature which caused Peek’s observations, it would be expected that the impulse ratio for sphere gaps should be rather greater than 1, that it should be greater than 1 for all periods of alternation less than a few seconds, and that, for such periods, it should be de- termined by the wave-form rather than the frequency; for the chance of an intense electric field coinciding with the presence of a casual ion in the right place would depend on the fraction of the whole period during which the peak lasted and not on the absolute period. None of these ex- pectations are fulfilled. Again, the time-lag should be extremely irregular, the potential at which the spark passes should be extremely variable and it should sometimes happen that a spark passes at practically the steady sparking potential ; further, the time-lag should be abolished by ionizing the air in the gap. Peek denies that in his experiments initial ionization had any effect upon the spark potential and makes no mention of any irregularity in his measurements of that potential. In this last matter, Mr. Peek’s experience seems at first sight inconsistent with some facts familiar to all magneto manufacturers. They realize that the point or needle gap is utterly unreliable for measuring the peak potential of a magneto, not only or mainly because its sparking potential depends on the frequency, but because its indications are so very indefinite; there is always a very large range of magneto potentials over which the gap will sometimes spark, but will not spark at every break. Thus a point gap may Phil. Mag, 8. 6. Vol. 38. No. 224. Aug. 1919. Q 218 Dr. Norman Campbell on easily be found which will sometimes pass a spark when the magneto peak potential is no greater than 7000 volts, and yet will not passa spark at every break unless the peak potential is 12,000 volts; the sparking potential of sucha gap for steady voltages might be 5000 volts. This indefiniteness of the spark potential is in accordance with the presence of a time-lag due to the necessity for casualions. Further, it is known that the indefiniteness of the point gap can be very greatly diminished by the device introduced in the arrangement called the ‘“‘ 3-point gap.” This arrangement is shown in fig. 1. A and B are the main electrodes connected Fig. 1. Cc pA B ee eae <__ ee 0) | L To high potential 7o Earth to the magneto terminals; Cis a pointed rod insulated from both A and B with its point very near A; B is usually earthed. As the potential of A changes a charge is induced on ©, and the difference of potential of A and C is sufficient to cause a spark to pass across the small gap between them. If C is carefully adjusted by triai, a position can be found in which regular sparking (2. e. sparking at every break of the magneto) begins at a peak potential much lower than that at which it beginsin the absence of C; for instance, the regular sparking potential may be reduced from 12,000 to 8000 volts. The obvious explanation of the changes is that the ions produced by the “pilot” spark between C and A diffuse or are projected into the space between A and B, and reduce the ‘‘ lag” in just the same way as ions produced by the action of ultra-violet light on the electrodes. These experiments then seem consistent with the expla- nation of the lag by casual ionization rather than by Peek’s theory ; while Peek’s observations seem consistent rather with his own theory, which asserts that the lag is not a variable quantity depending on chance, but the time required Time-Lag in the Spark Discharge. 219 for a definite process which must be completed before the spark can pass. Some recent experiments seem to throw further light on the matter; and though like all experiments undertaken primarily for practical objects, they do not serve to elucidate the matter completely, they are worth recording if they give any information on a matter of some pure ‘scientific interest. Liffect of preiiminary ionization. If the action of the third point in the 3-point gap is due simply to the preliminary ionization of the main gap which it causes, then any other source of ionization should produce the same effect. A gap of 5 mm. between pointed rods 2°5 mm. in diameter was found to have a steady sparking potential of 4800 volts. It was connected to a magneto, giving a potential of which the frequency was about 5000 per sec. and the magnitude could be altered without change of wave-form by altering the strength of the magnet. Regular ‘sparking was obtained with a peak potential of 11,000 volts, while a spark sometimes passed when the peak potential was as low as 6600 volts. The addition of the third point, suitably adjusted, reduced the peak potential for regular ‘sparking with the same magneto to 7200 volts, while the steady sparking potential and the peak potential which would ‘sometimes give a spark remained unchanged. The third point was then removed and a tube containing a few milli- grams of radium brought near the gap ; the peak potential for regular sparking was reduced to 8100 volts, while that which sometimes gave a spark and the steady sparking potential again were unaltered. The action of the radium is then precisely similar in nature to that of the pilot spark, though it was rather smaller, doubtless because the ionization produced by the radium was too small. It would be expected again that anything which would decrease the number of casualions present would increase the peak potential required for regular sparking. Here again the -expectation was fulfilled. Ifa pair of plates char ced toa potential difference of 10,000 volts were placed on either -side of the gap and at a distance of 2 em. from the electrodes, regular sparking could not be obtained with the highest peak potential which the magneto would give, 13,200 volts ; but again a spark sometimes passed at 6600 volts. Again, if casual ions can thus be removed by a field between external electrodes, they ought also to be removed by the field between the sparking electrodes. It is to be expected on il 220 Dr. Norman Campbell on that if the time occupied by the rise of potential of the electrodes from zero to the peak is increased, the number, of casual ions present when the sparking potential is reached will be fewer and the sparking be made less regular. Some observations in which the wave-form of the potential given by the magneto was changed slightly, either by i inserting a. high resistance between the magneto and the gap or inserting a capacity in parallel with the gap, seemed in decerdenee with this expectation. On the other hand, if the rise of ootential were made sufficientiy slow the opposite effect was obtained, and regular sparking occurred at lower peak potentials than before. Such a change must, of course, occur as the condition of a steady applied potential is. approached. But it should be noted that when the regular sparking potential was decreased in this manner, the potential at which a spark would sometimes pass was also decreased ; while when the regular sparking potential was increased by changing the wave-form, this potential was unaltered. It appears therefore that tne difference between the recular sparking potential and the potential at which a spark will sometimes pass can be accounted for completely by the theory which attributes that difference to the necessity for the presence of casual ions to start the discharge. On the other hand, that theory will not account for a difference between the least peak potential which ever gives a spark and the steady sparking potential. It is of course difficult. to be sure of the limiting peak potentiai of the magneto, below which a spark will never pass, but all evidence pointed to a clear difference between this potential and the sparking potential with a steady source: in particular, consecutive determinations of the limiting potential were very consistent. The conclusion is indicated that there are two kinds of “time-lag” involved: one an irregular lag connected with the presence of casual ions, the other a regular lag which causes the difference between the limiting peak potential and the steady sparking potential, and sired may possibly be explained by Peek’s theory. The fact that Peek did not observe the irregular lag and the change of this lag with initial ionization may plausibly be attributed to his use of much higher frequencies than any given by a magneto. The regular lag inereases with the frequency and, when the frequency is as great as a million a second, may be so great. as to obscure entirely the irregular lag which, as has been suggested, should not vary much with the frequency Te iG exceeds some quite low limit. But though the irregular and regular lags seem distinct. Time-Lag in the Spark Discharge. 221 and different in origin, there is certainly a connexion between them. For it was found that in a sphere gap, as indicated by Peek’s experiments, there is in general no regular lag, and the limiting peak potential of a magneto which will sometimes cause a spark is the same.as the steady sparking potential. (Some exceptions to this statement will be noted later.) And when the limiting peak potential and the steady spark potential are thus identical, there is also little or no irregular lag; the sparking potential with the alternating source is definite, and the limiting potential which will sometimes pr oduce a spark is also the peak potential which will produce regular sparking. In such gaps if a spark sometimes passes, it always passes, and no effect is produced on the sparking potential, either with steady or alternating sources, by increasing the initial ionization of the gap. The two kinds of lag appear together and vanish together, when the change is made from pont to sphere electrodes. The actual sparking potential. Some curious results were obtained in experiments directed to determine what is the potential between tae electrodes when the spark actually passes. If there is a time-lag, whether regular or irregular, this potential will not in general be equal either to the steady sparking potential or to the peak potential of the alternating source. It can hardly be less than the former, but it will usually be less than.the latter. If the time-lag is regular, the actual sparking potential will be constant so long as the wave-form is constant ; but if the time-lag is irregular and due to the casual ionization, the potential when the spark actually passes may presumably have any value greater than the steady sparking potential ; it will simply be the value which happens to obtain when the casual ions are present at the right place*. Accordingly a determination of this actual sparking potential should indicate whether the lag is regular or irregular. The determination is made possible and easy by the results described in a recent paper by Mr. C. (> Paterson and eee It is shown there that if a spark gap is connected * The “right place” will depend to some extent on the potential prevailing. For the ion has to be in a field of oiven intensity in order to start the discharge ; and the region in w hich such a field exists will be the larger the greater the potential difference between the electrodes. But this consideration does not alter the conclusion that the actual sparking potential should be determined by chance, t “Some Characteristics of the Spark Discharge and its effect in igniting explosive mixtures,” Phys. Soc. Proc. “Mareh 14,1919. (The paper is not published yet.) 22? Dr. Norman Campbell on to a source of steady high potential through a large resist= ance (10 megohms or more}, a discontinuous stream GF sparks passes across the gap; Q, the quantity of electricity passing in each of these sparks, is the product of V, the sparking potential of the gap, and C, the capacity in parallel with the gap. In these observations there was no time-lag, and the actual sparking potential was the same as the steady sparking potential; but if the two are not identical, then, so long as the discharge still consists of successive individual sparks, V in the relation Q=CV is doubtless the actual sparking potential, that is mine potential between the elec- trodes when the spark starts ; for the time that the individual spark lasts is so short that, w “ith the highest frequencies inves- tigated here, the change of potential during its passage is too small to be appreciable. The experimental devices described in the paper to which reference is made enable Q and C to be measured ; accordingly V can be determined. This method of determining the actual spark petential was applied to a gap between needle-points, the steady spark potential] being varied between 2000 and 6500 volts by variation of the distance. The steady high potential, to which the gap was connected for a time long enough to allow one spark to pass, was of course always greater than the steady spark potential, and could be increased to 12,000 volts. It was found that, owing to the irregular lag, the spark did not always pass even when the greatest potential reached ex- ceeded considerably the steady spark potential, and, owing to the regular lag, never passed unless this potential was some- what greater than the steady spark potential. On the other hand, “Gf the spark did pass, V turned out always to be the steady spark potential, and to be the same as it was if there were substituted for the point gap a sphere gap, free from time-lag, with the same steady spark potential. This coincidence of the actual sparking potential with the steady sparking potential was found for all the conditions investigated. No theory can be suggested which explains it fully, but it receives a partial explanation on Peek’s theory of the lag. Onthat theory the period of the time-lag is occupied in the development of the corona. Now when the corona is formed, the current through the gap is not passing between the electrodes only, but from the high potential electrode to all the surrounding earthed conductors of which the low potential electrode forms a small part. The quantity of electricity passing during this stage will not be included in Q, which is the quantity received by the low potential electrode. Accordingly, if the discharge starts in the form Time-Lag in the Spark Discharge. 2a of acorona ata potential considerably higher than the steady spark potential and continues for a finite period until it is ultimately transformed into a spark; and if, during this period, the potential difference between the electrodes falls continuously: then the measured value of V, which is the potential at which the spark as distinct from the corona starts, will be lower than the potential at which the corona starts, and lower than the potential required to cause the spark to pass ultimately. Now, for a reason to be given immediately, it is probable that in these observations the potential between the electrodes did fall continuously after the discharge began, and it is therefore possible to explain why V is always less than the potential required to start the discharge. On the other hand, it is not obvious why it should always coincide with the steady sparking potential ; that is, why the development of the corona and the preparation for the true spark should never be complete until the potential has fallen to the steady sparking potential. Nevertheless the observations do provide some support for the theory that the process which occupies the period of the time-lag is one in which current is flowing to other places than the low potential electrode. However, it is not probable that this process consists in the Full development of the ordinary corona. The period during which the potential was applied to the spark gap was about -005 sec. and very nearly the same in all these experiments. It was divided into two parts, in the first of which the potential rose from zero to the maximum, in the second of which it remained constant. It was found that the frequency with which the spark passed decreased with the time occupied by the rise of potential between the electrodes from zero to its maximum value. This observation accords with that de- scribed on p. 220, and the explanation given there is probably correct. But it was found also that, if the potential was raised very slowly, then, even if it were kept on indefinitely, no spark passed, but a visible corona was formed on the high potential electrode. Owing to the presence of a very high resistance between the gap “and the source of potential, the current conveyed by the corona reduced the potential between the electrodes so much that, even when the potential would have risen to 12,000 if no current had been flowing, the potential did not actually rise above 4000 volts, the steady sparking potential of the gap. (It is for this reason that it is probable that the potential always fellas soon as the discharge began at all.) It appeared then that if the full corona did develop, the spark would never pass owing to the drop in my 224 Dr. Norman Campbell on potential along the high resistance. Accordingly the faet that, if the potential were raised to its maximum more quickly, the spark did sometimes pass indicates that in these ‘ases the corona did not develop fully. The full corona cannot be a necessary forerunner of the spark, but itis quite conceivable that seme initial condition which may develop either into the corona or into the spark is such a necessary forerunner. Vheory of the regular lag. And it is not difficult to suggest what this condition may be. Attention has so far been concentrated on the fact that there is a lag, regular and irregular, in the needle gap ; but from one point of view it is much more remarkable that there is no time-lag in the sphere gap, even when the frequency is as great asa million a second and the sparking distance as great as 10 em. For if, as the theory of casual ions suggests, the discharge is started at one point in the gap and spreads thence to the remainder, it would be expected that, apart from the irregular lag, there should be a regular lag equal to the period required for the ions to spread from the starting-point throughout the whole gap. Now the sparking field is about 30 000 volts per em.; the velocity of the ions about 1°5 em./(sec. volt); and the time required for the ions to spread over a gap of 10 cm. should he of the order of 10~* sec.* How then can a spark occur when the whole duration of the potential is little more than 107° sec. ? The answer is, of course, that the discharge does not start in one place only in a sphere gap. It starts at a great many places at the same time, and the lag which must occur is only the time required for the ions to travel from one of these places to the other. But if this is so, the difference between the sphere and needle gaps is at once explained; for in the needle gap the field midway between the needles is usually much too small to cause ionization by collision. The discharge cannot be initiated there; it has to spread to such regions “from the neighbourhood of the electrodes where the field is more intense. It is the time re equired for this spreading which is doubtless the regular time-lag ; it will increase, as is shown by Peek’s observations, aa the distance between the electrodes ; it will increase also with the humidity which reduces the speed of the ions. * This estimate might be reduced about 100 times if it is supposed that the ions concerned are electrons which remain free the whole time. But, according to the accepted theory, the spark discharge cannot stait without ionization by the positive fone , so that these must have time to travel through the whole gap. Time-Lag in the Spark Discharge. 225 This very obvious view seems to explain qualitatively the whole matter. - It shows the connexion between the regular and irregular lags, for the occurrence of both is due to the fact that the region occupied by the field in which ionization by collision can occur is much smaller in the needle gap than inthe sphere gap. It explains why, during the period which leads up to the spark, the current Howing from one electrode does not all arrive at the other, and it shows also that the aischarge during this period need not be the same as the fully developed corona. The matter needs, of course, much fuller investigation, and there are many ‘lines of experi- mental research which might be followed up. But since it is unlikely that I shall pursue the subject further myself, it has seemed well to record these fragmentary observations. Hard and soft gaps. It is not only in point gaps that there is a time-lag and that the sparking potential depends on the frequency; the same dependence is sometimes found in sphere gaps and others in which the sparking distance is not large compared with the radius of curvature of the electrodes. Sparking plugs are often found which are “hard,” and will not pass the spark from a magneto, although to all appearance they are precisely the same as “soft” plugs which will pass the spark, and although they have the same steady sparking potential as such soft plugs. At first sight this difference between hard and soft plugs seems essentially s similar to that between point and sphere gaps. A large amount of information on the subject was obtained in the course of routine tests on numerous types of sparking plug. In these tests the sparking potential of the plug was measured, both with a steady potential and with a magneto, in air at atmospheric pressure and also in air at a pressure of about 5 atmospheres. (By the sparking potential with a magneto is always meant the peak potential which will cause a spark.) The following facts were established :— (1) Plugs could be divided into two classes, according as their sparking potentials were definite or indefinite. When the sparking potential was definite the difference between the potential which would sometimes give a spark and that which would always give a spark was “scarcely greater than could be read on the voltmeter. At atmospheric pressure, when the sparking potential was about 2000 volts, the difference would not exceed 50 volts ; atthe higher pressure, when the sparking potential was about 9000 volts, the ee eat ———s lee es we Pri eon aes - Se ee to 226 Dr. Norman Campbell on difference would not be greater than 100 volts. On the other hand, in extreme cases of indefiniteness the difference at the lower pressure might amount to 1500 volts; at the higher pressure it was never found to exceed 600. (2) lf the sparking potential was definite, the sparking potential with the magneto was equal to the steady sparking potential. Jf it was indefinite, it was hight with the magneto*. In this matter, then, hard and soft gaps resemble point and sphere gaps; irregularity is associated with an ‘impulse ratio” greater than 1. (3) The “impulse ratio” was much less at the higher pressure, when the sparking potential was greater ; in fact, ut these higher pressures it could never be established cer- tainly that the impulse ratio was really greater than 1. In this matter hard and soft gaps are unlike sphere and point gaps, for in these the impulse ratio increases with the spark potential. (4) The cause of “hardness” was connected with some very easily variable surface condition of the plug. It had nothing to do with the geometrical form or material. The hardness was liable to change with the most trivial alterations ; it usually seemed much easier to make a soft plug hard than to make a hard plug soft. These observations were supplemented by others on a specially constructed gap in which one electrode was a plane. of which the material could be varied, the other a steel sphere. At the same time a very convenient and sensitive: method for detecting hardness was introduced, which de- pended on the application of the potential for a very short time: only. It was a modification of the device of the ‘ ‘auxiliary gap,’ which has been applied to making the spark pass across a “leaky” plug, that is one in which the insulation has become covered with a conducting layer. The gap under consideration was shunted by a resistance. of 10,600 ohms, representing the ‘‘leak.” If the distance between the electrodes is now set to represent a steady sparking potential of about 5000 volts, a magneto will not. produce a spark across the gap. But if, between the gap and the magneto, is inserted a second gap ‘which is not leaky, the passage of a spark across this auxiliary gap will often be accompanied by a spark across the leaky gap. In order that this may happen certain conditions, involving the capacity in parallel with the two gaps and their sparking * Some few exceptions to this statement were noted, the sparking potential being less with the magneto. These observ Bienes were never explained. Time-Lag in the Spark Discharge. 227 potentials, have to be fulfilled. The auxiliary gap will be more effective the greater is its sparking potential and the greater is the ratio of the capacity in parallel with both gaps in series to the capacity in parallel with the leaky gap. These relations were investigated and found to be in accordance with the obvious theory of the action of such an auxiliary gap. When the spark passes across the auxiliary gap, its equivalent resistance becomes small compared with that of the leak, and the whole potential to which the con- denser in parallel with both gaps has been charged by the magneto 1s thrown on the leaky gap. IEf the capacity of this condenser is sufficiently large to maintain this potential while the smaller capacity in parallel with the leaky gap is charged up, a spark will pass across theleaky gap, so long as its spark potential is less than that given by the magneto. In the experiments on the hard gaps the sparking distance of the main gap was always set at the same value, so that the sparking potential for steady potential was the same ; for this potential depends only on the geometrical form of the electrodes and not on the material*. The capacities in the circuit were also always unchanged ; but the sparking dis- tance of the auxiliary gap could be changed. A steady source of potential was used in place of a magneto to cause the spark to pass the auxiliary gap, and the length of this gap was increased until] this spark was accompanied by a spark across the main shunted gap. This length was thus a measure of the sparking potential of the main gap when the potential was applied for the very short time (probably less than a millionth of a second) between the start of the spark in the auxiliary gap and the dissipation of the charge conveyed by it through the shunt. Accordingly the harder the main gap, the greater was the necessary sparking distance of the auxiliary gap. The method was extremely sensitive : a degree of hardness barely perceptible in the difference between the steady sparking potential and that with a magneto would increase the limiting value of the auxiliary gap by 50 per-cent. Hardness, therefore, like the difference between point and sphere gaps, if measured by impulse ratio, increases creatly with the frequency. By this method the conclusions already established were confirmed and it was shown that hardness (measured again by impulse ratio) decreased, instead of increasing, if the spark potential was increased either by increase of pressure * The sparking potential is probably slightly smaller for aluminium electrodes than for those of less easily oxidizable metals, but such differ- ~ ences were too small to be appreciable here. - TELS me oS ee SS ———— ae 228 Dr. Norman Campbell on of the gap or of the sparking distance. It was again shown that electrodes of Pt, Zn, Cu, Ph, Ke, Ni, Co, Cr Aiea He, C, Sn, W, were all liable to hardness, but also that they might all be soft; no difference in respect of liability to hardness could be discovered. But one new proposition of importance was discovered. (5) Hardness is a property of the cathode only. No treatment of the anode will ever change a gap from soft to hard or vice versa: if the cathode of a hard gap be sub- stituted for the cathode of a soft gap, that gap becomes hard ; but the anode from a hard gap will not make a soft gap hard if still employed as an anode. A gap may thus be hard in one direction but soft in the other. The circumstances which make a cathode hard are ex- ceedingly various and difficult to define in detail ; but their general nature was obvious. Thus a cathode could always be made hard by rubbing it with fine oily emery- paper 3 it could usually be made soft again by heating to redness in a Bunsen flame; but such heating always makes a platinum cathode hard, while it could oenerally be softened by boiling in nitric acid. A freshly turned surface was usually soft. Hardness is therefore due to some adherent film on the surface which usually consists of oily matter; but it seemed that the state of the atmosphere exerted some influence, for on some days it was impossible to make any gap really soft. If hardness is due to a surface film, the reason why it affects only the cathode is apparent; for the liberation of electrons from the cathode is an essential part of the mechanism of the discharge, while the anode usually plays the part ofa mere terminal. Time-lag in hard gaps. Hardness of this nature, then, is essentially different from that which distinguishes point and sphere gaps. By the measurement of the actual sparking potential in different conditions, by the method mentioned on p,. 222, an attempt was made to determine whether such hardness is due to the existence of a definite time-lug. Measurements of Q and © were made when the rate of rise of potential was varied. The apparatus was adjusted so that the potential rose atarate nearly uniform until a value slightly greater than the actual sparking potential was reached, and then remained nearly constant. If the rate of rise is known, the time-lag can be deter- mined as the difference between the time when the potential ‘reaches the steady sparking value and that at which the actual sparking potential, deduced from Q, is reached. No great accuracy in determining the exact rate of rise was Time-Lag in the Spark Discharge. 229 attempted, but the following figures show the nature of the results with a hard gap of which the steady sparking potential was 3120 volts. The first column gives the time taken by the potential to rise to the steady sparking potential, the second the actual sparking potential, the third the time-lag. 0:02 — see. 3180 volt.’ (SILIE "REG 0-008 3540 0030 0-0004 A050 00017 <= 0:00003 4540 < 0:00008 It thus appears that the actual sparking potential is greater the morerapid the rise of potential, but that the lag,and also the whole time which elapses from the beginning of the rise to the passage of the spark, decreases as the rate of rise increases. In another series of experiments the rate of rise of potential was kept constant, and the relation between the actual sparking potential and the steady sparking potential deter- mined as the latter was increased by increasing the sparking distance. The results are given in the following table, in which the first column gives the steady sparking potential, the second the actual sparking potential, the last the excess. of the latter over the former. 3090 volt. 4780 volt. 1690 volt. ZOU) 6700 1470 8040 8700 660 These figures show that the hardness, measured by the difference between actual and steady sparking potential, decreases as the latter increases. The time-lag was difficult to measure in this case because the rate of rise of potential was so rapid; this rate decreased somewhat as the potential rose, so thatit is not certain that the time-lag decreased with the spark potential, but it certainly did not increase. It is reasonable to suppose that the time-lag in these cases consists of the period required to remove the film on the cathode and to bare its surface for the liberation of electrons. If this is so, all the facts may be simply interpreted in the statement that the greater the excess of the potential above the steady sparking potential, the more quickly is the film removed. Itis not pretended, however, that the phenomenon has been thoroughly elucidated. Investigation of it is dif- ficult because it is so variable. It is difficult to be sure that successive observations are made ona cathode in the same condition. But the observations do suffice to show that hardness due to surface condition is a phenomenon perfectly distinct from the dependence on frequeney of the sparking potential of point gaps, and that it is due to a perfectly ditterent cause, phy | 930 Time-Lag in the Spark Discharge. They may serve also to show why it is so difficult to measure accurately comparatively low alternating potentials (up to 5000 volts) by means of a spark gap. A definite sparking potential in such circumstances can only be obtained if the gap is completely free from hardness ; but if it is so free, wonderfully definite measurements can be made even without any initial ionization of the gap. An accuracy of 3 per cent. at 3000 volts can certainly be obtained either with a steady source of potential or with a magneto or induction-coil; though the method is always inferior to that described in a recent paper*. However, since a spark gap is often used for such a purpose, two practical hints for abolishing hardness may be given. First it isa good plan to allow a strong spark to pass between the electrodes for a few seconds immediately before observations ; second it is a very bad plan to clean electrodes by rubbing with fine emery cloth. Unless the surface is actually pitted, discolo- ration due to previous discharges is positively an advantage ; to rub the surface with fine emery is to rub the “ hard ”’ film in, rather than to rub it of, and so to prevent its removal by other means. SUMMARY. It has been shown by Peek that the peak potential of an alternating source which is required to send a spark across a gap between needle-points is greater than the spark potential for a steady source. This dependence of the sparking potential on frequency is attributed to a time-lag in the initiation of the discharge. Two theories of this time-lag have been proposed or suggested. One, due to Peek, that the formation of the corona must precede the spark ; the other, that the presence of casual ions is necessary. Hach theory appears to account for part of the facts. Some additional facts are recorded, which show that there are two kinds of time-lag, one regular and one irregular. The latter is accounted for by the necessity for the presence of casual ions, the former byaslight modification of Peek’stheory. In normal gaps between spheres the regular time-lag is inappreciable, and in many conditions the irregular as well. But if the surface of the cathode is not quite clean, a time-lag may occur and the sparking potential may be dependent on the frequency. This time-lag becomes inappreciable if the _steady sparking potential is sufficiently great. It has not been determined fully what constitutes “cleanliness” in this matter, but some practical methods of obtaining it are suggested. * C. C, Paterson and N. R. Campbell, Phil. Mag. March 1919. Pamel> | XIX. Note on approwimations in the Theory of Probatilities. By Tt. J. VAS Bromvicn, Se.D., 7 hS.” 1 the April number of this Magazine t Lord Rayleigh has evaluated the formula known as Bernoulli’s theorem in the Theory of Probabilities up to terms of order 1/n’. In connexion with the revision of my book on ‘ Infinite Series’ for a second edition, I have recently noticed a formula —given in (2) below-—which enables us to push the approxi- mations in this and similar problems as far as we please. The result obtained in Lord Rayleigh’s problem is given in (6), below: this really gives the logarithm of his formula, which may be found rather easier to work with in numerical ealeulations. It can be proved that when z is large (real and positive) we have the pues formula t log {T(14+.2)}= “log EdE+ Slog wth log (277) wo By B, B, lO es FG ees (1) where the series is not convergent but the error committed in stopping at any stage is less than the next term in the series. Write here s=m-+yp, where m, p are both large, but p is of lower order than m3; in most cases it happens that the order of p does not exceed / m. Then we can write m+p m Set log EdE = { log Edé + {eee log EdE + - e (0) = (m log m—m)+ c log (m+t)dt. iE the last integral expand by the Noeariuimic series and integrate term-by- term ; we then obtain the formula ee log EdE= (mlog m—m) 0 hp? Le Te get oO a a ee oe moe Aion aes * Communicated by the Author. + Phil. Mae. (6) vol. xxxvii. p. 321 (1919). It may be of interest to state that Lord Rayleigh had seen the results of this note just before his death. t See, for instance, Arts. 182,176 of the first edition of my book on “Tnfinite Series.’ The fonmale given there differs from the above in appearance, because the integral is evaluated in the form 2 log 2—2. AO al Hl | HH | 232 D-. T. J. Va. Bromwich on approwimations Ht . Also log (m+p)=log m+ ” ae : 1 Pp ; mM 2m | 3m i Thus on substituting in the original formula (1), we obtain ! the result i ‘ | log el +m+p)}=(m +p+4)logm—m+4$ log 27 i | i : oe ee ate be i | as 1.2m 2.3m? 3.4m | ep Vp i | 2m 4m?" 6m q \ if yy? A De ee 4 oe (a m a ie ) i | q ' ae els a 3p | ee ee :) 4 fo wl. 1 In the actual Spphano it is not necessary to go beyond ; terms of order 1/m?, and regarding p as of order /m, we find 4 from (2), the derived for mula q log {L.+m +p)} +log {TU +m—p)} =(2m+1) log m—2m + log (27) + p?/m ‘| A | ae. Lo 4 = G om. 6 =) q (Se ' ieee 4m? "6. ne q " oy ae et il eae gD? (es Fie | | ec G m- 6 mé& ne 6m 180 —) de - (3) i : . | | where the last three brackets are respectively of orders 1/m,. mS 1/m*, and 1/m’. To obtain the font ees corresponding to Lord Rayleigh’s, | we write m=4n, p=4wv; then, with the restrictions of the | | roblem, m+ p and e are both integers, so that we find i Pp ) foo) i a ia! in the Theory of Probabilities. 233 (correct to order 1/n’), log {(gn—4u) !} + og {(4n+ $0) !} = (n+1) log Gn) —n+ log (277) + 422/n His os hee ype) 2 2 ae ) 4 Lee a & i} =) sy Oe on? Ieee 1. ge ea 5) (4) 56n? 6n®&' 3Bn® 45 73)° The formula to be evaluated contains also log (n!) which is given at once by writing a=n in (1) i 1 log (xn !)=(n+3) log n—n+ 3 log (27) 3 joraa 36072 ° (9) Tf we subtract (4) from (5) and also subtract n log 2, we obtain, on reduction, n! “3 12h — gu) + (ant gw) J 1, 2 a? 7 ee te) Ten a see) —(35 Neen, Aon 1 Met 1 a a (3 po. ae 3) Aye We? 1 at Mi ik 56? 6 no ae oe eee =i (6) where the last three lines are respectively of orders 1/n, in? Lin’. This is readily seen to be equivalent to Lord Rayleigh’s formula (15) (/. ¢. p. 327). In fact call the last three expressions in brackets «, 8, y respectively ; then, neglecting terms of order 1/n® because y is of order 1/n°, agen) | (4a?—). Phil. Mag. 8. 6. Vol. 38. No. 224. Aug. 1919. R 234 On approximations in the Theory of Probabilities. 8 6 4 2 i ee ee iA «oe (4 = (ra i A + 24nt Eni Tee oo tie er 4n4 — a 8 Be eo oe Ia aS ~ 988n®& 40n®* 4874 2403" 32 n?? which completes the verification. But the formula (6) above appears to be rather simpler than Lord Rayleigh’s for purposes of actual numerical work; it can be adapted at once to ordinary logarithms by multiplying the correcting terms by logy e=" 43429. . Further, we can readily determine the later terms in the series, if it is desired to push the govern further in any particular example. 17th May, 1919. Since the above was sent to press, | have been asked by Dr. H. Jeffreys to record a similar formula for the terms near the maximum in the expansion of (a+0)", where x is large anda+b=1. The maximum is readily seen to be the term nearest to r=an,n—r=bn, and accordingly we wish to calculate the value of ‘@ Rh jap t fe) =a ae . PG@>1)T@=r+i) ae where * r=ant+e2, n--r=bn—xz Applying (2) above we now find that log P@r+1)+log P(n—r4+1) —log P(n+1)—rloga —(n—r)log 6 S50 og(2mabn) + 5 —(< +;) ee 1 ee 3) ee ) ae ae ae ) it if 1 ) 1273 Pca 4 n? tp TT i.e - (8) * Note that x here is not the same as in Lord Rayleigh’s forms: the present « corresponds to 32 of hiswork. It should also be observed that an is no longer an integer. Equations for Material Stresses, and their formal solution. 289 Thus we can write finally se na ahineaon( a) |: HO =a ain)> fee) where G(w) is nearly unity, and more accurately log G(z) ==) 4 (L212 aie Qn iy 6 7 NGZ 02 22a ib Rear a aaa Lee 1 ae ala .+7)- igletm tint l it). ay The terms are arranged so that when 2 is of the same order as ,/n, the first term in (10) is of zero order, the next { } is of order 1/x (or n~2), and the last { } is of order 1/2? (or n~1). This remark may assist workers in selecting the order of approximation required in special problems. a ee (10) reduces to (6), as it should, on writing a=t 2 b=+4 (and allowing for the fact that x here corresponds to +z in Lord Rayleigh’s work). It will be noted also that (10) contains only terms corresponding to the first and second correction-terms. XX. The Equations for Material Stresses, and their a “mal solution. By R. EF. GWyrHeR * N this communication I offer statements of the Stress equations based on the “ Doctrine of Material Stresses ” contained in your issue for June 19184, and the formal solution of those equations in Cartesian coordinates. In the Cylindrical Polar system, and especially in the Spherical Polar system, the expressions become long and complex, and I have not dealt with those cases in full. The special Stress equations of which I treat are derived from the three Mechanical Stress Equations by eliminating ‘separately all but one of the separate stresses by means of the six stress conditions arising from the doctrine which ] have proposed. It is to be noted that this mode of treat- ment includes the Elastic Stresses, although not necessarily confined to these stresses only. As is indicated in the paper referred to, I propose to include among the forces acting per unit of volume of the } * Communicated by the Author. + Phil. Mag. vol. xxxv. p. 490, June 1918. R 2 See Se eee 236 Mr. R. F. Gwyther on the Equations for stressed body not only natural forces, but also reactions against acceleration in case the body is in motion. Whatever the forces may be, it will always be possible to express their Vy VON Ov; 0 components as p-— ay: respectively. Pas , and p ae namely, Cartesian Coordinate System. We shall have, firstly, the Mechanical Stress Equations, OP) Zo) One OV; San Ee Oe Ov Su ow etait OS OV, a a ol Bs) Siena OV; A Sy ene Bei = (). Ree teeta (a) The six conditional Stress Equations, obtained by the method I have indicated, are * :— m(m+n) vip oman) Vuaee + ®) = peng sat Ss) P+ O+R) +2 S90 V2Q4 MEY 7B + Q4R) --"(S+3) (P-+Q+R) +2p 5%! =0, VR + MEY FP 4Q+R) es ye (S + $.\P+Q4R)3 2p OMe =0;, () and Vist imeayae FOE Pay “-(V.+V,) <0 2m Vit 5° QZ 5p P+ O48) +e. (Vit Nel WU aaa 2m lols 3m—n Ov Oy a) folk Oaaat (Vi + Vo)=0. (y) * Manchester Memoirs, “Specification of Stress,” Part iv. (continued), vol. lx. no. 14 (1916). Material Stresses, and their formal solution. Zar In obtaining the equations (8) and (¥), the equations («) have been made use of, but it remains to secure that those equations are satisfied. By operating on the several areas ue and substituting from (@) and (y), we find that ava 4 oan O°V; Ov? ) Oy 7 32 epmidependent of cy, and 2) *5...). - 2.1.) (6) Also by adding the three equations (8), we find that mtn Or Vier On NaN Y epe oT at) =0. (e) ps OG Cale ot ye +o. © m(m+n) n(3m—n) VAP +Q+R) +p( V7(P+Q+R)+ 2p (Sat i It therefore follows that both V?(P+Q+R) and are independent of w, y, and z, unless m=n. A condition is thus imposed upon the nature of the force- system under which this stress-system can be maintained. The formal solution. In what follows, | shall write P+Q+R=V’"O, pVi=V'h1, PVe=V ho, PVa= Vb, (1) so that, from (8) and ( a aie Pa Ninn) VP analgyet ga) Ot? Se 5372; 320; hiOee). . Ov tater 4} mom+n) ; 2m oe I n(dm— 2) jm—n Os 0 Oe OO Foe Palit O22” ’ pa mont) o ae 2m (25+ S2)O+25 3 ni dm—n) 3m—n Oat | ‘Os ip Ons) | 0" Os Ia (2) se Oa? Ow? palin i Seen ss Se te A ee a 238 Mr. R. F. Gwyther on the Equations fov and Zn ViO7® 0? 3m—n OY 02 i OY Ae ee >? aa bE 4 (OQ, +0;4+,), 2m O° Boon eR = (pi + $s) “3 (OF BO on: L 2n 0° ag mane ee v OY +525, (it os =5 5 (QO; + O24 98), . : z : (3) where the ‘six O-functions are all arbitrary harmonic functions, given in a form selected to suit equations (@). By substituting in equations (a) from equations (2) and (3), we obtain the condition m(m-+ 1) g ‘a O¢, , Od; n(3m—n) Vor a OY ce 02 0°) , 0O, , OW = ye aans ea +C.7{4) By adding the three equations (3), we obtain also m+n Ord four Ops OF Vo + 2 Ne 2 +o%)=2 aie “; (20, + 0,) + 2, (20, +9) + ei 20; +0,). (5) We are to regard the functions di, po, and a as known, but subject to a condition. Then, from (4) and (5), (am +n)(m—n) O72: 0’Q, n(3m—n) Mae 3a ae Gy a 02 The arbitrary functions are to be such as satisfy the surface-traction conditions in any particular case. The whole scheme is approximative and to some extent tentative. It is not anticipated that the requirements will, in general, be capable of being absolutely satisfied, and any case in which complete disagreement arises would appear to be of 20a) Material Stresses, and their formal solution. 239 particular interest for.examination. The special case when the forces acting per unit of volume are negligible compared with the impressed surface tractions only needs consider- ation from the simplification which is made in equations (B). and (vy) by the null values of Vi, Vos anduneew hor, thre consequence is that \77(P+Q+R) is also null. Thi is hardiy necessary to emphasize further the simplifications which follow in equations (2) and (3). Cylindrical and Spherical Polar Coordinate Systems. Provided the points in the material body which we are considering are sufficiently distant from the polar axis, or from the pole, as the case may be, the solutions already given may be readily adapted to either ‘of these systems. But if this is not the case, further points will arise for consideration. It will naturally be noted that no definition is given of “sufficiently distant.” I shall content myself by stating the conditional Stress Equations corresponding to equations (98) and (vy) in the Cylindrical Polar Coordinate system, as follows :— Oi Wal kon Bagi) ria ne aes 2 ‘ae n(3m—n) VP+O+R)— 2m ( 3m —n Ay. + <3) (P+Q+R) 0: ee (P- Q+297)= 0. Our m(m-+n) 2m role me n(3dm—n) Van) 3m—n Se + 2,)(P+Q4R) DOV aleve 2 aon Ele ae aeace) + 3 (P—Q4255 )=0 m(mi + 2) PATON bay ao aan Ca ea n(3m— Dee) — (eee + ap) (P+QER) +2p 5 =0, 3 RS ene MERI A ol (8') 240 Mr. D. L. Hammick on Latent Heat and 9 2 2 7384 (Pe a ee 3m—n r 00 Oz 2m 2 Vines 3m—n 0? o (P+Q+R) +p5°. ae (V,+ V3) aa 2(285 +1) =0, ZIn yy 5 fey eee ee 2 ect (EQ) ata If if a oe Nag 5 aSO7 ES) = Wi+V)- 359 +2 (° a —2U)=0. @') The terms written last in five of these six equations are the terms constituting a material difference. Terms of a similar character, but less simple, present themselves in the Spherical Polar Coordinate system. This communication deals only with a proposed formal solution, and no application to a particular problem ‘is offered. The difficulties which will arise in respect to surface conditions are of a quite different character and are probably always greater than those of obtaining a mere solution. WaUEE XXI. Latent Heat and Surface Energy.—Part I. By D. L. Hawick, Chemical Laboratory, The College, Win- chester *. HAT the surface energy of a liquid and its latent heat must be intimately related has long been recognized, and in connexion with the familiar demonstration that wane Hog of the magnitude defined as ‘‘ molecular surface energy” > pvi (where p is the surface energy in ergs per sq. cm.and V is the gramme molecular volume) with temperature is aconstant, a relationship between surface energy and latent heat is hier met with (cf. van t’Hoff, ‘ Lectures,’ vol. iii. p- 77, Eng. Translation ; Nernst, “Theoretical Chemistry,’ * Communicated by the Author. and Surface Energy. 241 276; also Stefan, Wied. Ann. xxix. 655). Seeing that the internal latent heat of a liquid is presumably a measure of the work done against the internal pressure, and that the work done by molezules in getting into the surface of a liquid (2. e. halfway out of it) is measured by the potential energy they acquire there as surface energy, it is argued that one-half the latent heat must be equal to the molecular surface energy, which is proportional to pV? *. Apparently no attempt has up to the present been made to give the above general relationship precision or to apply it to the data. In the present communication a definite connexion between the two magnitudes (internal latent heat and surface energy) is put forward and shown to be in very fair agreement with the available data. As has been pointed out by Matthews (Jour. Phys. Chem. 1916, xx. 555), a molecule in passing into the surface layer of a liquid does work only against the component of the internal pressure directed "perpendicularly to the surface towards the interior, the lateral attractions on the one side of the molecules being balanced bv identical attractions on the other. It would thus appear that the work done by a molecule in getting into the surface layer is not one-half the orks it nt do in passing altogether from the liquid state, but 1x42 or one-sixth, since it is doing work against one only of the three components of the internal pressure. We can therefore conclude that the work done in passing all the molecules in a gramme molecule of a liquid into the surface is equal to one-sixth of the internal latent heat, 7. e. to one- sixth of the work to be done in moving all the moteeules apart from one another until the liquid is transformed into vapour. We have now to devise a method for measuring the potential energy the molecules will have acquired when they are in the surface layer—in other words, we have to deter- mine the true molecular surface energy or energy due to all the molecules in the gramme molecule. We can do this by imagining a gramme molecular volume V of liquid to be spread out in a layer of thickness equal to the molecular diameter on the surface of excess of the liquid. If we assume as a first approximation that the molecules in such a layer are oa contiguous and approximately spherical, * Van t’ Hoff (toe. cit. above) actually equates pVi= ts ee —- 2 of - eee: Se 242 Mr. D. L. Hammick on Latent Feat then the area of such a layer will be ue where d is the eee molecular diameter. The product a will then be approxi-- mately the “true” molecular surface energy, since every molecule in V ¢.c. is contributing to it and we expect to find that the relationship. We. 1 als, A ° ay = ies . ° : = . ° (1) will hold (J is the mechanical equivalent of heat, L, being the internal latent heat per gramme molecule in calories), though in view of the assumptions made great precision cannot be looked for. It is also fairly certain that the density change on passing from liquid to vapour is not abrupt, but that the molecules must pass through a ‘capillary layer.” It would seem reasonable, however, to take the potential energy (7. e. surface energy) associated with the molecules in the actual surface layer as a correct measure of the work done in getting there, even though that work has been done by passing through a succession of surface layers (cf. Matthews, loc. cit.). But the surface. energy as measured is the potential energy at the boundary between liquid and vapour, which has also a “surface energy’? due to its own internal pressure. The surface energy of a liquid not in the presence of its vapour would thus be greater than when in equilibrium with vapour. At present it does not seem possible to compute or measure this vapour surface tension; hence equation (1) must be tested at temperatures where the vapour effect is likely to. be a minimum, 2. e. at temperatures as lowas possible. Low temperatures are also necessary if the assumption of approxi- mate contiguity of the molecules is not to cause very large error. Unfortunately, most of the latent heat data refer to the boiling-point of the liquid. A further difficulty lies in the determination of the molecular diameter d. There is no satistactory method available for deriving d from measurements of properties of the liquid itself. The diameters used in testing (1) have therefore been obtained by taking the mean of the values calculated from measurements of various physical properties of the vapour, suchas thermal conductivity, refractive index (or dielectric constant), the critical data in the form of van der Waal’s “6,” and the limiting density of the most dense form. Values derived from viscosity measurements. and Surface Hnergy. 943. have not been used, as such values frequently exceed the “qaximum’”’ values obtained from the limiting density (c7. Jeans, ‘ Dynamical Theory of Gases,’ p. 345). The ~values of d obtained by the various methods are given in Table I, In Table If. a comparison is made between pv | Ly, ey aay and >? values taken for p, the surface energy, are, for the various organic liquids, those found by Jaeger (Zeit. Anorg. Chem. i917, ci. 1-214); his determinations of s, the specific gravity, are used, where available, in calculating V. Values of L, are derived, for the most part, from mean values of L found in the usual collections of physical constants. Some estimates of the molecular diameter will be found in Table Il. that do not occur in Table I[.; these values are derived from van der Waal’s “0.” J is taken as 4°18 x 107 ergs per calorie. the various data used being shown. The TABLE I. Values of d x 10° cms. Rate of Limiting nemo Refractive 3 : Substance, Teor. eae Conduc- Tages O27 Mean. iffusion. Density. vey. MFENZENE 5.0 ....00.. +00. ie 58-4 By 4°28 4°53 . 4:88 LO TNCIE” "Sa ae a ce Be Be 4°12 470 441 Hthyl formate ...... val i se 3°79 4°38 4:09 Ethyl acetate .....,... ee Le Bik 4:02 4:58 4:30 Methyl acetate ...... en ian ily 3°74 431 4:03 (OSS calc a aaa ie ir ne 4°42 394 418 Chloroform . 20.20... ie Ma ae 4:07 460 4:34 Carbon tetrachloride. ie ihe ue 4:34 4°69 4:52 INES csede eae a a ae 2°54 3:04 2°79 SIO) 5. etl iE He me se 4:02 Soli no he SC) Skea nee us 3°46 4°20 2°88 SiGe eon il! (CO) a ae by 4:04 an 280[E] 3°40 3-41 LO): Nee Sl cee 3°72 4:22 3°80 252(/E] 3:08 3:47 (OS AAR e ae Sa set 4°60 Bt 3°28 496 4:28 SURG Sen eee mene Bn 4-02 3°60 2°36 2 C6 Vol 1S bcc. A ee 3°80 3°96 3°82 2°40 3:58 ~' (3°50 Oslo cottee ie Sata a eee aee 3°62 3°70 3°60 2°24 no el! Ethyl] alcohol ......... ue se ive 344 403 374 Milethy ly... ce. ese ens = ate an roy 3°06 SO" oe | K]=from dielectric constant. { | 244 Latent Heat and Surface Energy. : TABLE Te 3 pV 1 G; 8 di) TAG | o ostaage, eee T,° ©. a oe Se L, ©: cals. cals. { Oncen ene Jee 13°23 —183 1-138 321 17248 —188 276 987 Nagropen ee 8941 --196 850? 3:50 1210! —196 201 202 | eon: ea eee 11:00! —186 1-404! 321 1284 =186 226 214 NAOT eon csc et 963 —89:3 1:226 $71 3658° —s9 606 609 \ NEC) ee ee 41-75: 29 - 672 (279° 5817 +—83:5 '905- * eR6 RO Sick cee 33:82 —25 1509 377 5684. —10 930 939 COME ite ae 28°5 Q \ 1-632. 4:52) 82545 —.05MI4o> Raa | 20:2 W642 1:48 710. 16439 76:4 1106. 1072 c CECI yi ee eee 32:5 @. 1555 «4°84 7454 0. 1374 11242 , 21:8 Gle2y 1408) 4 GSO 617 HOIsha0at | Cgt cui aoe ee 93 + 46 1-22 418 5852, “467 Spas i 33°9 OW 1-292 ee se2ee 0 1136 1047 N=NEXAME wee eee 18°54 82 ‘69 453 7132 QO 1074 1189 4 Benzene ......00.06 29-2 175 “88 488 7760" . 20 =il@esmeiees 20°7 805 “sl 6739 80" TOT mes Hole cy eee 195 109° 778.484 6926 109) iitenmenia | p. Xylene meee 191 1862 -754° 5:16 7777 ~~ 139) alge i Mesitylene ... ........ 158 1605 °737. 535 7728 163 1145 1288 if Biher oe ee 19:2 02 735 441 63800 0 1043 1050 1 159 348-695. :.. . 5803 40 /erea among } Methyl acetate ...... 25°14 10 986 4°03 7865 “ODOR i Ethyl formate ...... 19°5 54 873-409 6166 54 «=: 962-1028 : Ethyl acetate ......... 172 771 73) 4°31 6768. - 7a, SIs | 255 0 924 ... 80365 . 0 “eomiieaa r Propyl acetate ...... 2484 10 ‘891 510 9168 0 1327 1445 ; Methyliso-butyrate. 166 913 806 503 6972 91 995 1160 | Aniline y 66000 2436 18h 875 4°70 9190 185 “isieqmeds if Chlor-benzene ...... 20°1 122 ‘995 483 7294 . -150) 20s a) Ethyl iodide ......... 92-11 7821811 4:20 .6612 71. aso : Amino Reto 5. 198 151-7 -865 498 7910 151 1182 1319 4 Waterss, 2. 75°8 0-4 100 2:88 10194 0 1133 1699 i Methyl alcohol ...... 23:5 0 810 3-48 8708 .0 647 aaa i Ethyl alcohol ...... 23:3 ) 807 «= 33°74. 9615 0 850 1603 iq Acetic acid... ... 23°5 90105: 436 4457. 20 | 7aaeemee is Ohiorine 13... 5 33:59 | =72 9249 4-28 4250 99 agomeneees 1 Baly & Donnan, J. C. 8. lxxxi. p. 907 (1902). 2 Dewar, Proc. Roy. Soe. xv. p. 133. Mt 3 Mean: Hstreicher & Shearer. 4 Mean: Alt & Shearer, > Nernst, Theor. Chemistry, p. 273. ® From Trouton’s Rule. * Griffiths & Marshall, Phil. Mag. [5] xivi. p. 1 (1896). Considering the simplifying assumptions made, the rela- pve alee nals tionship pee * fits the facts remarkably well. Measuring the Capacity of Gold-leaf Electroscopes. 245 At the end of Table II. certain liquids definitely regarded as associated are dealt with. The considerable excess of 2 over a : > found is clearly to be attributed to work done (heat absorbed) in splitting up the associated molecules: as they pass into the vapour phase. In this connexion the agreement found with acetic acid is interesting, in view of the fact that the vapour of the acid is known to be associated. Seeing that d, the molecular diameter, varies but slowly with increasing molecular weight, we may use the relation to make a rough estimate of the “association factor,” n, for liquids the vapours of which are completely dissociated, by writing : where M is the gramme molecular weight of the simplest or vapour form of the molecule. In this way the following values are found for n :— SANT ArS 1 nN Substance. T. dus Ramsay-Shields. \OY SHI Roe ae 0 1:5 er Methyl alcohol ......... 0 2°3 2:3 Hithylealcoholl:..:.4.05.. 0 19 L7/ Chlorine (as Cl) .. ... --72 2°6 The value of n calculated by the Ramsay-Shields method is recorded for comparison. XXII. A Method of Measuring the Capacity of Gold-leaf Electroscopes. By A.T. MuKxersrs, M/.A., Patna College*. ABSTRACT. THE arrangements for using a JDolezalek electrometer for accurate work, devised by V. H. Jackson and the author (con- tinuation of earlier work described in J. A.S.B. vol. x. No. 6, 1914) and shown in working order at the fifth session of the Indian Science Congress at Lahore, 1918, have made it possible to develop a quick and very accurate method of measuring extremely small capacities, such as those of gold-leaf electroscopes. Using a standardised (Gerdien) sliding condenser, the absolute value of the capacity of the quadrant system, including a specially designed connecter, is first determined by the method of mixtures. ‘The gold-leaf system of the electroscope is then charged toa known voltage, adjusted so that wnen the charge is shared with the quadrant system the final potential, as measured by the electro- meter, is not far from one volt. As the capacity of the quadrant * Communicated by the Author. LE ee SS ee EET Ss PS MEH fei pa 246 Mr. A. T. Mukerjee on a Method of Measuring system can be varied by the sliding condenser, the capacity of the electroscope can be measured at different voltages. There is a smal] but definite increase of capacity with voltage. The method is compared with those of Lester Cooke (Phil. Mag. vol. vi. p. 410, 1903) and T. Barratt (Proc. Phys. Soc. London, vol. xxvill. pp. 162-171, 1916). The capacity of a small G-ray electroscope charged to about 100 volts has been measured by the three methods with the following results:—Present method 0°78 + -01; Lester Cooke’s method, as improved by the author, 0:75 +:013; and Barratt’s method 0:73 + -03 E.S. U. FHXHE modern gold-leaf electroscope is extensively used by workers in radio-activity and atmospheric electricity. Many cases naturally arise where a knowledge of its capacity becomes necessary. ‘The capacities of such electroscopes, being extremely small, of the order of one micro-microfarad, cannot be determined by the usual methods. Ordinary commutator methods lose their sensitiveness for such small capacities. Even the ordinary method of mixtures cannot be employed with a Dolezalek electrometer, as the probable error of a single observation is usually of the same order as the quantities to be measured. The first suggestion for measuring such small capacities was made by Borgmann * and Petrowsky f in 1899, put the method was complicated and uot very satisfactory in practice. C. T. R. Wilsou { measured the capacity of an Exner’s electro- scope in 190L by charging it and determining the fall of potential resulting from contact with a brass ball suspended by a silk fibre. This method, again, can only be regarded as an approximation, and Harms. § mentions that individual values obtained by this method ditfer by as much as 25 per cent. Several other and much less simple methods have been described by Harms || (1904) and Lichtenecker J (1912), but none are sufficiently accurate to justify a detailed discussion. The best methods hitherto used are those of Lester Cooke ** (1903) and T. Barratt Tf (1916). I have used both these methods myself, and the results, which are detailed later on, show that Lester Cooke’s method is quite good, prov ided that satisfactory arrangements are made for ‘insulating and Phys. Zeit. vol. ii. pp. 651-6538 (1901). ‘omptes Rendus, vol. cxxvill. pp. 420-422 (1899). Roy. Soe. Proe. vol. Ixvin. p. 157 (1901). Phys. Zeit. vol. v. p. 47 (1904) & Ann. d. Phys. vol. x. p. 816 (1903). Loe. ett. { Phys. Zeit. vol. xii. pp. 516-518 (1912). ** Phil. Mag. vol. vi. p. 410 (1908). i cere: Phys. Soc. London, vol. xxviii. pp. 162-171 (1916). SSteursay ap Re the Capacity of (rold-leaf Electroscopes. 24 completely shielding the outer case of the electroscope. This may not be convenient or possible with all types. It is also essential to undertake a series of measurements of the capacity of the outer case, thus insulated and shielded, before the actual measurement of the capacity of the gold- leaf system ean be begun. Barratt’s method is to charge a standard condenser and reduce the potential by a sufficient number of charge and discharge operations, using the same electroscope to measure the initial and final potentials. The method is a valuable one where a quadrant electrometer is not available or is not in proper working order, but it is slower than Cooke’s and still slower than the method to be described, thus increasing the effect of any leakage of the electroscope itself. Partly in consequence of this, the probable error has been found to be from two to four times as large. The present method, which is only a slight modification of the ordinary mnetned of mixtures, gives quicker (aud therefore more accurate) results, and is applicable to any conductor of extremely low capacity, while avoiding the necessity of constructing any special subsidiary apparatus for each electroscope to be measured. — Its practicability depends on the fact that, as shown by V. H. Jackson and tie author, very great accuracy can be obtained in quadrant electrometer measurements of capacity by adopting certain modifications. These were described in a preliminary form ind. A.S& B.vol.x. No. 6 (1914). The finalform, as shown at the fifth session of the Science Congress at Lahore, has not yet been described in detail, but the diagram given below shows the essential arrangements. The usual troubles due to failure of insulation and zero creep are notably lessened by reducing the number of insulators as far as possible, and by enclosing all working parts of the electrometer in an earthed metal ease. Ii all be seen that there are only two insulators required in this arrangement, one the usual amber supports of one pair of the quadrants and the other for connexion with the outside. The air inside the case is kept constantly dry by means of sulphuric acid, which can be easily ea eel if necessary. ‘his we have found to be the only satisfactory drying agent. ‘The case itself can be instantly removed or replaced when any adjustments are necessary. All connexions are established electromag- netically on the key supported by the quadrant insulator. - which brings different mercury cups into contact by well- amalgamated copper points. In the paper referred to, it has been shown that the capacity of the quadrant system ‘of the electrometer itself at constant voltage, when compared with 248 Mr. A. T. Mukerjee on a Method of Measuring a fixed capacity enclosed under the same conditions as the electrometer itself, can be measured with a probable error not greater than 0:05 E.8.U. When the comparison is made with a condenser placed ouside the case, the probable error is much greater owing to insulation troubles which cannot be avoided, but is still within 0°6 E.8.U. In all the experimental arrangements dealt with, the needle of the quadrant electrometer was suspended by a 6p quartz fibre and charged to 20 volts only. The sensitiveness was such that when the quadrants were connected to the poles of a standard cadmium cell deflexions between 350 and 300 mm. were obtained at about one metre. The average capacity of the quadrant system for needle charges between E lectromagnets these limits was determined. ‘The variation of the capacity between these limits is not more than one H.S.U., as will appear from the table on page 239 of our previous paper. The form of the electroscope used in the following work is the Capacity of Gold-leaf Electroscopes. 249 shown in the diagram. The gold-leaf system was insulated in the usual way “by a bead of sulphur inside the case, and potentials are put on it by means of a metal charger mounted on a plug of amber. It is necessary to distinguish between c the capacity of the gold-leaf system itself and ¢’ that of the charger including the mercury cup mounted on its end. Method, The following method is used :-— (i.) The capacity of the quadrant system K as far as the connecter is determined, as shown later, using a stardardised Gerdien condenser. (ii.) ¢’, the charger only, is charged to V volts and shared with KX. The resulting potential v is measured by the deflexion produced. c’ is given by the equation eV = (K-26 Jv Be Cl OR es couse etal boul ee) (iii.) ¢+c’', the gold-leaf system plus the charger, charged to V volts and shared with K. The resulting potential v’ is measured as before. Then (ctc)V = Ce Tee je af! i or e+e = ee PN Minaya ah Cz) Subtracting (1) from (2) the capacity of the gold-leaf system 1s eed (iv.) To determine the capacity of the gold-leaf system at different voltages, a sliding condenser is added to the quadrant system andiihe: walkie of Ke adjusted so that the resulting potential atter sharing with the quadrant system is not far from one volt. Discussion of Errors. The equation from which ¢’, the capacity of the charger, is computed is Vv \ aa where K, V, and v are independent variables. Phil. Mag. S. 6. Vol. 38. No. 224. Aug. 1919. S ea=Kk i 5 ee er ees et a ne See nee 250 Mr. A. T, Mukerjee on a Method of Measuring Let Ey, E,,:E,, and. H,' be the probable errors of the corresponding quantities. Then the probable error of tne result Ey is given by the following equation : wore [Hed + [ed +(e] v Do Sega Ki see KV 2 =[vasBe) + [oa]. + Lege Taking the approximate numerical values K=162, V=110, v=1, E,= +0°6 (determined experimentally as shown below) ; K,,, being the combined errors due to the voltmeter used and to the leakage during the interval between the charging and the sharing of the charge, may be taken at not more than 0°5 volt; E,, the error in measuring the final voltage by the electrometer, is certainly not greater than +0°002 ; (Be)? = (-006)? + (007)? + (002)? = (009), Determination of K, the capacity of the quadrant system up to the connecter :— The system was insulated and charged by a standard cadmium cell and the charge shared with a capacity of 92-1 H.S.U. on a standardised sliding condenser having a very accurate Reichsanstalt certificate. It was necessary to attach a mercury cup to the connecting wire of the standard condenser. This introduced a slight correction to the values read off from the calibration curve of the condenser. This has been ealled « and determined as shown below. The de- fiexions of the electrometer before and after sharing are as follows :— 40:9) .333:3_ Baer 339°0' 333-0 alee 99°2 | 99:0 eae - 92-8. “Oba ois We get from the above 9214+ oe = 9°497, 2417, 2°413, 2-424, 2-444, 2-435 — 2427 (mean): . = 73am The standard condenser was then shifted to a position corresponding to a value of 31:0 H.S.U. on the curve and the measurements repeated with the following results :-— a00°0 - 328°0 foemo 326°0 (324593250 TTT -O , 170 0, alee, TASTER 3G the Capacity of Gold-Leaf Electroscopes, ZoL These give the equation a 0-864, 0°867, 0-865, 0°866, 0°864, 0-867 = 0°866 (mean). Meets G2) solving the equations (1) and (2) we get Ka 39-2) We Said: 3" 0c Beseu. Probable error of a single observation = +0°6 E.S.U. Determination of the capacity of the charger alone :— The charger was charged from a battery of 56 lar ge storage- cells, the potential being measured by a standardised Paul millivoltmeter and 150-volt shunt. The capacity of the quadrant system was adjusted by the sliding condenser to 162:2 E.S.U. The results obtained are Pula below : V = Potential to which the charger was raised, d,=deflexion by one cadmium cell, deflexion after sharing the charge on the charger with the quadrant system, d,=deflexion by one cadinium cell again, c'=capacity of the charger calculated from the equation lg = MG EVOR sy 1-02 Utd; Darcn J. No. 1 2 38 7 cid Wintel 6 Tes Oya AO his 12 OG LOST NOOR O! SL LOR KOA TOs CORE alerme Lell0: \ ae 1OFr 109 ee 3166 315°8 315-2 3148 314-0 313-0 312-2 311-0 310-0 309-2 3084 306-4 i 305-2 3060 305°7 3042 305°5 306-4 3038 3023 302-0 301:3 301-0 298-8 d,..... 3158 315-2 314-8 314-0 313-0 312-2 311-0 310-0 309-2 3084 3074 305-4 .. 148 149 149 148 1:49 1-49 1-48 148 148 1:48 148 1-48 Mean value of c’=1:483. Probable error of a single obser- vation = +0:004 H.8.U. | In the same way the capacity of a gold-leaf system plus charger, c+c', was determined. The capacity K being this time ‘adjusted to 242°2 E.S.U. to bring down the potential after sharing te near one volt. TaBLeE II, INO. ONT Oo eM GL ROOD. TO kh dee ea Byes. Oe atOn LO: et Oe Oa TOS: POS TOS TTOh BON tO EO Be... 391:6 320°2 318'8 3182 317:0 3162 3154 3148 3140 3813:0 312:0 3811:0 Pe... 3905 319-0 3180 3181 3162 3160 315-0 3126 313-7 311-9 3116 3105 a 320°2 318°8 318°2 317°0 316°2 315-4 3148 314:0 313-0 312:0 311-0 3100 Sto . 2:26 2:26 2:26 297 296 227 297 295 297 295 297 297 252. Mr. A. T. Mukerjee on a Method of Measuring Mean value of ¢+c'’=2°264 E.S.U. Probavle error of a single observation= +0°004, c, the gold-leaf system, therefore=0°78 E.S.U. Probable error of a single observation= +0006 E.S.U. A second set of twelve observations by an M.Sc. student gave the following statement :— C40 = 2°26, 7¢ =LAQ: Ao c=0°79, with the same probable error. Final value of the capacity of the gold-leaf system may be taken as 0°785 + 0-006 E.S8.U. or for all practical purposes 0-78 + 0°01 E.S.U. Determination of the Capacity of the Electroscope by Lester Cooke's Method (Phil. Mag. vol. vi. p. 410, 1903). The method used is best described in his own words: ‘To determine the capacity of the gold-leaves the electro- scope was set on an insulating block of paraffin, and the outside cylinder connected to the quadrants of a very delicate Dolezalek electrometer, which was connected in parallel with a standard capacity of ‘002 microfarad. The leaves of the electroscope were charged, and the reading observed. The leaves were then discharged, and the charge which had been attracted to the case of the instrument thus released and allowed to charge up the electrometer and standard capacity. The deflexion of the electrometer was then read, and was a measure of the charge which had been on the leaves of the electroscope.” Our experience with paraffin as an insulator in delicate electrostatic measurements, especially when in large blocks, warned us against it in 1913. Hence the outer case of the electroscope _ was insulated by four very small pieces of freshly cast sulphur and the whole case was completely shielded from electrostatic disturbances by an earthed metal cover, leaving a small window for the connecter on the quadrant system to work through. A small mereury cup was mounted on the electroscope case in front of the connecter. The capacity of the case with the mercury cup was deter- mined by a set of measurements with the method of mixtures, - the mean value being 37°77 E.S.U. The gold-leaf system was charged to 85 volts, discharged, and the charge attracted to the insulated case thus set free was shared with the quadrant system of capacity 392 E.S.U. The resulting potential was measured in the usual way. the Capacity of Gold-leaf Electroscopes. 253 Discussion of errors. Taking the general form of the equation in this method where K’ is the joint capacity of the outer case and the quadrant system, V the charging potential for the gold-leat system, and v the final potential, we find that the probable error of the result EH, is obtained from the equation ; i v 2 K'a 2 / K’ ; Z (E.)? = |e } ns We Ey rp + Ly bef Taking approximate numerical values 2 2 52 y= [506] + [F053] + | 50-002 | 85 oD Bl = (007)? + (005)? + (002) or EK. = +0°009. The probable error by this method should thus be about the same as in the previous method. A set of twelve observations taken on the 27th October, 1918, is given below :—Inserting actual values in (1), ¢ the capacity of the gold-leaf is calculated from the equation ; 2 ds ; R45 — Gg e ° e rete Ms “ e Y x 85 = (39°24. 37 EN a) SE, dy, dy, and d; being electrometer deflexions as in the previous tables. TABLE ITI. No. 1 2 3 4 5 6 7 8 9 10 11 12 eS. - «= 303°2 3016 350°0 349°4 348°8 347-4 345°6 3442 343°0 342°2 541-4 3400 Bal. 273°0 290:0 2780 288:2 279°8 2646 272:1 287°3 269°6 263°5 2785 278-0 eae. 3516 350°0 349°4 348°8 3480 3468 3442 343°0 342°2 3414 340°0 339°0 Baer ds Ou OG O13) (C7On OFA 0 70 -ON2 Ont O72) OF L- Ofo) (O75 Mean 0:°735 ; probable error of a single observation = +0015 H.8.U. A second set of observations gave the mean value of 0°73 ; two other sets, taken by an M.Sc. student, gave mean values of 0°76 and 0°78, with the same probable error, the difference being due to personal equation in reading the final electro- meter deflexions, the damping being such that in this method the final position is rendered rather indefinite by any leakage Se ee SaaS SSS EEE eee 254 Mr. A. T. Mukerjee on a Method of Measuring on the external insulators. Taking the mean of all these four sets, the final value for the capacity of the gold-leaf system by this method may be taken as0°75 + :015 E.S.U. at about 100 volts. Capacity of the Electroscope by Barratt’s Method. The method depends on obtaining the relation between the divergence of the leaf and the potential as applied to it, and the observation of the fall of potential of astandard condenser when it shares its charge a convenient number of times with the electroscope. If V isthe original potential of the con- denser and C its capacity, vp the final potential of the system after n alternate earthings and re-chargings, and ¢ the required capacity of the electroscope, then iM @ " Un =(——,] ? V and », being read off the calibration curve of the electro- scope after observing the corresponding deflexions. . To compare the accuracy of this method with the previous ones, the probable errors. may be discussed. If E., E., Ey, and EH», be the probable errors of the corre- sponding quantities, | en 2 Com ~C-1) Fata (E.)? = {(r)'=1 Bey pe! n By Un - Un” fa . he te fe It will appear from Barratt’s paper that, as he used a travelling microscope with vernier reading only to }, mm. to read the deflexions of the gold-leaf, E,, or E,, could not have been less than 0°8 volt*. E, may be taken as 1; then, taking the values given by him in Table III (a) of his paper, it will be seen that | (E.')? = (0°021)? + (0:039)? + (0-049)? = (0-064)?. The probable error of the method as used by him may therefore be taken as +0°06. Ba tin * The ordinates on his calibration curve for the electroscope should probably be cm. instead of mm. In the latter case the probable error would be 8 volts, which seems unusual. the Capacity of Gold-leaf Hlectroscopes. - 255 In applying this method, the gold- leaf electroscope was mounted on a stand and a microscope with micrometer scale focussed on the gold-leaf. The calibration curve was first obtained fora range of 60 to 150 volts. The Gerdien con- denser witha capacity of 95°3 E.S.U. was taken as standard of capacity. The electroscope was charged to 135 volts, and after ten charge and discharge oper tions the final potential was read off ‘by the electroscope. Two extra short wires were soldered on the charger to facilitate the charge and discharge operations. The following values were obtained :— Charger plus leaf system, SIO oem 2s) SrOAL BeOS Pidosd oly S4sOGrve0O,: 3 94 Ki o*9G: Mean 3.95 Kes.U: Charger only, ! Sale onOien wor lem tole ok On: woraen Omar oO imo 20, \ oO ae oy Mean = 3°20 H.S.U. The capacity of the gold-leaf system, therefore = 0°75 +°04 E.S.U. at about 140 volts. A second set of observations gave 0°71 + 0°02 Hi. g. Ge A final value may be taken as 0°73 + °03 E.S.U. Owing to the fact that the deflexions of the gold-leaf could be measured to within + volt, the probable error works out at about half that deduced above from Barratt’s own observations, but it is still. considerably larger than either ot the previous methods. 7 The Capacity of an Hlectroscope at Different Voltages. ©. T. R. Wilson found the capacity of his Exner’s electro- scope to be sensibly the same for 100 and 200 volts, the variation in capacity, due to the change of position of the gold-leaf, being too small to be detected by his method, Barratt found it to be practically independent of the amount of divergence of the leaf. Lester Cooke, whose method is more sensitive, found the capacity to be 0°85 tor 200 volts and 0°91 for 300 volts. Tt ae that there should be an increase in capacity at eee voltages on account of (1) Proximity of the gold-leaf to the EAS case cee to increased diver gence, and (2) Extra work done in lifting the g@old- Teat to a greater ~ height at the higher voltage. 256 Mr. L. C. Jackson on a Mathematical Investigation The difficulty in such measurements is that any leakage of charge off the insulation of the gold-leaf system. produces an error which tends to increase the result obtained for the capacity. ‘The leakage and the consequent apparent increase of capacity will be greater at higher voltages. In modern electr oscopes the insulation of the gold-leaf system itself is so good that this leakage does not affect the result appreciably, “put in most of the methods of measuring the capacity of the leaf system the insulation of the charger also comes in and this is usually much less satisfactory. Preliminary measurements show that at 10 volts the capacity is 0°71, at 40 volts 0°73, and at 110 volts 0°78. In order to investigate this point more carefully by the method described in this paper, I have constructed an electro- scope without any charger, and work with this is in progress. I have to thank Mr. V. H. Jackson for much helpful criticism and advice. Physical Laboratory, Patna College, India. XXIII. A Mathematical Investigation of the Stability of Dr. A. W. Stewart’s Atom. By Lueonarp C. Jackson, F.P.S.L., Science Scholar of University College, Nottingham ™*. i a recent paper (Phil. Mag. Oct. 1918) Dr. A. W. Stewart set forth the structure of the atom trom the point of view of the physical chemist. This atom proved so successful from the chemist’s point of view as to distinetly merit the attention which the author invites from physicists. The atom, in its simplest form (the form treated in the present paper), is as follows :—‘‘ At the centre of the structure is a group of negative electrons travelling in closed orbits which, for the sake of clearness, may be assumed to be circular. Closely surrounding this negative group lies another series of orbits occupied by positive electrons. .... These orbits are assumed to be circular also... . . Further still from the centre, other [ negative] electrons move in orbits of an elliptical character, the ellipses being much elongated, so that the electrons travel in paths like those of comets in the solar system.”” The figure given later will explain the idea. The present paper is a mathematical investigation as to the stability or otherwise of such an atomic system. The * Communicated by Prof. E. H. Barton, F.R.S. of the Stability of Dr. A. W. Stewart’s Atom. 257 following limitations, facts, and assumptions are used in the investigation :— (1) The case considered is that in which all the inner negative electrons are in a single ring, similarly the positives in a single ring and the cometary electrons in a single ellipse. This would probably be the case for the atoms of simplest structure and lowest atomic weight. (2) Since the presence of elliptical orbits requires that the law of force should be that of “inverse squares,” this law will be used as the foundation of the investigation. (3) The total number of negative electrons in the atom is equal to the atomic number of that atom. (4) The number of cometary or valency electrons is as given in the following table : | Group in Periodic System. O 1 | 2 csr i | OF iets | ee aad RR Paina WEB EAE No. of Valency Electrons... 0 1 | 2. 8 oe | Br BEAN A =! | a Ane ——=—— : 0 41/42 48 44 -3 -2)-1 — | \") STV a a ee 0 = Ze og | a5, Fag +7 — The theory is developed generally and_ conditions for equilibrium and stability are obtained. ‘These conditions are then applied to special cases on the assumptions (3) and (4) given above. . Conditions for Hquilibrium. Let no. of positive corpuscles in ring == Ne no. of negative electrons in inner ring =n, no. of negative electrons in outer ellipse =1, then =n+n, radius of inner ring of negatives ik radius of positive ring =i, major semi axis of ellipse =e and eccentricity of ellipse =e. Consider the equilibrium of an electron in the inner ring. It is in equilibrium if the net force due to the attractions and repulsions of the other components of the atom is equal and opposite to the centrifugal force on the electron. 258 Mr. L. C. Jackson on a Mathematical Investigation General view of atom. | Centrifugal force i} 2 if =m 0.0)". i ‘ Repulsion due to electrons in inner ring : i repulsion on electron at C due to electron at B | = BOP | ij radial are | er | cos BCO= 4 OB 4a? sin $BOC ~ : Hence ae repulsion (forces towards centre positive) if e” al 20 (n—1) eo — — cosec + cosec~- + .....cosec — 4a? n n n 1. 9 i a rerisis ° e e ° . 5 e e . ° ° (1) qn Yoo of the Stability of Dr. A. W. Stewart’s Atom: 2 Repulsion due to cometary electrons : repulsion of electron D on electron ( 9 “ e = Gp» ; e radial component = ~,,,,cos DCO CD 9 aE at+d-— Saas (2 at é) a—d, cos (77 + o) Dar Aaty, w+d?e— —2ad, co (* + ) is ny xX total repulsion due to ee electrons et ae) iG 4- dy? — 2ad, xa z 5) i A(a e,, cos (7 + $)) fe +d; — 2ad, cos et fi ye 2(a—d,, cos (27+ ¢)) Sao Oe as {a = open Cos (29 af hb) }2’ (2) Gl pet where d, is written for b(1 a ) "| : 1c), Yy | dy Eee | 1—ecos(77 +4), (3) Ny b(1 —e?) ” tlm ss Es 1—ecos(27r+ 9) | | oy) or) Fic pot (w,—@)l, Z =a tS" a 200 Mr L. ©. Jackson on a Mathematical Investigation Attraction due to positive corpuscles : attraction of positive corpuscle Ii on electron U = ar radial component : e ao a? +77 —2arcos i —_ + @ N O46 eos + 8) ideal {at +7? — 2ar cos (< ~~ ay yt =+ total attraction due to Rositine corpuscles Seas 9) {attr —2ar cos ( +6) b 4 e? *(a —r cos +6)) {a +7? —2ar cox(S +0)t 4 Cl Gm C08 (27 4-0) {a* +7? —2ar cos (27 + @) }2’ =+ where @ is written for 09 +(@,—o)t. Hence ) . é(a —d cos a +¢)) Oi an oe a fl ae ap a f 2 L207 l= a + d°—2ad cos G +- 6] § 1 >) , é(a-r toy A ee ae oe —maw*=0. This determines a state of possible ete inner ring. It is now necessary to find whether the equi- librium is stable or not. To do this, it is necessary to find (5) for the of the Stability of Dr. A. W. Stewart's Atom. 261 the forces called into play by an infinitesimal displacement and see whether the direction of the force is such as to cause the displaced electron to return to its original position. This must be done for radial and tangential displacements in the plane of the rings and for displacements perpendicular to the plane of the rings. Conditions for Stability. Radial Displacements. Suppose an electron to be displaced radially so that its distance from the centre of the orbit becomes a+da. The force called into play 2a*— bad cos (= + $) + 6d? cost((7 +4) de + d?—2ad cos +4) i ee ey ig 2a? — bar cos (N \ +8) + 677 cos i +8) Bee —> z ee = ((6)) { +7? —2ar cos (WF +0) | | For radial stability, the quantity inside the bracket (taking into account the proper sign for each term in the summations) must be positive. @ Sa ae a 2a? as hi) Tangential Displacements. Suppose an electron is displaced so that the radius through it turns through an angle dy. Then the tangential displacement is qi and the force called into play 9 deos(e +¢) =e'dy > i SER ee ~ sin?(*7) fa | ee c03( 727 +o) h Vt i reos( ~-+0 a - Gr ’) se age Ie ey 29 war ye {a + 7 2arcos( "x ne lt \ For tangential stability the quantity inside the bracket moust be positive. —_ —— 262 Mr. L. C. Jackson on a Mathematical Investigation Nsplacements perpendicular to the plane of the rings, Let electron C be displaced distance z perpendicular -to plane of rings. Force due to inner electrons — component perpendicular to plane = 9 eZ i Il 1 8a’ T y/o Ti 5 tt le) a Slt Sa | —— Sl i rn 2 ics Edge?" ta noe a ~~C~«~“C< 1 ! ! oe | 1 SN U ; A I N : i J at L \ ! ean at aa ae pat Ree F wl Hee ‘ 1 . ‘\ / ‘ , o - - ~ = : \ re) “tS gs m — a = QO 4 V Sores ees 4 ‘ ‘. — s / Sate a s R a is / ~-@- / . 4 nN / XV 7 , Plans, 4 2 . x 7 ~ ee ~e. ay ~ toe borat Displacements from orbital piane. Lorce due to positives , — component perpendicular to plane u == ob 7% f r 3 9 9 <7 2 V4 —2ar cos(< = +6) } NS NEN y i: rant nnn (EEE eee 4 3 . {a +1 —2ar Cos low + @) | 2 Force on displaced electron due to cometary electrons — component perpendicular to plane 1 \ + dy? —2ad, cos(-™ +) 1 = ¢@72 \ 14 +— ii = ae {ae + d,? —2ad, cos a =F ») } iL Ole Se Ne — 2ad,,,COs (Airset ae wa of the Stability of Dr. A. W. Stewart's Atom. 263 Hence for stability ez —> A — +> — oe : > sint( 27 Ey 0 Se 8a° sin ( _) {a +7*°—Zar cos( N su) i —- by : a es ee 4 a! + dt—2ad cos ss + o) be | (3) must be positive. SPECIAL CASES. I. Helium—group O—no. of electrons = two; no. of valency electrons = 0. Consider the configuration of the atorn in which the two positives and two negatives are in one straight line. Then apply equation (5) for the condition of equilibrium. 1 1 iL 7 | tee re 3 E ae 2 ca e ( Wee ae ae ae ant) maw: — ae @) But the quantity inside the bracket is always negative (r>a). Hence for the particular configuration considered, equilibrium, cannot be maintained. Thus, when the atom in the course of its existence arrives at this configuration, the inner ring breaks up spontaneously. A note might be added here concerning the a particle. Of this Dr. Stewart writes as follows :—‘‘ With revard to the expulsion of charged helium atoms from radioactive elements, it is assumed that the @ particle consists of four positive and two negative electrons: the pair of negative electrons being situated at the foci of an ellipse around the circumference of which two positive charges revolve. The extra pair of positive charges travel in longer, ‘cometary ” orbits; so that they are easily detachable when in aphelion.” (The italics are the present author’s.) There are serious objections to sucha view of the con- stitution of the e@ particle both from the dynamical and physical standpoint : (1) It is difficult to see how the two inner positives could revolve in an ellipse of which both of the foci were occupied by a negative electron. ‘They certainly could not, if the law of force was that of “inverse squares.” It would have been more in accordance with the previous part of the paper if Dr. Stewart had assumed that these positive and negative electrons revolved in concentric circular orbits. ee er SS eS Se Se eee eee 264 Mr. L. C. Jackson on a Mathematical Investigation (2) The « particle would then have atomic dimensions,, say 10-' cm. It is inconceivable how a system of atomic dimensions could travel through air distances up to 7 cm., and on an average pass through 500,000 air molecules before being seriously deflected from its original direction of motion. The photographs of C. T. R. Wilson show very clearly the long straight path of an « particle flying through air, and the experiments of Rutherford on the structure of the a particle are in accordance with the view that the diameter of the a particle is of the order 10716 em. (3) The change of an @ particle into a helium atom would on this hypothesis be due to the loss of two positive charges and not to the gain of two negative charges. Rutherford’s view that the « particle is a positive nucleus of charge +2e and that it becomes a helium atom by gaining two negative electrons seems far less. artificial and more in accordance with the facts, It might be mentioned that Dr. Stewart’s helium atom cannot explain the « particle asa helium atom minus two negative electrons, since the remaining two positives would immediately fly apart. II. Lithium—group 1—no. of electrons three, no. of valency electrons = one. In this system also we can find a configuration, at which the constituents of the atom will arrive in the course of their revolutions, and for which equilibrium cannot be maintained. Lithium atom. Consider the equilibrium of the electron C in the configur- ation shown in fig. 3. of the Stability of Dr. A. W. Stewart’s Atom. 265- Then applying equation (9) we have (-a I OTe a) 4a? (a —ry (@? +7"? + ar)a (a@—6(1—e) )? —maw*=0, (10). but 2a+r on iL (V@+r7+ar)s 4a (a-r)? is always negative whatever values a and r may have (a<7). Hence the quantity inside the bracket is still more negative, and the equation cannot be fulfilled. Hence the configuration is impossible for equilibrium and the atom would have no features of permanence. Til. Beryllium—group 2—no. of electrons = four, no. of valency electrons = two. Consider equilibrium of electron C in configuration shown. in fig. 4. Beryllium atom. Apply equation (5) : we have : Te Ug Le ils AIAN A EO A ‘ (- 4a? (a—r)* ‘i (a+r)? (a*+7?)2 (a—b(1—e) )? 1 emia 22 TERE. oe ce \ eae MG: ——\ (Damen eet) but 1 2a il 1 Pill. Mag. 8. 6. Vol. 38. No. 224. Aug. 1919. E __— 6|lUlUu =S == 266 Investigation of Stability of Dr. A. W. Stewart’s Atom. is always negative for values of 7 greater than a, and 1 ae 1 (ate (a —b(1+e))? 4b*e — 4abe =-((, ~b(1—e))*(a—6 ETS] : since ) >a, the above is negative. Hence whole quantity in bracket in first equation is negative and the equilibrium condition is impossible. Here again the inner ring would decompose spontaneously when the above configuration was reached. SUMMARY AND CONCLUSION. (1) Conditions for the equilibrium and stability of Dr. Stewart’s atom are obtained. They are then applied to several special cases among the simplest atoms. (2) It is found that in each case tried, equilibrium cannot be maintained indefinitely for the inner ring of negative electrons, even in the entire absence of external perturbing forces. Thus, on the atomic theory considered, instability is placed to the account of those elements which are found to be the least unstable in reality. (3) Dr. Stewart adds a note to his description of the atom to the effect that it is not of material consequence to his theory if the relative positions of the inner ring of negatives and the ring of positives are interchanged, or even if these positives and negatives form a kind of double-star system. The equations obtained will still hold (due attention being paid to sign) for the former alternative arrangement. The inner ring (in this case the positives) would still break up, and this arrangement seems more unlikely than the original one, since a copious supply of separate positive electrons would be continuously given off. Such an effect has not, up to the present, been observed. (4) The author hopes to investigate the second alternative given above at some future time. The success of Dr. Stewart’s atom in explaining chemical phenomena makes it reasonable to hope that, by some such modification as that mentioned, the atom may be made more amenable to ordinary dynamics and more successful in explaining physical phenomena. Physical Department, University College, Nottingham, May 12th, 1919. ke Cae XXIV. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. (Continued from vol. xxxvii. p. 464. ] November 20th, 1918.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. (pats following communication was read :— ‘The Geology of the Meldon Valleys, near Okehampton, on the Northern Verge of Dartmoor.’ By Richard Hansford Worth, M.Inst.C.E., F.G:S. The area dealt with lies between the London & South-Western main railway-line, from a point a little east of Meldon Viaduct to near Sourton, and the ridge of Dartmoor occupied by Black Tor, High Wilhays, Yes Tor, and West Mill Tor,—being the greater part of the valley of the Redaven and a portion of the valley ot the West Okement. The southern extreme of this area is occupied by the Dartmoor Granite, north of which are shales, in which occurs a patch of limestone, and these are intersected by numerous bands of igneous rock. The shales as a whole, with but slight local deviations, strike north-east and south-west and dip north-westwards, the mean angle of dip being about 50°. The sedimentary rocks are divisible into :— (1) An alumino-arenaceous series, extending from the granite northwards for a breadth of somewhat over half a mile. (2) A calcareous series, abruptly but conformably succeeding the last. (3) A limestone, which occurs a short distance south of the railway. (4) Radiolarian cherts a little above and a little below the horizon of the limestone. (5) An aluminous bed north of the railway. Of these, (1) consists of impure grits, which, being well within the aureole surrounding the granite, have developed secondary mica, a little tourmaline, and small well-formed rutiles. In some places, at contacts with granitoid veins, andalusite is also found. (2) Consists mainly of porcellanites with beds of black chert-like rock. The characteristic mineral of the porcellanites is wollastonite, but at contacts with the Meldon Aplite garnet, idocrase, scapolite, axinite, and lepidolite are also developed. (38) Shows little sign of metamorphic action. (4) Are cherts of the character already well known as occurring in the Lower Culm- Measures, and described by the late Dr. G. J. Hinde & Mr. Howard Fox. (5) Is a dark-grey rock, almost black, the characteristic mineral of which is chiastolite. All these rocks sueceed each other conformably, and there is no evidence of folding or repetition. In the sedimentary series planes of weakness have developed, the surface-traces of which are broadly coincident with the strike, but which frequently he counter to the dip. These planes have been more or less successfully invaded by at least three series of igneous 268 Geological Society. rocks, the order of which, commencing with the earlest, is as follows :— (a) A felsite with phenocrysts of micropegmatite, and quartz which shows good rhombohedral cleavage. (b) A series, hereafter called the ‘ dark igneous rocks.’ (c) Granitoid veins, subdivided into: (1) The Meldon Aplite and its associates ; (2) Fine-grained granites of the ordinary Dartmoor type. The evidence on which this chronology has been based seems fairly clear. The felsite with micropegmatite occurs as inclusions in the ‘dark igneous rocks.’ 'The ‘dark igneous rocks’ occur as inclusions in the Meldon Aplite. The Meldon Aplite occurs as veins in the ‘dark igneous rocks.’ No evidence is available as to the relative age of the Meldon Aplite and the granite-veins. A marked feature of the ‘dark igneous’ rocks is that they are locally agglomeratic, as such they have been identified as meta- morphosed tuffs. But, on the other hand, every exposure is also in part homogeneous and compact, with clear flow-structure. The inclusions, where present, are always in part fragments of the contact-rocks of the walls of the sills or dykes. Some of the agglomeratic rocks are certainly dykes and not sills, and as such cannot be interbedded tuffs. Every exposure at some place irregu- larly invades the contact-shales. For these and other reasons their identification as tuffs is dismissed, and it is sought to explain the occurrence of the included fragments by successive injections of the same fissures and the break-up of previously-consolidated injected material. The geography of the Meldon Aplite is described: it occurs in several dykes, the principal of which extends from east of the western wall of Okehampton Park to the old Ice House on Sourton Tor, a distance of nearly 2 miles. ‘There are other minor dykes north and south of this. The texture of the aplite is microgranitic. The principal minerals are albite, orthoclase, microperthite, quartz, lepidolite green tourmaline, and topaz. Blue apatite is almost entitled to be classed with these. Fluorspar, montmorillonite, and axinite are accessories. Although, in conformity with other observers, the author has described this rock as an aplite, he uses the term with reservations. The rock is neither more acid than the normal granite, nor does it approach freedom from mica, and he submits that the true description, even if cumbrous, would be lepidolite-soda- granite. The whole of the mica is apparently lepidolite, and of 8°70 per cent., the total of the alkalies, roughly five-eighths are soda and three-eighths potash. Some veins of true granite occur, always of fine grain: in these andalusite is locally developed. It is noted that topaz and anda- lusite have never yet been found side by side in any Dartmoor granite or granite-vein, but topaz may occur in granite which is in contact with slate in which andalusite is present. In one and the same rock the minerals appear to be mutually exclusive, or, in other words, when the conditions are such that topaz may form andalusite is not to be expected. on. J Ly 3 x Mirra. Phil. Mag. Ser. 6, Vol. 38, Pl. 1V, Illustrating Large-Angle Diffraction. ApENEY & BECKER. Phil. Mag. Ser. 6, Vol. 58, PI. Y. Photograph of Bubble in Motion up the Tube. SF een % Phil Mag. Ser. 6, Vol.438, Pl. VI, | | | | "hil. Mag. S. 6. Vol. 38. No. 225. Sept. LOL: in ae i } i ah pies en lca! ae, ” Barron & Browninea. pees 3 ce / } i a} Me | ery : 5 4 ' a5 < Me: na * THE LONDON, EDINBURGH, anv DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. Ae Sk ’ [SIXTH SERIES.] "SEPTEMBER 1919. XXV. On the Molecular Theory of Refraction, Reflexion, and Extinction. By Lavistas Naranson, Ph.D., Professor of Natural Philosophy in the University of Cracow” N two communications brought under the notice of the Cracow Academy of Sciences in January and in June 19147, [have been concerned with following out the cireum- stances of the mutual action which must be supposed to take place between an advancing wave of light and the ultimate fragments of matter lying in its path. The leading assump- tions adopted in this investigation appeared conducive to clearness and consistency, and 1 believe they may possibly still commend themselves to the theorist’s attention. In the meantime, however, Science has progressed. Owing to excellent researches { the treatment of the subject has been notably extended and deepened, so that the interest of the papers published in 1914 may now perhaps be regarded as having passed awav. Acceding, nevertheless, to {riendly advice I venture to lay before the readers of the Philosophical Magazine the present essav which contains the substance of the papers to which T allude. Having found advantage * Communicated by the Author, + Bulletin Int. de V Acad. d. Se. de Cracovie, Cl. d. Se. Math. et Nat., Series A, 1914, pp. 1-29 & 335-352. t C. W. Oseen, Annalen der Physik, vol, xviii. pp. 1-56 (1915) P. P. Ewald, Annalen der Physik, vol. xlix. pp. 1-88 & 117-148 (1916). Fr. Reiche, Siaen der Physik, vol. 1. pp. 1-52 & 121-162 (1916). Ragnar Lundblad, dAnnalen der Physik, vol, lvii. pp. 185-202 (1918). Phil. Mag. 8. 6. Vol. 38. No. 225. Sept. 1919. U ere wera ZO Prof. L. Natanson on the Molecular Theory from a gradual mode of procedure in performing the caleu- lations I have confined myself, in this communication, to the consideration of some simpler aspects of the problems at issue. Thus we shall avoid, as far as we can, analytical intricacies, and the meaning and bearing of the fundamental principles involved will become apparent. The effect of small particles in diverting portions of incident light from its regular course was investigated by Lord Rayleigh in a series* of admirable papers. It would carry us too far into scientific history to give an account of his work as well as of later inquiries f that fall more or less directly within our province; but I desire to repeat here the ecknowledgment, made more than once in the course of the papers, of my indebtedness to Lord Rayleigh’s writings. I also desire to take this opportunity of referring to Mr. W. Esmarch’s paper cited below; with this author I find myself in closest contact and agreement in respect of many con clusions. § 1. Let us consider a linear simple vibrator maintaimed in vibration by the operation of a train of incident electro- magnetic waves. It is natural to begin the calculations by considering the expression of the secondary field that diverges from the vibrator. An effective method of working it out is as follows. Let V(@o, 7, 2y) be one of the points oceupied ay the vibrator, M(a, y, 2) any point where the secondary field is to be estimated. If we introduce an auxiliary thie VY whose components V,(2, y, 2, ¢), VjG@l gee W(x, y, 2, t) satisfy equations of type 9 On oy, . c being the velocity of propagation zn vacuo, the secondary * Phil. Mag. (4) vol. xli. pp. 107, 274, 447 (1871); (5) vol. xii. p. 81 (1881); (5) vol. xlvii. p. 375 (1899). ‘Scientific Papers,’ ‘vol. i. pp. 87, 104, 518; vol. iv. p. 397. + G. Sagnac, Recueil de Travaux offerts a H.A. Lorentz, p- 877 (1900). Lord Kelvin, Baltimore Lectures, p. 301 (1904). M. Planck, Sztz.- Ber. d. Kgl.-Pr. Akad. d. Wiss. for 1904, p. 748. L. Mandelstam, Annalen der Physik, vol. xxiii. p. 626 (1907). A. Schuster, ‘ Introduction to the Theory of Opties,’ 2nd edit., p. 8325 (1909). A. Einstein, Annalen der Physik, vol. xxxiii. p. 1275 (1910). H. A. Lorentz, K. Akad. v. Wet. te Amsterdam, Proceedings, vol. xiii. (1) p. 92 (1910). P. Langevin, Bulletin d. Séances de la Soc. Frrang. de Biekiue, p- 80* (1910). Les Idées Modernes sur la Constitution de la Matiére, p. 97 (1918). L. Vessot King Phil. Transactions Roy. Soc. of London, A. vol. cexii. p. 375 (1913). W. Esmarch, Annalen der Physik, vol. xlil. p. 1257 (1913). (*This paper vnly came under my notice after my first communication on the present subject was already printed.) : of Refraction, Reflexion, and Extinction. P(A field E®, H® initiated by the vibrator is known to be ~apressible in the form ee ois ian | «AMR Mra esd tay. C2) Eh omen en! /Ob)a vcnieen, mae) (a) where ac=1. To utilize this principle we have only to adopt for Wa suitable form of expression. Let 1(/,, ly, lz) be a direction unit vector belonging to a given vibrator. We will suppose that the direction of 1 remains invariable ; and we will call it, for shortness, the aazs of the vibrator. If r denotes the distance between M and V, we will assume NY ss NOG CCC Ma ste ls ot i aC) where Wr, t) represents the magnitude of V apart from direction. Let r be a unit vector pointing in the direction of 7, from V to M. Substituting from (4) we get diy was oe (5) s( ) being the symbol of the scalar product. Let us now write Po VG ae OVGse) a Oe SR 5 Seas ua ©) ONG hOViGe B= nae oP . one =§ Bo aan ee CO) and let (5) be substituted in (2); then this equation becomes a (CAC emu apa 4s) (8) The projections of E® on land r are \ Le eee (a (rl) 2A We A tt 2h,18(9) GOA B) ci eens ola. | LOD We now proceed to obtain the expression of H®. Com- bining (3) with (4) we easily find LS LCs cy sR CED) where the vector product is denoted by the symbol y(_) and CaO Gs t)/OrQteh) cans Mod Le) It appears from ($) and (11) that E® is always in the plane through r and 1, and that H® is perpendicular to the U2 ee Prof. L. Natanson on the Molecular Theory same; so that E® and H® are at right angles to each other. The direction of H® being at right angles to r, we are justified in calling H® a transverse vector ; with regard to the electric vector, however, equation (10) shows that H,. cannot vanish unless g(rl)=0, a condition fulfilled only in the equatorial plane containing the vibrator and perpendicular to its axis. At any point in the equatorial plane the direction of E” is parallel to that of +l and its magnitude is B. For points . situated in the (positive or negative) direction of the axis, H® vanishes and the absolute value of E® is | A—B |; the direction of E®) in this case is concurrent with that of +1. At any other point (B?)?=B?+(s(rl))?A(A—2B) . . . (13) H?”)2= (1 — (s(I))*)C*. en (4) so that, in general, the electric and magnetic energy per unit volume have different values. The following particular relations may be deduced at once: from the preceding formule :— 1 || 2. evel mn 1]| z. hO= = Boer rtyA ye os fs iy = iB 2 Be rA YyV,A EY = r.r,A yess, a Se H®? = 0) —rC +r,C . EL +r2C 0 we iE == =i +r,C 0 Consider a spherical surface S, of radius 7, with centre at. V (205 Yoo Zo). Calculate the amount of energy, emitted from the (perfectly isolated) vibrator, which crosses an element dS of the surface in unit time. Using as before r to represent a unit vector drawn outwards in the direction of the normal to dS, we have for the flow of electromagnetic energy across. dS, in accordance with Poynting’s theorem : S [— BOs(rv (iy (r1))) + s(t) ACs(rv(rv(tl))) J. (15): The value of the second term in square brackets is zero, so. that (15) reduces to cds oe — G BC sin’ Gh). Jw (os of Refraction, Reflexion, and Extinction. 273 which makes the rate of transmission a maximum for directions contained in the equatorial plane and zero along the (positive or negative) direction of the axis. The amount which crosses the area of 8 in unit time is — 2 c7°BC. § 2. To put our results in a form suitable for use in further deductions, we follow a well-known method of procedure, due to Rowland and Hertz. We _ beyin by assuming ) A (2) Wr, t) ee LAN cin(t—ar—h20+9), ; sphigs i Gp) Yh where n is the frequency of transmitted vibrations and 0, A”, and g are new constants, characteristic of the secondary radiation. Hvidently (1) is an admissible solution. Let us first consider the part of the field that lies nearest to the vibrator ; in this region aur may be supposed to be a very small fraction and (2), § 1, reduces to : BO edi Ws. hi alse oo) In order to write out the full expression take (5), § 1, and substitute from (1) above. If we write L= —1A” ein(¢—02,4 9), a UE etany reie ae (3) the resulting formula, holding true in the immediate proximity of the vibrator, is as follows : B=V5(u.V(-))3 syria she) and from this we conclude that the electric moment of the vibrator is correctly represented by the vector L localized in the axis 1. | Reverting now to the general case when M(a, y, 2) 1s an arbitrary point, situated anywhere in the field of the vibrator, we propose to calculate the values of the quantities A, B, C on which, as we have seen, the determination of the secondary field ultimately depends. Using R to denote l/anr we easily deduce azn2A) . RH He F big " Pee eon) 4i( Ni a aa ! (5) Dae Ae)! : gpa TEAM inlt—ar—te,+9){(1—R)i+R},. . (6) , an2A®) . a : - MS ee gt Sat OR ah. ot en OR heey he (7) ? —— 274 Prof. L. Natanson on the Molecular Theory It will save needless repetition if it is understood that im obtaining these results no approximation has been employed ; they are exact consequences of the initial equations from which they are derived. The importance of a distinction between exact relations and those which (at great distances. trom the vibrator) hold true approximately does not appear to be always fully appreciated. § 3. Suppose we had a material medium, that isan immense: collection of very minute, equal, simple linear vibrators dis— tributed at random through space. Consider the surtace of the medium as plane ; taking it for the plane z=0, refer everything to rectangular axes a, y, 2, the positive part of z being measured vertically upwards through the medium. Below the boundary pli ine conceive a void region extending to infinity in the negative direction of the axis af 2, Suppose also that the material medium is unlimited in the positive direction of the axis of 2. Let an infinite train of homogeneous electromagnetic waves: (proceeding from vacuum in “the positive direction of the. axis of z) be externally incident on the surface of the medium. Suppose the components of its electric and magnetic field to: be simply periodic with respect to z and ¢ and write them in the form BS) = AMent—aztp) i 0 E® =0, (1) HY? =0 H” = AG) tut—ae + p) He=0 (2) Tae) Uo ae ; & ° - Here p and A“ are constants; n represents as before the frequency of transmitted vibrations. In what follows we propose to give, to the disturbance defined by (1) and (2), the name of incident or primary wave. According to the present view, the primary wave is con-. ceived to get undisturbed across the aggregation of molecular vibrators ; its progress continues in fact over indefinite depths. in the medium without any mitigation or attenuation what- ever. The only effect of the medium is to superimpose, on the field propagated with the incident wave, a secondary field arising under the joint operation of the vibrators. Fronr each vibrator that is struck by waves travelling along in the medium, an infinite sequence of secondary waves is prco- ceeding: when all these combine, a resultant tumultuous: secondary radiation becomes established. The steady state is regarded as that which will not be further disturbed owing to adjustment and readjustment of the primary wave of Refraction, Reflexion, and Lutinction. 275: and the extreme variety of innumerable secondary 0. Secondary -electrom nagnetic waves, emanating from various strata of the substance, are uninterruptedly transmitted through the plane. Let us calculate the electric field (measured along x) which arises at M under the simultaneous operation of secondary waves travelling in the positive direction of the axis of z. ‘Things having become steady, this field, in accordance with § 4 above, is —2ran WA dg) He Oe ey) @)) J0 A convenient way of explaining the meaning of the ex- pression (1) is to remark that it represents the cumulative effect of concurrent secondary waves, advancing in the fimeetion im which the imeident wave traverses (the) medium. For simplicity and definiteness, we shall use the terms positive and negative to distinguish effects due to secondary waves which are propagated, relatively to the medium, in the Berre Wivection ag the wedent wave andl in the opposite direction. Putting then Ga). for the secondary positive -electric effect, we have Qa NA® . ence . 4 (B®), al ae at be+q) — @in(t—az+q)) (2) A similar argument may be applied to find the value of the secondary negative electric effect (E®) _ which is gene- rated, in the plane <= < in the medium, by the operation of so -— 978 ~~ -~Prot. L. Natanson on the Molecular Theory secondary waves crossing the plane in the negative direction of the axis of z: 2) \ eee 2aNA®? ein(t—bz+g ; i > en C—t+9) See The actual electric field induced at M, say (E,), may be considered as compounded of GU ie (Hae and the primary effect ES” ied due to the incident wave : daN AG (i, ie (U2? Cc" “T° 2aN A® ACC) 22229 oe) ioe, as cieaiee ante a E é ea) In this equation it will be observed that the occurrence of the term containing ¢’’"@-“+® points to-a train of waves transmitted through the medium with vacuum _phase- velocity c and without any loss or enfeeblement whatever. Among the observed phenomena of light an effect of this. kind is unknown; and if we confine ourselves to the con- sideration of steady states, we are justified in denying the possibility of its realization. In the subsequent treatment we will assume that electromagnetic waves are always. ‘obliterated as they advance in the medium. To satisfy this condition it is requisite that terms such as those to which we allude should remain znoperative in our solutions. The word is so significant of our meaning that there is presumably no need for apology in using it for the sake of brief terminology. We have accordingly ein(t— bz +9) aa i (4). . 2a NA” (1) ,in(p—q) — 2. 5). ADD! (eniy) 0 (5) and An@NA® , (Haar el in(t—bz+q) ee \ ) (be? —1)° (6) Let us determine the corresponding magnetic effects. generated at the point M, supposing as before that the state of the system has feonme thoroughly steady. Reverting to §4 we easily find, for the secondary positive magnetic effect 20N A”? i(be- — —1) c and for the agers negative magnetic effect (Hy int—be-+9)— gint—azt9y, | (7) aN A (H®)_ = i Pag: ee of Refraction, Reflexion, and [atinction. 279° Hence the expression of the component along 2 y of the actual magnetic field at M becomes AmNA®@. be ., i See fee eR ano tg) (Hy) i(b?.? — 1) . 2arN A®) hs OD CU DO) a al [A 7 i(be—1) lata : (9) ate prevent the occurrence in (9) of terms representing vacuum-wayes ” (which the medium is incapable cf propa- a precisely the same condition must be fulfilled which is already satished in consequence of equation (5) above. This should be observed in corroboration of our previous. result. Finally we have toNA™ . (Hy) =~ gaa 5 FeO), sine, Cy § 6. Let us consider now an electron which is set vibrating to) to and fro under the overation of an electric force E. Write | e for the charge on the electron and m for its effective mass. Let € be the component, in the direction of «, of the dis- placement of ue electron, reckoned from the position of equilibrium ; let m represent a constant and 3 a definite period of attic of very short duration which in most cases. of interest may be pub =2e*/3me*. Confining ourselves to vibrations in which $ is a very small fraction of the periodic time 27r/n, we assume as the equation of motion in the direction of wv: m(£ —SE+ n2£) =cE,. Semper ae tlt) [The second term of the left-hand member arises here: from damping due to radiation. In a celebrated paper, Professor H. A. Lorentz has shown A. c., pp. 103, LO4) that this simple form of expression, a SE is Aileatiate whenever the molecules of the substance, as in gases and liquids, can be supposed to be irregularly distributed. The third term is generally regarded as representing the component along & of the “ quasi- -elastic”” force supposed to connect the electron to the system to which it belongs. The introduction of this concept has not escaped criticism; by some writers it is considered as a gratuitous supposition, devoid of physical significance. Arguing from analogy we might perhaps invoke in its favour the well-known propositions deducible from the theory of small vibrations performed, about an equilibrium configuration, by a dynamical (holonomic) 280 Prof. L. Natanson on the Molecular Theory system. But from the point of view of the present paper the question evidently is of secondary importance. | It remains for us (wit the view of investigating forced vibrations) to evaluate the force E acting on the electron. The component E, of this eficacious force is made up of (H;), the electric field calculated in § 5 and a further additional part whose importance was insisted upon, many years ago, by H. A. Lorentz and Sir Joseph Larmor. In-order to express this term in our notation, we must recur to (3), § 2. With use of this relation we can write * E,=ae?(—b20+P)__ FarN AM ints tae : (2) where a is a constant. When we bear in mind that the expression (6), $5, with 2) written instead of z, will exactly represent what in the foregoing equation (2) is denoted by agin(t— +P), we find Aa@NA® = i(b2e?—L)aci™(P-9, 2. (3) We now require another equation connecting A” with a; for this purpose recourse must be had to the equation of motion. From (1) and (2) we deduce 9 . 4 e- cS 7 a J y 5 ae {n?— n? +715n?\e&= ae Jael(P 1) —SaN : ZA} girl? bzo +9), ae and substituting from (3), § 2, we have the result 2 2 é . ‘ Cx ,, F G ~n?—SaN Bar iSn? AQ =i — ge —2) een : m We come therefore to an important conclusion. Listwe assume be=p—ix,*. . 2 + * 5) Se y will represent the refractive index, « the coefficient of extinction of the medium. Equations (3), (5), (6) now give So) 2 (ee a—4nN@ + i8n°)((p—ie)*—1) 4a. (7) This is in agreement with experimental evidence and has been known to hold true ever since the Theory of Dispersion in its contemporary form was elaborated. In the domain of Optical Dispersion, the basis of the usual * A slight change in the original calculation is introduced here ; ef. /.c. pp. 10 and 22. of Refraction, Reflexion, and Extinction. 281 procedure rests on a scheme of differential equations deter- mining the laws of propagation of light in material media. On the view here being developed, these laws of course continue to apply and therefore must be derivable from the molecular standpoint. It will then serve to bring out more clearly the scope of the foregoing calculation if we show that our solutions are consistent with the fundamental Maxwell-Lorentz equations. To examine this point it will be convenient to recapitulate some of the results previously obtained. We have: NT A (2) (B,) = 4aN A“ Ee ONG 3 0) oer ms >) i(b2c2 —1)° ? (ii,) =0, (H.)=0, (8) Atr7NA™be . 3 (H.)=0, (H) = yaaa" *, (H)=0, (9) ee INA Che +9), (P,) =0, CE) a0, AO) These equations merely reiterate our foregoing results : (6) and (10) of $5 and (3), § 2; (H,), (H,) etc. represent as before the components of the actual electric and magnetic field at a point in the medium and (P,) etc. are the com- ponents of the polarization electrically induced. It may now be immediately verified that (8), (9), and (10) satisfy the well-known Maxwell-Lorentz equations as usually formulated. § 7. A slightly different form may be given to the fore- going equation (7), $ 6, namely : 4 e 9 Do Bes Ny —n“ — am N— y 3 nO y>—K?— 1 a) ye cay 7) 2 5) >» 99 ° ° é e p-—K-—1)* + 4y*k- Aa N — ( Met me Sn? QvK Ve F072 ae dp??? © * ; . (2) Aa N — : WK Equation (2) includes as a particular case Lord Rayleigh’s celebrated formula of extinction ; for assuming « to be very small in comparison with v—1 we have from (2) *, if we retain the leading term only : vi, ani 873 ; ee Ty = SS LS v*—1)- a7), ONE N ( ) 4mcN — n v * Of Bulletin Int. de l Acad. @. Se. de Cracovie for 1909. p. 92.0. 282 Prof. L. Natanson on the Molecular Theory where X is the wave-length and h=Amn/r represents Lord Rayleigh’s ‘“ coefficient of transmission.” Thus divesting our equations of perhaps unnecessary generality (for it is doubtful in how far we may consider the calculation applicable unless « is small) we revert to Lord Rayleigh’s fundamental theorem. It may be well to observe, by way of caution, that (2) and (3) are quite independent of the form assumed by the term which appears in the numerator of the left-hand member of (1). It.must be borne in mind, in this connexion, ‘that we can adopt a view of the damping action (experienced by the electron) which departs widely trom that here contem- plated without necessarily inv alidating the general form of (1} and (2). J£ we do so, the expression for the left-hand member of (1) may noe become substantially modified, whereas (2) will in general be quite different. This con- sideration is serviceable in illustrating the intimate connexion which exists between Lord Rayleigh’s equation and the hypothesis from which we have started as to the nature of the frictional effect generated during the vibration of the -electron. If we were to adopt, for instance, Helmholtz’s ‘tentative supposition (beautifully exemplified in H. A. Lorentz’s theory of impacts) according to which the vibrating electron experiences a resisting force in simple proportion to its instantaneous velocity of motion, we should have to put A’, instead of A‘, in the denominator of the right-hand member of (3) (and to modify besides the value of the constant). 8. We now proceed to consider the electric field 22 vacuo where z<(). This field consists of the effect of the incident -waye EM = AM in(t—ae+p) 0 els and the cumulative electric effect leo) — IaanN A”) ( dzpelé — bzg+a(z—2)+- 9] 0 L nd 2QarN A? alae 2) u(be+ 1) ; produced by negative secondary waves which are emitted, in ‘the negative direction of the axis of ¢, by every stratum of -the material medium, from z.=0 to z=«. We will assume that (2) simply represents the reflected wave thrown back of Refraction, Reflexion, and Extinction. 283 into vacuum, seemingly from the surface of the body. Hence, if the reflected wave be He gene 2)5,) | Seaton. (C3) (where A* will be supposed to be real), we get | oe NA AN — i(be+ 1) ° ° ° e . . (4) To connect A* with A” it is only necessary to resort to equation (5) of $5; the result may be expressed be ~ A*=— ae CLE el) 2k aa le G1) be+1 which gives the well-known relations, applicable in the case of external reflexion : Ata V ea] ae Ke TACO n(q—p) San C= ae ne ae Ser CONC) ny 2 2« i aoe n(q —p) =-+ (p+ 1) a we ar ed (6 b) It may be easily verified that the component along y of the magnetic field of the reflected wave is Mente 10), Da laCz) In concluding this subject it may be well to consider in a similar manner the refracted wave, propagated within the material medium (z>0). In order to represent this wave, we write Beier be i (8) supposing A** to be real; having regard to (6), § 5, we obtain mole seal He gin(g — 7) 3 A =iee 1): mute Soomro COR By (5), § 5, used in (9) we dedue 1) yen in pr) € *K * ee ) & be+ 1 ‘. (10) and therefore we find Ax* Pye) A) cos n(r—p) = wer (1a) Ax** i sin n(r— jp) = (11d) “Aw ( eae 284 Prof. L. Natanson on the Molecular Theory For the magnetic field of the refracted wave let us assume. Berit), ay where B** is real; the application of (10), § 5, and the preceding equation (9) gives Btt=bcA**e?C 9...) (13) This shows that RBx* “Awe C8 n(is—r)=y5 . 7 Saleen a sinn(s—7)—=—«. - 5) eee Our discussion has thus far been confined to the case of external reflexion. The general conclusion at which we- arrive is that, availing ourselves of the principles established in §§ 1-5 of the present communication, we are led in all. cases to correct results, although we have taken no account whatever of the requirements of boundary conditions. It should be remarked that the merit of having called attention to the advantage that can thus be secured belongs entirely to. Mr. W. Esmarch. 9, Let us imagine a plate of thickness Z and of optical behaviour (v, «) different from that of the void region on euch side of it. As before, let the axis of z be measured vertically upwards. The plate is supposed to be bounded by two parallel planes which for distinctness of conception we. suppose horizontal; let the lower plane be taken for the pline <=0 and the upper one for the plane z=Z. Conceive a series of plane waves of linearly-polarized light propagated f vertically upwards through the region of negative values. | of z: Ihe in(é—az 1 1 1) BO = AVE aed) BE; = (), D}S en (1) I : ; | HH =0, HOS AVC —etd) 7OS [ Under the influence of the penetrating wave the molecules. 1 constituting the substance of the plate are thrown into. i sympathetic vibration and may be regarded, at least very , approximately, as centres of diverginy disturbances. Our aim and object is now to specify a system of these secondary i wave disturbances capable of maintaining themselves steadily in the plate. To effect this, let us consider in how far, for- of Refraction, Reflexion, and Kixtinction. 285 the purposes of the present problem, the scheme of relations laid down in our preceding calculations reyuires modifica- tion. It may be readily seen that all the results established in § 1 above will here still hold true ; but in order that they may thus preserve their generality, we must replace (1), $ 2, by the following equation of definition : V(r, t)= bis’ : (Pe 170 + (OVaIA Za) eae i) aig (3) where P and Q are new constants, to be determined here- after. It follows that the quantities A, B, C (on which the eilculation of secondary effects ultimately depends) are expressible in the form 2,2 : : Hh, + A= —" (Pe inde + Qeinho) elt -47)§(1 _3R2\743R}, (4) 7, an” —inbz inbz,) in(t -- ar) 2); pee (be) tt Qe ie (i—R)jr+Ry, . (5) es ON (Pe inden 4 Qeindeoyeinl’—ar) 5 4 RI, thee ae (6) oa where R=1/anr as before. We now pass on to the calculation of the secondary effects to be expected, in the present case, upon the principles established in §§ 4 and 5 of this communication. Imagine a thin stratum in the plate, at right angles to the axis of z; let dz be the thickness of the stratum. We have to evaluate the resultant electric field, at a point M(a, y, 2) of the field, due to the presence of vibrators occupying the stratum. Let [EY] represent the component along w of this resultant field; proceeding as in § 4 we easily find: (Be |= —27ran dzyN (Re~ 1% + Qet??%) elt - 4 | 2-25 |]. (7) By alike process, for the component along y of the resultant magnetic field excited by the stratum, we obtain | ns TE ON) dey N (Reg ar Qe ayer 4 age (8) the upper sign is to be taken when z—zg is positive and the lower sign when this difference is negative. Let the point M be chosen within the substance of the plate, in a plane z==< at right angles to the axis of z. We proceed to consider the electric field (measured along 2) which arises at M under the simultaneous operation of secondary waves travelling across the plane in the positive Phil. Mag. Ser. 6. Vol. 38. No. 225. Sept. 1919. X | 286 Prof. LL. Natanson on the Molecular Theory direction of the axis of z; or, which is the same thing, of concurrent secondary waves advancing in the direction in which the incident wave traverses the plate. Using the symbol (Hf), to denote this quantity (which, as in § 5 above, we propose to call “the secondary positive electric effect”) we shall have by (7): le Q in(t —az (BY), = —20N Eeeaare |: oe e+) 2ZaNP gin(t—bz) _ 2nNQ eet bz), (9) i(bc—1) —1) i(be+ Ly : It remains to find the secondary negative electric effect (E)_ which arises at M under the operation of secondary wayes crossing the ae under consideration in the negative direction of the axis of 2 Going back again to (7) we find Pe 22 Qe c f VN Oa). cae | —inaZ ,in(t +az) eat lca I aa 10) . 5 2m eit —be) 4 2 Reser tea git + bz) (10) i(be +1) i(bc —1) «Des For the actual electric field at M we obtain, by means of (9), (10) and with the assistance of (1) and (2), an ex- pression which represents several superimposed sets of waves ray elling in the plate along the positive and negative direction of <. But if we assume (as we do throughout these calculations) that the state of radiation attained by the system is already thoroughly steady, we have to remember that the substance of the. plate i is Incapable of transmitting slepirbus ge ie disturbances without enfeeblement and with veteeny C to secure the inexistence of such waves the << inoperative” terms in the expression for the resultant secondary effect must be balanced by the field of the incident wave. We therefore have the equations [ P a Q ] — Al) end 2aN li{be—1) i(be +1) = AN? ane and Per mez QentZ ee = anda > Z be+1 bc—1 Y (‘) and the component in the direction of w of the electric field at M is simply AnN 1(b?¢2—1 Se ie 1 Pe in(t— be) 4 Qein(t+b2 Vi. of Refraction, Refleaion, and Extinction. 287 From (11) and (12).it appears that u(be+ 1) Oe? —1)A%e in(d + ba) a (be +1)e inbZ _ (be— 1) .—inbZ’ * i(be—1) (Be 2 ST AU AC 0) (be -+1)%e inbZ — (be —1)2¢— 02" i 27 NP= 27NQ = (15) All this may be corroborated, as a little consideration will show, when our knowledge of magnetic effects is turned to account. We have indeed to assume that the “ inoperative ” terms in the expression of the resultant secondary magnetic effect are destroyed owing to the cooperation of the primary magnetic field ; clearly this is a condition of things which is consistent with steadiness and permanency. To attain our object we need only evaluate (H\”)+ and (H™)_. We find: 2 le () - CD \ eae af we | 07) (haz) oN EF i(be —1) cee ; 27NE 2nNQ | é u(t —-bz) © in(t+ bz). i(be—1)° a(be+1)° siaets (16) (H®) 2 eer ms ey : Yes uUbe+1) tbe—1) YaNP F 2a7NQ) , SNe (0) ed gilt +bz) ( a Taos =a Ok: (a) enlt +a(z—Z)| By application of (2) we conclude that the component along y of the actual magnetic neld at M is as follows: An Nbe te , a we 2 a, (PME DG Ue Oa Be eh (18) and, on the other hand, that our present equations (11) and (12) are fulfilled. Thus, whether electric or magnetic effects are considered, the inference is, as before, that (11), (12), and accordingly (14), (15), must hold true. It is of importance to assure ourselves that the intensity of the wave apparently reflected trom the interface 2==0 can be correctly deduced from our investigation. For this purpose we calculate the cumulative “electric effect. of secondary waves emitted downwards, into the region of X 2 Ve gee ope 288 On Refraction, Reflexion, and Hutinction. negative values of z, by all the vibrators contained in the plate. If this effect be represented by E,, we have by (7): Z E,= —27anN bi ( dzyeitlt— Sot Ue— 0) + Qf day hret ee], ae (19) From this we easily obtain, making use of (12), (14), (15), ie (Pe—1)\(e"4— e— mb) A Vegint+az+d) 20 Be bet 1a (be yee The last application that we shall make of the results obtained is to investigate the intensity of the transmitted beam of light. We begin by calculating, at a point M past the plate (z>Z), the cumulative electric effect that arises under the influence of our system of vibrators. By means of (7) we get the result selfs ec eae aa Fear i(be—1) Pea inbZ Q einbZ mn 2| s ; wet l (¢—az) aa Clea on ‘ 5 ee The effect of the vibrators in the plate is thus to increase by the quantity (21) the component along w of the electric field of the incident wave (1) conceived to pursue its course un- disturbed. Accordingly, for the complete expression of the actual electric field at the point M past the plate, we have with use of (11), (14), and (15) : fe Abe A Dein? —uWz—Z) +d] at om (be + 1)2¢04_ (be —1)2@— 2" It may serve as a confirmation of the legitimacy of the method employed to mention that (21) and (22) are in accord with the results which it is customary to obtain by the aid of certain assumptions respecting boundary con- ditions. While constructing our argument we have convinced ourselves that, on the view here advocated, boundary conditions could be altogether ignored. ae XXVI. On the Lhe Diffraction by Apertures with Curvilinear Boundaries. By Sisir Kumar Mirra, M.Sc. Lecturer in Physical Optics in the University of Caleutta® [Plate IV. } Introduction. TEXHE scattering of light by a rectilinear diffracting edge has been studied by many experimenters, among whom may be mentioned principally Gouy T, Wien tf, E. Maye 9, and more recently also Kalaschnikow ||, who has used a photo- graphic method. The rigorous theory for the case was given by Sommerfeld { in the famous memoir on the Mathematical Theory of Diffraction in which he dealt with the case of a seml-infinite screen and developed a general expression for the intensity applicable both for small and large angles of diffraction. The analogous phenomenon of the large-angle diffraction by a curvilinear boundary does not, however, appear from the literature of the subject to have received much attention. Experimenters, as for instance W. B. -Croft** and more recently also Arkadiefft and Gordon ff, who studied the diffraction phenomena of the Fresnel class due to various forms of aperture, have practically confined their attention to the parts of the diffraction field for which the angle of, deviation is small ; and in fact the photographs reproduced with their papers Shee only the areas of the field which are relatively intensely illuminated and lie within or very near the geometrical region of transmission of the rays passing thr ough the aperture. It cannot be claimed that the results obtained by these workers give an adequate picture of the phenomena of diffraction by apertures of the forms considered. Judging from the analogy with the case of the semi-infinite screen treated in Sommerfeld’s paper, the in- tensity of the diffracted waves should be sensible over an area enormously greater than that of the cross-section of the geometrical pencil of rays passing threugh the aperture, and we should expect the distribution of “luminosity over * Communicated by Prof. C. V. Raman, M.A. + Gouy, Ann. de Phys. et de Chem. (6) vii. p. 145 (1886). t W. Wien, Wied. Ann. xxviii. p. 117 (1886). § E. Maye, Wied. Ann. xlix. p- 69 (1893). | Kalaschnikow, Journ. Russ, Phys. Ges. xliv. (1912). {| Sommerteld, Math. Ann. t. xlvii. p. 317 (1895), ** W. B. Grotu Phil. Mae. ser. 5, vol. xxxvili, (1894). tt Arkadief, Phys. Zeit. xiv. (1913). tt Gordon, Proc, London Physical Society, 24 Oct., 1912. 290 Mr.S8. K. Mitra on the Large-Angle Diffraction such large areas to depend on the precise form of the aperture used in a much more striking manner than is poxsible within the limits of the geometrical region of trans- mission. If the analogy could be stretched a little further, it might indeed be said that we have to proceed outside the limits of this region in order that the true “waves of dif- fraction” having their origin at the boundaries of the aperture might be separately observed. An investigation of the distribution of luminosity in the fainter outlying regions in diffraction-patterns of the Fresnei class is thus evidently of considerable interest and importance ; and the present work was undertaken with a view to studying a few typical cases of the kind, and thus ascertaining the general features of the phenomenon. In order that the best results might be obtained, it is necessary that the edges of the apertures used should be smooth and highly polished. The cases for which the large- angle diffraction have been studied in the present investi- gation are the following :—(a) Hlliptic apertures ; (6) semi- circular apertures, and other forms of aera bounded by arcs of circles and straight lines ; (c) apertures or screens with undulating or corrugated boundaries. ‘The experi- mental method adopted was to use a brilliantly illuminated point source, the light from which after passing through the diffracting aperture falls upon a photographic plate held at a suitable distance from it. By using prolonged exposures, the fainter regions of luminosity surrounding the Fresnel pattern may be recorded on the plate, which is then studied at leisure. The brighter parts of the field of course become greatly over-exposed and cease to show any detail. && 8 = (3) oO I 5 ial =| 4 3 we Eg Yes = ee i aS ; i= Sb EE 7 © — i a 3 FS Ss 10 oO & s cob) 2 ~ ec 8 iets ¢ = wm o S -_ s ia) oO it) $272 wer Ss . 2 age “UGI}EINILY JO aSvj,usd1ag ‘value, for when the experimental results are expressed as percentages of saturation, the curves from the different experiments are almost coincident, as shown in fig. 5. Solution of Atmospheric Nitrogen and Oxygen by Water. 333 The results of the ¢xperiments show that the bubble of air in passing up the tube continually exposed fresh water- surface to the air, and at the same time kept the water mixed. In order to confirm this view, a test experiment was made with a tube 5 feet long. This was filled in the ordinary way, and the inversions carried out until the air- content was about 60-70 per cent. of saturation. ‘The water in the tube was then drawn into the pump in two equal portions, each of which was analysed separately. One of the analyses represents the air-content of the water in the upper halt of the tnbe, and the other that in the lower half. Upper half. Lower half. ORs Cath. O09) xe 0:075 c.c. Oe as CEA 0235 er. 0°247 c.e. IN een ORAM @yoh 0°466 e.e. Mortals ns.) 0-776 c.c. 0°788 c.e. These figures show that the difference in air-content between any one portion of the water in the tube and any other is not appreciable on analysis; hence the assumption that the water is well mixed seems to Ibe a reasonable one. In order to keep a check cn the manometer readings, the air-content of the water was determined after each ex- periment by boiling out im vacuo and measuring the gases, using the apparatus described above. A comparison of the results obtained by the different methods is given in Table 4. The area of the surface of the bubble was caleulated from the measurements of its length when at rest and in motion and the known internal diameter of the tube. The volume of the bubble was 15 c.c. Its length when in motion was 16°04 cm., and when at rest 13°30 cm. The diameter of the tube was 1°20 em.; hence its sectional area was 1°13 sq. cm. The total volume of the water supported around the bubble was the amount which collected when the bubble burst, een (OOo: 5 lr die. == 3; 10\¢.c. This volume is made up of the portion round the hemispherical cap of the bubble and the cylindrical shell below. Bri Mag. S,i6, Vol. 38. No. 220. Sept. L919.) 2A 334 Prof. Adeney and Mr. Becker: Determination of Rate of TaBLE 4.—Comparison of Results of Analysis of the Dissolved Gases with Manometer Observations. | | | Volume of gases | Analyses of gases dissolved. : No: of | T absorbed Saturation BF) ee eg | ea values P rature. : ment. | from from |Oxygen+ Carbon m4.) (Dittmar). Addition., Graph. |Nitrogen.) Dioxide.) ‘ | aS ee Na 2 ES —— = | = — — "022 ce. | -028 ec.) -050 cce.*| — — 12 | 11°4C. | 2:076 ec.| 2-060 ec.) 2°039 ce.| -034 ec.| 2:078 ce. Be Distilled 13 11:3 C. | 2°186 ee.| 2°115 ce.) 2°134 ce. | +101 ce.) 2-235 ce. |2°142 f water. | 14 12°6 C. | 2177 ec.) 2130 ce. 2-007 ec. | “868 ee.) 2°375 ce. | 2-079 | Vartry | 15 =| 121. | 2121 ce. 2:10 ce.| 1-932 ce.| °375 ce.) 2°307 cc. |2°106 ; water. | a = — | — /0°015 ce, | 3-984 ce.| 3:999 ce.*; — a 16 11-8 C. | 1°750 ce.) 1-700 ec.) 1°632 cc. | 4:179 ec.) 5°81 1 ce. oie Sea- | lid. 12°3 C. | 1:776 ec. 1790 ce.) 1°634 ce. | 4°119 ec.| 5°753 ec. | 1-700 | water | | | * These figures give the initial air-contents in the case of distilled water and sea-water réspectively. The volume shown dotted in the diagram (fig. 6) is equal to Fig. 6 GEN Gi a eee (ad, h l (t) 5 See 8 DY? and the volume of the cylindrical shell is DEN oe =m{ Ge ) 2 3) 2 2 Fs ee == (CRC: dy>—6ld? + 6ld?="984 x 24, d,>—96°2 d,?+114:93=0. By trial it was found that the value of which very nearly satisfied this equation Having found the in- ternal diameter of the bubble, it was com- paratively simple to calculate the surface area. This is made up of the top hemi- spherical area+ cylindrical area + base area: was d,;=1:0993 em. Ue: ae d d,? 2a + 7d, ((-F).+079 d =7d, (14-7) )j dy = 1:0993 x 3°1416(16-04 + 0:2748) =96°344 sq. cm. ’ Solution of Atmospheric Nitrogen and Oxygen by Water. 3395 Hence the area of water air-surface was 56°34 sq. cm. When a gas is dissolving in a liquid, we may assume that tthe rate of passage of the gas into the liquid is proportional to the partial pressure of the gas and the area of liquid exposed. Hence:— Rate of passage of gas into liquid =S.A.p, where p=pattial pressure of gas, A=area of surface, and S=rate of solution for unit area. As solution goes on, the gas in the upper layers of the water escapes into the air, and the rate of its escape is pro- portional to the amount of gas in solution ; hence if w= weight of gas per c.c. in upper layer, then the rate of escape of gas from liquid=f.w.A. ‘This gives us as the net rate of solution— S.A.p—f.w.A; and when equilibrium is reached, 2. e. at saturation, S.A. Deak: w.A or Sp=fw. eats of 0? is generally unknown, since the gas diffuses rapidly from the surface layer of the liquid and the exact gas-content is uncertain. If we keep the liquid mixed, we render A the area uncertain in general. But if a method of mixing the liquid which would leave A still determinate is possible, then we can calculate the rate of solution for a given area, assuming that the gas remains at constant density. These conditions are complied with in the case of a cylindrical bubble moving up a narrow tube. If V=volume of liquid and p=density of the gas (assumed constant), the rate of solution is:— dVw ; i SAp—fwA =SAkp —fwA GID Se i= y Ley syhichh-—a— 10). een). when @= she and b=} = ee. 2) | 1(w i)" a ad Oe a Ce-%, w=Owhent=0, hence c=— i? a a a=; (l—e7%), whent—co, w= i the saturation value. ) ) . dw Hquation (1) shows that plotting the values of wie against at DEN 2 336 Prot.Adeney and Mr. Becker: Determination of Rate of w should give a straight line, and when the actual obser- vations are plotted in this way, a straight-line graph results ince 7) Fic. ‘al Ue 70 60; 50 — 30 Rate of Solution (in ces, per minute), 20} “5 0 V5 2-0 Air-content in ccs. From the graph Ai gun - against w, we can obtain the initial rate of solution into air-free water. In the diagram (fig. 8) the intercept OA gives the total amount of air absorbed which agrees closely with that calculated from direct observations. The intercept OB gives the initial rate of solution at the beginning of the experiment. To obtain the actual rate of solution into air-free water, we must continue the line AB back to meet an ordinate thr ough ©, where OC is the quantity - of air in solution initially. Then GD represents the initial rate of solution into air-free water. Solution of Atmospheric Nitrogen and Oxygen by Water. 337 Fig. 8. The table shows the results of the experiments treated in this way. | 9 | | Values of “a. No. of | aes || MN Experiment. In per cent. | Values of “0.” | ‘In ccs. per min.| of Saturation | per min. 12 | 645 31°3 315 | Distilled | 13 | ‘675 31:9 '-319 [ water. 14 685 32°2 | 322 | Vartry 15 | 687 32°7 | 327 J water. 16 “510 30°0 "300 Sea- 17 | 510 30°0 “300 water. Taking the mean of the results in the case of sea water and ‘distilled water, we arrive at the formula os = 30°5—"308w, which gives the rate of re-aeration, in percentages of saturation per minute, for an exposed area of 56 sq. cm. These figures represent the mean effect over the area ‘exposed. Experiments are being made with bubbles of various sizes, and it is proposed to deal with these in a further communication. The authors desire, in conclusion, to express their in- debtedness to Dr. Hackett, Lecturer in Physics in this College, for the advice and assistance that he has generously given them in the mathematical treatment of the subject of this communication. Chemical Department, Royal College of Science for Ireland. XXIX. A Syntonic Hypothesis of Colour Vision with Mecha- mcal Illustrations. By Prof. E.M. Barron, F.R.S., and H. M. Brownine, M.Se.* [Plate VI] CONTENTS. 1. intreduction. 2. Some fundamental facts of Vision. 3. Three vibratory responders postulated. 4. Mathematical theory. 5. Experimental arrangements. 6. Results and their significance. 7. Summary and Conclusion. 1. Introduction. a E conception of tri-colour vision due to Young, and further developed by Helmholtz and Maxwell, has proved so successful as to be almost universally accepted. But in one respect it appears incomplete, since red, green, and violet sensations are always referred to without any indication being given of the type of mechanism to which the initiation of those sensations is ascribed. Theories of audition often involve some kind of resonance, but are occasionally attacked as demanding a larger number of separate vibrators and nerves than are actually present. An attempt was recently made ft to reduce very consider- ably the number of separate aural mechanisms and associated nerves needed on such an hypothesis. This led us to imagine that a similar view for the explan- ation of colour vision might be successful provided the number of vibrating responders for each element of the retina could be reduced to three; being one each for red, green, and violet. Probably the formulation of any such syntonic hypothesis has been deferred owing to the impossibility of imagining vibrators of such high frequencies in any molar system and the very considerable difficulty in the way of such high frequencies for molecular or atomic vibrators. These diffi- culties are distinctly lessened by the advent of the electron theory and the ‘‘ resonators”’ of Planck. But in adopting for the eye the so-called resonance theory started for the ear, this hypothesis cannot by itself suffice * Communicated by the Authors. t “The Resonance Theory of Audition,” Phil. Mag. July 1919. A Syntonic Hypothesis of Colour Vision. 339 for the former as it may do for the latter. It may be, how- ever, that syntonic responses initiate changes of a physio- logical or chemical character which in turn aitect the nerves. Experiments are here described with a set of three vibrating responders in the shape of pendulums whose relative vibrational frequencies correspond to red, green, and violet. ‘This crude experimental representation is allowable and may be highly useful because the mathematical theory of forced vibrations is almost independent of the exact form of the vibrators themselves. These three pendulum responders are found to be sufficient to imitate most of the fundamental facts of colour vision. Others, such as persistence of vision and irradiation, are then supposed to be due to the comparative slowness of the physiological changes and their spreading round the spot on the retina where they originated. 2. Some Fundamental Facts of Vision. For the present purpose we may summarize the most important facts of colour vision as follows :— (a) The range of visual sensations corresponds to rather less than an octave, or the wave-length varies from about 0°76 w to 0-4 p, the frequencies being nearly (4 to 7:6) x 10% per second. (b) The possibility of making white and matching any colour by the true addition of red, green, and violet. (c) Tie true addition of red aad green give the appearance of yellow. (d) The true addition of blue and yellow do not give green but white (or pink). (e) The true addition of red and violet give a purple which is not found in the spectrum. (7) About thirty colours can be discriminated in the spectrum, each one appearing to be a monochromatic patch. (g) A period of about one tenth of a second is needed for fully acquiring or iosing visual sensations. (h) The occurrence of irradiation or the spreading of very bright images on the retina. All the above are for persons with both sight and colour vision normal. For colour-blind persons one (or more) colour sensation seems to be absent. It is well here to contrast the eye and the ear. 340 Prof. Barton and Miss Browning on a Syntonic Tie visual analysis of colours into their components is practically lacking, but t! e spatial properties of the eye and associated judgment have great power in perceiving direction and estimating range. The ear, on the other hand, has an astonishing develop- ment of analytical power, but its perception of direction does not approach that of the eye. 3. Lhree Vibratory Responders postulated. As shown in a previous paper on ** The Resonance Theory of Audition”? *, we saw that the resonators must have a range comparable to that of distinct recognition of pitch. Similarly, if a “resonance theory” is to play any part in accounting for colour vision the ‘‘resonators” must be supposed to have a range approximately equivalent to that of the visible spectrum. The exact relations between the range of perception and that of the resonators may be slightly different in the case of the ear and the eye because of the different dampings which are probable in each case. Obviously if the facts of colour vision depend upon syntony the number of responders (for each element of the retina) will probably be three only, since the tri-colour theory of Young-Helmholtz has been so successful. Accord- ingly if only three responders are provided for about one octave, the resonance of each must be very much spread instead of very sharp, as the latter would involve gaps in the spectrum seen by the eye. From this it follows that the damping of the responders (as measured by logarithmic decrement) must be greater for the eye, than in the case of those that have any measure of success in explaining audition. These considerations point to the postulation of a set of three vibratory responders tuned respectively to red, green, and violet, their precise frequencies and dampings being chosen to accord with the facts of the case. Having previously used, for the imitation of the ear’s mechanism, a set of pendulum responders tuned to each semitone of an octave, it now seemed well to adopt, as a trial for the eye, three of the same responders, namely, those then referred to as OF, F%, and B. For the purpose of imitating vision these responders may be taken as corresponding to wave-lengths of 0°76 wu, 0°55 mw, * Phil. Mag. July 1919. Hypothesis of Colour Vision with Mechanical Illustrations. 341 and 0:4 uw, the colours being red, green, and violet respectively. Possibly a nearer imitation would be afforded if the range were still less. 4. Mathematical Theory. It is obviously desirable to associate some specified values of the dampings with the “resonators ” postulated, and then derive mathematically the consequent resonance curves. The equation of motion of forced vibrations may he written dy dy : 9 Pe We J 9 2h Tp y= fsin nt, 14h. (all) where y is the displacement of an element of the responder, 2k is the frictional resisting force per unit mass per unit velocity, p? is the elastic force per unit mass per unit dis- placement, 7 is the maximum value per unit mass of the harmonic impressed force of frequency n/2a; the frequency natural to the responder, if devoid of friction, being plan. The complete solution of this equation may be written fsin (nt — $) | pce a on WA ae te 2)2 + (2kn)?} + Ae“ sin (gf + «), (2) (Displacement) (Forced Vibration) (Free Vibration) where 2kn tan Can ae yy? Ct ee en ate, Ge) and A and « are me constants depending upon initial conditions. Weare here concerned with incident radiations of variable frequency n/27 throughout the visible spectrum, but with only three assumed values for the frequencies of the responders. These may be denoted by p,/27, po/2a, p3/2a respectively, and their wave-lengths by Aj, As, A3. Then, if v be written for the speed of light, we shall have pimy/fv= 2 2a or Qrrv 2arv Pi === inch Se ot Acad SR 4 P1 NE i (4) where w is written for the variable wave-length of the incident light. From equation (2) we may obtain the value of 6, the Aelytive amplitues of forced vibrations (=) a 5 = 342 Prof. Barton and Miss Browning on a Syntonic logarithmic decrement per half period natural to the responders. Thus, for the indices of e in (2), we have ky =k, (ry/2v) =, whence 2v6 zg k, Ay ( ) Referring again to equation (2) and using (4) and (5), we may write for the amplitude of the forced vibration ve vi ; WA ( pn?) + (2k,n)*} = Me Bv'dy / {1 Ao (= 7 2) } ASN Ne 1B and similar equations for Y, and Y3, the value of 6 being assumed the same throughout. For the shape of the “ resonance” curves we are concerned only with relative responses, so may fitly take the ratio of Y; to its maximum value Yo attained for v=),. Then, from (6) we have se | Mo == Sarv2d e ° ° ° > e (7) Thus for the ordinate w, of the relative curves we have may —e ~ zn = ee ae ae el r ANON Fig. 1 V/OLET 0-40 0-45 0-50 0-55 0-60 0-65 0:70 0-75 0-804 Wave length of incident radiation Resonance Curves. The three “resonance” curves shown by the full lines of fig. 1 are plotted from equation (8) by giving wu the three Hypothesis of Colour Vision with Mechanical Illustrations. 38438 values corresponding to wave-lengths 0-4, 0°55, and 0-76 w respectively, thus representing violet, green, and red ; the value of the log. dec. per half period being retained throughout 6=0:2. It will be seen that these resonance curves in their general features are very similar to those current as repre- senting the various degrees of excitation of the colour sensations. The following contrasts are. however, to be noted, (a) these resonance curves do not fall completely to. zero, {b) there is no special convexity of the curve for the red “responder” for very short wave-length. From the curves shown it is thus seen that the damping has to be considerable even for the half period. Accordingly the absolute time to reduce the natural vibrations of one of these responders to a negligible quantity would be extremely short. But vision persists for about a tenth of a second and also the full normal perception is not reached under about the tenth of a second. Hence in addition to the syntonic vibration as a primary response to the stimulus of incident light we must also postulate a secondary effect executed by another mechanism whose operation requires time of the order of a tenth of a second for its completion. We may therefore suppose that some atomic or electronic vibratory mechanism first responds in syntonic fashion to the stimulus of the light received, and that this response initiates some physiological or chemical change, the com- pletion of which occupies an appreciable fraction of a second. The nerves are then supposed to be excited by the results of this change. Further, it may be supposed that, after the light is cut off, the vision persists until the ordinary physio- logical operations reinstate the normal condition of things. And this process may be imagined as occupying about the tenth of a second. Referring again to the “resonance” curves plotted in vane inane drooping ends might be continued to zero, on the supposition that when the vibratory response fell below a certain limit the physiological change initiated by it fell still more abruptly and soon became imperceptible. Such hypothetical continuations are shown on fig. 1 by broken lines. ‘Thus the full line curves may be taken to represent both the syntonic responses and the physiological changes supposed to be directly proportional to them. “The broken lines, on the other hand, represent only the values of the phy siological changes supposed to be much smaller than would result from simple proportionality to the vibrations that initiate them. The whole of the curved lines, full and 344 Prof. Barton and Miss Browning on a Syntonic broken together, are then clesely like those current as representing colour sensations. It may be well to note here the flexibility of the present hypothesis of syntonie vibrators of three frequencies. The trequency of each responder and its damping are entirely at our disposal ; thus giving six variables which may be chosen to accord as closely as possible with experimental facts. It should be remarked that the full line curves give amplitudes of the responding vibrations postulated. Their intensities would be proportional to the square of the ampli- tude multiplied by the square of the frequency. Conse- quently, curves of intensity would be much more sharply peake i and exhibit some asymmetry. Any further details as to the possible mechanism 1 of such syntonic responders or that of the subsequent physiological changes we do not presume to enter upon. 5. Huperimental Arrangements. Among the facts of vision already enumerated, the early ones show that the range of the responders should be slightly within an octave, and that their relative frequencies should correspond to those of red, green, and violet. Further, to leave no gaps in the visual spectrum necessitates considerable damping in the responders in order that their “ resonance ” should be suthciently spread. Fig. 2. Experimental Arrangement. The mathematical theory of forced vibrations holds for almost any type of syntonic responder. Hence we here adopt as an experimental test and illustration of the hypo- thesis the simplest mechanical vibrator. The arrangement consists essentially of pegsuluins hanging from an overliend cord, as shown in Mie. 2 2. : i stout horizontal cord ACB has a pendulum with heavy Hypothesis of Colour Vision with Mechanical Lllustrations. 345 bob D hanging from C. A second cord AFE has a second pendulum hanging from F with heavy bob H equal to D. Hach pendulum. is adjustable in length so as to represent any incident radiation desired. It may be observed that the virtual lengths of these heavy pendulums are O’D and F’H respectively. The set of three responders, supposed present at every part of the retina, are represented by the pendulums Rr, Gy, and Vv, with very light bobs consisting of paper cones. These are made of such ‘lengths as to correspond, in relative frequencies, with the three colours red, green, and violet. These light-bobbed responders have a logarithmic decre- ment of the order used in plotting the resonance curves of fig. 1. The heavy driving pendulums have negligible dampings and so can represent the sustained amplitudes of ineident radiations. In order that each responder may be equally influenced by the heavy pendulum these light bobs R, G, and V are arranged along a straight line through A. (See “ Forced Vibrations,’ Phil. Mag. Aug. 1918.) Tf only one incident radiation is to be imitated the cord ACB and pendulum CD are sufficient. In order to imitate a second radiation simultaneously incident, the second cord AFE and pendulum FH are provided. This second pendulum eo the responders by means of a wooden bridge near v, connecting the two cords ACB and AFE. In order He record the response of these light pendulums under different conditions of stimulation, it is necessary to photograph them when in action. To insure that all these pendulums shall appear to hang from the same point in the photograph, the lens of the camera must be on the line AC produced, as shown in fig. 2. 6. Results and their Significance. Bighteen photographs are here reproduced (PI. VI.) showing time exposures of the responders vibrating under various conditions of excitation. Photographs 1 to “6 show tests as to colour mixtures. Photographs 7 to 18 show the responders under the action of a single driver, changed in frequeney by a number of small steps. In dealing with these effects it will be convenient to designate the pendulum or pendulums by the names of colours : thus a driver of such length as to equal that of the red responder will be called a “ red” driver, and will imitate the emission of red radiation. Pendulums of other lengths 346 Prof. Barton and Miss Browning on a Syntonic will in lke manner be referred to under the names of other colours. Colour Matures. Red and Green make Yellow.—Photo 1 shows the response to a yellow driver. Photo 2 shows a response t» two drivers, | red and green, acting simultaneously. It is seen that the responses in 1 and 2 are precisely alike in character as might be expected from the resonance curves of figure L. Thus, on our hypothesis, it is shown that red and green make yellow. Thatis to say, this set of only three responders red, green, and violet, cannot differentiate between (i.) the simultaneous stimuli of red and green drivers, and (11.) the single stimulus of a yellow driver, whose frequency is inter- mediate between those of red and green. Photo 3 shows the response to drivers representing red and bluish green. Blue and Yellow fail to make Green.—1t might be supposed that on our hypothesis any two colours of the spectrum being used as simultaneous stimuli, the system would give the same response as if under a single stimulus of intermediate frequency. If so, blue and yellow would make green, whereas, with the eye it is not so. On the contrary, the true addition of blue and yellow give the sensation of white or a faint pink. It is therefore a crucial test of the present hypothesis to ascertain the response of the three vibrators when under the action of blue and yellow drivers. This is shown in photo 4. It is seen that all the responders move, the middle or green responder least of all, and the red one most. Thus, so far as the colour departs from white, it is certainly not green but pinkish. It therefore accords with the known facts of colour vision as exhibited by colour tops and converging beams. It is easily seen why with this set of three responders, blue and yellow drivers do not give the effect of a single intermediate frequency as the red and green did. ‘The latter have each a special responder, neither of the former has a special responder in tune with it. Blue and Red give a Colour not found in the Spectrum.— Photo 5 shows the effect of blue and red drivers acting simultaneously. As might be expected, we have here a eee that could not be due to any single spectral colour, . e. to the vibrations of any single driving pendulum of Tosti anywhere in the range of the responders. This result accords with the known fact that the true addition of red and blue gives a purple which does not occur in the spectrum. Aypothesis of Colour Vision with Mechanical [liustrations. 347 Hering’s Theory tested—On Hering’s Theory, in addition to red, green, and violet, there are also yellow, black, and white cencitions. To test this, as regards the yellow, a fourth responder was used of relative frequency to correspond with yellow. Then with drivers representing blue and yellow light, the effects shown in Photo 6 were cbtained. Here the amplitudes of responses are unequal but the yellow predominates. It is difficult to believe that this would correspond to the impression of white light or pink as was the case in Photo 4, where only the red? green, and violet responders were used. Fineness of Discrimination of Colour.— Dr. Edridge-Green states that in the spectrum the eye can distinguish eighteen to twenty-seven different regions each of which appears monochromatic but different from its neighbours. It was therefore necessary to test the present hypothesis as to the adequacy of three responders to provide for a fineness of discrimination of this order. This was done by starting with the driving pendulum of period equal to that of the middle responder representing the green. The effect of this is shown in photo 7. The suspension of the driving pendulum was then repeatedly lengthened by small steps till in photo 18 it equalled that of the red responder. The inter- mediate effects are shown in photos 8to17. The differences of relative amplitudes of the responders are usually clearly marked as we pass from plate to plate. Hence, there are twelve states shown in about half the range of the spectrum, which would give a possible twenty-four in the whole spectrum, and this approaches the order of discrimination stated by Dr. Hdridge-Green as being actually accomplished by the eye in the most favourable cases. A glance at some of the photos (e.g. 7, 8, and 9) will show that still finer discrimination than that actually exhibited would be possible. So that, on the present hypothesis of vibrating responders of only three frequencies, provision is made for a fineness of discrimination quite equal to that known to exist. This is on the supposition that the relative amplitudes of the vibratory responses can be appreciated through the changes occurring in the eye just as well as one can estimate the relative dimensions exhibited on these photographic prints. In the light of these results the present hypothesis of colour vision may be compared with that put ‘for ie ard by Dr. R. A. Houstoun (“A Theory of Colour Vision,” Proce. Roy. Soe. ser. A, vol. xcii. no. A 642, pp. 424-32). 348 A Syntone Hypothesis of Colour Vision. Ih Summary and Conclusion. The hypothesis is here advanced that each element of the retina may possess syntonic vibrators of frequencies corre- sponding to those of red, green, and violet light respectively. These responders are supposed to be of molecular, atomic, or electronic nature somewhat like the resonators of Planck. The adequacy of three such responders to yield the great variety of effects corresponding to those of visual experience is tested by an experimental arrangement. This consists of oo) a horizontal cord from which hang three light pendulums. These are sympathetically stimulated by one or more heavy pendulums hung from the same or an associated cord. The light pendulums represent the vibrators supposed present in the eye. The slight displacements of the cord from which they hang repres sent the light incident upon the eye. The heavy pendulums which produce these displacements are like the luminous sources. In order that a vibration of any frequency throughout the range corresponding to the visible spectrum should appreciably stimulate these responders, their “ resonance ” must not be too sharp. The responders must accordingly have considerable damping. ‘T’o account for the persistence of vision and some sluggishness in its inception it is further supposed that the responders in the eye do not act directly on the nerves. On the contrary it is imagined that they only start some physiological change which lasts about a tenth of a second. And this change in its turn is supposed t» stimulate the nerves. The behaviours «f these three responders under various conditions of stimulus are photographically recorded and compared with the facts of colour vision. It is thus found that on these suppositions red and green would make yellow, blue and yellow would make a pinkish tint, and so on for other facts of normal colour vision. The cases of colour-blindness would be met by the supposition of the absence of “resonators” of certain frequencies. No claim is made that the resonators postulated have been proved to exist. The consideraticns here put forward simply show that if the mechanisms postulated were present, colour vision would be in the main as we now find it. Nottingham, May 30, 1919. gen o's eso) XXX. On the Derivation of the Lorentz- Einstein Transformation. By Haroup Jerrreys, M.A., D.Sc.” ie we have two systems of reference, S and 8’, and space and time co-ordinates be measured relative to each of them, the usefulness of the Lorentz-Kinstein transformation lies in the fact that it provides a means of determining either set of measurements when the other is given, and the relative motion of S and S' is known. Let the space co-ordinates and the time referred toS be a, y, z, ¢, and those referred to 8’ be 2’, 7, 2’, t’. One datum we have to start with is that provided by the Michelson and Morley experi- ment, recently confirmed bya different method by Majoranaf, namely, that the velocity of light is always the same relative to both systems of reference. Now if any object (a particle or a wave-front, for instance) moving with the velocity of light relative to S, be compared with a particular luminous wave-front starting at the same point and moving in the same direction relative to S, we see that the two coincide at all times. They must therefore, by the fundamental principle that makes it possible to refer objects to space-time co-ordinates at all. coincide at all times relative toS’. Thus if ¢ be the velocity of light, it is true, whatever the moving object may be, that de? (dy? (da? ,. ,. _(de'!? (2s! 2 de (a) “+ e. Ga) =c? implies (Ga) + Ag a (ea) ==. (dl) Put «edt=du, tcdt'=dw', da’ +dy* +d2+dwv=ds’, dx’? + dy? +dz'? + du? =ds", Then (1) becomes simply ds—Ontmplies,ds’ On) yo... (2) Most writers on the principle of relativity have assumed in deducing the transformation from the fundamental principles that TS Re 2 uIP IN al) but this is clearly more than the proposition (2) directly warrants, for (2) would hold equally well if ds’ were equal to kds, where k may be any function of 2, y,z,andt. It the two sets of co-ordinates were related, for instance, in the same way as the co-ordinates of a point and its inverse with regard to a four-dimensional hypersphere, & would be * Communicated by the Author. t+ Physical Review, May 1918. Phil. Mag. 8. 6. Vol. 38. No. 225. Sept. 1919. 2B 350 Dr. Harold Jeffreys on Derivation of the a function of a, y, 2, and t, so that the available data are not sufficient to prove the constancy of k*. With the further eae that z', y', <’, and t’ are linear functions of «, y, z, and #, (A) we see at once that & must be a constant; but there is no more reason for this assumption than for the other. If, however, instead of (4) we make the assumption that any object moving uniformly with regard to 8 is also moving uniformly with regard to 8’, . . .. |=) 2 ay) it is possible to infer from this and (2) both (3) and (4). For believing (5) there is a good a prior reason, afforded by the relativistic modification of Newton’s first law of motion. This asserts that for an “isolated” particle referred to axes in an isolated non-rotating body the differential coefficients of the measured space co-ordinates with regard to the measured time are constants. Thus for any such a ae di ymeidem ae dye de i dt) Meunier at? i ai! By comparing any unifermly moving object with an isolated dz dytade dt’ di? “de are all constant. Now if this are all constants. particle, we therefore see that whenever dg ay mdz Gu il” ae be so, = is a constant, for it is equal to dx\? dy\* dex? Ve yp) {oop (aig) Wea (ass ‘< (ai) a | (a) } Also ut 3 constant, and thus ds? ds? de Gems all are all constant, constant. Cx dy aes du dst ds? ds? ds? The argument is reversible and therefore leads to the proposition : Hence =(). If Cee C2 ea ase des to ds? ds then 9) adap wy) ds ue 6 asi mds? =| ds!" 7 ds2) sua pene (6) Thus (5) is equivalent to (6). * BE. Cunningham, ‘The Principle of Relativity,’ p 89, or Proc. Lond, Math. Soc, 1909.. Lorentz- Kinstein Transformation. d51 GAN, Oz au Gissmmas: ds?) ast and sufficient condition that \ds, taken along a path between two fixed points in four-dimensional space, shall not be altered by small variations in the path. Thus we have. the proposition that if \ds is unaltered by small variations in the path, Jas! also is unaltered. But ds’=kds; therefore when ds is stationary, fhds is stationary. Suppose now, if possible, that & is not constant. Take a path of integration such that \ds is stationary, and & is not the same for all points near the pathas for neighbouring points onit. If then the path be varied slightly, the ends being kept fixed, but in such a way that the new path is always on the side where & is greater, {ds will be unchanged to the first order, and fk ds will be increased. Hence the hypothesis is violated, and k must therefore be a constant. To pass from this result to k=1 requires merely a permissible change of units. There- fore we can assert that ds and ds’ are equal. Now the vanishing of is the necessary Next, peri | Oumeon da, Ouids Sm Ocds Oude. Oz ds Ou ds” immaye 1az du Wig ai : : C d Sem cule a But when ds? ds? dg? ano dq ave any constants, ds’ 8 On’ 02’ Oe’ Oa’ a constant by (6). Hence ae Sy Ae ak are all con- Semis, andy is) a linean function of a) 4,02) and 2, Similarly y’, 2’, and w’ are linear functions*. Thus the postulates (3) and (4) are proved, and the Lorentz-Hinstein transformation can be obtained from them by the method given by Cunningham f. * It may be noted that this cannot be proved from (6) alone without (2). For (6) merely asserts that every straight line in the S-hyperspace ‘transforms it into a straight line in the ‘S' -hyperspace; and this is satisfied by any transformation of the form Beale tg fie Hw: ) y= F? z a B, ul = a = where p,, Pr, P3, P4 are any linear functions of 2, y, z, and wv, and f may be any function of x, y, z, and « whatever. + ‘The Principle of Relativity,’ p. 49 (1914). 2B2 XXXI. A Method-of Measuring without Electrodes the Con- ductivity at various points along a Glow Discharge and in Flames. By Bare. vAN DER Pou, Jun., Docts. Se. ( Utrecht) *. a stationary waves are induced in a pair of parallel wires tuned to the frequency of a generator, the current in this Lecher system will increase till a stage is reached where the energy supply from the exciter is just balanced by the energy dissipation in the wires. The principal part of this dissipation is to be found in the Joule losses, while the radiation from the Lecher system wiil in general be very small, due to the currents in opposite parts in the two wires being 180° out of phase. But when at a point in the field between the wires some small body is introduced having a finite conductivity, energy will be dis- sipated in it as well. Under a constant excitation the current (and Potential Difference) will therefore in this case only reach a lower final value, and this latter current limit is a measure of the energy dissipation in the substance brought between the Lecher wires. It is, however, not a unique measure for the conductivity o of the body con- sidered, as is obvious from the consideration that both when . GMa) o7— co no absorption will take place. Hence a second factor must come in, and this is the alteration in the period caused by the presence of the body in the electric field. The above considerations are further equally valid whether the body brought in the field between the wires is electrically connected to them or whether it is not. The procedure is therefore of value when the absorption in a substance is to be measured where the presence of separate electrodes would either disturb the natural state of affairs (as is often the case with test electrodes in a glow- discharge) or where a strong polarization at the cathode makes the direct-current electrode measurements complicated or uncertain (as in flames). For mathematical treatment the simplest way is to con- sider the body to be located in one single point. As our experiments were confined to the body being placed near the free ends of the Lecher wires, where a potential loop occurs, we will only consider here this case. When the body * Communicated by Sir J. J. Thomson, O.M., P.R.S. Measuring Conductivity along a Glow Discharge. 353 is not electrically connected to the Lecher wires the theory can still be reduced to the case as shown in fig. 1, where a Rigel system B having a capacity C and resistance R in shunt is connected to the Lecher wires BD. for when the body is disconnected from the wires as in fig. 2a, this system can always electrically be replaced by that of fig. 2b, 2.e. by a Fig. 2 a. Fig. 26. C, C; R, Cc R To: condenser © and resistance R directly connected to the Lecher wires. If the capacities Cy, C., C3 and resistance R; of the loose system of fiy. 2a are given together with the angular frequency w, U and R of the equivalent fig. 26 will be uniquely determined ne the conditions : 1 ua | +3(Gt ta) 1 joC + R jolt Ps" where j= / — ae i, When the resistance in the Lecher wires is, for the time, neglected and the distance from the inserted system B to tie first potential node A (fig. 1) is called /, the resonance value of J will be given by * Anl 2b tan oun = 7p aa : : 2 ° : (1) where the numerical quantities a and } represent c Ci RG SOW eral ® (= Se : AC eC * Drude, Wied. Anz. xi. p. 466 (1897). 854 Dr. Balth. van der Pol on Measuring without Electrodes Here Cand Rare the capacity and resistance respectively concentrated at the end of the wire system, C the capacity per unit length of the Lecher wires, and c the velocity of light. Further, the ratio p? of the reflected energy and the incident energy at B (fig. 1) is found to be 2 Gaia aa 2a 2) ~ (a+1)?+0? (@ Fl)? 0? ee @ and it is seen from (2) that total reflexion will take place (p?=1) when a=0 corresponding to R=a, and a=o2 08 99 99 as previously stated. When therefore the concentrated capacity © of fig. 1 is kept constant but the resistance R is increased gradually from infinity to zero, we must, in order to keep the Lecher system in resonance with the exciting E.M.F. of constant ere eno} gradually shorten the distance ! from one given by a a 2arC- Xr rAC to l=0. At the same time the current in the parallel wire system at all these resonance points will first, when R=a, bea maximum, it will then gradually fall till it reaches a mini- mum ; next it will inerease again, till, when finally R=O, the current reaches a maximum again. If further, C is kept small, it will be seen that during the first stage of the process just described, 7.e. from where R= toa point where R has still a considerable value, the numerieals a and 6 are very small and the resonance len oth of J is principally determined by the value of the added capacity ©, for here (1) reduces to | Anl _ tan esac TAY From (2), however, it follows that during this stage the reflexion coefficient p? and, at the same time, the current in the Lecher system is principally determined by the value of Conductivity along a Glow Discharge and in Flames. 355 the resistance R, viz.: the greater Kh the smaller the absorp- tion in the Midad sy stem and the bigger thereforé the current in the parallel wires, for here (2) reduc ‘es to pe=1—2a. As the resonance curves at this stage (current squared in Lecher wires plotted against /) become more and more flat the smaller the resistance R, it 1s practically possible to leave the length 7 constant and to measure the absorption in the added system B (fig. 1) by the fall in the current in the Lecher system, and at the same time be sure that for the said range of the resistance R all current measurements are made under resonance conditions. Therefore over the range considered an increase in the conductivity of the concentrated system manifests itself in a fall of the current in the wires. And a very simple means of making sure that one 1s working within the conditional range is afforded by the fact that little or no change in the length / is necessary to keep the parallel wire system in resonance as R changes from toa big finite value. The working condition is therefore similar to that of an oscillatory circuit of a con- centrated capacity, resistance and self-inductance, where the capacity or self-inductance is shunted by a big resist- ance, the current being measured in the main oscillatory circuit. The apparatus used for the measurement of the absorption in various parts of an independent glow-discharge is described in another paper. In fig.3 ADBC is the parallel wire system. At B (BC~}A) the leads E to the Duddell galvanometer end freely over the Lecher system, while at A the dischar ge-tube FG is brought between the wires through a hole in the ebonite piece H, supported in turn by the slider K. By the aid of a string running over a pulley the discharge-tube could be lowered gradually from a distance, while ‘sliding through the hole in H. This movement of the iicehar ge- -tube has to be effected from a certain distance, as the approach of the hand near the free ends of the Lecher wires disturbs the resonance. A rubber tube at L connects the apparatus with a Gaede pump and drying and filling apparatus. V indicates the position of the generator. The glow- discharge, started, if necessary, with an induc- tion-coil, 1s eniGained bya 1000- volt storage-battery with a S ifemeecictamce and microampere meter in series The total length of the parallel wire system was never longer then 356 Dr. Balth. van der Pol on Measuring without Electrodes IX, a resonance position of the slider K being always possible in this range. The discharge-tube had a thin paper scale stuck to it over the whole length so that both the positions of the various Fig. 4. parts of the glow-discharge and the positions of the tube with regard to the Lecher system could be determined. In this way it is possible by sliding the discharge-tube vertically through the parallel wire system at A and at the same time taking the galvanometer readings, to measure the conductivity at different parts of the tube with a simple apparatus and without the disturbing effects due to separate test-electrodes brought into the discharge. Conductivity along a Glow Discharge and in Flames. 357 Our first experiments were made with trains of damped oscillations generated by a Blondlot exciter producing waves of 150 cm. length. When with such an exciter galvano- metric measurements giving the M.S. current in the resonator are made, it is advisable to make the spark under oil as short as possible, so that an early quenching sets in permitting the oscillator to vibrate with a small decrement in its own natural period. The induction-coil was fed with alternating current, and water-resistances were inserted between the coil and the spark, making the latter as constant as possible. oii 1 as ic a | Lt ie Lae 1 D>» As an instance of the results thus obtained we give the curve of fig. 5 obtained with a striated discharge ‘through Hydrogen at. 1x10-*amp.). Here the ordinates of the curve M represent the galvanometer deflexions, while the ab- scissee give the positions of the various parts of the discharge Bs: wa wails anaes es (eeeeeenes . as NS i\fwy VV v7] 4 & Wil se | a) F 358 Dr. Balth. van der Pol on Measuring without Plectrodes between the Lecher wires. The shaded curves at the bottom of the figure indicate the approximate distribution of the light in the tube, while the upper dotted curve N was ob- tained in the same way (by sliding the tube again vertically between the wires) but without a glow-discharge passing. The latter observations were taken in order to examine the influence of the disk electrodes on the tuning of the re- sonator. It is seen that only when the electrodes came very near the plane of the Lecher system did a small detuning take place, observed as a slight decrease in the galvanometer deflexion, so that this effect can in general be neglected. Turning to the curve M we see that the absorption of the waves increases up toa point in the negative glow where the absorption is maximum and therefore the current a minimum. In the Faraday dark space the absorption gradually de- creases in the direction of the positive column, while over the striations the current is constant; near the anode an increase of current (decrease of absorption) occurs, and as soon as the anode reaches the plane between the wires all absorption has disappeared. In their general aspects these observations agree with the measurements of the cross currents under a constant E.M.F. between two test-electrodes at various points of the discharge * However, some differences still occur. In the first place, no variation of absorption in the different parts ot the striations was obtained with our electrodeless method, while the electrode method yields a maximum of conductivity at the bright parts of the striations. Again, the maximum conductivity found by the electrode method is at a point much nearer the cathode than that found by the electrodeless method. The reason for the first discrepaney is obviously that with the Lecher wires 42 mm. apart and a tube diameter of only 18 mm. the electrodeless method must yield, for the conductivity, values averaged over a bigger distance than will be the case in the electrode method. The second discrepancy can only be explained by a further in- vestigation of this special point and will he left for the moment. In order to concentrate the field between the parallel wires more to a small region with the view of detecting the varia- tion of absorption over the strice, we fixed a pair of wires, * See e.g. J. J. Thomson, Cond. of Electr. thr. Gases (2nd Ed.) p. 561. Conductivity alongia Glow Discharge and in Flames. 359 each forming an are of about 90°, to the ends of the Lecher system, closely fitting round the disvharge- tube in the way as shown in fig. 4. It was then found, however, that through the more intense action of the waves on the olow-discharse the latter was disturbed by the presence of the waves. The Faraday dark space appeared in this respect to be a very sensitive part of the discharge. When, namely, the Faraday dark space was brought in the plane of the Lecher wires and the waves were started, the striated positive column suddenly increased in length, thus producing one or two more stria- tions. If thereupon the tube was gradually moved so that the anode approached the parallel wires, the marked tendency of the positive (striated) column was invariably observed to keep the last stria just between the Lecher wires. The first abnormally extended column could in this way be com- pressed like an elastic body till it extended over a length of one or two striations smaller than normal. If the tube was moved still further a point was reached where the com- pression went too far and the first striation suddenly jumped through, the positive column having again its normal length and number of striae. The first few big amplitudes of each of the damped wave trains account for this interference. The Blondlot exciter was therefore replaced by a three- electrode thermionic tube generating continuous oscillations of a very high frequency, as described elsewhere. At the same time another discharge-tube was blown having a bigger diameter (about 32 mm.) with the aim of getting the stria- tions at greater distances apart. Again two pieces of wire, each forming an are of 90° round the new discharge-tube (fig. 4), were embedded in a groove in the ebonite support A and connected to the parallel wires. Special care was given to make this arrangement accurately symmetrical with respect to the plane of the Lecher wires. ‘The continuous waves produced by the thermionic generator had a wave-length =376 centimetres and were very steady. The use of the continuous waves proved to be a great improvement. Though it was still possible for the waves, when the generator was brought very near the oscillator, to disturb the discharge, this disturbance could be eliminated by reducing the coupling between oscillator and resonator, and hence a state could be reached where, while the galvano- meter deflexion amounted to 15-30 centimetres, no direct action of the waves on the discharge could be discovered. In order to make sure that this was not the case, the aspect of the discharge was carefully watched while the waves were 360 Dr. Balth. van der Pol on Measuring without Electrodes switched on and off (by closing and breaking the filament circuit), and no change in the position of the various parts of the glow-discharge was observed, neither did the steady current through the discharge vary. _Hven when the Faraday dark space was brought between the Lecher wires, which part of the discharge was found before to be the most sensitive one, no disturbing effect of the waves on the dis- charge could be detected. On the other hand, the effect of the discharge on the waves was very marked, and the absorption of the latter in the various parts of the discharge could easily be measured. An exception is to be made, however, for the brighter part of the negative glow and the regions near the electrodes. The elec- trodes, namely, were turned out of brass, covered with aluminium foil, and fitted closely inside the discharge-tube. When now the latter was lowered till one of the electrodes was in the plane of the Lecher wires, the big capacity be- tween the electrode inside the tube and the circular arcs directly outside the tube, being only separated by the glass wall of the vessel, upset the tuning and the slider K had to be moved over several centimetres, and the Lecher wires therefore shortened, in order to attain resonance again. The same effect of detuning was caused by the big conduc- tivity of the brighter part of the negative glow. However, the complete positive ‘column, whether striated or not (with the exception of the region close to the anode), together with the Faraday dark space, could easily be measured out. The wave-field was further sufficiently localized to find the variation of absorption over the distance of one stria, as shown by fig. 6 (these observations were taken without the bent wires of fig. 4). Here the discharge passed through hydrogen taken from a steel cylinder with a little turpentine vapour added to get — the striations well marked. The striz had a sharply defined concave boundary facing the cathode and the parts between the successive striations were very dark. The discharge current was 7=4:0x107* amp., while the pressure was p=0-'50 mm., as read from a MacLeod gauge. The galvanometer current is again represented by the ordinates, and the abscisse show that a region round the central part of the discharge was measured out, the anode being placed at the point 0 of the abscisse scale, while the cathode is found at the point 42-2. The resonance length of the parallel wires was first care- fully determined and found to be constant over the range of the discharge represented in fig. 6. The big absorption (high Conductivity along a Glow Discharge and in Flames. 361 conductivity) in the Faraday dark space is again very well marked, while, with the exception of the last stria, where the absorption has probably been influenced by the near B50 Dae. 20. ow ye 720M. proximity of the highly conducting dark space, the maxi- mum wave-amplitude corresponding to minimum conduc- tivity always occurs at the negative front near the brightest part of each stria. The same minimum of absorption was always observed in numerous measurements on glow-discharges, both in air and hydrogen. Again, the observations of fig. 7 were taken with the wire attachment of fig. 4, the dischar ge being again in hydrogen, of pressure 0° 145 mm., while tie steady discharge calsent was 0°6x 10-7? amp. The cathode is here at the abscissa 0 and anode at 42°2. The action of the discharge on the waves is here more pronounced, and again, as in the former case, the maxima of wave-amplitude (minima of absorption and conductivity) are found at the negative heads near the 362 Dr. Balth. van der Pol on Measuring without Electrodes bright parts of the striations. The same distribution of absorption over the striee was obtained whether the wave- amplitude was strong or weak. This fact may justify the extensive use here of the term “ conductivity.” A series of measurements was finally made of the conduc- tivity in the positive column when the discharge current, by means of an external resistance, was kept constant at i=2%5x10-3 amp., while the air pressure inside the tube was varied from p=0:400 to p=0'140 mm. At the higher of these pressures the positive column was continuous, and as the pressure was lowered, it gradually broke up into striations. In a preliminary experiment the absorption in the un- striated column was found to decrease only slightly tewards the Faraday dark space. Therefore only a few observations were taken in the unstriated column, these being sufficient to indicate the position of the absorption curve. The result is shown in fig. 8. It is seen from the measurements that when the pressure is relatively high (0-400 mm.) and the column is continuous, the conductivity inside the tube is small. When the pressure is reduced to 0°215 mm. the conductivity, for the same dis- charge current, is considerably increased. At this pressure some faint striations began to appear at the cathode side of the column. A further reduction of the pressure (to 0:170 and finally to 0:140 mm.) causes the column to break up into strize while at the same time the conductivity increases Conductivity along a Glow Discharge and in Flames. 363 still further. With a still further reduction of pressure it was not possible to keep the discharge current to the same value as before. Fic. 8. an. x es 32 tent Some S40 Sole 77. It appears to us that the chief interest of these experi- ments lies in the variation of the conductivity found over the striations. With our electrodeless method the minima of conductivity were always observed close to the brightest part of the stria- tions, while on the other hand, ih the electrode-cross- current method maxima ot conductivity are obtained at these points. It is however possible, as suggested by H. A. Wilson, that in the electrode-transverse- ton method secondary ionization due to the illumination of the testing electrodes might increase the cross currents at the luminous parts of the stries * This possible cause of error is of course eliminated in the eleotronuiens method. Moreover, the results * See H. A. Wilson, Phil. Mag. [5] xlix. p. 505 (1900); also J. J. Thomson, Cond, of Electr. thr. Gases (2nd Ed.) p. 562. 364 Measuring Conductivity along a Glow Discharge. yielded by the latter method are in good agreement with the measurements of the electric force at various parts in the strize made by Sir J. J. Thomson with the aid of a pencil of cathode rays sent at right angles across the axis of the strize * Here the maximum electric force was shown to occur at the bright parts of the striations very near their negative end. Now, if at the pressures used the velocity of the ions is assumed to be proportional to the electric force at — the spot, a minimum number of ions and therefore the mini- mum conductivity can be expected to occur at a maximum of electric force f, hence the agreement. In conclusion a number of very preliminary experiments were made with flames. They were simply brought in the space between the ends of the Lecher wires, and the reduc- tion of the wave-amplitude thus obtained again afforded an indication of the conductivity of different flames or of dif- ferent parts of the same flame. In the electrodeless method the uncertainties arising otherwise from the polarization at the electrodes and from their electron emission if hot are obviously again eliminated. The following are a few of the results obtained :— When the unabsorbed wave-amplitude corresponded to 49 centimetres thermogalvanometer deflexion, the latter was reduced to 43 when the flame of an ordinary Bunsen burner was brought between the wires. If thereupon a small piece of asbestos soaked in a concentrated NaCl solution was brought near the bottom part of the Bunsen flame, a con- siderably greater absorption took place and the galvanometer deflexion fell to 29. Again, when some cesium-aluminium-sulphate was brought in the pure Bunsen flame the deflexion decreased from 49 to 26. This is another proof that the increased conductivity of salt-laden flames, as measured with test-electrodes, is not wholly due to the presence of the electrodes, as is sometimes supposed, but it is a peculiarity inherent in flames. In conclusion, I wish to thank Sir J. J. Thomson for his valuable advice and kindness in putting at my disposal the instruments of the laboratory. Cavendish Laboratory, Cambridge. * Phil. Mag. xviii. p. 441 (1909). + J. J. Themson, Cond. of Electr. thr. Gases (2nd Ed.) p. 561. r 365 J XXXII. On the Propagation of Electromagnetic Waves round the Earth. By BALTH. VAN DER Po., Jun.,.Docts. Se. (Utrecht) *. | HE theoretical solution of the problem of diffraction round a sphere, surrounded by a perfect insulator, for electric waves emitted from a source close to this sphere and having a wave-length small in comparison with the circum- ference, is of considerable physical importance. For, where a comparison of measurements taken at various points on the earth’s surface of the amplitudes of the waves emitted by a transmitting station for Wireless Telegraphy would show that they are not in agreement with the calculated values, it would be obvious that other phenomena than pure diffraction have to be accounted for. That e.g. the electrical conditions of the atmosphere have a pronounced influence on the wave amplitude at big distances is undoubtedly revealed by the fact, that incoming signals from a far distant station are as a rule stronger in night-time than during the day. An annual variation of signal strength in overland transmission has further been found, while very small variations during a solar eclipse are also observed. The measurements of signals received during the dark hours show further appreciable fluctuations, while the daytime measurements are usually more steady. This latter fact suggests that we should con- sider the daytime values of wave-amplitude as “normal” in the sense of probably not being affected by changes in the lower atmosphere which are naturally of a more or less accidental nature. It is the object of this paper to compare these observed normal values with the magnitude to be expected according to the diffraction theory. It is, however, obvious that such a comparison. cannot be made unless a definite theoretical formula is obtained show- ing the decrease of wave-amplitude with distance in a form admitting numerical interpretation. Now expressions for the magnetic force at the surface of a sphere for various orientations from an oscillating dipol were obtained (a.o.) by Nicholson | and Macdonald {, but the numerical values these formulee yield differ to a very great extent. | An expression for the amplitude of the magnetic force, however, in the form or a rapidly alternating harmonic series * Communicated by Prof. Sir J. J. Thomson, O.M., P.R.S. + Phil. Mag. (ser. 6) vol. xx. p. 157 (1910) ; vol.:-xxi. p. 62 (1911). { Proc. Roy. Soe. (ser. A) vol. xc. p. 50 (1914) Phil. Mag. S. 6. Vol. 38. No. 225. Sept. 1919, 2¢ 366 Dr. Balth. van der Pol on the Propagation of was readily obtained* (1903), but the summation of the series presents difficulties, and this summation is to be con- sidered the chief cause of the discrepancy of the final formule: of different investigators. This led Nicholson in 1910 to consider the problem under consideration as one in the whole field of mathematical analysis about which most divergent views were held f. Up to a short time ago a definite agreement on the numerical side of the problem could not be said to be arrived at, and consequently very divergent opinions were expressed. on the possibility of explaining the observed values of waye- amplitude at various orientations from a transmitting station by way of pure diffraction only. But recently Watson ¢ has effected «2 new summation of the said series, and, as will be shown below, agreement can now be obtained with the other solutions within the ranges of their validity. When the oscillating dipol is placed outside the sphere at a distance } from the centre with its axis pointing to the centre, a Hertzian potential function II can be defined such that, using the ordinary notations, the components of the electric and magnetic forces outside the sphere expressed in spherical coordinates are given by where @ is the angular distance and » is the number of - oscillations in 27 seconds. IJ has then to satisfy the wave equation (V7 +e) T1=0, : @ 27 k being equal to Pama wa * Macdonald, H. M., Proc. Roy. Soe. vol. Ixxi. p. 251 (1903). | Jahrb. der Draktl. Telegr. u. Teleph. iv. p. 20 (1910). t Proc. Roy. Soe. (ser. A) vol. xcv. p. 83 (1918).. Electromagnetic Waves round the Earth. 367 When the sphere of radius a is supposed to have infinite conductivity and the consequent boundary conditions are taken into account together with the prescribed pay oe of II near the pecillanor the solution is : ie (kb Ria, \=— 75 = (Qn +1)P,(m aes Mae els) where Boake »)2 A? n(@) = ($72) we 5) H®, being the second Hankel function which is zero for 2 2 ° ° ° infinite real argument. The ratio of the circumference of the sphere to the wave-length, viz. ka, is moreover great in comparison with unity. Now Watson transformed this series into a contour integral which in turn he modifies into the rapidly converging series P = kb in Anas yeah He, _ af ) bad . : od , COS V7T {oo Os where the v’s are the (complex) roots of CoN 1 (ka). This series is also valid wlien b=a, and the dipol iis therefore placed on the surface of the sphere as required in the problem of Wireless Telegraphy. It is further shown that when 2 is written for ka (2) s=yV Vyp=2+p,ee *, 2 where Come (3&,)°; é., being the pth root of Y-4(6) —33(6) =0. Further, an approximation for is found to be so that when the oscillator is placed on the earth’s surface 99 368 Dr. Baltn. van der Pol on the Propagation of the final expression for II(a, @) is prea) el) Ne ae which is the form in which the solution is left by Watson. In order to obtain numerical values for E, and Hg at the surface of the sphere (He being zero there) an accurate knowledge of the values of the v’s, and therefore of the &’s, is required. Now Macdonald has already calculated the first three &’s, viz. the first three roots of J_,(€)—J,(&) =0, and he finds . ; } E, => 06854, E> = 3°90, £, = 7-05. Of these three roots the first one is of dominating import- ance as far as a numerical calculation is concerned. We therefore determined &, independently and obtained 0-6855, in close agreement with the value given above. From these valucs of & the first three p’s follow at once: pi 0.s0s3) Dey = ey Ol - p3 = 2°83 ; hence v, = v + 0°8083 28(4—i V3), y= a+2577 a3(L-iv3), Va = OG w(4~iV3). In view of w being a very large quantity, viz. the ratio of the earth’s circumference to the wave-length of the sending antenna, the second real part in the v’s of low order (these only being of importance) can be neglected in comparison with the first, so that these quantities can be written in a more simple way : Pere y= 2— Biz’, a Vz = 2 — Bot2*, o) al V3 = 2 — B3123, where PB, = 0°7000, (Ey = PRD. (So SD Electromagnetic Waves round the Earth. 369 cos v7 is further very approximately equal to 1 Cos pra bebura tien Further, for all values of @ not near 0 or 7, that is for the whole sphere except the zones near the sender and its antipodes, Laplace’s approximation for the zonal harmonic | Bite #) of complex order yields 2) vq 7 Sel —p)= Veen { uae) — 7 } 9\2 _ (rn) ~ 27sin 8 ae { nCRONee a ae T site “ 48281 —8) 5 ces) and the second term within the brackets can be neglected mMIpeIDiS so ucar to a that et 7 is comparable with unity. | The neglect of this second term amounts physically to the neglect of that part contributed to the total value of the potential function (or the wave amplitude) which arises’ from the wave that has travelled round the angle (27—8@), and therefore started in “opposite” direction from the sender, passing on its way the sender’s antipodes. That this amount contributed to the total value of I is negligible, except for points very near the antipodes (@~77), is obvious. When those points are for the time being excluded from consideration (which is already necessary as for this region the approximation used for P _, is not valid), + é Br Dia aca is found to be COS Var or va) ,—Bxt0—i(2947 SS SSS é V sin 0 \ i), (9) Hence the expression for the Hertzian potential function II becomes Lae ay SEONG iN si Tae f2\2 e =e (2 ) m TI (a,6) = — (=) e aps 1 —SBxrs@ ‘ eae é 260 a \@ V sin 0 ; (6) Z p 370 ‘Dr. Balth. van der Pol on the Propagation of and by simple differentiation, bearing in mind that «>l ° dwt By and that the time-factor e”” was everywhere omitted, we arrive at the approximate expressions 13 V2qr x6 ii — px30 ere a’ A/sin@ Pp = due 5 (OLLI Y . Waa ead mm Rl} Bee ~ » os a? Op a 13 JQ «8 IP ear a —— C , Gar aising 8) which shows that the same expression is obtained for ie and | H,|. As of the series (7) and (8) the first term is by far predominant over the greater range of 6, the expression for the wave-amplitude takes the form —-u0 A TS 2 5 Vsin 6 where A and wu are independent of 0. Lhis form of solution is consistent with the following physical : A interpretation :—-The first factor ( vaca) shows that the waves follow closely the sphere’s curvature, with a divergence in accordance with the aea of the sphere over which they spread, the radius of the parallel circles being proportional to sin 0 ; but, while travelling, they continually give off energy at a con- stant rate to the surrounding medium as shown by the second factor (e~“”). The equality of | E,| and | H4| points to the fact, as was to be expected, that at the surface of the sphere the waves behave like plane ones, and the calculation of the amount of energy absorbed in a receiving system may be carried out as if the waves were really plane. For wave-lengths A not too long and angular distances 6 not too small, the series still occurring in (7) and (8) is seen to converge very rapidly. In fact we have 51,60 p 1 07000259 1 _90390%9 , 1 —3-82a%0 8083" F517 + 333° Te 27a. where — ’ Electromagnetic Waves round the Earth. 371 27a being the circumference of the sphere representing the earth, and therefore in the case of Wireless Telegraphy equal to 4. 10* kilometres. . If an error of 10 per cent. 1s allowed, only the first term of this series can be retained for those values of 6 which satisfy 6 > 0:746a * = 0:021825,,, where X is measured in kilometres, or, which is the same thing, from distances d (in kilometres) from the sending station such that iene Se etOl YN coe When an error of only 1 per cent. is allowed this condition has to be replaced by rene > 420 N, Noie For regions on the sphere nearer to the oscillating dipol more terms of the series are required. Moreover, with the first term only a fairly good agreement is already obtained at a distance 0=0°746xz * with the formula directly derived for the plane problem (Hertz, Abraham), viz., ae) a) where | Hz| is the electric force at right angles to the plane boundary surface; here the factor 2 arises solely from the reflexion due to the dipol being placed near or on the con- ducting plane surface. The ratio of | H,| calculated from the first term of the spherical problem to the |H,]| of the plane problem for 0=0:746a7* (corresponding to d,.. = 140 WrAx.«,) is about 0°8. When we now examine the electric and magnetic force near the transmitter’s antipodes, that is near 0=7, Laplace’s approximation for the zonal surface harmonic has to be replaced by Mehler’s Pr(cos o) = Jo{ (n+ t)o}. Near 0=7 we have therefore UAC a a | F,-y(—#) _ 23olv(m—6) 2008 | om Olas cos vir Boia "| aidan Ole nee 372 Dr. Balth. van der Pol on the Propagation of or when we introduce 6’ as angular distance from antipodes, i. e., 0’ =7—8O, we have Nea ie = ( — p) (2) — Bata (0 Sit é ,1cosvTr W/O! — 178! BO Ae 10 4 +e i 4 . e e ( ) analogous to (5), but near 0=7 the second term within the brackets cannot be neglected as here the effect of the waves which travelled over the longer distance 7+6! (i. e. the other way round) is of importance. The effect of both waves, 1, coming round the are @, and 2, round the are 27r—8@, interfere near the sender’s antipodes and give rise to inter- ference rings in the region near 0=7. The region in which (10) is valid is bounded by two con- ditions: The zonal harmonic P,_4(-P) may be replaced by the Bessel function of zero order when a(7—0)?=20%<1. The approximation subsequently introduced for J,, however, is only valid for #(7—0)=a20'>1. It therefore follows that (10), which could further be derived at once from (5), has only a very limited region of validity. In order to discuss the behaviour of the waves in the immediate neighbourhood of the antipodes, |E,| and | Hg | must separately first be found. We have for this region : LT RS ve) ){v(r— Oy pa p COs v7r l= — in which series again the first term is predominant. The magnetic and electric force therefore become, after omission of terms only affecting the absolute phase : 8 2a? 1 _ Bret s—-é rp The electric force, however, is o= Jifvr—6)). . See Bee eo SG LO ap It is seen, as might have been expected, that near the sender’s antipodes the waves coming round “ both” ways interfere and produce interference zones. The magnetic force in the neighbourhood of 6~7 is again of the same \y S Electromagnetic Waves round the Earth. 373 order of magnitude as the electric force. It only happens that at 0=7 the interference zones are such that Hg has its minimum there and equals zero, while E, just attains a maximum. In the accompanying diagram (fig. 1) we give Fig. 1. to) AVNET 10 n 10 AD ee /00 antipodes an illustration of the amplitude of the theoretical value near 0 = 7 of the electric force for a wave of length 10 kilometres travelling over a sphere of 40,000 kilometres circumference. The ordinates of the curves max. and min. denote on an arbitrary scale the maximum and minimum amplitude of the electric force, while the abscissee give distances from the antipodes measured in wave-lengths. Between these two curves the actual electric force oscillates, forming inter- ference zones with about $2 distance between them. Very near the antipodes the distance between two successive zones is a little bigger than 42 in accordance with the successive zero’s of I,(v@’). In order to show how these interference zones are formed a few have been drawn in the small inset at a larger scale. It is further seen that an increase of wave- amplitude is to be found near @=7r, but it does not attain such a value that the point @=7 can be considered to be a focus in the optical sense. Moreover, the figure shows clearly the gradual transition of nearly pure interference with regions oe zero-intensity to the zones farther removed from the point 0=7, where the interfereuce becomes less pronounced till it finally reaches a stage where only single waves are present and where the effect of the disturbance that travelled over the are (27—@). is negligible. That 374 Dr. Balth. van der Pol on the Propagation of these regular interference phenomena will be found in actual practice of Wireless Telegraphy cannot reasonably be ex- pected in view of the irregular distribution of land and sea over the surface of the earth. Moreover, as will be pointed out below, the aimosphere has such an important influence on the amplitude of the waves that it is hardly likely that these regular diffraction zones will actually present them- selves. With regard to the important increase in distance over which signals nowadays are received, it is, however, of some interest to consider to what facts pure diffraction would lead. It is now possible to compare these results with the formulee obtained by Nicholson and Macdonald. Nicholson, who determined the magnetic force and thereof only the first term of the series, arrives at a formula which in the present notation can be written ; Vir = sind em | 4 1.5 2-280 ates where 6' =0°696. The formula obtained above for the whole region for which @ is not near O or 7 was ia 18 a a” Vn0p 7? ne |E,| =| Hg| = where 8,;=0°7000. The difference between Nicholson’s 8’ and Macdonald’s f, is very slight indeed, and is due only to a closer approxi- mation by Macdonald to the first root of Ja(Ej}—J_3(€) =0. >) The unimportant factor —= in (13) does not oecur in (7, 8). mo | As the complete solution is now at hand, it is an easy matter to find the source of the discrepancy between the ik factor \/sin 0 occurring in Nicholson’s formula and ~=—= 4/ sin 0 in (7, 8). This discrepancy is found to be due to page 531 of vol. xix. of the Phil. Mag., where in formula (47) is written for 2 an oversight in the very complicated analysis by Nicholson, where the factor —sin @, thus intro- duced, has been retained throughout the further analysis. With this correction therefore the above formule (7, 8) are practically in agreement with Nicholson’s result. Electromagnetic Waves round the Marth. 375 Macdonald’s formula* can be written with the present notation V/27 8 cost0 = geevoeia ) a ee GE org theme PS Bx3(2sin 39) 14 Fava a? Vinee 7 p ; on and this is seen to be equal to (7) for those small values of 6 for which cos3}@ may be replaced by 1, and 2sin3@ by sin 8, which is just the limited region of validity of (14) as shown by Watson fF. Love {, who further calculated numerically the values of Hg for distances 0=6°, 9°, 12°, 15°, and 18°, found results in close agreement with Macdonald’s formula (14). From a comparison between (7, 8) and (14) or from direct caleu- lations it is obvious that Love’s numerical results are in agreement just as well with (7, 8), and as the latter formula has a much wider range of validity it will further be used in the derivation of an expression suitable for comparison with expe1imental results. So far all formule discussed here have been derived for a boundary condition Hyg=0 at r=a, i. e. for a sphere of in- finite conductivity. Approximations have also been obtained for the magnetic force at various distances by Love §, Macdonald ||, and Watson§ for a sphere having a conduc- tivity of c=1.10-!! (Love, Macdonald) and o=3:77.107" (Watson). All three investigations arrive at the same result, viz. that a finite conductivity of the order of ¢=107" such as sea- water actually possesses, only has a small influence on the wave-amplitude, increasing it a few per cent. in comparison with the case of infinite conductivity. The same result is found by Watson for a conductivity of dry earth. When discrepancies therefore occur between experimental values of wave-aimplitude and theoretical formule the cause is not to be sought in the finite conductivity of the earth’s crust. Wa now turn to the practical side of the matter. The formulee (7, 8) derived above give the wave-amplitude to be expected due to an oscillating dipol of unit moment. * See paper quoted. + Proc. Roy. Soc. (ser. A) vol. xcv. p. 84 (1918). { Roy. Soc. Phil. Trans. (ser. A) vol. 215. p. 105. § See paper quoted. || Proc. Roy. Soc. (ser. A) vol. xcii. p. 498 (1916). {| See paper quoted. 376 Dr. Balth van der Pol on the Propagation of The moment M is defined as the finite product of the infinitely increasing amplitude of the charges gp (in electro- static units) at the ends of the dipol into its length / in centimetres, while / shrinks indefinitely. If, however, the wave-amplitude is only considered at points at a great distance from an oscillator of finite dimensions (great in comparison with these dimensions), M can be defined in the same way by the finite quantities g) and /, so that also in this case M = Gol. As it is usual to measure the current amplitude I, at the bottom of a transmitting antenna, we introduce the relation between I, and g,. We have _——— and Xr qo Fz LOESv. = a -Ioamp. When the antenna height is not small in comparison with the wave-length emitted, the current distribution over the antenna is not quasi-stationary. This fact can simply be allowed for by introducing a form-factor e (Zenneck) giving the ratio of the mean current amplitude over the sender height into the maximum current, which is usually found near the bottom of the transmitting aerial. The moment of a sender station can therefore be ex- pressed as My= zl duipt hia om. > where /, is the actual height of the transmitting antenna. The amplitude of the electric force at the earth’s surface set up at an orientation 8 from such an oscillator is accord- ing to (8) 13 ae a ‘ sin sin 6 pi 200 — pat b Ky E.S.U. a ahi] amp. é where all lengths are measured in centimetres. Further we have Ko volts/e.m. = 300 Ky E.S.U. If in this radiator field a tuned receiver antenna is placed of height h,., form-factor x, and total equivalent resistance Electromagnetic Waves round the Earth. ot R ohms (including radiation resistance), a current I, will be set up in it of the approximate amount (Ruedenberg) aie /es d Lee URED REE NaS tdi) The reradiated energy is here accounted for by including in R, the radiation resistance of the receiving system. The ratio of the current amplitudes or R.M.S. currents in the receiver to the same quantity in the transmitting antenna (all quantities relating to the transmitter have suffix 1 and to the receiver suffix 2) is therefore 13 I, bi 300 V 20 oe A ayhyeghy —Bxr9 I, 207 p,r/sinda® Rk, a : or, when the earth’s circumference is taken as 40,000 kilo- metres, this reduces to _ 23:94 Cher MP iia eeaet MnCl) Ibs il a hyayhs et 5868. pete I, “sin O rE RS where all lengths are in kilometres, I, and I, are expressed in the same units, R, is in ohms and @ in radians. lf the linear distance d in kilometres is introduced be- tween the transmitter and receiver, measured along the great circle, the last factor occurring in (15) can also be written — 0:00376d eae This formula gives the ratio of the received antenna current to sender antenna current under the assumption of a perfectly insulating atmosphere and a perfectly conducting earth. A similar formula for the case of a plane is I, 377214 C ashy I, IU where d is the linear distance between the oscillator and resonator. Formula (15), in which only the first term of the series in (8) is employed, is valid over the whole sphere with the exception of 1, a zone near the sender such that ad<140 4/ (otherwise more terms of the series in (8) have to be employed) ; and 2, with the exception of a zone near the sender’s antipodes where diffraction zones oecur, as explained above. A comparison of formula (15) with experimental obser- vations must now decide whether actual wireless trans- mission is established by diffraction only, the atmosphere being assumed to be a perfect insulator, é 378 Dr. Balth. van der Pol on the Propagation of The experimental data published up till now are very few in number. Practically all the work in this field has been carried out by Austin *, who measured the received signal in a telephone-receiver by shunting the latter down to such an extent that dots and dashes could just be discriminated, and then the value of this shunt gave an indication of the amplitude of the wave at the receiving spot. Now the numerical relations entering in the action of crystal detectors and thermionic valves are not yet com- pletely known, and a certain variety of explanations of their working has been given, while a definite numerical theory of their action has not yet been generally adopted. What was therefore actually determined in these experi- ments is the strength of the sound in the telephone-receiver, representing a quantity of energy which was ultimately drawn (wholly or partially) from the energy of the radiation field. But it only appears in the form of sound waves after two transformations: (i.) in the detector, and (i1.) in the tele- phone-receiver, and neither the one nor the other trans- formation can completely be calculated. Consequently the calibrations had to be carried out under conditions approach- ing as near as possible those of the actual measurements. Unfortunately the description of the experiments and methods is occasionally very short and not quite clear, so that, while reading the accounts, one often regrets that several important particulars have been omitted or not dealt with. What makes a full appreciation of these experiments and complicated calibrations further difficult is the fact that it is not always clear from the descriptions whether certain values or constants were obtained experimentally, by calcu- lation, or by mere estimation. This at once precludes before- hand any suggestion for a different interpretation of the results than that given by Austin himself. ‘The smallest energy in the receiving antenna producing a barely audible sound is given as 1-2 10> watt. It is obvious that the measurement of these amazingly small alternating energies 1s a matter of considerable difficulty and that variations in the observations of a few hundred per cent., as actually obtained, are to be regarded as relatively small, especially in view of the atmospheric disturbances which occasionally prevent the observations altogether. * See Bull. Bureau of Standards, vol. vil. p. 315 (1911); Bull. Bureau of Standards, vol. xi. no. 1 (1914); Proc. Inst. of Radio Engineers, vol. iv. no. 3 (1916) (New York). Electromagnetic Waves round the Earth. 379 When the value given above is accepted as the limit of measurable energy, an estimate of the oscillating electric field near the receiver antenna can be made on the basis of a total antenna resistance of about 25 ohms and an effective receiving antenna height aaliy of 146 metres, which are the figures given for the recelving system at Deion (Canal zone), where a part of the Picaanerieume irene carric dione: From (14a) it is seen that the least amplitude of alter- nating potential gradient measurable in this way is of the order of 107?! voit per centimetre, being about 1071 of the normal static atmospheric potential oradient, and it is there- fore not surprising that proportionally small variations in the latter field occasionally prevent or disturb the measurements of the minute alternating field superposed on it. When Austin’s observations are accepted as giving the true attenuation of wave-amplitude with distance, it is evident that the pure diffraction theory based on the Maxwellian equations and the assumption of a perfectly insulating atmosphere yield values for this attenuation which are in flagrant contrast with the experimental data. We take as a single instance the measurement at Darien of the waves sent out by the Nauen station. In this case Austin gives =150 amp., A=9-4 kilometres, «,h,-=120 metres, aohy=146 metres, R,=29 ohms, d=9400 kilometres. This is the biggest distance up to the present over which received antenna currents have been measured. The ‘audibility’? was in this case 200, corresponding to a received antenna current J,=1:°3.10°° amp.; while the theoretical diffraction formula (15) derived above would yield I,=0°6 10~ amp. In this case therefore the actual value of the wave- amplitude according to Austin’s measurements is about two million times bigger than the value yielded by the pure diffraction theory. Even if the measured quantities are affected by errors of many hundred per cent., agreement be- tween theory and experiment cannot be obtained. As several independent theoretical investigations have undoubtedly proved that the finite conductivity of the earth’s crust cannot increase the wave-amplitude to such an enormous extent, and as the conductivity of sea-water has the same value for the frequencies used in wireless telegraphy as for steady cur- rents *, the cause of this discrepancy has to be found higher up—in the atmosphere. These considerations therefore * Phil. Mag. vol. xxxvi. p. 88 (1918), _ 3880 Propagation of Electromagnetic Waves round the Earth. strongly support the theory of ionic refraction put forward by Eccles*, who suggested a possible increase of phase velocity of long electromagnetic waves when propagated through an ionized gas, which would cause the wave-front to follow more or less the earth’s curvature. An increase with height in the concentration of heavy ions in the atmosphere, together with an increase of their mobility, will cause the rays to bend downwards in a somewhat analogous way as sound waves, emitted at a certain elevation, undergo refraction. A silent zone so often observed in the acoustical analogy has, so far as we are aware, not been found in the propa- gation of electric waves, and it does not seem likely that it will be found, for the energy distribution round an ordinary wire- less transmitter, unless special arrangements are provided, is such that by far the vreatest amount of energy is emitted horizontally, and therefore the part of the alternating field ata certain distance from the transmitter due to the rays emitted with small elevation, will generally be great in com- parison with the part due to a ray leaving the transmitter at a greater elevation, even if the latter were refracted down- wards without any absorption whatever. Roughly speaking the ionic refraction will amount to the same as if a layer were present in the higher atmosphere which is more or less impermeable to long electric waves, and this may account for the view, adopted by some writers, of a reflexion of the waves against a layer of high condue- tivity. Whether this latter interpretation of the phenomena is appreciably different from the adoption of a refraction obviously depends on the vertical density gradient of the ions in the upper atmosphere. It is, however, not likely that this reflexion could occur without any losses, for in a transition region energy would be absorbed, but to what extent it is difficult to predict. (After this paper had been written Professor Watson com- municated to the Royal Society a theoretical investigation of the electric field on the assumption of a spherical concentric shell of finite conductivity surrounding the earth. With the assumption of a sharp inner boundary of this shell and for a certain conductivity of the latter, values of the field could be obtained of the order of magnitude of the observations. It is, however, impossible to state whether a sharply defined inner boundary of this reflecting layer can be expected to exist in the upper atmosphere.) Cavendish Laboratory, Cambridge. * Proc. Roy. Soc. (ser. A) vol. Ixxxvii. p. 79 (1912). faeset XXXII. A Simple Theory of the Knudsen Vacuum Gauge. By Georce W. Topp, ).Sc., B.A.* 3 ROBABLY the most reliable absolute instrument for the measurement of extremely high vacua is the Knudsen gauge. The following simple deduction for the pressure in terms of the constants of the instrument may be interesting :— Consider two parallel strips A and B at a distance apart less than the mean free path of the gas molecules. Let A be at the same temperature I’ as the residual gas, while B is maintained at a higher temperature ‘l,. The strip A will be bombarded from the side away from B by molecules having a velocity of thermal agitation V corresponding to the gas temperature I. These molecules will rebound with the same velocity. ‘The strip A will also be bombarded from B, but with molecules having a velocity of thermal agitation V, corresponding to the temperature T, of the strip B. These molecules will rebound with a velocity corresponding to the temperature of the strip from which they rebound, 2.e. with velocity V. Thus A will be repelled from B. We will use the Joule method of dividing the gas mole- cules into six streams. The mass of the molecules striking each square cm. of A on the side away from B is pV/6 per sec., where p is the gas density, so that the momentum re- ceived per second is pV?/3, since an impulse is given on the rebound as well as on collision. On the side of A facing B the momentum per sec. due to collision is pV,?/6, and due to the rebound, pV?/6. This, of course, assumes complete interchange of energy. ‘’he difference between these is a pressure )P me ; (V,?—V?), repelling A from B. Since the gas prsssure is p=pRT, P= +_ (V,°—V?). r eT ; T Cue w Beale le) Now let each movable vane of the Knudsen gauge be a vertical strip of area A at an average distance 7 from the suspending torsion fibre. If the couple required to produce a twist of one radian be A, and @ radians be the actual But V?=3p/p=3RT, therefore P= 5p * Communicated by the Author. Phil. Mag. 8. 6. Vol. 38. No. 225. Sept. 1919. 2D 382 Dr. L. Silberstein on a Time-Scale deflexion produced, we have P.2A.r= A060. Also if the period of the suspended system be t, then t=274/1/X, where 1 is the moment of inertia. Eliminating 2X @ives P=27°10/rAt?. When D is the scale defiexion of a spot of light on a scale at distance d 277ID eee 9 Fe Ae ie i ee Combining (1) and (2) gives Knudsen’s expression 47°ID T BS pAPd T,—T" Royal Grammar School, Neweastle-on-Tyne. XXXIV. A Time-Scale independent 07 Space Measurement. Parabolic and Hyperbolic Kinematics. By L. SILBERSTEIN, Ph.D., Lecturer in Mathematical Physics at the University of Rome”. 1) Oa: principle of common, metrical time-scales amounts to this: —A certain kind of, say, rectilinear motion of a particle t is declared to be a uniform motion, which logically plays the part of an undefined term. The path of the particle is cut up, by means of compasses or other rigid traustferers, into a series of equal segments, and the instants of passage of the mobile through the divisions of this metrical scale of lengths are taken as ¢=0,¢=1, 2,3,and so on. The instants 0 and 1 being fixed arbitrarily, all other instants are constructed with the aid of a rigid length (or angle) transferer. The history of the moving particle (or of a light flash) is thus dissected by an extraneous process, time is subdivided by appealing to measurement of space, which process, moreover, implies the concept of rigid bodies. Without questioning the high practical advantages offered by this, the usual and only known, chronometrical procedure, 1 propose to show how a time-scale can be set up without the aid of space measurement, in fact, without any space scale of points whatever. For reasons which will appear hereafter I propose to call such an emancipated scale of instants a projective time-scale. Any three distinct instants, say marked by three bell-strokes, being arbitrarily chosen, an * Communicated by the Author. + In practical chronometry a motion of rotation round an axis is used. This, however, does not change the nature of the argument. independent of Space Measurement. 383 indefinite number of other instants of such a scale will be unequivocally co-determined, provided of course that certain (very inoffensive) axioms are assumed at the outset. Every logical (mathematical) theory is bound to start with some undefined terms and with a set of axioms or, better, assumptions. These, being enunciated about the “ undefined terms,” make these latter defined, in part at least. 2. Now, besides such concepts as ‘‘instant of time,” distinct instants, ‘earlier and later” (at a spot at least), and “ between ” two instants, which we will take for granted, let our principal undefined ere be “ unzform motion.” (The objectors will notice that this is also the undefined term of ordinary chronometry and enters as such, whether named or un-named, into every treatise on Kinenaltes. ) To be more explicit, we shall call certain, undefined, motions of a particle along a straight path, anton motion About this class of undefined motions we will enunciate our assumptions. With regard to space itself it will beassumed that it is the ordinary three-dimensional projective space (independent of any idea of measurement). Asa matter of fact, however, we shall require only two of its dimensions, that 1s hg ¢ say, a pro- jective plune in the common sense of the word, Thus our space-time, necessary to prove the determinateness of the proposed time- scale, will be, to use a now favourite name, a three-dimensional orld. ‘And, having proved this, the (actual or mental) experiment leading to the required time-scale will be all performed on a single, one-dimensional, space line. The passage of a particle through a point of the plane at a time instant will be called an event or sometimes a world-point ; and the succession of such events, a history or, more fashionably, a world-line of a particle * 3. Let a particle p be capable of moving along a fixed straight drawn in our plane. Let A, B be two points on the straight, and a, 6 two instants of time, say, 6 later than a. (Points will be denoted by capitals, and time-instants by small letters.) Let it be required that p should be at or pass through A and B at the instants a and ¢ respectively. Then, in absence of other requirements, there will still be an * The latter name is borrowed from Minkowski’s relativistic vocabulary. But it will be kept in mind that throughout only one and always the same reference system will be contemplated, so that our investigation will have nothing to do with Einstein’s theory of Relativity, whether “ classical ” (1905) or “ new ” (19138). 2D2 384 Dr. L. Silberstein on a Time-Scale infinite variety of possible histories or motions of the particle between these two events. In other words, there is an ee of world-lines joining the world-points «= A,a and B=B,b Let, now, our first assumption be : Among all the possible motions leading from A,a to B,b there is one and only one uniform motion. We may say, equivalently: between two world-points a and @ there is one and only one uniform or right world-line, to have a short name. As a consequence of this assumption we have at once : Two particles moving uniformly (along the same or distinct straights), and not permanently united, do not meet more than once. (Or, equivalently, two right world-lines do not cross. more than once.) For if they met twice they would, by the first assumption, never part company. It does not follow, however, that they will meet, or have met, even once only. It is scarcely necessary to say that the above assumption is simply a translation of the foremost axiom of projective geometry, the terms “ point ”’ and. ‘‘ straight’ being replaced by “event” or “ world-point ” and by ‘* uniform motion” or “right world-line.” Similarly let us assume the correlates of all the remaining axioms of projective geometry, well- known bv the name of axioms of order and connexion. Thus, calling shortly a, 8, etc. the events A,a; B, b, etc.: If y is an event of the world-line segment «8, then any fourth event 6 belonging to this segment belongs either to ay or to yf, but never to both segments, If y, an event distinct from £, belongs to «8 and if the event 8 belongs to 6, then y, and therefore also 8, belong to the seoment ad. (This gives the extension of the: right world-line beyond the terminal event Bape. and similarly A, a.) If a, B, y do not belong to the same right world-line, and if 6 belengs to By and e to «6, then there is always an event € belonging to «@, such thats e belongs to yf. Each of these assumptions can be enunciated in terms of places and time-instants relating to uniformly moving particles. This is left to the reader, who will find the use of representative drawings helpful, but by no means necessary for the purpose of logical deductions. If these projective axioms were valid only for particles moving uniformly all along the same straight path, we should independent of Space Measurement. d80 have a two-dimensional manifold of events or world-points (X, t) which would have all the properties of a projective plane. Many interesting theorems would hold for such a world-plane. Yet the Desargues theorem about perspective triangles (built up of right world-lines) which is the most important thing for our purposes, would not necessarily hold *. But if, as was announced, the said axioms are assumed to hold for every straight of (at least) one space- plane (X, Y) as the possible path of uniformly moving particles, then we shall have a three-dimensional world (X, Y,¢) endowed with all the properties of projective space. And the world-plane (X, t) being now “ part” of this projective three-dimensional manifold, the theorem of Desargues will hold in that plane. in other words, the world-plane X,¢ will bea Desarguesian plane. Such being the case, any three collinear world-points Beniasicy (2. e. belonging to the history of a uniformly moving mote) will determine a unique fourth point, their fourth harmonic conjugate to a, say. likewise, any three con- current right world-lines in the plane X, ¢ will co-determine a unique fourth line. 4. Now, this enables us to build up at once the desired ‘* projective” scale of time-instants. All that is required for this purpose is a number of uniformly moving particles which we will call 7, 2, p3, 4, etc. We shall use these symbols also for the histories of the particles or their representative straights in the world-plane CX, ¢). In fact, let all these Pe move along the same X-straight. Let ¢=0, 1, 7 be three conventionally fixed instants of time, say 1 later than O,and 7 later than 1. Let us represent the X-straight at various instants by straight Jines drawn in the X,¢-plane through an arbitrarily fixed centre which we will call ©,. Thus the X-straight at the said three instants will be represented by the three lines marked ¢=0, 1, 7 (fig. 1). Now, through any point of the first line draw two straights p;, p. crossing the second line in «, 8,and the third line in y, 6 respectively. Draw ps, the join of a, 6,and py, the join of Bandy. These will cross in e,and Qe will be a definite line through ©,, independent, that is, of the choice of the two lines p,, po. The line O,e will thus represent the X-scraight at a pertectly definite instant. Let us give to this instant the label ¢= * Since this, as is well-known, would require either three-dimensionality or the congruence axioms, 386 Dr. L. Silberstein on a Time-Scale This graphical construction shows us at once how to arrange what we may call the jfour-particles experiment. In fact, it is enough for this purpose to remember the meaning of the allegorical figure 1. Mx Thus, the three instants 0, 1, 7’ being fixed arbitrarily, let the two particles p,, p. start together at the instant ¢=0 from any point O of the X-straight. At the instant t=1 let p, and p, send off two uniformly running messengers ps, and py, such as will catch or meet ps and p;, respectively, at the same instant 7. Then, these two messengers js, p4 will meet with one another at a perfectly definite instant (no matter, where p3 meets po and where p, meets p,), and this will be the instant ¢=2 of our time-scale. A glance at the upper part of fig. 1 will suffice to see how to construct the instants t=3,4, and so on. ‘Thus, let p, send off, at ¢=2, a fifth particle ps so that it should meet p, at the instant 7. Then p; will meet p, ata perfectly definite instant t=3. Similarly a sixth particle pg parting from py at t=3 will meet p, at ¢=4, and so on. The proof that the instant 1 will precede 2, and that this will be earlier than t=3, and so on, may be left to the care of the reader. The construction of fractional-¢ instants and of negative ones (preceding ¢=0) will be obtained on similar lines. At this stage we may introduce the generalized archimedean postulate, z.e. assume that every instant of time between 0 and 7’ (and distinct from 7’) can be exceeded by a finite repetition of the process described above. The index to be given to the instant 7’ itself, as approached independent of Space Measurement. 387 through 1, 2, 3, etc., will of course be =o. The same instant 7, when approached through t=—1, ete., will ob- viously have also the index t=—s0. Thus, 7’ aieele will be a singular point of the projective time-scale. But, exactly as is the case with the familiar staudtian space- -scale, this formal singularity will give rise to no real confusion. 5. Having thus set up a projective time-scale, let us now construct a similar space-scale, i.e. a staudtian scale of points (places) on the X-straight. This can be done either in the well-known way, by means of von Staudt’s construction in a space-plane passing through the X-line, or—which in the present connexion seems more elegant—by an experiment performed upon the X-line, without invoking any other dimension of space proper. In fact, replace in the reasoning ae by fig. 1 the pencil of world- (rae of constant date by world-lines representing particles fixed at points «=0, 1, S, all these lines passing through a conventional world- point QO, as centre. ‘hen, if Pw Pz, etc. be again uniformly moving particles, the required experiment will easily be seen to be as follows: p, and ps start simultaneous ly from the same place e=0 ; when p, arrives at the place g=1, it sends p3 to meet py at s, and when pz arrives at e=1,it sends p,to meet p,at S. Then p3 and p, will meet with one another at a perfectly definite place (no matter when p3 meets po,and when p, meets p;), and this willbe the point e=2. Similarly, a fifth particle ps5 will give #—=3, and. so on. In short, everything will be as in Section 4, the words “instant”? and ‘simultaneously ”’ (same instant) being sub- stituted for ‘place’ (point) and ‘‘same place,” and wie versa. Having thus obtained a scale of instants of time and of points along the straight in question, we can now set up a system of projective “* coordinutes”’ wv, ¢ defining an event in the world 2, ¢, in much the same way as in projective geometry (the supplementary axioms of continuity being properly intro- duced). Thus, graphically, let O,, Ox (fig. 3) represent two fixed events or fol! -points, the above « centres,” and let a, a third fixed event, be the “ origin ” of our coordinates. Let the lines af;,, 2, or axes be provided with their scales as explained above. Then, if « be any event in the contemplated two-world, draw the joins Q,«, Qre. The former will find on the first axis the date t, and the latter, on the second axis, the place w of theevent « Any right world-lines through Ox 388. Dr. L. Silberstein on a Time-Scale Q; will be lines of constant date and of constant (or fixed) place, respectively *. There is no difficulty whatever in extending this coordinate system to the three-world (v,y,t) or to our full world (,y,z,¢). But our chief object being not so much space as time itself or in connexion with any one of the space-dimensions, we can continue to confine ourselves to a bidimensional world. With this system of projective coordinates the equation of any right world-line, 7. e. the equation of uniform motion, will obviously be a linear equation avw+bt+c=0, which we will conveniently write f= 2+, . . 5. ee %, v being constants. If the reader likes to have a name, he may call v the projective velocity of the particle w hose motion is represented by (1). More generally, if «=/(é), with f standing for a one-valued function, be the equation of motion, dxr/dt=/'(¢) will be the instantaneous projective velocity. Thus, uniform motion will be characterized by a constant projective velocity, as in metrical kinematics (if this be of the kind to be called hereafter parabolic). We can now say that equal ¢-intervals are those during which a uniformly moving particle covers equal numbers of * The coordinates of Q,, Q, and of any point of their join Q will be infinite. We could feuide for these world-points by introducing homogeneous coordinates. But we shall not need them for our purpose. independent of Space Measurement. 389 “staudtians” (as I proposed some time ago to call the projective steps). Notice in passing that the world-lines of two particles pj, )2 having equal constant velocities intersect on the Q-line (fig.3), and wee versa, every two right world-lines crossing on © represent uniform motion with equal projective velocities *. Fig. 3. Such particles p,, p, are always thé same number of staudtians apart. This singularity of the O-line in the graphical representation must be well kept in mind : two particles meet actually when and only when their repre- sentative lines cross outside the Q-line, but not on the Q-line (unless the particles are permanently united). A sub-case of this is a pair of constant-place lines, 7. e. crossing in Q,; the corresponding two particles, fixed for ever in two distinet places, obviously do not meet. It may be well to describe the four-particles experiment of Section 4 with the aid of equation (1) and thus to verify analytically the determinateness of its result. Let, for the * The proof of this statement follows at once from the fundamental properties of such a pair of lines with respect to the reference system. Cf. ‘ Projective Vector Algebra,’ more especially the paragraphs dealing with equal vectors. 390 Dr. L. Silberstein on a Time-Scale sake of simplicity, p,; and p, start from w(e=t=0). Then their histories will be given by Cab ands, & —stpte and those of p3 and p, by x = v,+(t—1)v;, “= 2+ (t-1)% Therefore the instant of their meeting will be determined by Vo— VU pave Ne 2 V3— U4 (2) Now, by the conditions of the experiment, writing t= T for the ““instamt, Mee m+ (PI = eT, + (P-ln = 00; so that, by (2), 9'T = 7H independent of 7%, vs, v3, v4. And since T=ax, we have =2. (More simply, since p3 and ps2, py, and p,; meet on the O-line, we have v3=v, and v,=%, om 1 tor the ratio on the right of (2), and therefore t=2.) Thus the instant of meeting of pz, p4 1s seen to be independent of the velocities of all ihe particles. Similarly the reader will find for the meeting of ps with p, the value t=3, and so on. Turning for a while to non-uniform motion, we can introduce at once the concept of projective acceleration, defining it by d?a/di?, and similarly for the higher acce- lerations. Thus all the material required for a system of what may be called general or projective kinematics can be put together, without ever appealing to ordinary clocks or to rigid mons ines -rods. 6. We have said in Section 3 that two uniformly moving particles do not meet more than once. This, of course, does not imply that they meet (or met) actually even once. The assumptions hitherto made leave in this respect three different possibilities, which may be stated shortly by saying that our world (w,t) might be either elliptic, parabolic, or hyperbolic. Let us discard at once the ‘elliptic ” case, not only because it would make every right world-line to be a closed one, and independent of Space Measurement. aoe the notion of a cyclical time (although logically not incon- sistent) seems repugnant, but also because it vould amount to assuming that every two right world-lines meet, whereas we cannot say, for instance, that two fixed particles ever meet, without straining very much our common language and ideas. But the remaining two possibilities deserve some careful attention. or the sake of shortness we will henceforth say simply *‘ particle’ instead of “ particle moving uniformly along the «x-line.” Parabolic. Wet p be one such particle, and g another, never meeting the former, say always lagging behind p. Then the parabolic axiom, which is also tacitly being assumed in ordinary kinematics, amounts to this : P. If q be sped up *,no matter how little, it will overtake p at some future instant; and if q be slackened, p will have overtaken it at some pastinstant. In other words, through a world-point # (not on p) there is one and only one world-line g not intersecting p. This axiom is simply the kinematical or world-analogue of the much debated parallel-postulate of Euclid. It has hitherto been tacitly assumed by all physicists. But, as in the case of Kuclid’s postulate, the axiom P is by no means a consequence of our previous axioms. On the contrary, it is in no logical relation with them ; it isa further possible assumption entirely independent of them. Without any real loss to generality we may consider the particle p as fiwed, say ata point A. Then the parabolic axiom will run, somewhat more drastically, thus : The only particle g which, being at.a certain instant at B (distinct from A), never met and never will meet p, is that which does not move at all, i. e. which is alwaysat B. Every particle which moves, uniformly, towards A (or away from A), no matter how slowly, will reach A in future (or was at A in its past). In this form it sounds almost asa truism. Yet it is but one more axiom or assumption added to the projective ones. Alyperbolic. The parabolic axiom is but the limiting case of the following, more general one, which it will again be enough to state for the case of one particle, p, being kept fixed, at a point A, thus : * That is to say, if g be replaced at a certain instant by a particle q’ which after that is ahead of g. In a similar sense we speak of “ slackening.” 392 Dr. L. Silberstein on a Time-Scale H. Let an infinity of particies g be at (or pass through) B all at the same instant 6. Some of them will reach A ata future date or have been at A in their past history. Other particles, among them gq) which is fixed at B, never were nor will ever be at A, ‘he class of the latter is divided from those of the former which move towards A by a limiting particle q’, and similarly, from those which move away from A, by a particle q'’, and these two linuting particles q', q'' have distinct histories. In other words, through the world-point B=(B, 6) there is a whole pencil of right world-lines not intersecting p, this pencil being divided from the intersecting ones by two distinct world-lines q' and q'’. Such is the axiom of what may be called hyperbolic kinematics. If both limiting lines q/, g'', which are but the world-analogue of Lobatchevsky’s parallels, coincide or represent but one history, e. g that of the fixed particle q, we fall back to the parabolic or usual system of kinematics. 7. Let this last Section be dedicated to a rapid comparison of projective with metrical kinematics of the parabolic, as well as of the more general, hyperbolic kind. To make the metrical kinds of kinematics complete let us imagine all the necessary «2, t-analogues of the usual congruence axioms properly enunciated and accepted. Then, in the case of the axiom P, the world 2, ¢ will have all the properties of the ordinary euclidean plane or, better, will be another “concrete representation”’ of the same logical theory. And in the case of H our bidimensional world will be faith- fully represented by a lobatchevskyan plane. Parabolic. Let &,7 be the metrical coordinates of an event, i.e. the ordinary length and clock-time, and let us retain our previous symbols «, t for the projective coordinates of the same event. As we already know, the equation of uniform motion of a particle is, in a, ¢, i Ly + Vimue ° . . ° ° ° (1) Let the origins of both systems of coordinates coincide, as well as their axes and units (2/e¢. let £2,= 72 Further, let &', 7’ stand for the metrical scale ‘ divisions ” corresponding tow=«,t=«. Then,as can easily be shown, the relations of the two kinds of coordinates are Ea ERE de iad ee 7 aaa: . 3 : (3) independent of Space Measurement. page Using these relations the reader will easily convince himself that the metrical equivalent of equation (1) is E=&+4+¢.7, c he We Ont neva salme (4) where & and ¢ are constants, that is, again a linear equation, as was to be expected. The value of &, is of no interest. The constant ¢ has the meaning of ordinary velocity. Its relation to the projective velocity v will be found to be BSS |) rer ei 1)" AM Shot) ess CO) The case 6=0 seems puzzling (giving apparently v=o). But it will be remembered that, by (4), &’=£)+7', so that (5) becomes ) ee) Oh DP) emma El a ok UD atten (4 ® EME fi), - giving v=0 for ¢=0, as it should be. If we take &=0, 2.é. also 2)=0, this relation becomes '—1] C= bem, In particular, if &'=0, we have v=q, as might have been expected. Hyperbolic. Let & 7 be the metrical coordinates of a world-point in the lobatchevskyan world-plane. Then, with appropriate units of & and of 7, these coordinates. will be related to the projective ones wv, t by coth & = : [a+(e@—1) coth &']; cotht = ; [a +(¢—1) coth7’], (6) where a=coth 1, or also ua a—coth&® = ss. a—coth 7! (6’) ~~ eothE+cothé’’ ~~ cothr+cothr = Let us again consider a uniformly moving particle. We may take its place at t=0 for the origin of «. Then its history in projective variables will be Or" Obs Substituting here (6’), we have, the equation of uniform motion in metrical variables, coth €+coth@ 1 a—coth& coth T 16 coth 7! an yy Gaacothin. 394 Time-Scale independent of Space Measurement. But for £=~» we have z=, i.e. for r=7', E=é'. Thus the last equation becomes tanhhé = y.tanh?, . 92. 2 where y=tanh &'/tanh7’= const. That this is the equation of a lobatchevskyan straight line (as it should be), passing through the origin of &, 7, can easily be proved *. The last equation can also be written 1+ytanht eS e “1—ytanht Such then is the metrical equation of uniform motion in hyperbolic kinematics. The constant y characterizes this motion. It is scarcely necessary to say that in the present - case the metrical velocity, d&/dt, is not constant. For T=0 we have tanht=1. Thus our particle, passing at 7=0 through €=0, never exceeds the point £ = 41 (7") : 1 F=}log, "7. a This will suffice to remove the puzzling impression left by the axiom 7. Ifyisasmall fraction we can write £—y(1 + }y”). If so, then for any moderate 7, & =y tanh zt (1+ Jy? tanh? 7), and if 7 is small (2.e. a small fraction of the “ radius of curvature” of the hyperbolic world-plane), we have, approx1- mately, | C= Weel)? | yal aia exhibiting the departure of hyperbolic from parabolic uniform motion. London, 4 Anson Road, N.W.2. June 26, 1919. * In fact, if a pair of perpendicular lines is taken as the system of axes of r, € in the representative lobatchevskyan plane, and if 1, m, and 7 be the lengths of the perpendiculars drawn from a point to these axes and the distance of this point from the origin, respectively, then the familar form of the equation of a straight passing through the origin is sinh/:sinhm=const. On the other hand, our 7, & are the lengths cut off by those perpendiculars on the axes themselves, so that cosh r.cosh/=cosh7=cosh €.coshm. These relations together with the familiar relation sinh? /+sinh?s=cosh?7—1 give at once tanh? €: tanh’ 7=sinh*/:sinh® m, which proves the statement. Wn 3o6: XXXV. Critical Speeds of Machinery placed on Upper Floors of Buildings, as related to Vibration. By A. B. Hason, iWA.* i YHIS article discusses some of the relations between the frequencies of vibration of floors and machinery placed upon elastic supports on the floors, and suggests the principles upon which one should seek to find suitable supports for the machinery in order to prevent vibrations being communicated to the building. The. sy stem under discussion consists of a motor resting upon elastic supports such as rubber, cork, or springs, upon a floor which is capable of deflexion. The mathematical problem is to find the critical speed of two masses coupled together by one spring, one mass being joined by another spring to an immovable mass. ‘The springs are assumed to be weightless, which of course does not correspond to practical cases, but the general solution of the problem is of use to those dealing with the problem of preventing vibrations from electrical or mechanical machinery pene- trating to various parts of a building. ‘The problem (fig. 1) Bie 1 FORCES P=P, Cos wt y=) MOTOR, mass ™ J FLOOR, mass M | \ is solved by Stodola (Steam Turbines, p. 355) and by Berger (Gesund. Ingr. vol. xxxvi. p. 433, 1913). We have a motor of mags m, subject toan nter nal unbalanced foree P= Pycosat, the angular velocity of the motor armature being @. Between the motor and the floor is an elastic support, represented by a spring, which requires a force a dynes (or poundals) to deflect the. spring ee unit distance, 1 em. or 1 ft. (If the foree =A ke. or lb, a= Aq.) The floor has a certain deflexion due to its w eight and ean be repre- sented by a beam of mass M resting upon another spring. * Communicated by the Author. 396 Mr. A. B. Eason on Critical Speeds of Every floor is elastic and bends under its own and super- incumbent weights, and will oscillate if displaced from the position of equilibrium. Let this system of springs and masses be displaced from the equilibrium position owing to the force P acting on the motor. Choosing the following symbols : «= displacement cf m from the point of equilibriuin at time ¢. y=displacement of M from the point of equilibrium at time ¢. a=force to compress the motor spring unit distance. b=force to compress the floor spring unit distance. #=angular velocity of the motor, assumed constant. @,=resonant speed of the motor resting on spring 1 alone. @,=resonant speed of floor vibration resting on spring 2. (These resonant speeds correspond to the angular velocity of harmonic motion whose period=the natural period of the masses on their springs. w =2arn, n= frequency per second.) ad=mo,”?, b=Mao,’. @),@ =the critical speeds of the motor, such that the system asa whole is in resonance, and dangerous vibrations may occur. ‘lhese are ordinarily neither @, nor @;. ky=,/@,, ky=@»/@>. a) =@,/@2, = m/M. The forces acting on m are P downwards and a(e#—y) upwards. The forces acting on M are a(e«—y) downwards and by upwards. The equations of motion including the accelerations of m and M are 4 d? x M <3 =a(x—y) —by, m Faq =P—a(a—y). iD _ These equations neglect air friction which would bring in a function of dy/dt probably of the form (dy/dt)?. We assume that the oscillations of m and M will only be the forced oscillations due to the impressed force Po) cos wt, as the natural oscillations will soon be damped out by friction ; the solutions are Mo?—a—b P= (no —a)( Mage) y= Po coswt(—afC), | 2). “ae eee z= P, cos wt Machinery placed on Upper Floors of Buildings. 397 where C is the denominator in eq. 2; when C=O, the amplitudes of displacement may become large and the Rey. per min. Fig. 2. Deflection of spring, f° em, or inch. critical speeds will be found; the solution for w? when the quadratic equation C=0 is solved, is gk al iy Ce fo lee o=3| ate ty ti[t ai | ‘ ke where L=(a/M+a/m+6/M), and so M This means that there will be two critical speeds, such that 20.7 =@,’@;", or kyk,=1:0; we shall now seek to find out in what relation , and @, stand to w,and w. ‘The natural period of a mass m oscillating upon a spring which requires a force a to compress or deflect it unit dis- tance, is 27r(m/a)? ; the oscillations per second =(a/m)3/(27), @,= 27 (frequency) =(a/m)?.. In our problem this gives the critical speed for the motor if it is connected to an Phil. Mag. 8. 6. Vol. 38. No, 225. Sept. 1919. 245 veh o 9\1 wat [ pect otto] +4(L¥—daso). . (5) 398 Mr. A. B. Hason on Critical Speeds of immovable body by the spring support 1. The floor may be considered to be an elastic beam which deflects under its own weight. If it is set in oscillation by a large enough impressed force with a period equal to its natural period, dangerous vibrations may occur ; this critical frequency is w@,=(b/M)?. The natural period of a weight M which deflects a weightless spring by / units is 2a{/f/g)?; the frequency of the oscillations per munute =187/f? (if fis in inches), or 300/f? (if fis in cm.) and g=32-2 or 981 respec- tively. The same law holds good approximately for any body on an elastic support, so that we get the critical speed related to the amount of displacement, shown in fig. 2; if one wants a critical speed of 3000 rey. per min., then the motor should be placed on a support which deflects 0-1 mm. in consequence of the load. Now let us return to the question of critical speeds for the whole system which we call @,=,o, and @.=kh,w, ; to assist the caiculations put m/M=, and w,/a,=¢. ‘lhen from eq. 5, after dividing by o,’, k, and kh=3[(1+wW)¢?+1]+4(1?—4¢2)3. (8) The way in which /, and ky vary with the various values of w and ¢ is shown in fig.3. The ordinates of the concave curve show values of k,, k, when m/M is the variable, but @,/@; OF Na/ns=1:0. The series of convex curves to the left show &, and ky with the ratio n/n, as a variable for particular values of m/M. In the curves n,/n; is put instead of w,/,, as the expression rev. per min. is more usual than the expression angular velocity. ‘here is no need to discuss the curves in fig. 3 as the effect of altering the ratio m/M or 2,/np is quite obvious. In practical cases we may have a motor of mass m placed on an elastic support™ such as cork pads, rubber, &c., which can be altered in size or thickness ; these pads rest on a floor of mass M, the deflexion of which is f, with a critical frequency of oscillation 7. Suppose that the motor causes obnoxious vibrations in the floor and that the motor speed is n, and that this speed is near the resonant speed for the whole system. If the motor causes big vibrations at all speeds the only cure is to improve the motor ; but usuall such vibrations will only occur at particular speeds. In the above case the one factor which can easily be altered is the elastic support under the motor, the characteristic quality of * See Elec. Rey. vol. 84. p. 689, 1919 (June 21st), for references to yarious supports. : Machinery placed on Upper Floors of Buildings. 399 which is a, which is the force in dynes or poundals required to deflect it unit distance, either 1 cm. or linch. If pads are being used—say six 4 inch pads each 6 inch square— a has reference to the whole area of 216 sq. in. (1340 em.) ; a would be 96g times the force in lb. required to compress the pads 2 inch. Fig. 3. Ratio of weights, m/M, with na=n 1 speeds, N2/ng4, for various M/M, Taking a particular case, let the motor run normally at 1000 r.p.m.; let the floor have a natural frequency of vibration, n,=800 ; let m/M=C-1; to avoid resonance we need to have n, and 7, well below 1000 r.p.m.: suppose we decide to have ng, less than 900 r.p.m., 4, < 900/800 < 1125 ; then fig. 3 gives $< 0°9 so nz<720 r.p.m. ; the deflexion of the motor support under the influence of the motor weight should therefore exceed 1°8 mm. (fig. 2). If this deflexion can be obtained by using 3 inch square cork pads each 2 inch thick instead of using 6 inch pads } inch thick, the obnoxious 22 400 Mr. A. B. Eason on Critical Speeds of vibrations might be avoided. ‘The effect of increasing the area of the pads will be to increase a and thus increase n, ; to increase the thickness of the pads will reduce a and n,. The question of whether n, should be increased depends altogether upon the relative values of n, nj, ng, or of o, @3, Da: Fig. 4 shows the critical speeds of the motor with m/M=0-1 for any two Speeds n, and m, all the speeds being Fig. 4. Fev. per Min. 2000 1600 4 -8 1-2 6 2:0 Value OF na/Ne referred to nj. The actual speeds will be m, multiplied by — the values of the ordinates for the particular value of under consideration. It is seen that the critical speeds are always above and below the natural frequencies, (1) of the motor on its support, and (2) of the floor on its supports. In order to avoid resonant vibrations in any actual case, one must know m, accurately or approximately. One can find m, experimentally thus:—Place a motor directly on the floor ; attach a small weight to its armature to unbalance it, start the motor running and notice if there are big vibrations in the floor at any speed up to the full speed. If such vibrations exist notice the approximate speed and then run Machinery placed on Upper Floors of Buildings. 401 the motor at about this speed until the position of worst vibration is found accurately. If the vibrations with the badly balanced motor become too great, remove the weight and allow the ordinary unbalance to create the resonant vibrations. If the point of maximum vibration is not easily perceptible to the feel, one might use an indicator of vibrations such as Digby’s Vibragraph as made by Siemen’s Bros., and watch its indications during the test (Hlectrician, vol. vil. p. ° 8, 1912). Suppose, now, we have found n,=1400, and that m/M=0°1, and that the motor to be installed will run at 1800 r.p.m., what sort of elastic support shall we use to avold communication of vibration? Here n/n,=1°285 ; to get a good result the motor speed should lie midway between the critical speeds, so make na/ns=1°5 ; nq then =2100 and the support should deflect 0-2 mm. under the weight of the motor. ‘The critical speeds are 1280 and 2340 approxi- mately. Another good result would be obtained if Nq/ny~=9'4 or less; na<560; the deflexion of the support should be 3 mm. or more; the critical speeds would be 550 and 1420, which lhe well away from 1800 r.p.m. If the normal speed of the motor is the same as that of the floor, viz. n=n,, then one should try to get n,=m, and the critical speeds of the motor would be 1°17 and 0°86 7. The conclusions to be drawn are :— (1) The critical frequency of rotation of a motor, mass m, joined by an elastic support S to a body, mass M, resting on another elastic support 8’, ditfers from the natural fre- quency n, of m oscillating on S, and from the natural frequency n, of M oscillating on 8’; unless the ratio m/M is very small. (2) The critical speeds ,, nz, or the critical angular velocities w,, . of the motor are related to the n: atural fr equencies, nq of the motor on spring S and m of the floor on spring 8’, thus Ny Ny = NgNp. (3) If the ratio of weights m/M<0:1 and also nq/nj <0°2, then n,=n, and no=n5. (4) Fig. 4 shows how n, and ny are related to n, and ng, ny, and nz always lie outside the range between nz and np. (5) Usually the mass of the floor M and the stiffness of the floor represented by 6, and the mass of the motor m are fixed, but the elasticity of the support represented by a is 402 Dr. R. A. Houstoun on a variable ; so that for practical work the elastic support under the motor should be adjusted until the critical speeds of the motor are nowhere near the normal speed of rotation. (7) The way in which the critical speed of a motor is related to the deflexion of its support is given in fig. 2. (8) If the elastic support of the motor is made up of pads or layers of cork, rubber, or felt, then n, is reduced if the thickness of the layer is increased ; n, is increased if the area is increased. i AXXVI. A Theory of Colour Vision. By Dr. Ki. A. Hovustoun, Lecturer on Physical Optics in the University of Glasgow *. ) § 1. ee the above title I contributed a short article to the Proceedings of the Royal Society Tf two years ago, in which I showed that it was not necessary to assume the existence of three fundamental sets of nerves or mechanisms in the retina in order to expiain the facts of colour vision. This article has not been understood, possibly owing to defects of exposition on my part, but more probably to the unfamiliarity of the ideas involved. In the present paper I develop my theory somewhat further ; on account of its nature I cannot hope to make it fully intelligible to all the psychologists, physiologists, and non-matheinatical physi- cists interested in colour vision, but I have included some numerical examples, and hope at least to show that my theory is as capable of giving correct numerical results as the Young-Helmholtz theory is. To put my theory as shortly as possible : (1) The eye is sensitive only to a limited range of wave- lengths, from A=4 x 107? cm. to X=7°6 x 1075 em. (2) To explain this we must assume resonators or vibrators in the retina. (3) No matter how regular the incident light is, the vibration set up in the retina will always be more or less - irregular ; there will be stoppages and changes of amplitude and phase owing to molecular disturbances. (4) In Optics an irregular vibration of this kind is equi- valent to a very great number of regular vibrations of different periods. This follows from Fourier’s integral theorem ; it is also the basis of the modern work on the * Communicated by the Author. t Vol. xcii. A, p. 424 (1916). Theory of Colour Vision. 403 constitution of white light and of spectral lines. Conse- quently, no matter how monochromatic the incident light is, the vibrations in the retina are not monochromatic. If sodium light is incident, the vibrations in the retina contain light on both sides of sodium, 7. e. orange and yellowish green, or even red and green. (5) Thus if we attempt to mix a monochromatic red and green, subjectively we are mixing two wide regions of the spectrum which overlap in the yellow ; and the effect is the same as if we allowed monochromatic yellow to fall on the eye. (6) The more irregular the vibrations in the eye which correspond to monochromatic stimulation, the worse is the colour vision of the eye. (7) The quality of a light impression on the retina is very conveniently represented by a curve such as fig. 1—the energy Riga d: the nee 400 500 600“™ curve, as I call it. The abscissee denote wave-lengths in up, but instead of wave-lengths any arbitrary spectrum-scale may be used. The area of the curve represents the intensity of the light impression, and the hue and degree of saturation depend on the position of the centroid and the shape of the curve. To find how much of the luminosity corresponds to, say, the region included between 500 um and 505 up, it is only necessary to erect ordinates there, and measure the area included between the ordinates. If two curves of the type shown in fig. 1, but the one displaced along the spectrum from the other, are super- imposed, the character of the resultant impression is obtained by adding the ordinates. Thus if the separation is small, the character of the resultant curve is not much altered. This is the case of yellowish green superimposed on bluish green to give pale green. If the separation is greater, the breadth of the resultant curve becomes greater in comparison with its height. This is the case of yellow superimposed on blue to give white; white is represented by a curve the breadth of which covers a large part of the spectrum. If 404 Dr. R. A. Houstoun on a the separation is greater still, the resultant curve has two maxima separated by a hollow. This is the case of blue superposed on red to give purple. Many educated people who have not studied light regard purple as composite, as something analogous to a chord in music ; this view finds its natural expression in the two maxima of the curve. The general rule for the mixture of spectral colours is, that if they are not far apart in the spectrum, the mixture has the same hte as an intermediate spectral colour, but is paler than the latter. If the separation is increased, the two colours become complementary, and the mixture is white. This is what we would expect if the effect of mixing colours is obtained by superimposing curves of the type shown in fig. 1. Se Lite is, however, not sufficient to explain the facts of colour mixing qualitatively ; ; a quantitative explanation must also be given. In order to do this, it is first of all necessary to find out what the facts are; this is not so easy, because the different investigators are not in good agreement. The principal papers are by Maxwell*, Konig and Die- terici f, Abney ¢ (1900), Abney § (1906), and Rayleigh |. Any colour whatever can be expressed in terms of three fundamental colours. It is the object of the first four papers cited to do this for the spectral colours. The results are given in different units and it is impossible to compare them. Konig and Dieterici, indeed, say that their results are in essential agreement with Maxwell’s, but no comparisun has as yet been made between Abney’s results and those of Konig and Dieterici. Maxwell gives two sets of results, one for an observer J (himself) and another for an observer K, Konig and Dieterici each give a set of figures, and Abney gives two, one in each of his papers. Asa preliminary to studying the results I found it necessary to reduce all six sets to the same units and exhibit them in parallel columns. * “On the Theory of Compound Colours, and the Relations of the Colours of the Spectrum,” Phil. Trans. cl. p. 57 (1861). + “ Die ,Grundempfindungen und ihre Intensitaétsvertheilung im Spektrum,” Berl. Ber. p. 805 (1886). { “The Colour Sensations in Terms of Luminosity,” Phil. Trans. excill. A, p. 259 (1900). Sued Modified Apparatus for the Measurement of Colour and its Application to the Determination of the Colour Sensations,” Phil. Trans. ecy. A, p. 333 (1906). “On Colour Vision at the ends of the Spectrum,” Scientific Papers, vol. v. p. 569. Theory of Colour Vision. 405 Konig and Dieterici and Abney take as their fundamental colours the extreme red and extreme violet of the spectrum and an imaginary green, more saturated than any actual green. Maxwell takes as fundamental colours a red at 630°2 pu, a green at 5281 zu, and a blue at 456°9 py, all actual spectral colours, chosen at points where the hue varies slowly with the wave-length. I think Maxwell’s procedure is undoubtedly the better in this respect: results in the first case should always be stated in terms of real colours. Also the extreme ends of the spectrum are unsuitable as standards, since they are faint and not visible to some colour-blind persons, and, according to Rayleigh, many persons with colour-vision otherwise normal are anomalous in the extreme violet. Konig and Dieterici and Abney give their results in the form of tables compiled from ‘‘ smoothed” curves ; Maxwell gives actual observations for sixteen wave-lengths selected so as to be equally spaced over a prismatic spectrum. I consider Maxwell’s procedure the better in this point also ; in view of the inaccuracy of the results 16 wave-lengths are quite sufficient to take in the spectrum. I have therefore expressed the other observer’s results in terms of Maxwell’s three standards for Maxwell’s 16 wave-lengths ; incidentally I have found Maxwell’s and Konig and Dieterici’s data incomplete, and I have had to heip their observations out by Abney’s. Maxwell determines his wave-lengths by what is now called Hdser and Butler’s method, and expresses them in Fraunhofer’s measure. Fraunhofer’s wave-length determi- nations were made in Parisian inches, and are quite accurate enough still for colour-vision work. I translated Maxwell's wave-lengths into wy by multiplying by -2707, then keeping the three fundamental wave-lengths unaltered I adjusted the others until the first and second differences increased continuously, and adopted the values so obtained ; the wave- lengths he gives for scale-numbers 20 and 48 are quite con- siderably out. Maxwell expresses the luminosity of his fundamental colours in terms of slit width, and adds his units in figs. 6 and 7 as if they were all of the same value, thus producing a very striking hollow in the top of his luminosity curves. But slit widths in the red, green, and blue have not the same value, owing both to the different intensities of the spectrum at these points and to the different degree of dispersion it undergoes in the prism. According to Abney (1900 paper, p. 285), the luminosities of a prismatic solar spectrum (taken graphically from fig. 10) should be Dr. R. A. Houstoun on a 406 Mise Ol. sel 261 =. pe i Le— 82— O0L™ 86-— ‘sl "e aa GL 19 8G ae GREP Sole OC oC meta a oe Cir 86am RdO=- 0O=— (G8i=- 761 96 4 0 eat rg 1g GG (Yer le SlbP Glee eG Sols eke 28: Oma WGea CP Cle Claw |) Ol 6 6G O& GG GG ae 88FP OOT OO OOT OOT OO O01 | 0 0 0 0 0 0 0 0 0 0 0) 0 | tO L9 69 14 69 69 SL | FP GP Tg bg 86 TL Pi ae Si ee ee ee 6GOP E GP bP OF GOL 9L | G6 BL L6 G6 LG GOT 96 0G=— hs ae Oe eee ' 6GLP GT FG LT 0G 1S L reek = Shs Hee Gi WO GIT | ee OS Ag I thc? ¢ 6 9 9 ¢ G oe Ae OMe Ci IO! CW Sem Os. ede eG) ee) " 1667 [ i G 0 G I ae AHL FAL, OO OE DL AS iSO Pa ae eee ale 0 0 ©) 0 0 O | OOf OOL OOL OOT OOL OOT 0 0 0 0 0 0 i.” [84g “s 0 0 88 18 F6 6 G6 66 Gl SI 9 ) ¢ Lie eer es T9#¢ eat ae as 22 0 me) CL GL 68 #8 68 61 86 66 SI 91 LT 1G Jeo 8E9¢ = ie zt 0 0 6F LY 6¢ 09 99 oy =s«*T@ ¢¢ IP OF PE Oras ee ” GPSS ; 0 0 61 61 96 9¢ GE Té 18 18 FL las) L9 69 |" €909 0 0 0 0 0 0 0 0 |00O. OOt OOL OOT OOL OOT | G0E9 I I OL 20S Cl Le G8 Oil. SOP ee weiner iy 6S¢9 OAL OO = Cbs Ui IE D1 OO hve .O0 Wee CE rE NM | 90V 00OV Cd OS aa alee aN oe ee <2 — ——————- —— a Ss nS a ‘on |g “Uda | “poy ‘S19ALOSYO JUSLAYIP 0} Surps0ooy ‘ontg pure ‘user ‘poy Jo sulsoy, UL wn.zoodg ayy, “T Wavy, Theory of Colour Vision. 407 28°6, 41, and 1°6 at Maxwell’s three fundamental wave- lengths. The dispersion of Abney’s apparatus was pre- sumably about the same as that of Maxwell’s, and Maxwell used the solar spectrum. I have therefore multiplied the three columns of Maxwell’s tables V1. and X. by 28°6, 41, and 16 respectively, added the three figures in each row, divided the sum into each ot the three figures separately, and expressed the results as percentages. I thus obtained the columns headed K and J in Table I. Unfortunately Konig and Dieterici also do not express their results in units of equal luminosity ; they choose their units so as to make the quantity of each fundamental colour present in sunlight numerically the same. I have not been able to find what factors they multiplied up by. So i have reduced the results on p. 819 to units of equal luminosity in the following manner :—I have taken A,, and A, (p. 820) as 573 and 496 wu for Konig, and as 570 and 493 pp for Dieterici ; at’ these wave-lengths the red and green, and green and violet present in the spectrum are, in their units, numerically the same. I have next ascertained from each of Abney’s papers what the ratio of the luminosities should be at these wave-lengths, and taken the mean for each wave-length ; at Konig’s X,, the red should be 2°095 times as bright as the green, and at Konig’s 4,, the violet should be 116 times as bright as the green, ale at Dieterici’s Ay, the red should be 1°94 times as bright as the green, and at Dieterici’s X,, the violet should be 144 times as bright as the green. J therefore obtained the values corresponding to Maxwell’s 16 wave-lengths from the table on p. 819 by graphical interpolation, multiplied the R and V eolumns by the factors cited above, added the three figures in each row, reduced each three columns to percentages, and finally evaluated Konig and Dieterici’s R, G, V in terms in Maxwell’s R, G, B. The results are given in the columns headed Kn and D in Table I. I had to extrapolate both the R columns on p. 819 to wave-lengths 475°9 and 465°9 pu in order to obtain smooth curves. Abney’s results were taken in the case of the 1900 paper from columns IV., V., and VI. on pp. 278-9, and in the case of the 1906 paper from columns IV., V., and VI. on p. 344, and plotted as functions of A. The values for Maxwell’s 16 wave-lengths were then obtained by graphical interpolation ; Abney’s R, G, V were then evaluated in terms of Maxwell’s R, G, B. The results are shown in the columns headed A’00 and A’06 in Table I. 408 Dr. R. A. Houstoun on a To explain the use of this Table take, for example, the wave-length 5842 and the columns headed Kn. Then the Table states that light of this wave-length, according to Konig, can be regarded as made up of 40 per cent. of red of wave-length 6302 and 60 per cent. of green of wave- length 5281. Or take wave-length 4488 and the three columns headed A’00. Then the table states that light of this wave-length, according to Abney’s 1900 paper, can be regarded as made up of 9 per cent. of red of wave-length 6302, —37 per cent. of green of wave-length 5281, and 128 per cent. of blue of wave-length 4569. The minus sign simply means that the colour is to be on the other side of the equation ; thus the colour in question superposed on the green ‘37 times as bright gives the same hue and luminosity as the red ‘09 times as bright superposed on the blue 1:28 times as bright. It will be noticed that Maxwell’s two observers differ from the others in stating that the extreme red has one per cent. of blue. This point has been tested speciaily by Rayleigh in the paper cited, and he has quite definitely decided that Maxwell is wrong, that the extreme red has no tendency to blue whatever. Maxwell’s observers’ last three wave-lengtlis are also extremely irregular; the values for K’s last wave-length are omitted on account of their being absurd. Maxwell himself mentions the accuracy of the figures for these wave-lengths as being “doubtful.” Also Maxwell’s own colour-vision was somewhat abnormal in having an excessive absorption in the yellow spot. I have therefore rejected the observations of K and J. The observations of Kénig and Dieterici agree fairly well, except for an abnormality in Konig’s at 5728, stated as due to absorption in the yellow spot. I have therefore chosen Dietericis in preference to K6nig’s. Abney states with reference to the difference between his two sets of .obser- vations (p. 333, 1906 paper) ‘‘some slight alteration in the sensation-curves was the result, and, though small, ought to be recorded.” His 1906 set must be regarded as superseding his 1900 set. The chief difference between his 1906 set and Dieterici’s set hes in the violet. Dieterici’s last three wave-lengths show a marked tendency towards the red, the last one having 72 per cent.of red. Abney’s last three wave-lengths do not show this tendency at all, the last one having —4 per cent. of red. Now this tendency of the violet towards red was investigated by Rayleigh in the paper already cited, and he has decided that there is no doubt about it; that Theory of Colour Vision. 409 according to the great majority of observers the extreme violet approaches red. It seems consequently that Sir Wm. Abney’s colour vision is not quite nermal in the extreme violet. I have therefore adopted Dieterici’s observations as the basis of my work. When we have three sets of variables the sum of which is always the same, such as the R, G, B in Table I., they can be represented on a diagram as the perpendiculars from a point on the sides of an equilateral triangle. Maxwell and Konig and Dieterici represent their results in diagrams of this kind, although each of their perpendiculars is in a different unit, Konig and Dieterici’s diagram being very well known; there is graph paper on the market for diagrams of this kind, divided into little equilateral triangles. I decided, however, after using seven or eight sheets of it, that trilinear co- ordinates are not to be recommended for colour diagrams. We can get on very much better with ordinary squared paper. Fig. 2 represents Dieterici’s results, the R being plotted as abscissa and the G as ordinate. The figure can be regarded as a trilinear coordinate diagram, the ‘fundamental triangle being RGB. If, for example, the point P is considered, ‘its perpendicular distances from GB, BR, and RG are respec- tively —22, 51, and 502 units. Owing to the triangle being right-angled isosceles the 50°2 must be multiplied by ,/2 giving 71 as a result. The amount of blue present in the colour represented by P may thus be obtained by taking the perpendicular distance from RG and multiplying by ./2. But it is simpler to add the R and G and subtract the result from 100. Fig. 2 is a Newton’s colour diagram. It has the property that if any two colours are represented by points and their intensities by the masses of particles placed at these points, then the colour of the mixture is represented by the centroid of the two masses. The curve representing the cclours of the spectrum is something like a parabola, although the seven points next the red end lie on a straight line ; ‘T have drawn in a parabola in a broken line, in order to show how much the observations deviate from it. All actual colours are represented by points inside the spectral curve. Sun- light contains 28 R, 69 G, and is represented by the point W in the diagram ; it is thus very close to the yellowish-green, the brightest colour in the spectrum. If other fundamental colours are chosen instead of Maxwell’s R, G, B, what effect has this on the shape of the spectral curve? It can easily be shown that the straight 410 Dr. R. A. Houstoun on a part remains straight and the relative spacing of the points on it is not altered, but that the curvature of the curved part alters, and the curve still retains the same rough resemblance to a parabola. Fig. 2. re) For the purpose of Newton’s colour diagram any three colours may be taken as fundamental colours. The advantage of Maxwell’s three standards is that they are convenient for experimental purposes. The upholders of the Young- Helmholtz theory, however, maintain that a certain three colours are distinguished from all others in being perceived each by a single sensation. There are thus three unique points on the plane. It is clear that the triangle joining these points must include all actual colours, because when a colour is expressed in terms of fundamental sensations, there Theory of Colour Vision. All must be no minus signs. This condition admits of an infinite number of triangles, and there is no unanimity among the adherents of the Young-Helmholtz theory, as to where they place the corners ; Konig and Dieterici, Helm- holtz, and Abney all obtain widely differing results. The corner must be fixed from other considerations than the facts of colour-mixing as recorded by Newton’s diagram. Exception may quite properly be taken to the fact that the results in the Table are given in terms of luminosity. It would be better in future work to state them in terms of energy—to say, for example, that a yellow was matched by a mixture of standard red and standard green, the energy of red radiation received per second by the eye being so many times the energy of green radiation, and to add at the foot of the table the luminous equivalents of the standard radiations for the observer in question. If we state the results in terms of each individual observer’s luminosity, they are not directly comparable, and it seems unnatural to state a colour-blind individual’s results in terms of another man’s luminosity. As the table stands at present, the results are stated in Sir William Abney’s luminosity values. I thought it better to leave them in this form, as I was not certain what multipliers to use to reduce them to terms of energy. If, however, we use the luminous equivalents given by H. HE. Ives*, namely R=278, G=889, and B=44, and reduce Dieterici’s results, we obtain :— TABLE II. rn R G. B G559us 108 aS 63026. he 100 ) GOG SH hie 90 10 5B42> 69 31 BESS Al 59 BANG et oe 17 83 OSI Ye ) 100 PIOSh us —50 111 39 ASG a. —42 69 73 ASGOWM inn: BS 33 98 Ar Oly Mawes ea 11 104 AG5O 2 ee narah 4 101 ABGOr 0 0 100 AAR SHOE Ie 4 —2 98 Aas a Ae 6 Bas O7 SADE Sv. 8 4) 96 * Phil. Mae. Dec. 1912. 412 Dr. R. A. Houstoun on a This table states, for example, that the percentages of R, G, and B energy in the mixture giving the same hue as N=5128 are —50, 111, and 39 respectively. The composition of white light according to energy is 41 R, 31 G, and 28 B, so that, when the results are represented this way, the close proximity of the white with the limiting curve is avoided. ‘The violet end, which is difficult to measure, is also repre- sented on a much reduced scale. §3. Let us suppose that two colours are mixed, and that it is required to determine the colour of the mixture. Let each colour be represented by its energy curve in terms of any arbitrary spectrum-scale s. Let the centroids of the two energy-curves be at s; and s,, and let their areas, i. e. the intensities of the two colours, be denoted by I, and I,. The question arises as to how the shape of each curve is to be specified. The most natural way is by means of the square of the radius of gyration of the area about the ordinate through the centroid. ‘Ihis is in accordance with statistical practice ; in fitting curves to observations the statisticians are accustomed to consider first, second, third, ete. moments ; when the area is known, the first moment specifies the centroid, and the second the square of the radius of gyration. The radius of gyration is the “standard deviation” of the statistician. ‘The colour-perceiving centre in the brain is so badly developed that it is not necessary to characterize the shape of the energy-curve any further than by the second moment. Let 4,7, kh? denote the squares of the radius of gyration for the two component colours, and let s, I, and k? specify the mixture. Then Vea, oy a tee and nf (8,7 + Ay?) Ti + (59? + he?) Te I, ahi Ils : Newton’s colour-diagram follows naturally from these equations. For let s,, s, specify the abscissee, and s,?+4,? and s 2+? the ordinates of two points on a plane, and place particles of masses I, and J, at these points ; then it is clear that s and s?+4? denote the abscissa and ordinate of their centroid. Let us suppose that k? is the same for all the spectrum colours. ‘Then if we plot them on the colour-diagram we have E=5, Yee, ly ey fee the equation to a parabola. This isin accordance with fig. 2. Theory of Colour Vision. 413 To make the agreement with fig. 2 perfect it is necessary to make k? vary from colour to colour, and this variation was determined in the following manner :—AN, the tangent to the vertex of the experimental curve was drawn, and perpendiculars were then let fall to AN from each of the sixteen points; thus the perpendicular from P was PN. The distance of the foot of the perpendicular from A in arbitrary units was taken as s, values on the violet side being taken negative, and values on the red side positive. The parabola represented by the broken line was next drawn to pass approximately through the sixteen points. It cuts PN in Q. PQ was then measured in the diagram, and expressed in the same units as NQ. The results are shown in the following Table :— | TABLE III, d. s. PQ. ke. PRO ssecanee 23:9 90 215 6302 ae 21-7 98 223 ROS een 19:0 94 219 BOA al 15:0 is 198 a es SORE TORE 38 163 BAIL oy, 11-0 15 140 ROS I .e>, 10:3 3 128 RID Sine oa 8:0 eon 101 AGC ON) Ween 61 m4 lll ASGOM ok. 17 eS 124 AGI el tee Be | Bas 112 LRG) ig - 19-7, 36 161 A5Bo oo! —18°8 25 150 AARSi —Y1-] 118 243 AAS! Wttee, — 93:4 179 304 A2AD ewe —24°4 247 372 PQ represents &? except for an additive constant. The latter must, of course, be sufficiently great to remove the negative sign. I have taken 125 for its. value; if the energy-curves have roughly the shape of probability curves, and red and blue are added, this value allows the resultant eurve a double maximum, but at the same time ensures that there is only a simple maximum when the component curves are closer together in the spectrum than red and blue. It will Be obvious now to a mathematician that any problem in colour-mixing can be solved by means of the s and k? columns in the above Table, and the result will be Phil. Mag. S. 6. Vol. 38. No. 225. Sept. 1919. ae at 414 Dr. R. A. Houstoun on a the same as if we had used Dieteriei’s Rand G. For the purpose of illustration I have added four examples :-— (1) What proportion of light of wave-length 6559 should be added to light of wave-length 5451 to give the same hue as wave-length 5842 ? Let the proportion be x of 6559 to 1 of 5451. Then if we use the ordinary method, since the three wave-lengths in question contain only R and G, we may consider R alone. We thus have #1124 6=(¢+4+1)41, which gives £= AY: Jf we use my method, since we are dealing only with the straight part of the curve, it is not necessary to consider ? at all. For the wave-lengths in question s has the values 23°2, 11, and 15. Hence e23°2 + 11=(#+1)15, which gives i Ae (2) Red of wave-length 6063 is added to blue of wave- length 4659, and it is found that the same hue and intensity ean be obtained by adding green of wave-length 5281 to violet of wave-length 4488. Determine the proportion of each. Let w of 6063 be added to 1 of 4659, and let y of 5281 be added to z of 4488. Then by the ordinary method =k Wear = Considering the proportion of R present 274 —22 = 229, and considering the proportion of G present, x26+51=y100—248. These equations give <='531, y=°935, and z=-596. By my method we have 2+l=yt+z, w19—12-7=y10°3—22111, and x(19? + 219) + (12-7? + 161) =y(10°3?+ 128) + 2(21°174 243). These equations give 7=°519, y="931, and z="588. The small differences are due to experimental error in taking sand k? graphically from fig. 2. Theory of Colour Vision. 415 (3) If white is specified by s=12°7, k2=210, find the complementary colour to 4659. This resolves itself into finding where the straight line through s=12°7, k*°= 210) and pie) (ee = Lol cuts the straight line through ams (es IG oy Ane. SA ok = N63 Writing y for s?+k? these straight lines are Yeo Saaleed 371—322 ~ 12°74 12:7’ y— 423 s—15 OR BI ee Their intersection gives s=13°8, and by the principle of proportional parts X=5748. The last two problems could be solved more easily graphically. (4) Light of }»=5842 can be matched by a mixture of X= 6063 “ad 9638 and also by a mixture of X=6302 and 5451. Show from the energy curves, that in each case we obtain the same value of &?. Since the shape of the energy-curves is specified only by 4, we shall draw them as rectangles in order to save trouble. The upper diagram of fig. 3 represents the first mixture, and the lower one the second. In the upper diagram 240 units of X=5638, 1. e. s=12°4, are added to i>oo units: OF “N= 6060,.2.0e) s=19:0. The area, of. the rectangle to the right is consequently 240 units and of the rectangle to the left 156 units, and their centroids are respectively at s=12°4 and s= 19: 0. It will be found that the centroid of the combination lies on the dotted line at s=15:0, i. e. N=5542. The values of k* corresponding to s=12-4 and s=19°0 are 163 and 219. The lengths of halt 2h 2 416 A Theory of Colour Vision. the base are consequently ,/3k, i. e. 22:1 and 25:6. If we consider the two rectangles as one system and calculate the square of its radius of gyration about the dotted vertical we obtain Te 240 (163+ 2°6*) +156(219 + 4?) 240 + 196 The correct result should be 198. The discrepancy is due to the values in Table JII. having been determined graphically. The mixture is represented by the dotted rectangle. } In the lower diagram 268 units of X=5451, 7. e. s=11:0 are added to 160 units of A=6302, 7. e. s=21:7. The area of the rectangle to the right is consequently 268 units and of the rectangle to the left 160 units, and their centroids are respectively at s=11:0 and s=21'7. It will be found that the centroid of the combination lies on the dotted vertical. The values of £? corresponding to s=11°0 and s=21'7 are 140 and 223. The lengths of half the base are consequently 20°5 and 25:8. If we consider the two rect- angles as one system and calculate the square of its radius of gyration about the dotted vertical, we obtain 268 (140+ 4?) +160(223+6°7?) _ 59. 268 + 160 | = 1960; }?= the required value. The dotted rectangle in this case also represents the mixture. At first I thought s would specify the hue of the colour and k? its degree of saturation, but this is not the ease ; k? certainly increases as a spectral colour becomes paler, but purple has a greater value of 4° than white. As regards the arbitrary scale s, in drawing energy-curves there is no a priori reason why the abscissa should be A or 1/A or any special function of ». So I allowed the observations to define their own scale ; I find that the general result is the same as if the curves were plotted against the index of refraction of some substance with a greater dispersion than water. It is not necessary, but it is a great simplification to have the origin for sat the vertex of the parabola—taking it elsewhere involves oblique axes for s and k”. § 4. To put the difference between the Young- Helmholtz theory and my view as shortly as possible :— The Young-Helmholtz theory represents the R and G of fig. 2 as linear functions of three primary sensations, m view as linear functions of s and k?. Hence there can be Surface Tension and Chemical Interaction. 417 no contradiction between the two explanations as regards normal colour vision. his should be obvious without the numerical calculation of particular cases. ‘The Newton colour-diagram is common to both explanations, but the Young-Helmboltz theory makes three pvints on the colour- diagram unique, and asserts that they represent primary sensations, whereas my explanation gives no point a preference over any other. The Young-Helmholtz is a “three elements” theory; my theory depends on three quantities, intensity, s, and k”, but these three quantities vary continuously. My theory may be regarded, I believe, as the mathematical formulation of Dr. Edridge-Green’s views.. The two theories lead toa different classification of colour- blindness. It should be possible to decide between them in this way and also by a statistical investigation of colour- vision. According to my view the colour-blind are simply the outliers of a “‘ homogeneous population,’ whereas the Young-Helinholtz theory appears to require, for example, that the red blind should be a homogeneous population of their own. I have dealt with this point in a previous paper *. [ am greatly indebted to Prof. A. Schuster for the benefit of his criticism, which has enabled me tv considerably improve some parts of the foregoing paper. By Prof. G. N. AWtoNorr lr | a a paper published in Phil. Mag. xxxvi. Nov. 1918, a theory of surface tension was developed. It was as- sumed that electrical and magnetic forces are acting between the molecules, which were treated as electrical doublets or as small magnets, the law of attraction in both cases being the same. It is difficult to settle the question as to the nature of molecular forces at the present time. Some evidence indi- cates that these forces are purely electromagnetic in nature. Lewis ft bas shown that the Obach- Walden relation (propor- tionality of dielectric constant and internal pressure) follows from the hypothesis that the molecular attraction is electro- magnetic, not electric, in nature. On the other hand, the fact that this relation is not wholly correct seems not to * Proc. Roy. Soc. vol. A, xciv. p. 576 (1918). + Communicated by Prof. J. W. Nicholson, F.R.S i Phil. Mag. xxvil. p. LOL (1914). 418 Prof. G. N. Antonoft on Surfuce support the above view entirely. The works of Harkins, Langmuir, and Svedberg indicate that the electromagnetic forces play a considerable role in capillarity, and, according to Chatley *, these forces may be both electromagnetic and electrostatic, which would oe the complexity of vec- torization. It is especially difficult to settle the question as to the law of molecular action. In the previous paper it was selected as the inverse fourth power law, which follows from the assumption that the distance between the doublets exceeds considerably the length of the doublets. If on the contrary the distance between the doublets is small, the forces would vary as the inverse fifth power of the distance (see Chatley, loc. cit.). We may take as an example of this, a case described in ‘Molecular Physic-,’ by J. A. Crowther (1919). See figure 1. Fig. 1. O 71 [@) e Oo ° ® @ (@) fe) O QO e e Ce) Oo fe) O e Cy e ® oO oO oO 5 ° « o e The outer rings of electrons in two adjacent atoms are repre- sented by the black dots, and the effect of positive electricity is regarded as equivalent to that of the charges concentrated in the white circles between them. If the atoms are brought together very close, they will turn so that the electron of one will face the positive charge of the other. In this case apparently the attraction will be inversely proportional to the fifth power of the distance. But whatever the actual law of molecular action, it is a question of secondary importance so far as our chief results are concerned. ‘The scope of the paper was chiefly restricted to the problem of interfacial tension All the calculations were done assuming the inverse fourth power law. It can easily be shown that the same would hold true whatever the law of molecular action. In the previous paper it was shown that the following equation must be true when two liquids are in equilibrium one above another (case of partial solubility) :— le — fee = eae — a) Py — agp", ° . ° (1) * Proc. Roy. Soc. Lond. xxxi. pt. 3, April 15, p. 92 (1919). Tension and Chemical Interaction. 419 where P, is the intrinsic pressure of the saturated solution forming one layer, IEG ¥ the other layer, IPs coal eee pressure at the interface, a, =surface tension of one layer, pi=number of molecules per ith volume in the same layer, a, and p,=corresponding quantities for the second layer. It was shown that the expression must conform with Pyo= f(t) are or A on gg Regma Mee" (0) The result (2) is only possible Hoi —oe It can be seen that the above results would remain the same whatever the law of molecular action, whether it varies inversely as the fourth, fifth power of the distance, or other- wise. It can be shown that assuming the inverse nth power law our expressions for surface tension a and internal pressure P respectively would be n+1 cys Ss | dine B=2ikp ° Thus the formula connecting both quantities will be the same as indicated on p. 382 of my former paper, 2. e. [ee hoor where k= 2. Thus the relation (2) does not depend upon the law of molecular attraction, but is a result of a certain chemical interaction between the liquids forming the solutions, which enables the heterogeneous system to become stable and pre- vent both layers from mixing. The essential condition of this equilibrium is the equality of molecular concentrations (pi=p2). ‘This is based on some physico-chemical evidence briefly mentioned in the previous paper. On p. 394 two typical curves were reproduced to this effect. They were borrowed from a theoretical paper by G. 'amman*, which reference was by mistake omitted in the previous paper. Some experimental results of that kind together with more complete theory will be published in a subsequent paper. 6 Featherstone Buildings, Fligh Holborn. * Z. An. Ch. xlvii. p. 274 (1905). [420° | XXXVI. The Travelling Cyclone. By the late Lord RayueEicn, O.M., F.R.S.* | Vote——The concluding paragraphs of this paper were dictated by my father only five days before his death. ‘The proofs therefore were not revised by him. The figure was unfortunately lost in the post, and I have redrawn it from the indications given in the text.— RAYLEIGH. | (os of the most important questions in meteorology is the constitution of the travelling cyclone, for cyclones usually travel. Sir N. Shaw f says that ‘“‘a velocity of 20 metres/second (44 miles per hour) for the centre of a cy- clonic depression is large but not unknown; a velocity of less than 10 metres/second may be regarded as smaller than the average. A tropical revolving storm usually travels at about 4 metres/second.” He treats in detail the com- paratively simple case where the motion (relative to the ground) is that of a solid body, whether a simple rotation, or such a rotation combined with a uniform translation ; and he draws important conclusions which must find approximate application to travelling cyclones in general. One objection to regarding this case as typical is that, unless the rotating area is infinite, a discontinuity is involved at the distance from the centre where it terminates. A more general treat- ment is desirable, which shall allow us to suppose a gradual falling off of rotation as the distance from the centre in- creases ; and I propose to take up the general problem in two dimensions, starting from the usual Eulerian equations as referred to uniformly rotating axes{. The density (p) is supposed to be constant, and gravity can be disregarded. In the usual notation we have 1 dps: Br bbice Du em}? De Jl dp 5 5 Dr: 9 pdy hapa aie be (2) where | D/Dixd/di-+-ud[/de+rd[dy.. 2. @ Here a, y are the coordinates of a point, referred to axes * Communicated by the Author. + ‘Manual of Meteorology, Part iv. p. 121, Cambridge, 1919. { Lamb’s ‘ Hydrodynamics,’ § 207, 1916. The Travelling Cyclone. 421 revolving uniformly in the plane zy with angular velocity w*, u and v are the components of relative velocity of the fluid in the directions of the revolving axes, that is the components of wind. We have now to detine the motion for which we wish to determine the balancing pressures. We contemplate a motion (relatively to the ground) of rotation about a centre (, fig. 1, situated on the axis of w, re the successive rings P at distance R from © revolving with an angular velocity ¢, which may be a function of R. And upon this 1s to be superposed a uniform velocity of transla- tion U, parallel to w and carrying everything forward. If initially © be at O, the fixed origin, its distance from O along Ow at time t will be Ut. Thus u=U—y, v= C(w— Ut), 5 5 : . (4) € being a known function of R, where eat Coa D) = Yt Sean elt os VC) These equations give wu and v in terms of the coordinates and of the time, and the values are to be introduced into (1) and (2). From the manner in which w and ¢ enter (representing a uniform translation of the entire system) it is evident that dfdt=—Ud/de. We have Gu GRY du. ¢ Cy? dx BR ea dacliny, ly & dv CEUX eran) C xy * In the application to a part of the earth’s atmosphere, @ is tie earth’s angular velocity multiplied by the sine of the datztude. 422 The late Lord Rayleigh on ¢’ being written for d&/dR ; and Du is CUE OTs Nis es be RY EY (gaff) gn ae —U(E+25 a )+u(e+ +p) tobe ey Hence Idp _ De ace 29, ane 2 oe Oe dy °! 2w(U —fy) + &y,. (6) and on integration . : = $0%(a? +y?)—20Uy + \(2e6+ 67)RdR. . (7) As might have been expected, the last term in (7) is the same function of R as when U=0O, but R itself is nuw a function of U and ¢. In the case considered by Sir N. Shaw, € is constant and may be removed frum under the integral sign. Thus Ze) [ks = Jo°(a? + 93) —2oUy + (b+ 3)? + (@ = Ul If U=0, R? identifies itself with w+ 4’, and we get pip=s(0+ C)*(a?-+-y"). 5 en A constant as regards « and y, which might be a function of t, may be added in (8) and (9). We see that if a+ €=:0, that is if the original terrestrial rotation is annulled by the superposed rotation, p is constant, the whole fluid mass being in fact at rest. It was for the purpose of this verification that the terms in w? were retained. We may now omit them as representing a pressure inde- pendent of the motion under consideration. In the strictly two-dimensional problem there is a pressure iucreasing outwards due to “centrifugal force.’ In tbe application to the earth’s atmosphere, this pressure is balanced by a component of gravity connected with the earth’s ellipticity. Thus in Shaw’s case we have P =Const. + (wf + 32? >. aay z— Ut)? mee st. + ( e440) 1 (y wee. + (z— Ut) ie (10) the Travelling Cyclone. 423 showing that the field of pressure, though still circular, is no longer centred at O as when U=0, or even at C, where x= Ut, y=0, but is displaced sideways to the point where a=Ut, y=oU/(of+467). Shaw calls this the dynamic centre ; it is the point which is conspicuous on the weather map as the centre of the system of circular isobars. As a case where the circular motion diminishes to nothing as we go outwards, let us now suppose that €= Ze- 8”, falling “ih slowly at first but afterwards with great rapidity. We ave J, SRAR=3Za%(1—e- PM), (YO RAR=3Z%a%(1—e Me); and thus from (7) : = Const. —20Uy—a(Le- M4 20)?, . (11) where, as usual, R?=7?+ («— Ut)’. May 17th. The completion of this paper was interrupted by illness. The two-dimensional solution requires a ceiling, as well as a floor, to take the pressure. In the absence of a ceiling we must introduce gravity, and since in the supposed motion no part of the fluid is vertically accelerated, the third equation of motion gives simply p (== CONSt. — gz. p g Thus (10) is altered merely by the addition of the term —gz. I had supposed too that the solution would remain sub- stantially unaltered even though p were variable as a function of p. But these conclusions seem to be at variance with those put forward by Dr. Jeffreys in the January No. of the Philosophical Magazine. Iam not able to pursue the comparison at present. June 25th, 1919. [The following note has been contributed by Sir Joseph Larmor.—Ebs. This paper was left incomplete on Lord Rayleigh’s decease on June 30. It may therefore be permissible to direct 424 The Travelling Cyclone. attention to its main conclusion from another aspect, by way of paraphrase. ‘T'wo questions are involved. If a vortical system can persist at rest, in an atmosphere rotating with the Karth, can it also persist, slightly modified, with a translatory velocity U? Andif so, how will the distribution of pressure in it be modified ? The equations of fluid motion relative to the ground are (1) and (2); in them the last terms Du/Dt and Dv/Dt express the components of relative acceleration, and these are clearly the centrifugal accelerations —€?X, —€?y in the relative orbits assumed to be circular, as found analyti- cally lower down. Onsubstituting these values, the equations give for 6p an exact differential form which is integrated in (7); therefore a moditied motion is possible, and the first question is answered in the affirmative, in agreement so far with fact *. The displacement of the pressure-system due to the progressive motion is then examined for two special cases by the formule (10) and (11), showing also general agreement with fact as regards displacement of the centre of the vortex. But the value of U is not determined by these considerations, which refer to frictionless fluid. When viscosity in the fluid is taken into account, the general argument seems to remain applicable ; tor the velocity of convection U, being uniform, will not modify the viscous stresses. But, in any case, internal viscosity is negligible in meteorological problems. It is the friction aguinst Jand or ocean, introducing turbulence which spreads upward, that disturbs and ultimately destroys the cyclonic system ; and the high degree of permanence of the type of motion seems to permit that also to be left out of account. As remarked in the postscript, the changes of pressure arising from con- vection involve changes of density, which will modify the motion, but perhaps slightly. There does not seem to be definite discordance with Dr. Jeffreys’ detailed discussion. | * The conditions of stability for flow of liquid with varying vorticity had been considered in a series of papers, for which reference may be made to the section Hydrodynamics of the catalogue appended to Lord Rayleigh’s ‘ Collected Papers,’ vol. iv. [ 425 cS) XXXIX. Proceedings of Learned Societies. GEOLOG:CAL SOCIETY. (Continued from p. 268. ] December 18th, 1918.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. TEIXHE following communication was read :— ‘On a Bed of Interglacial Loess and some Pre-Glacial Fresh- water Clays on the Durham Coast.’ By Charles Taylor Trechmann, Se, F:G:S. A few years ago the author described a bed of Scandinavian drift that was found filling up a small pre-Glacial valley-hke depression at Warren-House Gill on the Durham coast. This section and others north and south of it have been kept under observation at different times, and several new features have been noticed as the high tides and other agencies exposed parts of the coast. Towards the southern end of the old pre-Glacial valley at Warren-House Gill a bed of material, varying from 4 to 12 feet in thickness, was found overlying the Magnesian Limestone and also the Scandinavian drift. This material has been carefully examined chemically and microscopically, and proves to be identical in chemical and physical characters with a sample of the true Con- tinental loess. It is light brown or fawn in colour, very porous and extremely finely divided, and is devoid of plasticity. Towards the base, where it has not been disturbed since it was laid down, it contains a number of rounded and elongated, often very hard, calcareous concretions. In the cliff-section 1t shows little or no trace of bedding, but tends to break down along vertical clefts and eracks. It passes upwards into a few feet of material that consists of loess which has been partly redeposited by water, and is mixed with sand, gravel, and other material derived from the Scandinavian drift. The bed of loess and redeposited loess-like drift has suffered much decalcification and weathering ; near its surface there was a large boulder of Norwegian titaniferous syenite which was super- ficially rotted, and decomposed to a considerable depth. Smaller granitic erratics in the redeposited loess are generally very much rotted. The limestone rubble and stones beneath the loess are strongly calcreted, apparently by material leached out of the loess. In a fissure beneath the loess some mammalian bones were col- lected, including astragali of two species of Cervus. It is argued that the formation and subsequent decalcification of the loess deposit lying upon the Seandinavian drift indicates an Interglacial lapse of considerable duration, as great as that which Continental geologists call an Interglacial Period, before the overlying English and Scottish drift was deposited. 426 Geological Society. About 2 miles south of the Scandinavian drift-bed several fissures occur in the Magnesian Limestone cliffs and on the fore- shore, filled with various materials that were transported in front of the earliest ice-sheet that advanced upon this part of the coast. The author has already recorded the occurrence in these fissures of Upper Permian red and grey marls and dolomites with clay and peaty trees. Continued examination of two of the fissures where they are exposed between tide-marks on the shore, resulted in the finding of a quantity of freshwater mollusca, ostracoda, and fish- remains. Some mammalian remains also occurred, including those of an elephant (probably Hlephas meridionalis) and of a vole (Mimomys). Vegetable matter has been washed from various parts of the clay. A large number of seeds came from a single patch of clay, and prove to be of Teglian age: they seem to represent a pre- Glacial flora, half of the species of which are either exotic or extinct. Seeds from other parts of the deposit appear to indicate a later horizon, and contain mainly living forms. The deposit is a mixed one, and seems to have belonged to a series of late Pliocene and early Pleistocene beds that occupied part of the present area of the North Sea and were torn up by the advancing ice-sheet, like a great glacial erratic, and thrust into the fissures. The fact that the Scandinavian drift in Durham contains only stones of Scandinavian origin has been confirmed, and the marine Arctic shells that occur in it were further collected and a few additions to the faunal list were made. The most interesting of - these is Cyrtodaria siliqua Spengler, an American shell which has been recorded hitherto in Great Britain only from the Caith- ness Boulder Clays. ; All the deposits described above are overlain and overridden by the main mass of local Cheviot and Northern drift that caps the cliffs of the Durham coast. A suggested correlation of the Durham sequence with the European drifts is attempted, and it is concluded that the fringe of the Scandinavian ice-cap that reached the Durham coast pro- bably corresponds with that of the second and greatest glaciation of Scandinavia, which some Continental geologists correlate with the Riss Stage of the Alps. In that case, the main local drift of the north-eastern coast falls into the third and Jast Glacial Period of Northern Europe. The evidence for Interglacial lapses in the local drifts is very in- conclusive. All the observed features seem to point to the fact that the Scandinavian ice-sheet advanced on the east coast of England in the same way as it invaded Northern Europe round the southern shores of the Baltic, and gave rise to analogous climatic conditions leading to the formation of loess, a fragment of which is found protected from the erosive action of the later local glaciation in a small hollow on the Durham coast. aot, LORD RAYLEIGH. By the decease of Lord Rayleigh on June 30 the outstanding figure in British theoretical physics, and indeed in the physical science of the world, has been removed from us ; and it is fitting that the Philosophical Magazine, in which so much of his work first appeared, should take note of the loss. In an age when the majority of successful workers have to be specialists, his outlook was universal. There is perhaps no special popular discovery attached to his name: the detection of urgon, which immediately led on in other hands to the detection of the other inert atmospheric gases, was an incident which occurred naturally in his stride. But he enjoyed to the full the higher privilege of the greatest minds, in moulding the scientific thought of his age. He has been eminently the instructor of his generation. His works have been largely the material on which the scientific tastes and capacities of the younger theoretical physicists have been formed and tested. They carry on the same tradition of means adapted to the end in view, of the quest of what is practicable and the avoidance of inconclusive issues, that has ever been characteristic of his nation, which maintained reasoned physical science on a high plane even in the days when mathematical analysis was in stagnation. The Zheory of Sound, and the Collected Papers in six coherent volumes, will constitute an enduring landmark in British physical science. No writer of such fertility, perhaps of any time, can stand so well the ordeal cf complete re- publication of all his work. No such Collection has been so influential on the higher education of the men of science of the age to which it belongs. It has been the great antidote to undue specialization in mathematical physies. To reach anything comparable in extent, solidity, and concise sus- tained thought one has perhaps to go back to the days of Laplace and Young. Along with Kelvin, Helmholtz, and 498 Lord Rayleigh. Maxwell, and Stokes whom he claimed as his own special mentor, their author stands at the head of the general theoretical physics of his age. The second edition of the Theory of Sound, revised and enlarged, appeared in 1896 extended to nearly twice the original size; it thus became even more emphatically a treatise on the general dynamical principles regulating vibrational and undulatory phenomena, developed indeed with reference mainly to the simplest and most tractable case, that of pressural waves in air, but in close touch with the more complex problems of optics and electricity. Thus a special chapter on electric waves expounded very concisely the dynamical principles of electric resonance initiated mainly by Maxwell, which have become essential to refined electric technology in all its branches; though more than twenty years old, it covers the subject in a manner so concise and complete that at the present day hardly anything but additional and even more wonderful practical illustrations need be added. Everywhere fresh suggestions occur. Thus, for example, one of the additions is an exposition of the analysis employed by Kirchhoff to explain the fall of velocity when sound travels in narrow tubes or channels, which was traced mainly to the absorption of the reversible heat of compression in the waves by the walls of the tube. But Lord Rayleigh goes on incidentally (§ 351) to explain a far more important feature, the degradation of the energy of the waves that is involved; and the absorption by a curtain or other porous medium of sound that is incident on it thus receives its precise and most sugvestive explanation, developable on a quantitative basis and with application to practical problems relating to the prevention of echo and reverberation. Lord Rayleigh was born in 1842; he graduated at Cam- bridve in Jan. 1865 as Senior Wrangler and First Smith’s Prizeman, sending in papers to the examiners which revealed a lucidity and precision that foreshadowed one side of his future achievement. He succeeded his father in the peerage Lord Rayleigh. 429 as third Baron in 1873. He served as Cavendish Professor of Experimental Physics at Cambridge from 1879 to 1884: he had yielded to the universal desire of his contemporaries,. includirg his intimate friends Stokes and Kelvin, in taking up the duties of the Chair at a critical juncture on the lamented and premature decease of Clerk Maxwell. He was Secretary of the Royal Society from 1887 to 1896: his successors in that office recognized many evidences of improvement in method and efficiency that had been effected quietly by him: later for three years he served as President. In the same year 1887 he took up the Professorship of Natural Philosophy at the Royal Institution, which he retained until 1905; thereby he had access to a laboratory in London, while his lectures provided a model of how problems strictly scientific could be handled in a manner interesting to an instructed public without any need for adventitious experimental adornment. Hewas Lord Lieutenant of Essex from 1892 to 1901, and was made a Privy Councillor in 1905. From 1896 he was Scientific Adviser to the Cor- poration of the Trinity House, a congenial office in which he had share in many improvements in the lighthouses and sound-signalling apparatus around the coasts. He was one of the twelve original members of the Order of Merit. He presided over the committees in charge of the National Physical Laboratory from its first inception under the control of thes Royal Society, until recently when it was taken over in greatly expanded form by the Government. His work in that capacity was of high value in stimulating the zealous assistance of the engineering and other public scientific societies, and still more in guiding the directions of expansion and in pointing out the way to new scientific progress. As Chairman of the Advisory Committee on _ Aeronautics he contributed probably more than any one else to the opening out of a scientific treatment of the essential problems of aereal resistance, especialty by development of dimensional theory and the principles underlying experi- ments on models, starting from the stage in which they had Phil. Mag. 8. 6.. Vol. 38. No..225. Sept. 1919. 2G 430 Lord Rayleigh. been utilized for ships and the flow of me by Froude and by Osborne Reynolds. He appears to have been Me in what he once called the art of experimenting, which in. the case of a powerful and observant mind is perhaps the best teaching of all. He began mainly on the subject of sound and vibrations, in which the interpretation of apparatus is as important as its perfection ; and by aid of the ordinary musical appliances, and local help such as that of the village blacksmith, he managed that. the experimental side.of his subject should march parallel with the mathematical theory. Perhaps it is not too much to say that he discovered in this way directly, from his own experience, the wonderful degree of accuracy of which suitably designed experiment is capable: and when later at Cambridge he undertook the formidable task of the precise determination of the electrical standards, with aid of the resources of a laboratory, he came fresh to the problem with clear views of what could be effected and what it was worth while trying to achieve, untrammelled by habit or conventional forms of apparatus. The preference for simple direct means maintained itself throughout all his work, and constitutes one of its charms as well on the mathematical as on the experimental side. His mind may be compared in this regard with that of Young, whose work, in common with Helmholtz, he greatly admired; though he would hardly have gone with bim so far as to say that the greatest triumph of experiment is to be able to do without it, by force of reasoning. It was always a wonder to visitors at Terling that so much refined investigation could originate in so slight an equipment of apparatus. He was reflective rather than precocious, and thus was somewhat slow in starting off in his career of investigation, but was the more fully equipped on that account. The quality of judgment, of the calm unhasting probing of evidence, seems to have been always highly developed. A remarkable final deliverance on the possibility of direct psychical interactions was pronounced only last April in a Presidential Address to the Society for Psychical Research : Lord Rayleigh. 431 it appears that these manifestations had attracted his notice from undergraduate days at Cambridge, and had continued to be among his active interests, with the result that they remained to him an unsolved enigma, paradoxical but not on that account to be neglected by inquirers. The same caution seems to have affected his physical work, in inhibiting dis- cussion of problems depending on disconnected and shifting hypotheses: in treating of molecular physics his main weapon was general dynamical reasoning, far reaching principles of the strict classical type, and their statistical application. Though he may fail to attain to the desired goal, he is not under temptation to bridge the discrepancy by unproved suggestion: the analysis may be supplanted by a better one for the purpose in hand, but it retains the character of a true investigation and an addition to real knowledge. Molecular thermodynamics and theory of radiation are prominent, but special hypotheses and models of molecular structure are largely avoided, even where they would not imply discordance with normal dynamical theory. In subjects so well founded in many respects as the electron theory, he has refrained from giving public expression to his thoughts, perhaps in part for similar reasons, but possibly also from a sense that the problems belonged to others. He was gentle in criticism, and most anxious that ability, of whatever kind, should not be discouraged or overlooked. To his scientific friends he was most hospitable, although he shrank from prominence at public functions. There are probably few men of eminence in physical science that have visited this country without a stay at Terling: his colleagues at home were glad to recognize in him their ideal repre- sentative in the welcoming and entertainment of fellow workers from other lands. The relations with his neighbours in Hssex were simple and cordial. He discharged the work of the high office of Lord Lieutenant of the County for about ten years, until he found that the social and administrative duties interfered too much with his own proper sphere. All through the later years his neighbours in the county have been consciously 432 Lord Rayleigh. and proudly aware that living quietly in their midst they had a fellow citizen who, both in character and in intellect, belonged to the select circle of the great men of the age. | In addition to personal scientific work to an extent that has rarely been accomplished by one man, there is a con- spicuous record of public service. He was naturally much in request as a member of Royal Commissions and other official inquiries. As Chancellor of the University of Cambridge since 1908, in succession to brilliant tenures of the office by two Dukes of Devonshire, it became his duty to be the official interpreter of doubtful points in the Statutes which he had himself helped to frame as a member of the Royal Commission of 1878: he used to say that at any rate he had direct knowledge of what the Commissioners meant to enact, though that is not always the same thing as the legal interpretation of their Statute. As time passed he began to exhibit some passive resistance to additional service on public bodies: he has been known to plead that it is a debatable question how far a man should defer to his contemporaries as to the manner of his service to his generation, rather than follow the bent in which he is conscious that he can be most effective. On the earliest possible occasion, on July 7, the President of the French Académie des Sciences pronounced an éloge funebre on their Foreign Associate Member; and we can conclude with his final words: ‘‘Théoricien puissant, sachant user avec une rare habileté des ressources de l’analyse mathématique, il s’est montré en méme temps expérimentateur hors de pair. Son nom restera comme un des plus illustres de la Science contemporaine.” Cambridge, Aug. 18. J. THE LONDON, KDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. i ae RQ Y A if G P \ {PP [SIXTH SERIES. ] Oe: \ ' f ae \ 7 OC ROB 7919. ~ XU. Stream-line Flow from a Disturbed Area. By Lt.-Col. A. R. Ricnarpson, D.S.O., Imperial College of Science, S. Kensington, S. W.* FNHIS note is an attempt to find solutions, defining flow past a rigid boundary, distinct from those obtained on the electrical and discontinuous stream-line hypotheses. To fix ideas, consider the case of a plate placed broadside onto a stream. In order to represent experimental facts the solution ‘should: (i.) Make the velocity finite everywhere, and discontinuous nowhere. Gi.) Give stream-line motion everywhere except behind the plate where eddying motion occurs. Photographs show that behind, and close to the plate, the motion is stream-line in character. (Advisory Committee for Aeronautics, Report 58, 1912.) (iii.) Give reasons for the periodic formation of approxi- mately circular eddies. Giv.) Explain why the resistance alters with the shape of the underbody of the plate and with the initial conditions in the fluid behind it, and also why the pressure changes sign on the underside of the plate. ; The analysis shows that if condition (i.) is. satisfied * Communicated by the Author. Phil, Mag. S. 6. Vol: 38. No. 226. Oct. 1919. 2 434 Lieut.-Col. A. R. Richardson on Stream-line (ii.) and (iv.) require no further explanation but are con- sequences of (i.). Reasons for (i11.) are also suggested. The following is a short account of the results of the analysis. The boundary conditions, and (i.) above, show that there is irrotational stream-line motion everywhere except in a definite area, behind the plate, inside which two-dimensional non-viscous irrotational flow is impossible. Fig. 2 shows the shape of a stream-line as it enters this disturbed area. It curls very rapidly indeed about a point P, then unwinds. and, after cutting itself a finite number of times, passes away to infinity *. This suggests that the pressure distribution over the boundary of the disturbed area B, necessary to produce the stream-line flow, is the same as would be produced by a system of eddies at places, such as P, inside the area. The disturbed area does not extend up to the plate nor to the bounding stream-line. Its actual position, and shape, vary with the postulated velocity boundary conditions behind the plate. The first part of the analysis discusses the general problem and shows that the existence of a disturbed area is a necessary consequence of conditions (i.). The second part deals in detail with the flow past a plate and with the disturbed flow from a semi-infinite pipe such as: is met with in sensitive jets. In both cases the solutions may be made to represent an unsteady state, and not merely the uniform motion. An examination of the pressure shows that a turning value occurs on the underside but not on the upperside of the plate. I. Flow past a corner. Adopt the usual notation : ai w where g and @ refer to the velocity. * The curve is not completed near P as | dz| is exceedingly small— the maximum negative angle 6 being about 27 and the maximum positive angle || 7/2. In the vicinity of P | dz| les between ‘0006e—18 and ‘002 corresponding to dd| = 01. Flow from a Disturbed Area. 435 Ref. fig. 1, let w=0 correspond to the corner B. In general © will have a branch point at w=0 ; e.g. for a free stream-line behaves like w'”. Four cases are possible :— (i.) Discontinuity of velocity on opposite sides of the stream-line ~w=0. This includes the case of discontinuous flow where the fluid is at rest on one side of w=0. The objections to this theory, in problems such as the flow past a plate, are well known and will not be repeated here *. (ii.) The stream-line w=0 is an analytical boundary across which it is impossible to continue the expressions representing the motion on one side of it. This implies that Q has an infinite number of essential singularities along ~w=0. In the neighbourhood of these points © takes all possible values, and therefore g, the velocity, takes all possible values. (iii.) The electrical case: the stream-line y=0 wraps itself round the boundary. This implies an infinite velocity at the corner B. (iv.) No discontinuity of velocity on opposite sides of =O for which <0. In such a case the solution for <0 must be that obtained by analytic continuation, across y~=0, in a w-plane so modified by cuts that oo is a single-valued function of w. i In a practical problem these cuts must coincide with the part of the w-plane which represents the rigid boundary. If not, every stream-line, for which the corresponding line y= in the w-plane meets the cut, will experience a finite discontinuity. In what follows case (iv.) is considered. The object is to obtain an analytic expression giving an irrotational motion on both sides of the stream-line ~w=0 such that the velocity is continuous on opposite sides of AB m ie. 1. In addition the velocity of the fluid at any point in the finite part of the z-plane is to remain finite. * Kelvin, Collected Papers, vol. iy. 2H 2 436 Lieut.-Col. A. R. Richardson on Stream-line Il. € has an essential singularity for some value of w. It can have no zeros in the relevant part of the w-plane otherwise g=~« at the corresponding points in the fluid. Hence if € contains an algebraic factor a zero, We; of this factor must lie in the non-relevant part of the w-plane. Let w= wy be the stream-line through wo, and w, a point on Yo in the relevant part of the plane. The path in the z-plane corresponding to wy must pass to infinity as w approaches wp along w= from wy. Hence there must be either a pole or essential singularity of z at some point on Wy between w, and wp. Again consider the case of any disturbed motion of a free stream-line such that @ takes the same value an infinite number of times; e.g. a disturbance over the surface of a, jet. Hence € takes the same value an infinite number of times and has an essential singularity for some value of w. dz In such cases [= ee | (w) has an essential singularity for some value of w, say w=d. Hence < will also have an essential singularity at this peint. It is therefore necessary to examine the solution with a view to seeing if any physical meaning can be attached to the integral. Til. Hvamination of the integral of “= fw). i “ aw To simplify the discussion suppose w=d is the sole essential singularity of f(w) which is holomorphic elsewhere. Suppose furiher that the whole of the w-plane corresponds to the whole of the <-plane. To one value of w corresponds one value of 2. The converse is not true, for in the neighbourhood of w=d z takes any value (not an exceptional value) an infinite number of times. Hence corresponding to any value ¢» of ¢ there will be an infinite number of values of w. The stream-lines, corresponding to the lines y~=constant which pass through such values of w, will all pass through 2p. The following” results show that this infinite branching arises solely from points w in the vicinity of d, the essential singularity. Flow from a Disturbed Area. A437 (i.) About d describe a small domain B, in the w-plane. Of all the stream-lines through Zo in the z-plane only a finite number lie on lines w=constant in the w-plane which do not pass through By; and all except a finite number of the points w which correspond to 2 lie inside Bo. For if not there will be a point of condensation 1 Ont = 5103 outside Bo. Hence w, is the limit of a set of values w, at each of which z=, i.e. w, is an essential singularity contrary to hypothesis. i.) A stream-line can have only a finite number of double points over a length AB for which the corresponding points w do not lie inside Bo. For if an infinite number exist there will bea value w,, such that near the corresponding z, there is infinite oscillation « f the stream-line. This is contrary to the assumption that /(w) is holomorphic near w,,. Gi.) Lhe domain By may be extended so that for values of w outside it the corresponding portions of the stream-lines. have no singularities. Describe a curve W in the w-plane not a locus of double points to cut every line w=constant but not passing through Bo. Let C be the corresponding curve in the z-plane. To the intersection of C with any stream-line W will correspond a finite number of points, in the w-plane, not lying on W but which are outside Bo. From these points blacken out that part of every line ar=constant remote from W. Hence on any line y=constant there will be a finite interval between W and the blackened part of the w-plane. Since the transformation is holomorphic, to a small defor- mination of C will correspond a small deformation W/ of W (except where al or co) and of the blackened part of the w-plane. This deformation on any stream-line can be taken so small that W' does not enter the blackened area unless it passes through a double point on the stream-line. Hence a finite domain can be determined about W such that inside the corresponding domain about ©, in the e-plane, every stream-line is free from singular points except such as arise from places w inside By supposed extended to coincide with the blackened areé 438 Lieut.-Col. A. R. Richardson on Stream-line IV. Application of the solutions to physical problems. Exclude the essential singularity d by a domain By in the w-plane as in III. (ii1.) above. Let Tbe the curve in the z-plane corresponding to the boundary of By in the w-plane. Postulate the pressure distribution over T which will give the stream-line motion outside P. Inside T’ is a region in which stream-line irrotational flow is impossible and where a disturbed eddying motion is to be expected. When the fluid consists of a thin film, the solution may be regarded as an approximation to a three-dimensional flow in which movement perpendicular to the z-plane is small. In such a case the domain By may be restricted to a small area round d, and the branches in the z-plane looked upon as defining flow in the various folds of the film resembling motion on a Riemann surface. These solutions indicate regions in which even a small amount of viscosity will allow vortices to form ; for as I is approached, the stream-lines begin to curl in such a way as to give‘a reason for the formation of circular vortices. It must be remembered that, in problems such as the flow past a plate, there will be an infinite number of solutions of the type here considered, for assumptions must be made as to the nature of the flow over the boundary I. These remarks will be illustrated by examples. V. Flow past a plate. (Fig. 1.) Let e(w—d) dz ab + wl) assis ies dw (w—1)2 , «eet where O41, c>0. To ensure that € is single-valued the w-plane is cut along the positive part of the real axis. The factor — is the Schwarzian factor at C. The term 6+ w!? is introduced to make the velocity finite at infinity and to ensure the termination of the thin rigid boundary at B. As previously explained, an essential singularity is introduced to avoid having an infinite velocity near BC’. The nature of this singularity depends on the assumptions as to the flow behind the plate. Flow from a Disturbed Area. 439 This is in agreement with experiments which show that the resistance depends on the character of the flow behind the plate. Bigel. Oisturbed Area corresponding to By a ee w=t00 G.) Motion along the bounding stream-linew=+0. (Fig. 1.) Start with those determinations which, at, some point A on AB, give _e(w— ae) ( dz b+iVv —w gotin : —w)* 2) alae ols toauen at 440) Hieut.-Col. A. R. Richardson on Stream-line the / sign denoting the real positive square root. —n<=w=— 0. =e c(w—d) ie eee glo tt N =a), dw” q Wee The curved stream-line AB is described, on which Gin ex Cae 7 <2 and Lo Te A = 0 O0 as w—>+1. The rigid boundary BC is traced out. It is important to notice that q steadily decreases as w increases: on the underside of the plate this is shown not to be the ease. Ih SW eg SO | ee c(w—d) dw Vw—l1 C=] 250, cand’, 9g, == 1a The straight stream-line CD is deseribed. Further, since 6+w'? has no zeros in the upper half w- plane, ¢ hee no singularities for p> 0. It will be proved later on that the domain Bo, of the previous work, does not extend into the upper half-plane. The fener therefore gives a possible irrotational flow for w>0, 2. e. outside the plate. (ii.) Flow along the bounding stream-line p= — Commence at A with the same determinations as in (2). The motion along AB will be continuous with that in (1.). Flow from a Disturbed Area. 44] O<—w=<+l. c(w—d) de) en 0- sor SS eee ele aie dw - q V1l—w (5), § = 0 and g,—>0 and the rigid boundary , BC' is described. In general C' will not coincide with C but, since 0 0 AS > 07, The point w=0? lies inside the domain B,. Hence the semi-infinite straight boundary OD ris: described. (ivis.) Mraengeen of the solution. Since the w-plane is cut along the positive el axis, and w=U, 1 and 0? are excluded . ell circles, by? + S so Te oe ee tag V VEE $, ( Ma iti, mn ei Lianne eee (2)! 5) (ww I= Big = VGH GD) + Vig —1)?+w—-(¢-D, (9) ee ie PR PROMI E. core SON pera et AL) Lieut.-Col. A. R. Richardson on Stream-line 442 eS) ae ee +o) Oa dw 4g (w—1)1? gives 9 2 6 = 7/2+tan=! — Pau ~ tant (Pa) (0+ +3) +9] - a7 WF? (? (11) Horse) tar {ere(o+ se) pS where 0 —H. S=> COM ie 28) *, O=7/2, i.e. all stream-lines are parallel at6=—o. {b) ve >. 0... Da oS > Osun pi 0—>r/2, i.e. all stream-lines outside the plate are parallel at infinity. The term e B(s2—d) (0+35) 9 2 2 sere) feHerhy Pe dare eee 4+ (0+ gh) b << ears Zi Hence if c<7/2 this term is < 7/2. Now a diagram giving w’? and (w—1)!” along a stream- line w~>0 shows that 8B eae —tan? = a2. = ie tan Hence, if >0, no stream-line can have a greater angular change than 7. Flow from a Disturbed Area. 443 Therefore no stream-line outside the plate can cut itself and the domain By does not extend into the app half ‘w-plane. At infinity the fluid is flowing at right angles to the plate with velocity e~°. (c) Flow behind the plate, i.e. <0. v If @ negative 7 — small, B=NV\0|, 7= ay, 1—@ approximately, 2cB(b?—d)b = a= Tea Sal (8? +6?) The angular change over this part of a stream-line is 0. When $>d and (6+4)?>", B is <0, and if > be small enough, A is >0. As. $ increases b+ a—>0, and B—>0, rises to a maximum and then finally tends to zero. 444 Lieut.-Col. A. R. Richardson on Stream-line At the same time A—>0 and then becomes negative, showing that the stream-line curls towards a point such as P,. fig. 2. Fig: 2. Showing stream-line ~=—‘1 from ¢=1 to g=17. b= c='033, d= o. (eV / ? / ¢ & tg é z é P 7 2 \ / \| PANS \ dq } jon R | Distance to <—bounding stream | { line = RS j i i FS =77 breadth of plate § Distance to plate = AS =! Further increase in ¢ causes A to become positive, to rise to a maximum and then tend to zero as eee Flow from a Disturbed Area, AAD _ Hence each stream-line w<0 for which | W| is small will behave a B OUT Os eae Fig. 2 has been plotted for the values Ale —= sill, b=1 2, Sep ys 2. Leda It shows that the-disturbed area may be expected to embrace part of C'D’ and to come close up to the plate. The dotted lines show on which side of y= —"1 a stream- line p< —'1 will lie. Hence equation (1) will represent the motion of a fluid past a plate except inside the area B corresponding to Bo. In this area.a very disturbed motion is to be expected. There may, however, be other places at which a slight amount of viscosity will cause vortices to form which in appearance may be regular in contrast: with the motion inside the area corresponding to Bo. This will be seen in the next paragraph. (d) Choice of the constants b, c, and d. The ratio CB: C’B (fig. 1) gives one relation between them. In order to give results which may be expected to have some resemblance to actual conditions, the velocity at the ed ‘corner B, viz. q=7, e* must not be allowed to become small compared with Geer’. Along W=0 and $<0, B= V —4, ig B 2cBb(B? + d) bt (4 Be dea «0 i 2ch(38? + d) My Sch 8?( 8? +d) dp hak 6? + /s? (6? + 87)? (b? + 87) 2 — tan (11) For a turning value B41 —2¢) + 287(0? + 2cb? —3cd) + bt + 2cdb? = 0. (12) Now since )>1 and 01 @ will have one maximum value somewhere between A and B, This single maximum corresponds to the right- angled bend in ihe. electrical solution, and the above equations m: Ly ‘be looked upon as giving a motion intermediate between the electrical case ant the discontinuous stream- line. The dotted curve in fig. 1 illustrates this point of view, 446 Lieut.-Col. A. R. Richardson on Stream-line Equations (2) and (11) show that |dz| increases very rapidly except when w is small, so that the parts of the stream-line w=0 with the greatest curvature lie near the corner B. It is in this neighbourhood that an eddy may be formed, owing to viscous action, if ¢ be sufficiently large. A closer approximation to the observed motion will result, if a more complicated expression is used ; e. g., ire AG Ao An Crs COT Fig © (6+ wl)? Te. “Tb olay SP occ would result in several turning values on y=0. Within the limits stated above one of the constants may be taken a function of the time, and so the effect of disturbing an existing motion may be traced out, e.g. increasing or decreasing the velocity of the plate. In addition to indicating the manner in which an existing motion may break up, due to viscosity, these solutions give possible stream-line shapes for the afterbody of a plate such that, for a particular velocity q=e-‘, there will be very little turbulent motion in rear. They also show that the pressure may be expected to change sign at some point on the underside of the plate, and that the maximum velocity on the underside is greater than that on the upperside. VI. Calculation of the resistance in a particular case. Let c be small compared with (b—1). CB =| Dea = 26+ 7/2, 0 V1l—w IW? Pars | O15) = ( (oe dw = 2b—7/2 approximately. Jo Vl—w 1 Pressure on CB = P). CB+Jof dae dd 1 - Py. CB— 3p) (“-7) a¢. Jo? Pressure on C’B= P).C’B—Jp ( (5-$) dd. . « 0) * I low from a Disturbed Area. 447 Resultant Pressure a dw "1 Ww dw a a Dies Cae tate P= xP, 2p\ y — +005 ny ee = 7™P) —7p4 a, . This has been obtained on special assumptiens as to units of length and time. ve __ 2mpqo lL em / 7 27 Pol P= LP —b/l?—1} + ies ress) 46+ 7 4b where go = Velocity of undisturbed stream, js b = 1/4." * 1, where r=CB: C'B, Hence (13) gives the resistance of a flat plate with a straight after- body. ihe b—>l1, Darel eine P—- 4+ or + (Twice the pressure on a plate of length L as given by the discon- tinuous stream-line theory). This is in better agreement with experiment than the discontinuous stream-line results (Kelvin, Collected Papers, vol. iv.). As b increases the resistance diminishes. However, since this will soon make the velocity, relative to the plate, 9-5 % small compared with gq, the solution soon ceases to represent an approximation to the actual motion. VII. Hifect of the shape of the after-body. That the resistance will depend on the shape of the after- body can be seen as follows :— Replace ¢ by a suitable function of w, Cr SCA aa to?) (7 peg) Ne e, d, and fall >0. c(w) has no singularities in the finite part of the upper half-plane. 448 Lieut.-Col. A. R. Richardson on Stream-line Several cases arise : Case (a). b>d>f>1. w= —0. Wee Sern (Ga N/a) w cd? om oe Vw a dw ae le a ee ov d= We) ONE a dw ns od << 7. M (Nw a Nw—f) ema = dw 1 Ihe after-body will be shaped as in fig. 3. Fig. 3. — as EAA MAIDAS oe Ny Sy g \ a) q Sa ee eer Case (b). b>1>d>f. w=-—O0. O< wal. = V d—VJVxr (, —J/w dz b— vw , ene ed dw Va ae fa WG ry, (4- Vw Vv wo— eo a dw a ; jp roar Mee Vw—d)\(V we - ) _ b= s/w v ( aa dw Iw Wem dz Flow from a Disturbed Area 449 The under-body will be as in fig. 4 Tig. 4. Velocity at infinity, @ = e~*. o Ndf Velocity at B, i= 3, OO. VIII. Flow through a semi-infinite pipe. lt has been pointed out in para. I]. that the disturbed motion Let of a free surface leads to consideration of a function having an essential singularity. _— pwiesinge!? | 1 dw = —¢ q sou Wy KG) —w =log(1+2). meee we |) 1. eco SL > 0 == ete sin B Vt dw ; a free surface over which @ lies between +c. It is of semi-infinite extent since ran = I e7 tesin B Naa dt 1L+t¢ E \|d: (Sica O¢>—1. ds ay = ¢ sinh B Yai dw ; = 0 and) q@ soestivomel to e~-°oKs, dz = | ef sinh B Nt dt 1+2 ? and z—>—o along a rigid boundary. Phil. Mag. 8. 6. Vol. 38. No. 226. Oct. 1919 450 Lieut.-Col. A. R. Richardson on Stream-line Along the stream-line ~=7, i.e. ¢< —1, _ de = ef sinh B Ne. dw 0=0 and g goes from pe ee amale Hence the equation represents the flow of a fluid through a semi-infinite pipe in cases where the issuing stream is disturbed. (1.) Description of the motion. t=(—1+e-Scosw)—iePsiny, . . (17) == —e-fsin NOS EFig, 69 = — v 9 ’ a a eee e7 sin B(E+1) dw Sp ile — ef cos Bé sinh py ve sin BE cosh By — — Ee, q is finite everywhere except at ~. @= —csin GE cosh By, .. . . 2 Gis Now pes it hy e-F sins bs V2 ~/ e cos Wr)? + e-°9 sin? yw + (1—e-* cos W) <3 VV T= 26F c08 be + (1—0-F 005), 7 — Hence, as ¢ is large and positive, Honea ll, E—=2E oda). 2. €. all stream-lines are parallel at the entrance. Also g=e-¢#"58 everywhere over the cross section. Owing to the presence of the exponential ¢~° this state of affairs holds close up to the mouth of the pipe. The stream-lines begin to curl as the mouth of the pipe is approached. Flow from a Disturbed Area. 45t If ¢ is large and negative, [6] » 2 COS af write |) =o; p g 2 = Es (Be? cos = cosh (ae?sin). ane MAELO 2 2 Hence there is a tendency to steadiness when w—>0 or 7, e. near the centre of the jet and near its bounding surface. Hquation (19) shows, however, that the stream-lines begin to curl back on themselves, and to develop double points, as soon as @ assumes even moderately large values. The domain By will be close to the mouth of the pipe, and vortices will be formed there or else the motion become sinuous (Rayleigh, ‘ Sound,’ vol. 11. pp. 406-408) as in the case of a sensitive jet. When |) is large, and passes through values for which sin BE=0, the angle changes rapidly ‘for small changes in |). The velocity g=e~°ccrésmhin will remain practically constant so that in this neighbourhood the stream-line is circular. Hence there will be a tendency for circular eddies to form The steadying effect of the boundary is very noticeable. Fig. 5 shows the shape of the bounding stream-line near the mouth of the pipe. Fig. 5. As one of the constants c, 8 is at our disposal if may be taken a function of the time. 2V.2 452 Capt. J. Hollingworth on a The motion resembles that observed by O. Reynolds in his experiments on flow through pipes (Scientific Papers, “Olle, Ts))s Conclusion. The view put forward in this note is that solutions of the type considered may be used to indicate where existing motions may be expected to break down ; to suggest where viscosity will cause vortices to form ; aun to enable an estimate to be made of the effect of alteration in shape on the resistance of a body moving at a given speed. My thanks are due to Prof. A. R. Forsyth, Mr. €. & Kebby, and Prof. A. N. Whitehead, of the Tmperi ial College of Science sae Technology, 8. W., for their assistance. = —— a = So XLT. Ona New form of Catenary. By J. Houtineworrn, M.A., B.Sc., Capt. R.A.F* | ae advent of aircraft during the war has naturally given rise to a large number Ue wer dynamical problems of great variety. The particular one under consideration here has been sug- gested by the application of wireless telegraphy as a means of communication between aircraft and the ground. For this purpose the aerial structure, which, as is well -known, on the ground consists of an aueacament of wires on high poles, is “replaced on aircraft by a long flexible wire lowered by the operator from the machine as soon as it has gained sufficient height and having generally a weight on the end to assist the lowering. Owing to the high airspeed of aircraft this wire trails behind the machine as the result of wind- -pressure on its surface. The actual curve formed by the wire, which has a considerable effect on its electrical properties, is, determined by the wind-pressure and various physical constants of the wire, and it is this curve which is investigated in the following paper. It must be understood that the calculations which follow are based on the assumption that the machine is flying ina straight line at a constant speed. Sudden variations in the movement of the aeroplane, even if they could be expressed mathematically, would produce transient effects of extreme complexity. A problem like this is almost incapable of direct experi- mental iny estigation, as the state of affairs cannot possibly * Communicated by the Author. New Form of Catenary. AD} be imitated in a labor atory. The trailing wire can of course be photographed from another machine fying alongside, but to’ obtain a sufficient number of such photographs to enable anything to be directly deduced from them would be an excessively long and costly process. It was therefore thought advisable to inv estigate the problem mathematically and then obtain a few photographs by which the theoretical results. could be tested, it being assumed that if the photo- graphs obtained gave results in fair agreement with the theory, the theory could be accepted in the general case. A few attempts had previously been made to measure the angle of emission of the aerial from the machine by means of a protractor fixed directly on the end of the guide-tube, but such results were not very satisfactory. It is in this case difficult to judge when the machine is flying exactly level, and also it is highly probable that in some cases the shape of the first few feet is affected by the slip-stream of the machine. This will have very little effect on the total shape, but may cause such a reading as that mentioned above to be largely in error. Like many problems of this type, the full and accurate mathematical analysis of the problem is far too cumbrous for any practical purposes, and it is continually necessary to make assumptions with a view to reducing the algebraical labour and giving a workable result. Such assumptions must of course be made with discretion, but the final test of them is that the result’ obtained by their means does not differ larg gely from the true state of affairs. It must always be borne in mind that in practical cases a simple result of moderate accuracy is of far more value than a highly com- plex and cumbrous mathematical expression representing the . true state of affairs. The primary assumption will be that the wire, which is actually a thin stranded cable, can be treated as a flexible inextensible string. As the forces involved are small and the radii of curvature of the curves are everywhere large, this is not likely to cause much error. Dynamically the wire will be treated as a cylinder whose diameter is ‘small compared with its length, and also the effect of air skin friction will be neglected. The former assumption will be justified later ; the latter is necessary for the simplification of the analysis and appears to be justified by the results obtained. Proceeding in the usual way, the first process is to form the equations of equilibrium of an element of the string. Let S be the distance of the clement ds measured along 454 Capt. J. Hollingworth on a the string from its lowest point, 6 the angle which the element ds makes with the vertical, and w “the weight of the wire per unit length. | The normal wind-pr essure is proportional to the square of. the normal component of wind-velocity, the tangential effect. will be neglected. From certain results which are not available at the moment, it appears that the ratio tang. force /normal force is pall except when @ is nearly 90°, and then its chief effect will evidently be rather to increase the total than to modify the shape. Direction of i w as For equilibrium of the element ds we have by resolving along the tangent Ol=wdscos0,. ... .« =<: aoe and by resolving along the normal Tdé= Kv? cos? Ods—wsin dds. . . . IL Eliminating T between I. and II., Zi sin 8@— Kv? cos? @) +2 cos o° a(u + Kv’ sin 0) =0,. Te ee ee, Ww | or writing Ke 7? us 9 ds ae p+sin . ao? dd cos” @—p sin @ To solve this : Pat <1 We Ce. Dh ae a, p+sin g d@ cos” @—p sin 0’ dz _ COS O(pt+sin@) dé | 22° cos? O—psin 8 New Form of Catenary. 455 t€. integrating ; cos 0(p+sin 8)dé log C-+log (2)! = | par eine slag aete AN he V- where C is some constant. Now cos? @—p sin @=1—p sin 0 —sin? 0. Let the roots of this be # and —£, so that 1—p sin @—sin? 9 = (a—sin #)(6 + sin 8), where ile CS tee Orn be: urs eer Wal Dc a ii Bs cos?@—psin@” e—sin@ 8+sinG@’ where A8+Ba=p, Aime ini ae We aval from V., : “A cos@ UR "B cos 0d6 log C+ log (2)? = “a—sind * ) B4+sind a (2—sin @) + Blog (8 +sin @) A (8 + sin @)8 — °8 (a=sin 0)4? gas (8 sim 0)?? i G) aa Cae ira: eames a aso Boh t oe where a, 8, A, B have to be evaluated in terms of p. Before proceeding to do this it will be advisable to in- vestigate the quantity p a little more closely. ee Le Tea K is a constant depending on the wire diameter and on the density of the air, and is of the form Ns, where s is wire diameter, and by Laplace’s formula, when the barometer at sea-level is at 30 inches, N at sea-level =°0027 : N,, 6,000 tt.=-00214 N ,, 10,000 ft.=-00179 Now w, the weight per unit length, is proportional to the square of the wire diameter, K to its first power; therefore 456 Capt. J. Hollingworth on a p is proportional to the wire diameter, and so the thicker the wire and the slower the speed the larger the value of p, It will be seen later that in the most unfavourable case, 2. é@., with the largest wire in ordinary use travelling at the slowest practicable speed, the value of p is slightly over 0°25 and is more generally of the order of 0°1; hence only small errors will be caused by neglecting squares and higher powers of p. This causes considerable simplification in the evaluation of equation VIII. For from VL., JV (p?+4)—p po — mer ————_ §_ == 1 —— 5) | as OS ee aa 4) + p pale i ee and from VIL., ap gne ep) 1 * oes . 2 v Pot) 2— ee 2° (Bs smmi@) ee oe et hoa (a—sin 0374 (a —-sin 8)?/?4} oe if ie eS ~ (a—sin @)(8+sin 6)’ \a—sin 6 as 1 {1+ (p/2 + sin 0) }P? cos? cos? O—p: sin 0° {l— —(p|2 + sin 0) tpi ab 1+ p/2sin @ ~ cos? 6 — —psin 0° 1—p/2sin @ 1+psin 6 ~ cos? O— psin 6’ since powers of » above the first have been neglected throughout. ds Leip sini eee dO cos? 6—psin @? - and this is the first integral of equation III. To determine C consider the lowest point on the wire. If the weight is heavy C? X. New Form of Catenary. ADT compared with its area it will hang approximately vertical, and for the lowest element of wire we shall have @),=0, W = weight of lead weight. from. Ll. W (3) 22 |0F 0 dQ x pee from X., C VW ili; Cue: Nv Hence full first integral of equation ILI. is ais’ 1+psin 0 eek : a XI: dO Kv" cos?@—psin@ The case in which the weight does not hang approximately vertical will be considered later. To integrate XI. further we have WwW 1+psin@ — ene dO (a— sin 0) (B+ sin 0) which by a process exactly similar to the previous one can be reduced to the form 2Ko? ~ta-—-sind @6+sin0 Dy Wy al i a ds= a—sin 0 B+sin0 which is in standard form and integrable. BR ae The integral is of the form Sp) = W fl+3p/?). tan 6/2 - ae Bee Ae 2Kv? 2 | Vp °8 tan O/2—1—p/2+ Vp Ly + tan 6/2 | a ee XLT. +(1—3p) tan7! This result is interesting but far too complicated to be of any practical value. Hquation XI. is, however, easy of evaluation, and from it, it is quite a simple matter to construct anv required curve by tangents. 458 Capt. J. Hollingworth on a Before proceeding to a numerical case it will be advisable to investigate the effect on the result if the wind-pressure on the weight is sufficient to cause it to hang not truly vertical. Let the angle at which it hangs be 4. Equation II. then gives (= Be ee i d0}.9 Kv? cos? 6,—wsin Oy but T= Wicosid7- W cos @ il 1+psin 4) Kv? cos? 0, —wsin 0, C2" cos? O)—” sin Oy’ ds__ Wcos@% 1l+psin@ d@- 1+psin@) cos? @—psin @ in this ease Owing to the irregular shape of the weight it is impossible to calculate the angle @. Experiments with a cylindrical lead weight about 2 inches long and 14 inches diameter weighing 1$ lb. have shown that the angle is 10° to 12°, at a speed of 60 M.P.H.; with a stream-line weight it would probably be less. The determination of this angle for any given weight is best done in a wind-tunnel, though as one was not available it was actually obtained by hanging the weight by a lead about 2 feet long from the upper plane of an aeroplane and observing and photographing the result from the observer’s seat. Eddies and vibration tend to make this method inaccurate. Returning to equation XI. there is another very im- portant point to be noted, which is that the expression for ds/d@ does not contain s explicitly. This means that for a given W and p the aerial form is independent of the actual length of aerial employed, so that it is merely necessary to construct one curve for each value of W and p and v, and having got this curve the actual shape of an aerial of any length can be found by measaring up from the bottom point of the curve a length equal to the length of aerial employed. A series of numerical determinations will now be given. Two kinds of aerial wire have been considered with two different weights and two air-speeds, making eight cases in all. The two wires A and B have constants p respectively : For wire A, K=:0000129 ” 39 B, K =:0000043. New Form of Catenary. 459 The two weights taken are 0°5 lb. and 1°5 Ib., and the two speeds 60 and 80 M.P.H. The eight cases are therefore :— Ww I. Wire A weight 15 Ib. speed 60 wv.u. For this p='267 fF, 3=382'3 CT, eS Ohi 60 ., » » = 267 =108 i A. OL ee sO... eg Sl eel Mar AS) aa Oy a me s0 , Ce elena Gal V. Bury. aeons G0. ;, ~ » = 0916 =96-7 ime Be Om. ~~ CO 5) 0916 ==823 ee Fae ees es 80, » yy 0517 «== 54-5 aoe 1B nice Om oe Pe SOi° ;, yo Sey Se For the purpose of plotting the results of equation XI. it is to be noticed that ds/d@ is the radius of curvature. This has been calculated below at intervals of 5° and for the mid- point of these intervals, 7. e., 295, 7°5.... To construct the curve it is merely necessary, starting from the lowest point (@=0), to describe an arc of 5° of a circle with radius first value of ds/d@ obtained. Joining this another 5° are with a radius =second value of ds/d@. ‘Towards the upper part of the curve, however, where the radii of curvature are incon- veniently large it is more convenient to multiply these by sO O(o expressed in radians), thus giving the actual value of ds, and set this out asa straight line at the appropriate angle. Again referring to equation VI., it will be seen that 0 increases from 0 to a, at which angle ds/d@=~«, and the curve becomes a straight line. It is therefore in each case unnecessary to carry the calcu- lations beyond the value of @ given by the real root of cos? @—psinO=0. The form of this equation which does not involve W shows that the ultimate shape of the aerial is independent of the value of the weight on the end which only affects the shape of the lower portion. These asymptotic angles have been calculated, and are as follows :— : Casesoy eva, -LE.612 16! A Cee: EVA CaO! Seem: Vii 70eoa! a Oe TAPS GS Oa? Capt. J. Hollingworth on a 460 let FG. ‘YsueT 68-6 LZ GOT 8.1 89-1 9.T £9. FF £6.96 81-61 96: ‘IySsue'y S861 6-FE F-8L FAG 16-98 ¢ L6-6F 91-6 18-F§ OF 18:16 C9.G OT-GG 0: O21 SI-F 60-€T L19-§ 96-6 68-6 60-9 90-€ 18-6 18-6 “TE oUary ‘Sp 461 New Form of Catenary. ¢-201 6-19 Ch eg 9L-8G QT-8% 3-61 91 eel 6-01 19-8 hg C6-F 83-8 9-1 “1Qauary 9 OF 6-91 OL 8-9 col 9.0¢ 6-66 6-06 C6-FI aSV,) “youo'y ‘Sp l~= ‘9 HSVD _ £6 9-81 g-eL cé-§1 cs IT c9-O1 68-6 G6 CLS Sp OGL 008 COP OCS QUG FG BLI Fel él 6el Sil 901 OOT 86 462 Ona New Form of Catenary. é —_—__—_ Direction = of motion A eroplane a Aerial wire as calculated ” re) ) traced An EHxperiment relating to Atomic Orientation. 463 The curves in fig. 1 have all been copied from tle results of Table I. In each case they are for an aerial wire 200 ft. long, and they have all been collected to a common point at the top in order to show most clearly their relative shapes. To check these results a few photographs of an actual trailing aerial complying as far »s possible with case I. were taken. It was found impossible to get the whole aerial (200 ft.) into one photograph, and as it has been shown that the shape of the lower portion is independent of the length, two photographs were taken—one of the upper end only with the full length out to obtain the angle of emission, and one of a length of 50 ft. of wire only which could be ‘obtained on one j)late to show the general shape of the lower portion. From the first the angle of emission was found to be about 62°, the theoretical angle being about 61°. Tig. 2 shows the second curve, the full line being the shape of the 50 ft. aerial actually traced from the photograph and the broken line the theoretical shape. Considering the difficulty in obtaining an accurate repre- sentation of the theoretical conditions when actually flying, these two show very good agreement. It was therefore taken that the assumptions made in the course of the calculations were justified and that the curves give the actual aerial ae: with a fair degree of accuracy. NEI. An a ee De to pMaeone Onis liie: By T. R. Merron, ).Se.* i modern theories of the structure of the atom the supposed arrangement is of such a kind that the atom may be described as having an axis perpendicular to the plane in which an electron, or a ring or rings of electrons are in rotation. Since it is well established that the atoms in a crystal are arranged in a perfectly regular manner, it is of interest to inquire whether the axes of these atoms are also oriented in a regular manner, or whether their directions are distributed at random. Of these two possibilities the former seems a prior: to be the more probable. Calling this the first assumption, an experiment arising from the appli- cation of a second assumption seemed worthy of trial. The second assumption is that the direction in which the e& and B particles are ejected from a radioactive atom is related to the orientation of the atomic axis. If both assumptions are correct there should be a difference in the e and 8 ray activities from the different faces of certain crystals of radio- active substances. It may at once be stated that no such * Communicated by Prof. Sir E. Rutherford, F.R.S. 464. An Huperiment relating to Atomic Orientation. effect has been found, but even a negative result may be of some interest, since it would appear to show that one or both of the assumptions are incorrect. The experiments need only be very briefly described. It is in the first place necessary to choose a substance whose disintegration products would not contribute appreciably to the radioactivity under the par- ticular conditions of measurement which are adopted, as it is obviously unreasonable to expect that an atom should remain unaltered as regards its orientation after a cataclysm which has changed it into a new substance to which the arrangement of the neighbouring atoms in the crystal is no longer a natural one. After a number of trials with sub- stances and by methods which were for various reasons unsuitable, the experiment was tried with some large crystals of uranium nitrate which were specially grown for the purpose. The « ray activity was measured by means of an electroscope essentially similar to the well-known instrument designed by Sir Ernest Rutherford for the measurement of weak a ray activities. Holes were drilled through the to of the electroscope, these holes being smaller than the smallest face of the erystals, and comparison was made of the rate of leak of the electroscope without the crystals, and with the three different faces of the erystals respectively placed over the holes. The crystals were removed from the solution in which they were growing and dried immediately before each experiment, with a view to avoiding as far as possible changes in the surface through exposure to the atmosphere. The radioactivity of the crystals measured in this way was found to be independent of the face from which the a rays were emitted, the accuracy of agreement being about 3 per cent. This was within the limit of experimental error, and the differences in the readings were not in any way systematic. Crystals of uranium: nitrate appear to offer a ver favourable material for the experiment from the radio- active characteristics of the uranium atom, and also pro- bably from a erystallographic point of view; they contain of course the two isotopic uraniums, but the 6 rays from the disintegration products are negligible in so far as the @ ray activity measured in this way is concerned. The expe- riment might be worth repeating under conditions in which the 8 rays from different faces are compared. A probable explanation of the failure to obtain a positive result is that the « rays are shot out from the nucleus without regard ‘to the orientation of the atomic axis. Balliol College, Oxford. An=(- Layee (a+ m—I1)! oe a Mista (I no (op) Cu ses XLIV. On the Potentials of Uniform and Heterogeneous Elliptic Cylinders at an External Point. By N. R. SEN, M.Sc.* } | Bae potential of an infinite elliptic cylinder at an external point is generally expressed in the form of an integral, and it is well known that a transformation in conjugate Functions would allow the same integral to be represented by a much simpler expressiony. It is here proposed to express the potential in trigonometrical series. The method followed is that of integration, which will be shown to be applicable also in certain cases of heterogeneity, It will be found that the potential is always expressible as Cos ee yt Ag| og ye DS} where A, in its most general form can be expressed by hypergeometric functions in e (eccentricity), reducing in two special cases to finite binomial forms. This happens when the cylinder is homogeneous or when the density (supposed constant along lines parallel to the axis) at any point on the elliptic section varies inv ersely as the focal distance of the point. We shall simplify our problem by considering only the logarithmic potential of the elliptic section to which the (Newtonian) potential of the elliptic cylinder is equivalent but for an infinite constant and the constant multiplier 2. 2 Before proceeding with the solution, of the problem pro- posed above, it would be useful to consider the expansion of (1 +e cos by", e<1 in cosines of multiples of ¢. Expanding (1+ecos ¢)~" by the Binomial theorem and replacing the powers of cos@ by cosines of multiples of ds it may be shown that ¢ (1+ecos d)-" = Ss An cos mg, m=0 where er aC »m+1; ), ect, ~_ (n-- Dye Pan! Qe" where IF is the hypergeometric function of the four elements * Communicated by Prof. D. N. Mallik, Se.D., F.R.S.E. + Lamb, Mess. of Math. 1878. * This expansion in another form is given by Gauss. Phil. ee Si6. Vol sou Nowea6. Oct Olav) 2K Mr. N. R. Sen on Potentials of Uniform and 466. Since Aj, is a Fourier’s Coefficient, within parenthesis. we have 9 cosmpdd _ A” SPAGLee cosh) 4 ie ras | m (n+m—1)! ce F (” +m ntm+i eae ) = (— LY oe ee 1 Qm—1" ome 5} and (~ cos nd dd hi ee Cle eos te) ae | a ee (a cae { B(n ane eieg2 = (—1) Grae et fe ere ales oe) De < ony, me” 1 oa Gee 29) a 2\n+4? : 1 ( ap 2) 2 also yi cos nd dd _ (l+e cos )"*! (Qn)? = (—1)” a i ae =: F@+ $, n4+1, nei m%, Soeznmion are” 1 = ar) n! nl° Qr-l’ (l—e?)"t+2? n being an integer and e<1. Be Taking the focus 8 as origin let the equation of the ellipse be given by l P= T+ecosd ) within the area of the ellipse Let P be any point (p, @) at a sufficient distance trom it. and A another point (7, Taking the area to be of unit density the potential el - 0g AP pdpd¢. Now AP? = p?—2prcos (6-8) +7" 2 and log AP = log r+ log Riess (7) cos (p—0)+(2) | le 7 \r a af p nN = logr— > (") Cos no—@), a Heterogeneous Elliptic Cylinders at an External Point. 467 yr being supposed to be greater than the maximum radius vector, 7. e. the length of the major axis from 8 to the remoter vertex of the ellipse. : Hence As == 2 i dd m2}: (le cose)? 2 Int? 1 (" cosn(d—O)dd a a n(n +2) ae (L+ecos d)"*? log r But me cosnodd (—1)" (2n+1)! sae ie meite cos)? (eI) t al 275% (eee te and 7; sind 0 sal pereas)"*2 = Hence ee ay = t—2 yr (2n+1)! 1 ( le y seen? ioe 2-1) n.n! G2) Qu-1 But See CS, where C is the centre, so that 3 ise ge == ¢ we have finally (1—e?)? o (2n+1)! IL c\” pie Beto. 7 at (eel) es jaded : =P V=log ai 1) | Ge (<) cos nO and 2V is the potential of the elliptic cylinder neglecting an infinite constant. 4. Let us suppose the cylinder to be heterogeneous and any line parallel to the axis to be a line of equal density. Let the density at the point (w, y) on the elliptic section be f(z, y) where f is a rational algebraic integral function in wand y. Such a function is also expressible in a series in p and @ of which the typical terms are p? cosq@ and p’sin gd. It will be sufficient for us to work out the case of these two densities. (i.) Suppose the density «=p? cos gd. Then as before V= (\. log AP pdpdd e the integration is to be carried over the entire area of the DK? 468 Mr. N. R. Sen on Potentials of Uniform and ellipse. Proceeding exactly as in the previous case, we have PY ("cos qbdd a (p+2))_, (1+ecos prt? = eae i cos qh cos n(hb—O)dh T log r SF n(ntpt+2),)_, (l+ecos 7 p+2 (Pir Pe a pee > a , log r (p aT 2) e/ mel: +eécos @)) Bae < TPP cosnd ("* [cos (n+ y)b+c08 (n—g)b] db. 22. (l+e cos b)'tP+2 and substituting the values of the integrals from § 2, Vn AE tog SARI TATE (Ly cocng at? (p+2) om 2n(n-+p+2) \ri where n~g is the positive value of the ditference between these two integers. ae When g=p one half of the series is expressible in a simpler form; since I re ace ((— IL a (2n+ 2p +1) ! oe, eu +p : 1B Us, (n+p+2)! (w+p) ! 2*e°) (| —e2ynrpts +p this part of the potential function can be written as | ee) 2 ee an) ES 9)1 log r Dee 2)? 2 PE (Cea S | (2n+2p+1)! in SY 2) \0 at) aes a 1) fie (Qe p42 ) ! (n+p) ! (x) Cas nd |, tie other part being expressed in hypergeometric functions. This is possible only within the limits in which such sepa- - ration of terms is legitimate. A similar simplification is possible when g=p+1. When g=0 and o=p? the potential is given by ee | eee mie (pu ce o> n(n+ p+ 2) =I Ea né. {Ro It may be noted that when p is a negative integer this formula is applicable with a slight modification. The terms beginning from the first up to the nth where n=—p+l Heterogeneous Elliptic Cylinders at an Haternal Poimt. 469 would have their coefficients in finite forms which it is easy to caleulate. A very interesting case of the above arises when p= —1 or the heterogeneity is of such a nature that the density at any point on the elliptic section varies inversely as the focal distance. Since f ee ye nel, weds (al eigen, 7 =(-1)" mel a QP (Va e2)242? we have in such a case V A oO. n+l ns —_ = Ao loo r— A, U COs nO Tt 2 i r é 2 n(n+ 1) n=l1 and making the above substitutions the potential function Is found to hes given by cas i | ea ID Cur ait ) co cos nO. aS) ann. nt (n+ 1)! (ii.) ‘Suppose that the e density c= oe sin qo. Then as before a lee AN sin god (pt 2) J-x(1+e cos gypee ai — 2 jeter? f sin g@ cos n(p—0) dp = i(naapee ea) -, (peices) tara Now SING ae. 1 co Da and replacing the product in the numerator of the other integral by the sum of two sines we have Aoi e ae Nea ay ees = re Anng n+q @ sin 0. a 2n(n-+ p+ 2) , The logarithmic term issabsent. -Lhe line @=0; @=a is a line of zero potential, as is obvious. Also at a great : iol Le: an be neglected distance from the origin where 2? 28 ete. can be neglecte 1 ee 470 Mr. N. R. Sen on Potentials of Uniform and in comparison with other terms the potential is approximately given by 1Ok — [atti Pi sin 7 N 726653) a +3) ep hence the corresponding equipotential lines are arcs of circles touching the major axis at the focus. oD. We can determine all the cases in which the hyper- geometric functions appearing as coefficients in the trigono- metrical series are expressible in finite forms. Two cases we have already studied where they reduce into binomials ; let us inquire if in any other cases such reduction is possible. The function F(a, B,y; e*) will be a binomial expansion if either y=«ory=8*. ‘Taking the most general case, (i.) § 4, we seek to satisfy either of these conditions in noe the functions Aji?*? and Ajt?*? by giving suitable values to p and y. AW = Ox F(nt14254, n nts +h age e); "hence for the required condition we should have n+q+1 = either n+ jee nt s4Pt9, or ils @ p = q!) or PETS | and Ate = el. Ox F(n+3 n—4q 3 = — n+5+4 i n—g+1; “), which in a similar manner gives b= ie aig! 3G) * The other complicated forms, e. g., F lot oe = Le; v to a © (Sr ie) are at once seen to be inapplicable here. Heterogeneous Elliptic Cylinders at an Eaternal Point. 471 So we are to find p and gq such that any of the four following sets of equations should be consistent : ao | P= 9 Veet 1 = ¢ pod Sg) p=3q[' pti=3¢)" p=3q)> p+tl=3qJ’ q being zero or an integer. I 2 e ; . From these four equations we get only two possible solutions, namely, gel g=0 | answering to the case of homogeneity and to that of the density varying as the inverse focal distance. ‘These are the only two cases in which the potential function for an infinite elliptic cylinder for the outside space is expressible in a trigonometrical series with binomial coefficients. "Gs In § 3 we have obtained a trigonometrical series for the potential function V for the outside space by integrating log r throughout the entire area of the section. But in course of our analysis, in order to make the expansion of the logarithm of the distance PA possible, we had to introduce a certain limitation, namely, that * should always be greater than p; this immediately marks out a circular area with centre S and radius equal to the maximum radius vector within which the point A must not lie. It will now be shown that the series V has a much wider area of convergence which extends even into the limiting circle, and consequently from considerations of continuity it represents the potential function everywhere inside that extended area. It is well known that the series }(—1)"a, cosn@ is convergent if a,,—>0 steadily. Considering the present series as a series of the same type, we have Ln = 2) ene oy ) ee | ee 2 ebb ee Cee ey Tn ee) ee | (n+1)! Dp mn 2 Peceoee.; . (2asb 1) a Tran $2) 2b Ohi: (Qn+2) VrJ ? 472 Mr. N. R. Sen on Potentials of Uniform and and this would be a decreasing monotonous sequence tending towards the limit zero if we take ie 2, 2), where S’ is the second focus. We can also show by applying the usual ratio-test that the series is absolutely convergent under the same conditions. This shows that in addition to the outside region the series is also convergent inside the area lying between the previous limiting circle (drawn for the purpose of integration) and a concentric circle whose radius is SS’. Consequently the present 'form of the potential function is valid at all points inside these two circles (and outside the elliptic area as ‘we are dealing only with the external potential). It would seem that we are incapable of accepting the potential function in the present form of the infinite series inside the circle of radius SS’. But in fact the region in which this trigonometrical series fails is much more limited. If we take 8’ as our origin and proceed to find the potential by the present method we get the same series, which ina similar manner may be shown to be applicable everywhere outside a circle radius S'S. In general, these two circles bounding the regions of convergence overlap outside the elliptic area, and it is only inside the two small areas common to the two circles and symmetrical about the minor axis that the present trigonometrical series fails. Hxcepting this common portion the present form of V would hold good everywhere. only we should take care to choose the origin properly—measuring r from 8 or 8’ according as. the point lies inside the circle cf centre 8’ or 8. | It is curious to note that the convergence of the series depends on the eccentricity of the ellipse. The two limiting cireles would have their common portion entirely within the elliptic area if SS' =< SB where B is an extremity of the minor axis, 1. @. 20e =a, ie (Q ex 4, This shows that when the eccentricity of the ellipse is not greater than 4 the function V gives the potential everywhere outside the elliptic area, with judicious choice of origin. This includes the important case when the ellipticity is small and the ellipse is obtained from a circle by a slight Fleterogeneous Elliptic Cylinders at an Haternal Point. 473 deformation. For the area within the two circular strips in which the trigonometrical series fails it is not possible to get by the present method a simple value for the potential function VY. Starting from the beginning, we have to divide the elliptic area into two areas by a cirele passing through the point where the potential is sought such that every part of the one area is nearer to the origin than the point while’ every part of the second area is further from it. We can use two logarithmic expansions in the two areas and find the potential of the two areas separately. The method of procedure is the same as in § 9. We get the potential both in direct and inverse powers of 7. Buta as the expression is not a simple one we do not propose to give it here. a A similar investigation is possible in the case of variable density. When the heterogeneity is of the nature we have assumed in § 4, we can show that at least outside the same two strips of areas between the two limiting circles the series V in § 4 is convergent. It should be noticed that a transfer of origin to the “other focus in the case of heterogeneity would entaila change in the law of density. But it we take density to bea wainonell algebraic, integral function of the coordinates of a point, a transfer of origin would involve a change of density of such a nature that the new distribution would still be represented by terms of the form p? cos gd and p? sin g@. So these two cases are sufficient for our purpose. As before, applying Dirichlet’s test to V in § 4 we get the condition of convergence by making the coefficient of cos n6 steadily tend to zero. This leads to such a condition as the following : | Li a a ae uo 7. (Waar) ae 2)! (n+q)! gn+a- Er mti+t me) n nists f Dae Oe ls ae) —>(). If pxy¢ every term ot F is less than the corresponding 1 gis er)rts a term in the expansion of ] so that rec 7 (1. Baye) s+P 3 474 Mr. N. R. Sen on Potentials of Uniform and Let p>q: the hypergeometric series is of the form H(a,a+3, 73 &) a(zts) , a(a+1)(a+4)(4+3) Lasoo 4s Ne er + ne ~ 2 4 castcciee Tae 112 yy Since a>vy (gq being positive), aA Ae at 2 0 ey Wee RS at 3 2 SO F<1+733( e+ he Bel 24 LAG) 1 < ae es (=2e) oy. when the series is convergent. Here Lt~=1; we can show as in §6 that V should converge at least (whatever p and g may be) if Lt CX — => 0, nto r( 1 —¢*) eis 1 where c is ultimately of the order — , n if mS ee —e~ Leen 20. Hence, at least outside the same restricted region as in § 6, V represents the potential function for the whole external space. : 8. In § 3 let us suppose that e is equal to zero. An ellipse of zero eccentricity is a circle and the semi-latus rectum is the radius. Making this substitution, we have the logarithmic potential of a circular area. V = 7a? log r = (area of the circle) x (log of the distance from the centre), and the potential of an infinite circular cylinder is twice this quantity, neglecting an infinite constant. Similarly, from § 4, when the density varies as the inverse focal distance we have V = (circumference of the circle) x (log Heterogeneous Elliptic Cylinders at an External Point. 475 of the distance from the centre), and the potentials of hetero- geneous circular cylinders can in the same way be deduced from the other formula in § 4. | Of course all these results admit of easy verification by direct integration. We shall deduce another simple result from the series for the potential function in § 3. Letus calculate the attraction of the elliptic cylinder at a point on the major axis produced of the section. On the major axis ON <7 = a. Differentiating the series of § 3 we have ee wl? af ie it @ n 2i2n+ 1)! eye eg? eee rae) el 0, and the attraction is (2n +1) en Ge n ! (n+2) ! ‘ gr-1 5 pet bo = 1 is (2p 1 or 76=0 T° a= CA wleene ag? 3 liege Oil ab C e\2 = aay: eg mate) Vad See Aor 2 ? (1 + | (1+2 | ] a == Ube ab 1B Gs, 1 Bcee al nee | h ds = dn [(e+r)— Verne] ab : ice EO = dns [E-VE-2 jy where & is the distance of the point from the centre of the ellipse. This is the total attraction of an infinite elliptic cylinder at an external point on the major axis of its section, a very well-known result. It is also interesting to note that when the cylinder is heterogeneous, the density at any point of the section varying as the inverse focal distance, the attraction at any point on the major axis is similarly expressible in a very simple form. Using the corresponding formula of § 4 we have as before 476 © Mr. N. R. Sen on Potentials of Uniform and OV Vip =0 on the irae: axis and nee i 3 2 Ont 20 eae 1 al or ) 0=0° aah 2) nl (nv +1) ie gn—1* gm+ 2 MDS a See .(2n—1) 2e n+1 = | =o ( 7 S 1)’ ea oy pear) a) i ire Hence the total attraction > it Zs tie JEP =o Ir | oN 3 © | SS Zee aie Se on) Site | — eae where & is the distance of the point from the centre of the ellipse. 9. We have so far considered the case of the complete elliptic area. ‘The method of analysis followed here is, however, applicable to the case of an area bounded by two elliptic ares. As any two arbitrary arcs would make the result cumbrous, we choose here for illustration a very simple case when the result appears in a rather symmetrical form. Let us suppose that the two elliptic arcs have the same focus and their major axes lie along the same line. Let 8 be the common focus and let tle two arcs whose equations are ie l+ecos @ - and ip es 1+eé cos d! intersect at C and D, and let CS make an angle @ with the line from which ¢ is measured. P is any point (p, 6) inside the area, and A another point (7, @) outside at a sufficient distanee from the focus. The area is divided into two elliptic sectors by the radii vectors SG and SD, and the potential of the whole area is the sum of the potentials V, and V, due to the two sectors. We shall suppose the area to be of unit Heterogeneous Elluptee Cylinders at an Eaternal Point. 477 density. Then rt den! held = ; ie: AP pdpd¢ J -8 l fo.0) "8 (itecos 1 n =| | NGF *[ log r—> “A2) COs n(p—8) | pdp dd, ania al ; . r being supposed to be greater than SC and the point A lying outside the limiting Semele deseribed with centre S and Paaiis SC ; ‘he |, ae ae _g oes cos ee ae Ge a (l+e cos fy" +?" - = 1)(8,e) log r—2 x lB; ) 2) (=) cos né, _ eet where "B cos nod dd LG e) =| (1+e COs ) ee and f oe dd I,(8, e) =') (1 +e cos $)? ‘) SN aR a or | tan’) x Bi ‘\-3 sin B LSE a 3 Ree ee? TE Cae We shall put [,(@, e) in the form of a series. Putting (1+ecos d)-“+?) = ¥ At” cos md, m=O where fe ee vm mee) ie” n+mt2 ntm+3 LA a (— ) (n+ 1) ! m! Qm-1 ° 5) p) 2 p) m+, ?) (e <1) O72. since a) cos md cos Np dp = 0 e | bole 1 m—n m—n Mm, 2 ‘ore (m 4-2) ean | +5 *B I yo cos mo dd Hi 1S 3 AnH HB, ¢ 0 (1+ecos h)rt? m Binns 478 Mr. N. R. Sen on Potentials of Uniform and we have o n+2q f n ub *=1)(8,e)logr— > & eA) cos n8. IP n=1 m=0 n(n + 2) Similarly V, Mice iN +29 je =I (w#—-B,e')logr—- = & (- LE ae (<) cos né. n=1 m=0- i un +2 WE) The potential of the complete area V=V,+ V3. It should be observed that the quantities /, l’, B, e, e are not all independent ; in fact 8 is determined by the equation — l s l' l+ecosB 1—e' cos’ When the point A lies within the limiting circle an analysis on the same lines is possible if we divide the elliptic area into two parts (by a circle with S as centre and SA as radius) in which two separate logarithmic expansions would apply. In the most general case of two arbitrary elliptic arcs the area may be looked upon as the sum of two elliptic segments each of which is the difference of an elliptic sector and a triangle. In the preceding analysis we have virtually given the potential of an elliptie sector and the potential of a triangle i is known. But as the results in all these cases are not simple or sym- metrical, it is unnecessary to deal with them here. 10. In this connexion we may also study the potential of the complete cylinder when the density 1s an exponential function of the vectorial angle @. Aswill be seen below, this may be considered as a generalization of the preceding cases. The method of analysis followed would be exactly similar. Suppose o=e*?, Then gh ox | tog r > (2) COs n(—8) |p dp dd a —7/0 i =i 4 : Bs ahs ext dd s ay S Bey 1 ek? eos n(d —@) dd. i +e cos $)? ent 2)rj_, U+ecos py Heterogeneous Elliptic Cylinders at an External Point. 479 Now ™ eB cos n(p—O) = SE ales | be (l+e COs pyr? = > EN ag [ cos (n +mo—né) + cos (n—ingd—n0) | dd and ‘ eXF cos (n +mo—n6) dd K cos nO— (n+ oy) sin n@ sinh «ar a (—1)"*" 9 s K+ (n+m)? and ( e? cos (n—mo—nd)dh s @—(n—m) sin nd 1y-” 2. K COS Nn el He! = (-1) Pa cary sinh «7. Hence V 0 2nA> eer 8m (? sinh «7 = a me 2" Oo © nim [K cos nO—(n+m) sin nO —> >(-1)"* oe ee N= m= L K ar (n ar m) «cos nO — (n—m) sin nO Nar BNE = ass aur 9 x yes K* + (1 —m) n(n+2) \r If we put «=0, the cylinder becomes homogeneous and the present series in this limiting case degener ates into the series of § 3. Moreover, the case of the density plex? can be easily worked out ina similar manner, and putting 1w%« (a= “ —1) for « we can deduce the formule of § 4. Thus this form appears to embody in itself all the preceding ditterent cases. My best thanks are due to Dr. Ganesh Prasad i his kind help and encouragement, and to my friend Mr. 8. N. Bose for his encouragement and useful criticism. Universitv College of Science, Calcutta. et S07 XLV. Note on Proofs of Elementary Theorems of Oblique Refraction. By A. EivERETtT*. igs text-books on Geometrical Optics, the elementary theorems of oblique refraction (7. e. where the ray is not in the plane of the paper), at a single surface, appear invariably to be proved by plane projectional methods. In drseussions of the question of deviation by prisms, however, the method of projection on the sphere is generally adopted. it might therefore be well if alternative proofs by this method of the three elementary theorems were given in the text-books, in order to smooth the way. Apart from this consideration, many people find it easier to form a conception of direction in space from the spherical diagram, and the proofs are extremely easy to remember, since they consist in hardly more than writing down the two commonest formule for solution of a spherical triangle. Since these proofs are not given in the books, perhaps an illustration may not be out of place. Suppose a ray of light to be incident at a point Q of a retracting surface. J'rom the centre of a sphere of unit radius draw radii as follows :— (i.) parallel to incident ray, and meeting sphere at I. (ii.) parallel to refracted ray, and meeting sphere at R. (iii.) parallel to normal at (), and meeting sphere at N. (iv.) parallel to any given axis of reference, and meeting sphere at P. : ic. 1 oir N Since the incident ray, refracted ray, and normal are coplanar, the points N, R, I lie on a great circle. Join also * Communicated by the Author. Proofs of Elementary Theorems of Oblique Refraction. 481 PI, PR, PN by great circular arcs. Tet o, 6’ be the angles of incidence and refraction, w, w the refractive indices of the first and second media. Then arc NI=@, arc NR=qd’. The angles ¢, ¢ being acute, from the nature of the case, and the refractive indices positive, the law wsin@=yp' sin ¢! shows that NI, NR have the same sign, hence I ae R hie on the same side of N, within 90° of N. I. From the spherical triangles NPI, NPR, we have cos PI=cos PN .cos@+ sin PN sin @.cosN, cos PR=cos PN .cos ¢/+sin PN.sin d!.cosN. To get rid of N, multiply the first equation by fs and the second by mw’, and subtract : then since wsind=yp’' sin d’, cos Pl= pcos BR e = PN =pucosd—p'cosd’,. . (1) which is independent of the position of P,7.e. of the direction vf the axis of reference. By taking as axis of reference in turn each of three co-ordinate axes, we get the general equations of refraction ple pM eM | eNE N l gn m oe n L, M,N, L’, M’,N’, l,m,n being the direction cosines of the incident ray, refracted ray, and normal respectively. (These three equations are, of course, only equivalent to two, owing to the relations between the direction cosines.) TI. If the axis of reference is perpendicular to the normal, PN=90°, andcosPN=0. Hence, since pcos ae cos is not inti, we have from (1) in lie case =pcosd—p' cos’, poosPI—p' cosPR=0, or .psing=p'sing, (2) where 7=90°— Pe 902 ele Hence the angles made by the incident and refracted rays with any plane through the normal obey the law of refraction (P being the pole of this plane). III. From the triangle NPI, sin N ne sin [PN Sine Be esi LN ° Hence, putting y= ZIPN, y’'= ZRPN, we have ne Nu 008- sin ¥ sin d Phul. 1 Mag S. 6. Vole sev Nov 326. Ove1919. pe fe 482 Proofs of Elementarg Theorems of Oblique Refraction. and similarly from the triangle NPR, / : / ; cos 7’. sin sin N= a ; vi sin } Hence since wsind=p' sin Pd’, 1 Cos. SINiy=p' cos7’. sin’. ~ = aay Putting PN=90°, the third well-known theorem is obtained as a particular case of (3). IV. The first step in the discussion of prismatic deviation may be added for the sake of continuity. Let P represent the refracting edge of a prism, N,, N, the normals to the faces, i, the incident ray, R the ray in the prism, I, the emergent ray. Let w''= the refractive index of the third medium, D=theltotalwdeyration— il. Do= theangle 1,PI,=the projection of I,I, on N, Ng. Since PN, = PN, =90°, the great circle N,N. represents the principal plane, and the arc N,N, is equal to the angle of the prism. Fig. 2. Pp From (2), «cos PI,=p’ cos PR=p" cos PI). Therefore if the first and third media have the same index The Relation between Uranium and Radium. 483 PI,=PI1,, and hence a the spherical triangle I,P1,, eos 1, 1,==cos? PI-esin? PI, .cos 1,P%,, that is cos D=sin?n + cos? 7 . cos Do, or 1—cos D=cos? n(1—cos Dp), sin} =cos. sin aL PEM eae.) 2 sake aed 2) (N.B. The two above diagrams are meant merely as rough sketches, to explain the lettering.) XLVI. Vhe Relation between Uranium and Radium.—, Part VII. By FREDERICK Soppy, M.A., F.R.S.* “WO sets of measurements on the quantity of radium now in the uranium preparations purified in 1905 to 1909 have been carried out this year. The results completely confirm those published in conjunction with Miss Hitchins in 1915+. In the two main preparations, III. and IV., con- taining respectively 0-408 and 3 kilograms of uranium (element), the growth of radium has proceeded according to theory, with the period of average life of ionium of 100,000 years, if that of radium is taken as 2375 years. This determination of the period of ionium now appears to be at least as, if not more accurate than the value for radium. ‘The product of the two periods, 237,500,000 years, which alone is involved in these measurements, may be accepted as accurate to within 5 per cent. The quantity of radium now present in the largest uranium preparation is already such that no increase in accuracy of measurement is to be anticipated on account of its further growth. The growth, 2.107! gram of radium, from 3 kilo- grams of uranium in 10 years, is equal to the quantity in one milligram of pitchblende containing 60 per cent. of uranium. With the methods employed a quantity much greater could not be measured so accurately. Indeed the measurements for the smaller preparation, III., containing (408 kilogram of uranium, in which the initial quantity of radium, relatively to the uranium, was smaller, and in which the quantity of radium measured has been from 0:1 to 0-2 * Communicated by the Author. + F. Soddy and Miss A. F. R. Hitchins, Phil. Mag. xxx. p. 209 (1915). 2 L2 484 Prof. F. Soddy on the Relation that in preparation IV., show a somewhat closer corre- spondence with theory than in the case of the latter preparation. The natural leak of the electroscope on corresponds to 7x 107 gram of radium, and the extreme variation during these sets of measurements has been only 6 per cent. It is quite a mistake to suppose that quantities of the order 10-! gram of radium are too small to be measured accurately. Given due precautions against contamination, the percentage accuracy of the measurements is at least as great when the leaf takes an hour as when it takes a minute to cross the scale. The results ave given in the following tables which are to be added on to those published *. Before leaving in 1915, Miss Hitchins made a number of new radium standards and determined: the quantity of radium in the radium-barium chloride preparation, referred to in the last paper (p. 217), as 652x107! gram of radium per gram. The standards employed were for the most part these, one new one being made from the same salt, which showed that the old ones had not changed. The methods einployed, the leak being taken with the leat positiveiy charged after three hours, are the same as those previously used and have already been fully described and illustrated. The constant for the electroscope, that is the number of units of radium (10~” gram) required to produce a leak of 1 d.p.m., for the first set of measurements is taken as 17°6 and for the second set as 16°83. Two measurements were done on each of preparations Nos. [V. and III.,and only one on each of Nos. II. and I. These last two are open to some doubt, owing to the presence of noticeable quantities of nitrogen peroxide in the gas after introduction into the electroscope, giving it a pi ronounced orange colour. Uranyl nitrate preparations, purified by ether, are apt to generate nitric oxide, which in absence of excess of oxygen, forms nitrogen peroxide when mixed with air in the electroscope. Unless the latter is perfectly dry the leaf system and silvering are apt to be destroyed by this cause. At the end of the Table, the measurements from the start of the growth of radium in the impurities, separated by ether, Soa some 1°5 kilograms of comarca uranyl nitrate, Heo for the earlier preparations, are included. It will be rec aniee that the first evidence of the production of radium was obtained, .* Loe, cit... between Uranium and Radium. 485 TABLE I. Standards. Date. Ra(x10—1!%¢.), Leak (d.p.m.). Constant. 0:1898 g. BaCl,... 17/4/1919 123°8 6:89 17-97 0:5683 g. BaCl,... 18/4/1919 705 21:68 17-09 J. Pitchblende aha se = HO me U. | 22/4/1919 354°3 19°76 17-93 0:0454 g. BaCl,... 29/4/1919 29'6 1:70 17°40 | Meanie yas) 17°60 0:5683 g. BaCl,... 29/7/1919 370°5 29-97 16:64 — 0:3981 g. BaCl,... 31/7/1919 259°5 15°14 17-14 0:0454 ¢. BaCl,... 4/8/1919 29:6 lerice 16-72 Mean ...... 16°83 Uranium Preparations. Date. Age (Years). Leak. Constant. Ra(x10 ~12). Wo. IV....... 19/4/1919 9:87 13:88 17-6 244-3 INOUE feet 30/7/1919 10-15 14:38 16:83 249-0 No. IIL. ... 1/5/1919 12°33 2:66 17-6 46°8 No. Ill. .. 1/8/1919 12:58 3:08 16:83 51°8 Nomis << 7/8/1919 12:98 2:57 16:83 43:2 ogi e.. 6/8/1919 13°87 3:02 16:83 50-9 Uranium Residues. (Exper. VI. Phil. Mag. xiv. p. 293, 1907.) SVG GOT eam OAM lla 4-0 22/5/1908 0:95 3-4 5:78 19:6 21/7/1908 ill 3:37 5:78 19:4 9/5/1909 191 . 696 57 40:2 29/8/1910 3-22 142 5:2 738 9/3/1912 475 19-4 535 103°8 1/8/1914 7-12 27-7 5:35 148-2 5/9/1914 7-24 24-6 5:35 132-0 6/5/1915 7:89 10:07 15-66 157-7 5/8/1919 12:14 13°85 16:83 233°1 long before ionium was discovered, in 1904 from uranium pre- parations purified from radium by means of barium sulphate. After the discovery of ionium this was to be expected from ionium contained in the commercial preparations, and not sepa- rated by this purification process. The corresponding graph is shown in fig. 1, and showsa steady linear growth of radium ‘with the time. The graph is of interest as showing that throughout the long term of years over which these mea- surements have heen conducted there has been no seriows 486 Prof. F. Soddy on the Relation error in the determination of the quantities of radium, due to the changes in the electroscope and reading microscopes, and in the methods employed. 260 | | | | 200be | La e I | 8 2 | | 1X 150 i = hs 2 1 i 6 < | 700% Ss : | | Pe) { j i | | | | | | . | | | YEARS.—> | | Sr a ro ee I KB Cutie 6 8 10 2 Production of Radium from Impurities separated from Uranium, Fig. 2 shows the graphs for the uranium preparations III. and IV. The quantities of radium are plotted against the time as full lines, and the growth of radium is plotted against the square of the time as dotted lines. The actual lines drawn are those theoretically calculated from the formula, used in the last paper, Fay USNs 1? x 34x 10 where Ra is the growth of radium in grams from U grams of uranium in T years, 3-4 x 10~“is the factor taken to represent the ratio between radium and uranium in equilibrium, and the product of the periods of average life of ionium and radium, 1/A,A3, is put at 237,500,000 years. between Uranium*and Radium. A487 ioe 2. =a ec | & = 12 ES e = = Production of Radium from Purified Uranium. TaBueE LI. Preparation LV. 3 kilograms uranium. Years. (Years). Ra(X107!*¢.), Growth Growth 1/A, (Years). of Ra. | (‘Lime)?" z 0-22 0°05 40 0) 2°66 (al 56 16 2°25 95,500 51D 26°5 104 64 2°41 89,000 5-49 30°1 104 64 2°12 101,000 578 yay) Ibe aah 2°30 93,500 5:91 34:9 116 76 2°18 98,700 9°87 97-5 244 204 2°095 102,600 10:15 100°3 242 202 2-014 106,600 Mean of last five ... 100,500 Preparation III. 0°408 kilogram uranium. 1:66 2°8 4 0 8:26 66°2 23 19 0:287 102,000 8°39 70°2 22 18 0:256 114,000 12°33 152 46°8 42°8 0-281 104,000 12°58 158°'3 518 478 0°302 97,000 Mean (omitting second) ... 101,000 488 The Relation between Uranium and Radium. Table IL., giving the later results for Preparations III. and IV., shows the values for 1/A, obtained from the individual measurements. It will be seen that, with the exception of one point each in the two preparations, the individual values lie within 6°5 per cent. of a mean value of 100,000 years fer the period of average life of ionium, assuming that of radium to be 2375 years. This figure has been retained unaltered from the last paper, though it is slightly in error. The half- value period of 1690 years accepted by Sir Ernest Rutherford as the most probable corresponds with a period of average lite of 2440 years. A new direct estimate of the period, by the Rutherford-Boltwood method of separating all the iontum from the mineral and measuring the rate of gr owth of radium from it in terms of the equilibrium quantity of radium in the mineral, has since been published*. As the mean of four experiments the value 2500 years was obtained, the mean of the last two on which the most reliance is placed giving 2393 years for the period of average life. The method involves the addition of rare earths to the mineral to separate the ionium, and it is unlikely that these rare earths would, if prepared from any known mineral, be completely free from ionium. Unfortunately the authoress does not give evidence of the amount of ionium so introduced and whether it was sufficient to affect the period obtained. Until fresh data are available, it does not seem worth altering the figure 23%5 years before taken. As regards Preparations I. and II., the much larger initial quantity of radium, relative to the uranium, makes their indications of little value in assigning the period of ionium. But the growth of radium from the start, assuming ionium to have been absent, corresponds in Preparation I. to the period of 100,500 years, and in Preparation II. to 130,500 years. No weightis yet to be attached to these preparations. Summar? ‘Y- The growth of radium in uranium purified initialiy from ionium and radium proceeds regularly according to the square of the time with the period, 237,500,000 years as the product of the periods of average life of ionium and radium. ‘This value is probably accurate to within 5 per cent., and the period of radium is scarcely known to this degree of accuracy. Assuming the period of average life of radium to be 2375 years, that of ionium is 100, 000 years. It is not expected that the further growth of radium from these preparations will enable the period of ionium to be determined with much greater certainty. * [Mlle. Ellen Gleditsch, Am. J. Sci. xli. p. 112 (1916).] XLVI. The * Slip-Curves” of an Amsler Planimeter. By D. M. Y. SommMervitte, .A4., D.Se., Victoria University College, Wellington, N.Z.* a a paper with the above title > Mr. A. O. Allen has discussed the loci of the tracing-point and the recording wheel of an Amsler planimeter when the wheel slips without rolling. ‘There are certain very simple geometrical properties connected with these curves which do not appear to have been noticed. It is the object of this paper to indicate some of these, and also to point out what appears to the present writer to be a mistaken idea regarding the role of these curves. Consider, first, the variety of planimeter in which the radial arm AH becomes infinitely long, so that the hinge H is guided along a straight line XX’ instead of in acirele. The slip-curve for the wheel W is then a tractrix with asymptote XX’ (fig. 1). The locus of C, the instantaneous centre for Fig. L. Fie. 2. ae Fl xX” HW, is a catenary which is the evolute of the tractrix, and the iotian of the rod HW ean be produced by rolling the line CW on the catenary. Also it is well known that the tractrix is the orthogonal trajectory of a system of equal circles whose centres lie on XX’. These results can be extended to the general case in which a line-seoment HW of constant length moves with one end * Communicated by the Author. t+ Phil. Mag. vol. xxvii. p. 645 (1914). 490 Dr. D. M. Y. Sommerville on the Hi on a given curve while the other end W moves always along WH (fig. 2). Let WH be slightly displaced to W’H’. I'he instantaneous centre C is found by drawing WC perpen- dicular to WH to cut the normal at H to the guide-curve. W is moving instantaneously in a circle with centre C, hence C is the centre of curvature for the locus of W at W. Thus the locus of C is the evolute of the locus of W; this is the space-centrode. The body-centrode is the straight line WC. Hence the motion can be produced by rolling a straight line on a fixed curve. Further, the locus of W at W cuts orthogonally the cirele with centre H and radius HW. Hence the locus of W is an orthogonal trajectory of the system of circles with radius equal to HW and centres on the guide- curve. In the Amsler planimeter, while the wheel traces a “slip- curve,” the tracing-point P describes another curve. It will be convenient to distinguish these respectively by the names primary and secondary slip-curves. It is stated somewhat loosely that in order to obtain the most accurate measure- ments the tracer should as far as possible cut the secondary slip-curves orthogonally. The essential requirement, how- ever, is that the wheel should as far as possible cut the primary slip-curves orthogonally, and this is expressed more sim)ly by the condition that the wheel should as far as possible move in a circle with centre H. At the same time the tracer would also move nearly in a circle with centre H. The curves which the tracer should as far as possible cut orthogonally are therefore not the secondary slip-curves for the point W, but the primary slip-curves for the point P itself. It may often happen, in fact, when the tracer is moved at right angles to the secondary slip-curve that the wheel moves nearly along and not nearly at right angles to the primary slip-curve. It is hardly necessary to use these curves at all since their function can be replaced by circles, but it is of interest to note that, since now HP can exceed AH, we may get slip- curves of entirely different character trom those of the wheel for which HW is alwavs less than AH. Taking as initial line the line AX in which AH and WH initially coincide, the angle AHW being then zero, let ZXAH=060, ZAHW=4, both measured in the same sense ; also let AH=a, HW=c, then when W describes a primary slip-curve we have, with Mr. Allen, a cdb ~ acos@—c ““Slop-Curves”’ of an Amsler Planimeter. AQT (1) When c% , d>a, and 7> Va?—c?. Both branches are therefore spirals with the same circular asymptote, the “base-circle.” Hach also has a cusp the tangent at which is the initial line. (Only half of each curve is drawn in the article cited, p. 646.) (2) Consider, now, the case ¢>a, and put c=acoshe. The integral is then @=—2 coth «tan7?(tan $¢ coth 4a), 1.e., tan $¢= — tanh $e tan (40 tanh @), where 02=0, 6=0 initially. There is no restriction in this case on the value of ; when $=180°, d=—mecothe. The distinction between inner and outer branches thus disappears. Fig. 3. ort\~ 2 & bole SS The circular asymptote has become imaginary, and the curve consists of a succession of continuous branches all lying between the circles r=c+a and r=c—a, and having cusps on these circles. (Fig. 3.) 492 Major W. T. David on the Origin of If tanhe or ¥ MS ale is a rational number, say =p/q- where p and q are integers with no common factor, the curve is algebraic. It has 2p cusps and is re-entrant a flee g—p rerolition: about the pole. (3) In the intermediate case c=a, the integral j is 6 = cot4d, | where 0=0, 6=180° initially. It @=0, ¢=0 initially, W is at the pole A and remains there. There is therefore only the outer branch remaining. It is a spiral tending to the pole, and having a cusp at a distance 2a. In each case the C -locus, or evolute of the slip-curve, has asymptotes touching the circle r=e and cutting the slip- curve orthogonally at the points of inflexion. Victoria Univ. Coll., Wellington, N.Z. Sept. 25th, 1918. XLVI. The Origin of Radiation in a Gaseous Explosion. By Major W. T. Davin, M.A., M.Sc.* 1. FT is now well known that inflammable gaseous mixtures emit radiation strongly both during explosion and subsequent cooling. In some experiments made in 1909 and 1910 the radiation emitted by exploded coal-gas and air mixtures was found to amount to as much as 25 per cent. of the total heat of combustion of the coal-gas present in the explosion-vessel fF. 2. The British Association Committee on Gaseous Ex- plosions were much interested in these results, and for a time their discussions mainly centred round the question as to the probable origin of the radiation. The chemists on the Committee were of the opinion that the radiation was wholly due to chemical causes. They believed that gases which are capable of emitting radiation can only do so when under- going chemical or quasi-chemical reaction. This opinion was not generally held. Many of the members held the view that the chemical activities which may be proceeding during the cooling of the exploded mixture were not sufficient * Communicated by the Author. + Hopkinson, Roy. Soc. Proc. A. vol. Ixxxiv. p. 105. David, Phil. Trans. A. vol. ccxi. p. 375. Radiation in a Gaseous Haplosion. 493 to account for the large emission during this epoch. They felt that much of this radiation must be due to thermal causes. All were agreed, however, that part of the emission recorded was due to chemical causes. 3. In aprevious paper * the writer examined this question in some detail, and he came to the conclusion that it was. difficult to account for the large emission during cooling unless the gaseous mixture was capable of emitting radiation in virtue of its temperature. He pointed out, however, that there was no experimental work which could be regarded as proving conclusively that a gas may emit radiation in virtue of its temperature alone. But on reviewing the matter recently he has come to the conclusion that the experiments he made with hydrogen and air mixtures do afford fairly conclusive evidence that water vapour has a pure temperature emission. Chemists appear to be firmly of opinion that the chemical reactions involved in the formation of H,O molecules are simple, and are completed at the moment of attainment of maximum pressure, or very shortly afterwards, so that the cooling of a hydrogen and air mixture after explosion would appear to take place undisturbed by chemical or quasi- chemical reactions. The large emission of radiation during the cooling of an exploded hydrogen and air mixture would therefore appear to be thermal in origin. 4. The molecules of CO, are capable of vibrations of the same order of frequency as those peculiar to H,O molecules ;. and it is considered that what is true of the latter in this respect is probably true of the former. There is certainty nothing in the writer’s experiments on radiation from exploded gases to indicate that the CO, molecules behave differently from H,O molecules, except that the former radiate more eee ail. (This, however, would be expected from a study of the infra-red absorption spectra of both gases.) At any given temperature the rate of emission of radiation from an exploded hydrogen and air mixture containing about 30 per cent. of steam (the remainder almost entirely nitrogen) is only a little less than that from an exploded coal-gas and air mixture which contains about 8°5 per cent. of ( Orn, 20 per cent. steam, and the remainder almost entirely nitrogent. 5. The radiation measured in the gaseous exple sion expe- riments was mainly of large wave-length (2 and over). * Phil. Mag. Feb. 1915, p- ¢ 256. + Phil. Trans. A. vol. cexi. p. 391. { Phil. Trans. A. vol. cexi. p. 386 (fig. 9). 494 = Origin of Radiation in a Gaseous Explosion. The energy in the radiation of shorter wave-length (which includes the luminous radiation) is small, and an analysis of the radiation emitted indicated that the emission of such radiation takes place only during explosion and in the very _ early stages of cooling*. It is probable, therefore, that the emission of the short wave-length radiation is dependent upon chemical activity. 6. This, together with the fact that the very large emission of radiation during the explosion period {i. e. the period between ignition of the gaseous mixture and the attainment of maximum pressure) is partly independent of temperature Tf, leads the writer to believe that the energy of chemical com- bination passes partly into the form of both high and low frequency vibrations of the internal parts of the combining molecules. Various considerations based partly on his expe- rimental work and partly upon Jeans’s theory{ lead to the view that the energy in the high-frequency vibrations is damped wholly by the emission of radiation, while that in the low- frequency vibrations (corresponding to radiation of waye- length 2 and over) is damped partly by emitting radiation and partly (and mainly) by a transference during molecular collisions from it to the other forms of molecular energy (translational and rotational). In the case of the high- trequency vibrations the damping proceeds until the energy in them is reduced to zero; but in the case of the low- frequency vibrations the energy will only be reduced to an equilibrium value (which depends mainly on the gas tem- perature)§. After this value has been reached the energy which they lose by radiation will tend to be restored to them ‘during collisions. * David, Phil. Trans. A. vol. cexi. p. 390. + David, Phil. Trans. A. vol. ecxi. p. 381. { See his ‘ Dynamical Theory of Gases,’ Camb. Univ. Press, 1904, chap. ix. § In this connexion see theory suggested in Phil. Mag. Feb. 1913, p. 267. [ 495 | XLIX. Notices respecting New Books. Lectures on the Philosophy of Mathematics. By Jamus BYRNIn SHaw. Chicago—London: The Open Court Publishing Co., Hons, Pp. vii 206. PricesGs. net. [)* SHAW, without entering very deeply into the subject, has undoubtedly succeeded in producing an extremely stimu- lating and suggestive little book, which every mathematician and physicist will tind very usefui. The author divides the subject- matter of mathematics into eight “ divisions” thus : State¢c mathe- matics containing four divisions, viz. 1. Numbers, leading to arithmetic; 2. Figures (geometry); 3. Arrangements, leading to tactic; and 4. Propositions (logistic), and Dynamie mathe- matics consisting of 5. Operators, leading to the corresponding calculus; 6. Hypernumbers (algebra); 7. Processes (transmu- tations); and 8. Systems, leading to general inference. These divisions are treated in as many chapters, Il. to LX., the first chapter being dedicated to generalities about the meaning of the Philosophy of Mathematics. Independently of these divisions the author considers the ‘‘ central principles” of mathematics to be classifiable under four heads: Form, Invariance, Functionality, and Ideality or “ Inversion.” Each of these central principles appears in each of the said eight divisions. The tour principles are treatedin Chapters X. to XIII. Three more chapters, on the Sources of Mathematical Reality, on the Methods and the Validity of Mathematics, together with a Table exhibiting the said divisions and principles, complete this work. General references and special quotations enhance its value. A very welcome feature ot the book is the emphasis put on the creative réle ot the mathematician. ‘The only spot on the sunny surface of this beautiful and carefully chiselled work is a slip of the pen on page 42: ‘“‘similar figures exist’? in Lobatchevskyan geometry. This will be read, of course, ‘do not exist.” Dr. Shaw’s book may be warmly recommended to all interested in pure or applied mathematics. L. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 426.] January 8th, 1919.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. (THE following communications were read :— 1. ‘On “ Wash-outs ” in Coal-Seams and the Effects of Con- temporary Earthquakes.’ By Percy Fry Kendall, M.Sc., F.G:S., Professor of Geology in the University of Leeds. The author differentiates two types of interruptions in coal- seams which have been confused under the general term of ‘wash-outs,’ ‘wants,’ ‘nips,’ or ‘dumb-faults.. One type he believes to be due, as geological writers have mostly held, to erosion by contemporary or sub-contemporary streams which eoursed through the alluvial area where the coal-material was. accumulating as a species of peat.. The channel thus cut was subsequently infilled with sedimentary materials. He describes a number of examples of this type in the Midland Coalfield, some being sinuous in course and traceable over many miles; others being of great width and of irregular form, and due possibly to shifting and meandering streams. Split seams of the type in which the seam rejoins are kindred phenomena, but in these cases the erosion was always con- temporary, and, after a channel was filled up with sediments, peat-producing plants spread completely across the infilling. Great diversity in the phenomena of. splits and wash-outs. arises from the differences in the ratios of shrinkage during consolidation of the various constituents, coal undergoing a. shrinkage variously estimated from ® to 4? of the peat from which it is formed; mud undergoing, as Sorby showed, a con- siderable though lesser degree of reduction; and sand undergoing almost no reduction at all. Thus the hog-back section of split seams is due to the shrinkage of the enclosing coal-substance letting down a relatively-incompressible infilling of a channel deeper in the middle than at the sides. In the process the lower surface of the sedimentary mass would flatten to adjust itself to the floor, and the top would consequently assume a curve corre- sponding generally with the original lower curve, but reversed. The upper element of the seam has some species of seat-earth which arches over the hog-backed inclusion. Cannel, which the author considers to be due to a kind of vegetable pulp that underwent most of its decomposition and chemical change coincidentally with deposition, acts as a substance of little compressibility ; and, whenever pools of cannel-pulp took Geological Society. 497 the place of an equivalent thickness of normal coal-stuff, they survive as swellings 1n the coal-seam. The infilling of the erosion-channels, usually of muds and sands, which often show current-bedding, sometimes includes masses of conglomerate with, in exceptional cases, boulders measuring up to 3 feet in length. The pebbles are almost invariably of clay- ironstone, never much rounded, and presumably the product of the erosion of the measures through which the stream has cut its way. Other disturbances of the coal-seams, commonly misealled ‘wash-outs,’ the author believes to to be due to earthquakes, and he holds that in Coal-Measure times earthquakes had an importance which has never hitherto been suspected. The area in which our Coal Measures accumulated he supposes to have resembled generally such alluvial tracts as. were the scene of the great earthquakes of Assam and New Madrid de- seribed by Mr. R. D. Oldham and Mr. Myron Fuller, save that in the Coal Measures peat-beds were piled in a much more numerous suite, and were on a vaster scale both of thickness and of area than in any part of the modern world where earthquake phenomena have been studied. Some of the effects of earth- quakes in Coal-Measure times might be expected consequently to be of a magnitude greater than the effects of recent earthquakes, but the types of phenomena are similar. The formation of permanent and transient ridges, troughs and fissures, the lurching out of place of belts of the superficial strata, great displacements by the subterranean flow of quicksand, traces of ‘sandblows’ and of the caving-in of river-banks have all been recognized by the author in coal-seams. Disturbances of this character are frequent along the margins of erosion-channels, just as earthquake-formed fissures and ridges are often marked beside recent rivers in alluvial tracts. A striking abnormality in coal-seams consists in the intrusion into the coal of sedimentary material, or the encroachment of masses of amorphous sandstone as ‘rock-rolls.’. The author attributes these to the invasion of sands rendered mobile by excess of water, and perhaps of gas, and moving under the impulse of waves of elastic compression produced by earthquakes. An earthquake-wave would tend to push forward the water contained in a peat-bed enclosed beneath a cover of laminated clay or mud. Where this cover was impenetrable the ettect would be merely transient; where the tenacity of the cover could be overcome, or where it came to an edge through erosion or failure of deposition, water would be ejected from the peat. If this passed into a sand-bed a quite small excess of water, whether accompanied or not by the gases generated in the peat by decom- position, would be sufficient to convert the sand into quicksand ; and, in turn, wherever the sand-bed itself was not confined within impenetrable laminated muds there would under the elastic strains of the earthquake be an extravasation of quicksand into adjacent Phil. Mag. 8, 6. Vol. 38. No. 226. Oct. 1919. 2M A498 Geoloyical Society :— beds, or its expulsion as ‘sand-blows’ at the surface of the ground. When impenetrable mud-beds occurred in a sufficiently yielding condition, such extravasation of sands might carry these beds with them in a more or less stretched condition, and so be perpetuated -as solid rolls enveloped in a wrapping of stratified shales. Lurching of the superficial strata took place on a considerable scale. The evidence is found in the gaps (often miscalled ‘ wash- outs’) of a type usually narrow and not sinuous, in respect of which the loss of coal is compensated for by swellings or folds of the seam, or by the overriding of the seam by great flakes of coal still retaining the characteristics of the seam. These flakes always show torn and ragged edges, which are sometimes splayed- out and interpenetrated by tongues of sandstone or of amorphous ‘clunch,’ and the fine lamine of the coal preserve their parallel arrangement to the extremities of the projections without con- tortion. In some cases the flake has been thrown in complicated folds, and in one instance completely inverted. The inference is that the flake of coal] was not moved (‘over- thrust’) by any tectonic stress, but that under the impulse of an earthquake a mass of unconsolidated, or but partly consolidated, peat-stuff or lignite was projected forward by its own inertia in a medium, usually of sand, which, through excess of water and gases, had only such resisting power as belongs to a fluid. Such disturbances are (with some doubtful exceptions) always limited to single seams and their contiguous measures, and there is cumulative evidence that usually the coal-stuff, and always the Measures, were unconsolidated at the time of the movement. In the overriding flakes the coal retains undistorted vegetable struc- tures in its excessively tender ‘ mother-of-coal’ layers. The ‘cleat’ in the overriding flakes follows the orientation general to the locality. The gap left by the projection forward of the belt of seam is filled with an unstratified sludge-like substance, commonly con- taining angular masses of stratified argillaceous or arenaceous material. A very finely-contorted specimen of sandstone from a true ‘wash-out’ shows quite plainly that the disturbance took place before the material was indurated. In harmony with the contention that the overriding masses are not due to tectonic ‘ overthrusts’ is the fact that reversed faults are almost unknown in the coalfield, are never of considerable throw, and many collieries have never seen one. In the roofs of many coal-seams and projecting slightly into the coal are very curious roughly-conical masses of sandstone, familiar to the miners as ‘drops’ (or by other names); but these have, so far as the author knows, hitherto escaped notice by any geological writer. They are commonly wrinkled on the surface, as though partly telescoped, and often have a flan ge on two sides, showing that they were produced on the site of a “eraek. They are commonly ranged in long rows. These the author interprets as casts of the funnel- shaped orifices through which the sands surcharged with Sandstone Dykes in the Cumberland Coalyield. 499 water have been expelled, an invariable accompaniment of earth- quakes in alluvial tracts. The shape of these drops and their grouping negative the idea that they are infillings of orifices occasioned by escapes of gas arising from the decomposition of the peat. Fissures filled with sand or other materials, the ‘sandstone dykes’ of American writers, are not so common in the Midland Coalfield as in some other coalfields, as, for example, Whitehaven ; but a number exist. They show contortion where passing through the seam—proving that the coal-substance had not undergone its full compression at the time when the fissure was produced. Trough-shaped hollows, called ‘swilleys’ or ‘swamps,’ to which some coal-seams are particularly prone, the author attributes to earthquake effects, such as the subterranean movements of sand, as quicksand. They are not tectonic, for exceedingly rarely, it ever, is more than one seam on the same vertical affected. Some- times the formation of the swilley was coincident with the forma- tion of the seam, as is proved by changes within the trough in the nature of the seam—particularly the occurrence of cannel. All the phenomena here described were produced prior to the production of the larger faults of this coalfield; but minor faults, some aifecting upper seams and not lower, others lower and not upper, are probably to be attributed to earthquake action. A large number of examples of each type of phenomenon, drawn from the examination of over thirty mines in the coalfield, are discussed. 2. ‘On Sandstone Dykes or Rock-Riders in the Cumberland Coalfield.’ By Albert Gilligan, D.Sc., B.Sc., F.G-S. The occurrence of these sandstone dykes was brought to the notice of the author when engaged in investigations into the interruptions in the coal-seams of this area. They have been encountered at various times in pits distributed all over the Coal- field; but those more particularly examined were met with in the workings of the Bannock Band and Main Band Seams at Lady- smith Pit, one and three-quarter miles south of Wellington Pit, Whitehaven. The pit-shaft is 1080 feet deep, and has been sunk through the St. Bees Sandstone, Gypsiferous Marls, Permian, and Whitehaven Sandstone to the productive Lower Coal Measures. Splendid cliff-sections of the Whitehaven Sandstone and succeeding beds, which dip southwards, can be seen in a traverse of the shore from Whitehaven southwards round Saltom Bay. The coal-workings have been opened up south of the shaft, and therefore pass under St. Bees Head. The dykes certainly pass through the Bannock Band and Main Band Seams and the intervening measures, which are about 54 feet thick; but their full vertical extent has not been determined. Their horizontal extent is variable: the longest has been traced for more than a mile. They all run practically parallel one SOO Geological Society. to the other in a direction approximately north-north-west and south - south-east. The inchnation of the same dyke is not constant, but the ‘greatest deviation from the vertical was 10° south-westwards, and in general the amount was very small. In only one case was a dyke found associated with a small fault, the displacement being 23 feet, and even this died out in a short distance. Sees OTe Changing to the variable « in (3) and (4), and then Wh 2 uae ? — ys dy al 4) — ‘B a(a— x) ” jo t+ Veta | oe OT) an i aly ( YJ 7 —v)—>5 + (a—23 x) +2 (1 | p=" () Swe ula ") dan? (« gt) an < / ’ oO differential equations (D.E.) of the hypergeometric {H.G.) form. ZN? HO Ans Sir G. Greenhill on the Whirled round ina plane about a fixed point, the D.E. for the vibration of the chain is again Legendre’s D.E. Lowered with constant velocity U, v=Ut, ons 0, (hs Ly = 0, 77 =9 (8) vy = dC G) = 60(5), with =. : ae eps ) za ncr() ie br (S), Y= tan =7G(2). ae 4. Calling D,(x) or D the associated second solution of the Clifford wD. BE. (5) § 1, then D,(7)=22"Yn(2 Vx), where | Y,, is the Neumann tree fan, second solution of Bessel’s equation. Thus Do(z) would be required, as well as C,(), in the solution of (2), (4), § 2, if the chain carried a weight W, whirled round at the lower end. Eliminating C,, D, between the two D.H.’s of the (91 form, a Oe 2 dx (Te Cr De= of J+ (ot ye ” D,—C =(0; ae and integrating, oni =P), or is 2 . Sopa d i) == CONS bss the equivalent of (36) p. 16, Gray & Mathews, ‘ Bessei Functions.’ Thence ad De aise A D ass AC te dx (3) — eS SSS aEs TaN 9 a 2 iisy= e e ip (On Gerd Os if pases Ope 5. In the linear differential equation of the second order, with arbitrary variable coefficients functions of <, the canonical form is written la?y — =i) St ee 3 : ydz* ae o (1) Bessel-Clifford Function, and its applications. 505 where I, called its differential invariant, is given by Tee PP ee ene ek ae (2) the original differential equation being I? vat ae ce + (jw =0, or RK, a function of 2; (3) and when I is some power of <, the equation is reduced at once to the Clifford form by a mere change of the independent variable, whereas the dependent variable requires to be changed as well if the reduction. is made to Bessel’s form. Writing the equation, with l=sz”, 1 d? ay y d?? and changing the independent variable to v=/z?/p?, the differential equation changes at once to the form in (5) § 1, with n= —1/p. The reduction is equally simple for the more general form dy dy et ly _ dy ale oh) hem Ue eas dz ie Tae by changing to the new independent variable 9 d?y ; 6 aka == 0), OF a pat kePy = 0, p = m2, (4) heer ye (Yo ((&)) x = ke™-I?] (m—@q +2)?, 7 = (m—9g+2) are a 2? a ; (6) Pe Jom (mg 42) (at ate ib sea nea ) i ae c tot hgeree ot I}w Fo ee Se OR aah AG) ey, q—1l Yo C,( 2), n= iene 42 Tie ts Mea ase veil pe (8) This is the diiferential equation arising in the stability of a mast or tree, a pile of books, or Hiffel tower; and for a vertical wire or uniform mast, g=0, m=1, n=—32, as investigated in the Proc. Cambridge Phil. Soe. vol. iv. 1881. The simplification in the results is evident, if the Clifford Function is used, instead of Bessel’s form. D 506 _ Sir G. Greenhill on the 6. The Bessel Functions of real or imaginary argument, denoted by ber and bei in Kelvin’s notation, are dis- tinguished here by a mere change of sign in the argument « of the Clifford Function. Thus the Clifford Function has no negative root: it would not be possible for the revolving chain in § 2 to stand erect. But by supposing each link to contain a coaxial flywheel in rapid ‘rotation, a gyroscopic stiffness can be given to the flexible chain, to enable it to hold itself upright, as in the reported rope trick of the Hindoo conjurer, and behave as what may be called a gyrorope (yupopoTn). Denote for each infinitesimal link, of length ds, at an angle @ with the vertical, the weight and moments of inertia, axial and transverse, by ies A, estimated per unit of length, so that o is in Ib/ft, CG and A in \b-ft?/ft, that is in lb-ft; and so C/o, A/o are in ft?; by R the angular velocity of each little flywheel. rs te eax Y+2dy| | ‘ The vector GH in the figure of the horizontal component of angular momentum (A. M. ) is given, in gravitation units, by GH =(C_ sin A® sin @cos 8) ds, Pacertio Ci!) and in steady bodily motion, equating its vector velocity, GH .o, to the couple due to the reaction at each end of Hie: Bessel-Cliford Function, and its applications. 507 link, with components, vertical and horizontal, X—4dxX, aa zaY at the top, —X—JdX, —~Y—4dY at the bottom (to split the difference), Gil vo — Nein — Yds sco g g tan 0. ‘ - (2) or X tan 0—Y = (C8? A® cos a) tan 6 ( — A cos a) : / | with G— lo ) Resolving vertically and horizontally for the relative bodily equilibrium of the link, (X + 4dX)—( See es DREN We die) (Y +4dY)—-(Y-]3dY) = dY=oa™ of yds = == 0,2 18, La Pe ‘. With the chain nearly straight, take s=x, © = 00s Ceslls then eliminating X and Y, ‘i = 1 oy oe o i Mua R Lae or, putting Oe —Az=ola, and differentiating (0-2) Toa | =O (6) = Ly nO, a as = bC : —*) pal =v NY) ies ET S : : ) Ne Oe) a) The first root of Ci: making ow — 0, 1s given by . AL ; ms = 3°8317, the first root of J; =0,° ) > = 3°67 Thus a length « of the gyroscopic chain ean be made to stand upright, given by #=a—3-67/, so that R 1 rs fad = (c 7A) oP must be made to exceed 3°67. @ With a weight W balanced on the top, the second function D w oul be required in the solution, and the constants adjusted for the relative equilibrium of W. jO8 Sir G. Greenhill on the 7. Clifford’s definition ad’ tog C,(2) = (—1)” —a)/H(n+h)Wk . (1) is generalised for fractional values of by interpreting Un to mean Gauss’s Gamma Function I'(n-+ 1). When »n is half an odd integer, the Clifford functions are the differentiations or integrations of cos(2V7x+e), changing to a corresponding “hyperbolic function for negative values of x. Then we can put _ sin (2V7a+e) Ca) = co 3 C.i(z) = —cos(QV ate. 5... eS Cs (a) _ COS evens) sinters, C_a(a) = —Vacos(2/r+e)+gsin(2Vaet+e), = 23Cs(x). (3) ce) im Bs Cee __ 8 cos os ene — sin ee C_s(#) = \ C_s(a) dx, = a Cs(@), srr and so on. Here the arbitrary ¢€ is at disposal, to give the two separate solutions, corresponding to C and D when 7 is integral; and C,;,:(—«#) is obtained by a change of the circular into the hyperbolic function. Vibration of an elastic sphere. 8. As an application of these functions of order half an odd integer, take the D.H. (Love, ‘ Elasticity ’) Oo ge Pape tana tp ru = 0, ob ou 2 ¢ au Q c e ot n= / 2 Bra or U=rusin nt, the radial displacement, m ith p res Bessel-Clifford Function, and its applications. 209 and put p?’=42, cee (B+ De St au Sn OMS) 04 (2) n= UC (0) =0CA(4 p), subject to the surface condition du dku op ?ku | de i er ae n*p and e=0 for a solid sphere, to make w finite at the centre r=0. Then pe OS LOY ] dr ax (4) or Bilge as Crtpiat) =S5 Calpe), | leading to the period equation _ 2p ha’ tan pa n?o PE os oe har ) 3P n2p 9. The differential equations of Riccati and Bessel were compared in the ‘ Quarterly Journal of Mathematics,’ xvi. 1879, p. 294; and starting from the canonical form (4) § 9, 2 ee —| = — kz”, be ilies a Veni Pedi ya (Gb) y dz this changes into Riccati’s form du : jixa ae + bu? = Cat, on putting a = U, and k= —be 3 (2) or into dv h 2 ‘ ° Orta 3 ee ot ct 0 = ce, putting vw =us,m+2=p; . . (3) or into . dv oa tet Oi Cee RC ees) == G00 6 == ae s) CE) Boole’s form. 510 Sir G. Greenhill on the In the more general case of (4) § 5, as 2 EY ntiny 0 2s 0) OY. Sie, ee oe © eee cde ~ +q(q— Lye], (6) 7 a dw oa tee a. +) g(q—1)+ht+kz?lw = 0, (7) of the form aw dw : 22 ies = : a OS a) 2 7 + a2 oe +(cetkz?)w = 0; | (3) and then, taking g(g—1)= —h, 2g-—1=V(Ci—4nA), , au dw Ze QGee = ew ) Tene @) Next, with a new independent variable e=h tia Ol eran (d) Denoting by T the complete period of the wave. g'T? au i=? VA = (9) ays Sir G. Greenhill on the where U?=gh, and U is the velocity and TU the wave length of the long flat tidal wave in epen water of depth h Taking an average depth of water of 12 tathom makes U over 48 f/s, nearly 30 knots, and the semi-diurnal wave length 12 x 30=360 miles, geographical (G). Then »=0, and there is no rise and fall of the tide, where ./,/(lh) =2°4, the first root of Jy=0. making =137 G miles from the end of the estuary. And &=0, and there is no tidal current where «/,/(lh) =3°8, the first root of Jj; =0, and e=218 G miles. These figures may be taken to apply to the tide in the Bay of Fundy. In an estuary, shallowing to nothing uniformly at one in 2, and given by b=c#” in ae EN ea ie d dn\ nen ef ned Sy m+ et me re ao / = |) ae B+ (m+1)5 a ae =i) i) = rae c.( : (10) - Tot = 9 E= Cnea("7")s (11) para 2 and the — considerations may be resumed. Thus with m=4, and parabolic in plan, the solution is given in finite terms by C, and Cy. For a canal of uniform breadth, m=0, and shallowing uniformly, g=1; and then n=G(=) €=C(7*). With m=0, q=2, the D.E.’s for & and y have an algebraical solution ; and with A=mi(1-4), n is given by P,(<). _ Generally, with m=0, and er Oma. din 1 dyn ; —1 ac A = ee Cg ES —q ds +n =0, = Cale n 7 9sge (12) and with uniform depth h, m=q, and a = (¢—2)?2. a) Ginn ee 1 keg ome ee 5g +e = Uy ey Ue) ice (13) In all these cases the vertical cross section of the channel may be taken rectangular or elliptical, but it ought to be supposed to vary slowly. Bei eas aac Bessel-Clifford Function, and its applications. 513. Figure of the Earth. 11. The investigation may be resumed here in the ‘Quarterly Journal of Mathematics,’ vol. xvii. 1880, of the “ Differential equation of the ellipticity of the strata in the theory of the Figure of the Earth,’ where a generalisation was made of Laplace’s assumption, leading to differential equations of the form we have been con- sidering. The differential equation for ¢ the ellipticity of a stratum of a liquid gravitating sphere, stratified originally in con- centric spherical surfaces, and disturbed slightly by a solid spherical harmonic of order 7, is given there: m dMe dar dp , ee Ube gp SA where M denotes the mass of liquid inside a spherical stratum of radius 7, so that aire =4rrp. Sie en each ora cts wae de oi (2) Laplace solved the equation for the Fis ure of the Earth, where the disturbance from the spherical form is due to the rotation about the polar axis and 7=2, on the assumption of A Ay? a ne a constant = —72 Suppose, . . . (3) “M adr : style dp the negative sign being introduced, as dy Teduires to be negative for the stability of the strata. Then son OIL 2, at P 2 lp 2 \ M=-—4za” Fe a = — 47a") aD Sia eae p, (4) eee eye Sfp at. 9 (5) ” dr? Me de Gee? | iO — Ae Sit (+e), Bib wa tee (ate) s in which e=0, to make the density finite at the centre. Then, in Laplace’s notation, writing ¢ for —, a sin sin qr dp cos gr sin 1”) Les = — ¢ > » » ,. . tamed Gr. ar por ( gr Oe Win dr Oda ( SIM) Ya OT COS Ga )ar kl Von CO) 514 Sir G. Greenhill on the The generalisation was made in the ‘ Quarterly Journal ’ by replacing Laplace’s assumption Amr* dp 2 ea) Ne : oa ie oe by -(") 2) laa (9) and then _4mardp dM Ara” d’?o9 4ar(n—A) ap \ = = SS = S= ! r-tilp? dp (pe ae ‘: pe wan = der? Ps (10) ya ce (1—4yr e+ (2) =6.. Re (14) a differential equation of a form (9) ehe3 considered here before ; and then putting GO Wo. nam Maher a pero ie toon Te lees i ie es ee adx* ane » + (13) the equation becomes 9 4 \ GON 6 \ Se Se rp 0, | p= Us (0) es 107 are and no change need be made in the dependent variable p to bring the equation into the Bessel form. Next, with dM k a Vl TE, = Aq Ps Pp => ree is 9 e 5 4 e (15) Worn lh PM iL dM dr Aaré dr? Qa dp ~~ Aart mal _ (1) 20M . aM rit ret mere EO V3 ct — ae en ale )M=0,. 5. seat 1a as in (9) § 9, changing with (*) = 772, Inito a 7M (1-3) ihre <0, M=0 se = eu Bessel-Clifford Function, and its applications. Next, with Me=y7 in (1), Cr — ~~ a” dn oat: ) -ii+1) = 0, ee 18) of the more general form of (5) in §9: a change is made here in the dependent variable, putting ri dn r\4 d?w 1=(2) W, 8 = (2) [ees gn + q(q— Dw), (20) Oe 4 gr a | g—1)-i6+1)+(7) Te se! (21) dr!" dp 4 all 3 Wah and taking g(g—1)=7e¢+1), g=7+1 or —i, 3 Oey. dy iB = (h, 90 dp? qv iL Jw = EN iautlen ume mie (12) w= Coo—1 (a). ° e (23) so that we may write, symbolically, i+] a+] Gael 7) pee Ce) or vn (=). oro (<) M, (24) i 21+ 4 = nN n The pressure equation is dp = —Mp-— 3 = -+-1 — AanCs(@) Cz (ea nda n fy 7 ») = —AC(r)C, (2)a* de, . . . > (25) for which Lommel may be consulted in Math. Ann. xiv. p. 910; but no simple general integral is obtainable, beyond Laplace’ s case of n=2. 516 Sir G. Greenhill on the 12. With n=6, we have | p = po cos gr, M = sipoc’ sin gr) ee and for i=4, Se oe AT a ap ee 3 zs Sie n = Me = A(sin gr? —qr* cos qr*), € = co(1 er at) (2) The subject has been investigated generally in the Bakerian Lecture, May 17, 1917, “Configuration of a rotating compressible mass,” ‘by J. H. Jeans; but on the adiabatic law of p=xpy, the only case so far, besides that of Laplace’s y= 2, is for y=1°2, given by A. Schuster in the Report to the British Association, 1883, and then fae ee) pam (=) = (2), Po. NG ashe, 0g CT! Po’ M= ‘Ee : :) Amvtdp der? (3) = SOE Nena M dro” (eee and the corresponding differential equation for » = Me, with n=r'tly, lid?n sade aoc nde eee)” dy ldery —_ nites ee Oe ese we ee Ce or with r=ce", y = = ze~ +d, 1 dz 3.0 EG = +IT=0, [=n(n+1)sech? u—(2+4)?’, n=. (9) a Lamé equation. This D.E. (4) becomes, with the new variable icf yiae (A 2) == BL 6° oe ery re vo) = (247) ° . . ( ) fy Os eee : e(loa) rat (itg—2a)e + 7y = 0,° 2.7 ee a H.G. D.E. (hyper-geometric-differential equation), having an algebraical solution. Thus, when the disturbing harmonic is due to rotation, Mente ime: cle) Of. = ea (8} r Va cr . 3 (c2+72)2 RS: € ce). Bessel-Ciiford Function, and its applications. 517 Then the D.K.’s for M and p become PM dM 15 w(1—2)— +(—$—-22) 7 + M = 0, nT (9) 3) eye) means) 3 dx? Die Cane ag Ora, (0) GaN nord N 15 eige ae 2 Dee Ree ce a f with © = M and (11) changed into (10) by a change of «x into 1—a; ; and these H.G.D.E.’s have an algebraicai solution, #2 for M, #: for N, and (1—.)3 for p. Tho pressure equation becomes dr is dp = —Mp “3 = —2rpc(1—-2)*da, (aoe 2mpo¢(1—«)° = 27,0" Ge ei (k2) in agreement with Schuster’s results. But in rotation, 7 and e become infinite at r=0, so that this conglomeration is unstable at the core. This suggests a generalisation, of ae Con UO | eeolge ne VG ar (c +72) ne : . : (13) replacing 42 in these equations by n(n+1), and then the H.G. D.E.’s have algebraical solutions. Raub2a 1 b(t, po or Nw, (10) and (11) assume the same form dy a But the integration of the pressure equation does not lead to any simple relation between pressure p and density p, corresponding to a physical law, except in Schuster’s case of n=. The change from M to N is (9) and (11) suggests that, in the gener: al H.G.D.E. for Ys: (1— zn! Att Wi OMe se) d*y w(l—a#) 75+ ly (a+ B+ J)a a0 "apy =0, . (15) a differentiation leads to the H.G.D.E. for wal, SaRee TaN a, B, y are changed intoa+1, B+1, y4+1. - Pints MagiS. 6. Vol. 38: Nov22 7. Nou. 1919, 2 © 518 Sir G. Greenhill on the Thus, as in (5) § 1, the D.E. can be replaced bya sequence relation. An attempt has been made by G. W. Hill in the Acta mathematica on the assumption of a linear law, p=Ap+B, connecting density and pressure. And other memoirs ma be consulted : Comptes Rendus, April 13, 1885, Radau “Sur la loi des densités 4 Vintérieur de la terre.” Bulletin astronomique, 1884, Tisserand and Stieltjes. 13. The pressure equation in (25) $11 does not appear tractable on the theorems given by Lommel, Math. Ann. xiv., for the integral of the product of two Bessel Functions. Changing to the Clifford Function, and adopting Lommel’s method, a CnCy ae pe 1C,,0,— 22 (Cha, + C,Caar ° (1) d 1 ; lx wen Orne Cra ae (p Le 1)2?CmyiCn 41 dx = 2? Cn yoCngi + Cngi€nys) 3 (2) and then, writing equation (5) § 1 as a sequence equation, tCnz2 = (n+1)C.4i—G, » op eee equation (2) becomes SPC Ong1= (P+ L)0°Ce Cay —x?(m+n+2)CniiCn41 + ee Cran On Se CAC) 3 wl (4) and adding (1) and (4), © (a? Caner + °C Cr) = pa? *CnCn + (p—m —n—1) C4 Thus with p=0, (m+n+ »| Cmp1Cnp1d@ = — #0m4iCn41—CaC,, (6) and with p=m+ntl1, (m+n+ my eC Gada = 2" (aC, One Cae ie easily verified by differentiation. But the pressure equation (25) $11 cannot be integrated by means of these reiations, except for n=2. Bessel-Clifford Function, and its applications. 519 14. The definite infegrals of the Bessel Function can be written, in our notation of the Clifford Function, Ag C_iiv cos? $) dd = pees (2,/w cos ddd = th a (cos )™*"dh She elit -2k-1) =S 5 Tae sco 2k ee a TOD). ie Nea | ch(2/ecos $) dg = a. evecoddd = m((—aj, . (2) 0 J0 e { ce COS OG Di = TROT) Ni Me etch) ays ad ay UCD) 0 and soon. Also ( Ge sin’) sin ddd = > TEE (sin 6)*+1dd kg > Gee Ae 2k Cie 3a Tore 1) es (—4.x)" sin Whe Teens es eo (4) and others derived by differentiation, such as i ice, ©) sin Ode Oa) he (5) i C\(@ sin? h) sin¢dd 0 — oan Ss aye jee Te a GL) anes dd Gio elng?) 2 4G. 2 es TAT +1)" SOG SCORE kg np A( oo i sas at Mea (2k + ae yet . — COS 24,/v =5 > IT(2k +2) uv be eae (0) 202 520 Sir G. Greenhill on the Diffraction. 15. Take the case of pure plane waves, either of light or 27 sound, arriving in a state of vibration ¢ cos Vt at a parallel plane AOB. In the absence of diffraction across AB, the waves advance: to a parallel plane PNP’ at a distance 6 in the state of vibration c cos =" (\ ft —b), 0. 3 25 with no change in the amplitude c¢ of vibration, taken as displacement or velocity. But suppose a screen is placed in AB, with a thin circular slit AB, of mean radius a and breadth da, through which the waves are diffracted. The vibration arriving at any point P of the parallel coaxial circle PP’, of Perias A, from an element ad@da of the slit at Q, will he mmbehe ciate Din cadéda cos = - (Vt—PQ), ee PQ? = A?—2Aa cos 6+a?+ 0, and with A, a small as compared with }, we may put Aa COS 6 20 Sie 2? 2 9 cos = (Vt— PQ) = cos = (Vt—b) cos ( enee cos 0) Ab + ane 5 (Vi=d) sin (7 a cos 6). .* Ge Integrating with respect to @ round the circular slit AB. for the resultant vibration at P, the integral {5 n (7 cos 0) ae — 0, : | and > (4): = /OmAa i | ; cos = xR 098 @) dO = } cos (2,/x cos 8) = 27C (x) a : 27Aa rae Ndi pa hs 0) 12/6: 0 = 2 v a — putting Wi NS 2h WEL Bessel-Clifford Function, and its applications. with a = ma?, 8=7A?; and the vibration at P is dy) ape 2mcadaC(a) cos i b= Dn ee nehs Aah ery on) of amplitude edaC(x), where dz is the area of the slit. If the slit Owe in breadth to an annulus between the radii a and a’, and circles « and a’, the amplitude of the vibration dittr acted through it to P heeor nes eae a GN ee Ne eNO? a’ 8 of 0( 28) de = ls 3 (Cag ae a) eee 92 — C001, 99 — Ca a6) 12527 es e C (6) A similar result will hold if the circular hole is filled with a lens, of focal length 6; and then J (a fone A iGYySs aN rb ) ( 1 n22 — TF 2rak ° e ° e ° ° (7) NG "he first root of J, =0 being 3 oes == 22 the) areal S of the first diffracted circle is given by “= (1°22 x 42)’, - 8 subtending a conicalangle 8 — (1:22x 4X)? ©) a y) in the sky, $2 — equivalent to an angular radius S, such that sin S = 1:29. RA ea yt ore, GQ) 2a In seconds of angie, takiny A=5 x 107° on the average, 2 2 2S gi 206265 x 1:22 x 5x 107! GES Maha ar 2a say S=1''} for an object-glass 10 cm in diameter. : For the pupil of the eye, take 2a=0°3 em, S=43 . (Schuster, ‘ Optics’). rf These considerations of definition and separation of image are required in the working of the Barr and Stroud range- _ finder. If a range of R yards is to be measured within AR yards, 522 Sir G. Greenhill on the from a base of B yards, with a magnifying power M, the angle of parallax or convergence must be measured within 8", given by sinS = MB ae Taking the gunner’s rule of one inch, or a halfpenny, subtending an angle of 1’ at 100 yards, 1” at 6000 yards, this makes the radian 3600’, or 216000", instead of the more exact value of 3438’, or 206269", and is equi- valent to taking w=3, instead of 3:1416; so that 1” is 0:00009485 radians, 12" is 0:0000582 radians. ; Then wath Me309b— 10, R= 20,000; Aino: 30x 10 x 80 AS a(S sins = =6x10°°,. S = 127:96; cayieme or more accurately, S = 6 x 206265 x IQ ae a Or nearly TO eae (‘ Nature” p. 404, July 24, 1919.) Whirling and lateral vibration of a bar. 16. Take Clifford’s function w=C,(#), and #w=a"C, (a) =C_,(#); then ae C R ({. ew dat" aC -n—2), —. — Crt 2 (x) —s e 20 ae (1) so that the differential equation aw ( ( a? d?w 1 x mi A 2 > tote ES +2 ee = n nr? ————— a Ww dz Or i Xv . = XW 2 dx? 4) dix? dx* i @) e has a solution ©,(r), with the associated functions C,(—2), D,( +2); it is required in problems of the whirling or lateral vibration of a bar of variable cross section, such as at the muzzle of a rifle-barrel, gripped at the breech ; or in the reed of an organ pipe or musical instrument. Change the independent variable, with #=e%, and (2) becomes d d ey oe ; (“a+ (54 an as 1 )w = ee, ae Bessel-Clifford Function, and its applications. 523 More generally, with a change from #=e® to z=e9, v=2k, oe . 1 @=kd, and writing D for eg dd (D+nk)(D+n—1.4)D(D—-f)w = ew; . . (4) and with a change of the dependent variable from w to y, ey — chy, (A) (D+th+nk)(D+thtn—1.4)(D+h)(D+h—h)y = ey, with the solution : emi O,(2") =n Cha). aly Co) Conversely, starting from the D.E. of the whirling or lateral vibration of a bar or tongue, of variable rectangular or elliptic section, varying as 2? broad and <% deep, (B) ad? (", ae TAX GAY) = 8 p aa —KpS, +: aes y=o =- - (BK 2) es with g=lo’, ope and with E=pc, so that ¢ is the hydrostatic head of the substance, density p, for the pressure E, dividing out Kp, and integrating, with \ydr=u, 12 d’u- u es dm 2/0) re a D.H. homogeneous in the lengths a, c, 4, 1, X, &, contrasted with the undimensional, equation (8) ‘ ie and combining the two qualities of the independent lateral vibration of the elastic rod when »=0, /=, and of the flexible chain of \-0,¢=0= ang apparently intractable in the general case. Bessel-Cliford Function, and its applications. 525 a*— x . ‘ . Replace meee (4) by kh for a constant tension length h; then for a length a between fixed ends, not clamped, ee y= sin mw cos wt, ma-=7, for a plane vibra- tion. And then if making N beats per eee meN2) ol BEIT Gs 47.2 1D) ——_ =-—- =m ck: = We he. ¥ Change the sign of / for a thrust, and let 4 increase gradually up to mck? ; N decreases to zero, and the vibra- tion becomes sluggish ; finally the bar ceases to beat, and is sprung permanently on Buler’s law ; X has become negative. or an upright rod, deflected slightly by vibration, or bodily rotation, ‘the D.E. (4) will change into du CU. ath at ere ce ee) he) ee eos dat ae WE? AX, 9) With c=0, the equation reduces to (2) § 2, but with the sign of w changed, for a chain hanging vertically downward. If the rod is drooping slightly under gravity, as in § 5, Bes cand with OO Ora ; d*p oe dx da?’ ke dx? ae with the solution, by (4) $5, p=00( a) of which the smallest root is about 0°88, say $, making the critical height Os z= 2( ck?) = (4ed?):, for a circular rod of diameter d. 18. Looking back, then, on the rival claims of the Bessel and Clifford function for employment in mechanical questions, it would appear that the Clifford function has the advantage of simplicity in the expression of results dealing with a linear extension or propagation, as not usually requiring an extraneous factor, a power of w. And in any emanation from a centre or axis, it 1s more natural to take 2? or spherical surface 4a7* as a variable, and not 7, as implying that a negative 7 could arise, to be inter- preted in a formula. But with 7? as variable, say m?7?=4a@, a negative x would imply 1 imaginary 7, and a new set of conditions, so that « or 7 is more appropriate as a variable. 526 Sir G. Greenhill on the In the passage in Lord Rayleigh’s manner of the tesseral harmonic P? (u=cos @) into the Clifford function of order p, as the order n is indefinitely increased, in its representation near the pole @=0 on the surface of a sphere, when the radius a is enlarged to infinity, the change is made through : Le? Lm x P ‘ 4(l—p) = hav @ = sin? 30 = eis on i — ie agreeing with the definitions of §1, in the limit n—>s ; and the Clifford function of order p is derived from the zero order by p differentiations with respect io «x, as the tesseral is derived from the zonal harmonic by p differentiations with respect to p. Take the various expressions in the recent Phil. Mag. July 1919, given by Dr. Bromwich on “ Electromagnetic Waves,” and apply this our principle. Then his equation for F=F,, por ie CO Tae —n(n+1)F = AE dpa —m'r7F, 2 in a vibration with v2 =— mcr, 2) becomes, with m?r?= 4a, and F,=a2@tC, C0 ea au? roe (n+ 3)e 7 + aG = 0, . - . (2). our equation for G = Cale) = => C2), . ain and Fy = aCx(@) = sin (24/a+e) ; changing for ee gent waves to the hyperbolic or exponential form, with m? replaced by ~m’ Thence the ex pression o the other functions with a= dm?r? as variable, mr=z=2 In Lamb’s Sry ued namics,’ yr, and VY, are both included in Ci43(4m"r*), when the phase angle € is introduced into Ci, as in FK,; and then his fr(Z) 18 Crzal —im’r?), for the. divergent wave. Bessel-Clifford Function, and its applications. \27 The solution of the equation (V/?+m’?)¢ = 0 is given in solid spherical harmonics S=8,, by terms of the form RS8, where 2 PTS + A(n4 lr + mePR = 0, Satis CAs) yo mea) —" Ona (10m ie eer al ce CO) This equation arises in the solution of CED ns ess dé — ( WW db, ° ° ° . ° . (6) y d | “fo =— meg, for the propagation of spherical waves : or else in the conduction of heat, with with a periodic time factor, so that du f an = LONUGUE . e e . 6 e (7) and an exponential time factor of decay at compound discount, u=e ”"*g@ ; and the surface conditions are to be adjusted. Then we take sin mr __ sin mr _ COS mu Ro eRe D) Lie ae ? mr mr my Ds ae 2 ine" sin (mr +e) —— i3) we have 1 d(dt+y) _ dd sin ¢ . > amar riage . 20 eee Substituting (3) and (4) in (2), we have the following well- known formula, in which h=7—R is height above the earth’s surface : 1 dn 1 ndh- psingd = 0... ne Equation (5) must yield the law of variation of refractive index with height, for any assumed form of the radius of curvature p. By (1) we may write (5) in the form india oF mn” ai pk Since the relation giving n in terms of h cannot depend upon the particular ray by means of which (6) has been derived, (6) cannot depend upon /, and so pk must be inde- pendent of the zenith distance at the ground, do, of the ray under consideration. Since pk is a function only of n or h, we must therefore have p=zitlma)t . ae where f(n, h/R) does not involve & and has no dimensions. Hence (6) becomes ldn : = Se eae st —o . . nee The simplest assumption that can be made about the radius -of curvature is that 1t is constant for a given ray. This amounts to taking only the constant term, g, in the expansion of f(n.h/R), so that (7) then becomes R? p=. 4 2°) ae =0...,. We then have Lon 8 2 yoo ls Terrestrial Refraction. 549 The solution of (10) is is he? -= a +i (htop) DO leery % being the value of n at the ground where h=0. For h<8 km., we may neglect h?/2R with an error not exceeding 1 in 1500. Then, approximately, gm | ght NN r ity This may be written with an error of less than 1 in 1600, mare CT Coie) 2 ale (12) Now by Dale and Gladstone’s Law, SLOP Ian een isk vavecies 3) (Clic) where mw is constant over wide ranges of temperature and pressure for a given gas, and @ is the density. Applying this to air, we have, if o is the density at the ground, Go = Ol \ a ° e e ° ° (14) Thus, the assumption of a constant radius of curvature for a given ray involves a linear law of density, at least for Tete up to 15 km. In the textbooks —for example, Winkelmann’s Handbuch— the assumption is made that the radius of curvature of all rays has the same constant value. This is obviously absurd, for (6) would then give a law of variation of density with height that depen: led upon the particular ray utilized ; that no definite law of density would exist. In deriving (14) the radius of curvature has been assumed to be R?/kg. Substituting the value k=nKsin do, the sufix “ »” referring to the lower end of the ray, at the ground, we have R ging 81n do = Sue. Aeon ur Clas) Hence p varies with the ray, being infinite for a vertical ray. Phil. Mag. 8. 6. Vol. 38. No. 227. Nov. 1919. 2Q 550 Mr. A. R. McLeod on 2. Empirical Formule for Density and Refractive Index. From the curve which accompanies the ‘ Computer’s Handbook’ (Meteorological Publication, M.O. 223 4/18) we have the values for the mean atmospheric density at various heights, which are given in the second column of Table I. These values are presumably for heights above sea-level. The densities are in grams per c.c. and the heights in kilometres. PAB Ik ole h. o (observea). o (caleulated). Odknar et esresee 001250 g./c.c. 001250 ¢./c.c. te ooaL nOR Grids 001126 __,, OOLT3G aie PAREN IEP Saale anm ar OOLOZ0 7, 00102975 CS EN Tb Nat ate ai 000207 7000928 __so,, Ay gee ee ese cae 000820 ___,, "000833 __,, tee A art oe Yet" 8 0C0739"( 000743 — ,, Outed ea eames ‘000661 —,, ‘000661 df ote eee ee 000591" \,- "000584 __,, BF hee eee (000525 _,, ‘000514 _,, OUD: Sieebs esate: 000466 __, 000449 TS pe Seed ee 000410 —,, 000391 ng Ie ae rte ee, 000356, 000339 __,, Deng 8 tele kee 000301 __,, "000293'-_,; DSS boas Saag a ete "000261 __,, "069253, A pelt BER Ropes 000220 __,, "000219 ., 1s eee eee FOUOT OT “000191 The values in the third column of the table have been calculated from the empirical expression a = (001250—-0001165/+ 00000306317. . (16) The constants of (16) have been determined from the observed mean densities at the heights 0, 6, and 15 km. It would have been rather better, as far as ordinary aeroplane or airship observations are concerned, to have taken the observed values at say 0,4,and 8 km. But the former values were selected as having a greater range of usefulness for meteorological work. To determine the constant w in (13) we have, for a wave- length »=:5000 A.U., at 0° C. and 76 cm. of mercury, av N00I2928 ee ee Sea Owes fo= Terrestrial Refraction. Hpk Substituting in (13) this value, and the value of o given y (16), we have for X=5000 A.U. n = 1'0002839—-00002646h+:000000696h?. . (17) To compare the merits of (14) and (16), curves have been drawn in fig. “00115 | (00095 | DENSITY (N GRAMS*PER C.C.* SQ eS) -c0075 | -00035 | HEIGHT IN KM.= representing (.) the observed data in the (1) Observed values of & (2) Empirical curve for + (3) Curve for H=:1736 | Radjus of % : Curvature i (4) Curve for MA=1AFZ20 of any cay C (SC urve for H=:/020 } H onstant. second column of Table I. ; (i1.) the empirical expression (16) ; (iil.) three of the straight lines (14) corresponding to the three selected values of g°1796,°1420, and:1020. These three values are determined, respectively, by the observed values of o at heights of 1, G,gamalaley kim, From the curves, it is seen that the empirical formula (16) gives a much better representation of the density, than any of the curves which are based on the assumption of a constant radius of curvature for a given ray. This we would naturally expect since (16) has one more available constant than (14). Along 3. Refractions. ray which traverses a continuous atmosphere, which has a constant density over the surface of the sphere r=constant, the relation n7sin@=k holds. At a definite point on the ray, the slight deviation of the ray due to the 552 Mr. A. R. McLeod on slight change of index, dn, is given by “+ cot dg = 0. or dn dd = —tan ¢—. ele The whole deviation of the ray is thus given by the usual formula 9; “tan ddn ee dn 5 ip — eS 2s Ag; i dp ie n i nr/ nev? — k2 or the suffix 7 referring to the upper extremity of the ray. W riting (17) in the form 0 N=Mm—phtgh’, . ... 2 ae hy; kip—2 h)e A = a e es . J if \ h AE, i 2bh ate ch? \ ) where jh ny Rsin b, = n(R + h;) cos Di, a= ny R?—K? = n,’ R’ cos? do, DS ylily—yolesy) a c = (m—pR)?+2mR(qR—p). In the second expression for f, Di=5—¢; refers to the dip, in the case of observation from the greater height, hi, and n; is the corresponding index. The factor 1/n in the integrand of (21) may be replaced by unity, with an error, Ban h=8 km., of not more than 1 in 4500. Hence we may give (21) the approximate form th; k(p—2qh)dh Mp Vat 2bh+ ch? This has the solution: Adi = (23) (— + Vch;+ Va+2bhit ch2 ) we | Agi = 70 Noes — + Vchy+ Vat 2bho + cha? | c a [Vat 2bhitch®— Vat 2h +ch,’]. (24) Terrestrial Refraction. DDd Tf ho=0, (24) becomes 6 2by lo oO fay RI Ts aa e b b at ered ae a ee ea = + Veh; + yat2bh,+ ohe | moll eS a DQ), 4 ie =s, — IE) a+ 26k + che — Ne ee eka) Substituting the values p = ‘00002646, g = (000000696, ny = 00028395 k= 6370: km, we find the following values for the constants involved : 1, as b — —* )= 0000207186 ; —— = 702-861: V/¢ ee ¢ VC Ve 75401 - 2b = 10599-28 ; ¢ = 56-8530; a = (000000244842. The values & and a involve the zenith distance, do, of the ray at the ground. With these numerical values, the values of Adi have been determined for the 8 values of h; given by h; =ikm. G@=1, 2, ....8) ; and for a series of values of d, ranging from 0° to 90° inclusive. These values are given in Table II. They were obtained with the aid of seven-figure logarithms, and are correct to the nearest second. They have been checked by taking successive differences, and in other ways. To show that the values of Ad; given by (23) represent the corresponding values of (19) to the nearest second, we have calculated the coefficients of A? and A* under the radical sign, when (20) is substituted in (19). These are, | respectively, d and e, where d = 2{ qn + (gR—p)(m— pR)} = -0162005, e = (qR—p)?+2q(H—pR) = °0000206. Taking h=8 in the biquadratic under the radical, we find that the terms of the third and fourth degrees contribute only 1 part in 11,000 to its value, and this in the case most favourable to them, when dyo>=90° and a=0. Jot Mr. A. R. Mcleod on Dropping these terms therefore makes an error of less than 1 part in 5500 in all cases considered ; which affects the value of Ad; by less than a fraction of a second. he integral (23) does not converge when h; =a, but, as has been shown, it determines A¢; to the nearest second for all zenith distances, when h;< 8 km. TABLE II. (Refractions, Adi.) Qo. Ae kane h,=2 km. h,=3 km. h, =f koe Sty tet ne eae Tas eee ta!’ 207 eh abe QOD SOA Ores: G3 1Oe 28 13 24 15 44 BON i we ce 4 18 7 29 LO teas SS YA Rok eee eed) + 30 6 20 7 56 SCN fi iter aiate 139 By a) 4 30 h 43 SO a ew saee a iecalhs) 2 24 3 Sys 4 26 rod Se yak ery pe Bice (0) 1 56 2 AS & 30 SIAL AR a peda 43 S25 Diem 2 ae COy 2a were 50 58 1b) 25) 1 AO TORR Tht Nae eee 20 39 56 Lt ae ROL Eas oe 15 28 4] D4 130) 3); dense pees 9 18 26 34 EM BR lad cae 2 6 12 18 23 AO) aa ee 5) 8) 13 16 Ot Vite ees 2 + 5 ih Ohi Ys Racer ) 0 @) 0 Po. h,=5 km. h,=6 kin. he" km. h,=8 int DOSE Nis ea eek 22RD 24 34" Doe OF pay ag eV SO SU ee tee 17 41 19 20 20 44 21 56 SO 2 a coe A a 1S, OF ho e29 16 48 lt 56 SS aay i dane as O20 LOW ll 41 12 oo Sie hd ps aoe ea 6 49 AS 8 43 9, at San Gms se HLS) (2 is) 6.52 i oa SHG hiaticaets Goa 4 20 elt Gy aie: Gb 21a REM cei aes ee Sy ag) 3 39 z Wal 4 32 SONAR caeasess ba 2 13 2 34 2 54 o 12 OM nie tue eae 2s ery i 56 2. 2G LORS OE ares: ie LG 1- 25 1 34 OE se wa 41 48 54 Lice DO) ee 28 33 37 41 AT Sit ee eee 20 23 26 29 QOH yp ea ee Y 10 Hat 13 Terrestrial Refraction. Hy) 4. Simpler and more approximate Formule. If, in integrating (19), we assume a linear law for the index Ri Ve. if we retain only terms of the first degree in h, which is accomplished by putting g=O0=c, the formula giving A¢@; becomes: Me (oS ee € Adi= vat 2bhi— Va]; oa eo) ho being taken zero, as befure. This equation was found to give the same values as (25) for h;<1km.; but the values for h;=2 km. were about 3 per cent. gr antler than the corre- sponding values of (25), while the discrepancy for 4;=3 km. was about 6 per cent. Formula (26) may therefore be used for heights up to 1 kmm., to give Agi to the nearest second. The ordinary, simple, Surve eyors Formula, MUS CO at li eet in which s is the projection of the ray on the earth’s surface, is a particular case of (26). Tor, writing (26) in the form 2pkh; = Sera ng hen | Ati wieno) Ag Vat 2bh;+ Va oe we have when 2bh; may be neglected in comparison with a, Adi = = PON Pitan eh, 9) Va Replacing h; tan ¢o by its approximate value s, we have ORG US Dee Ore Thus the value we vet for the Surveyor’s Constant m is given by = iy == 109) Formula (27) is derived by assuming the rays of light to be ares of a circle of constant radius R/m ; and they ‘alues of m calculated from observations vary according to the time and locality, ranging from °105 to *168 ‘according to Winkelmann’s Handbuch. 996 Mr. A. R. MeLewd on Formula (29) may be used instead of (26) for zenith distances not exceeding 85°, the error caused by the approxi-— mation not exceeding 1’. Since a=0 when ¢@y=7/2, it is- obvious that the assumptions under which (29) was derived are no longer valid when the rays pass close to the horizon. Formule (27) and (29) are, of course, subject to the same restrictions as (26), and cannot in any case be relied on to within 1" for heights greater than 1 km. The reason for the variation of the constant m in (27) is at once apparent. If the radius of curvature of a light ray is assumed to be constant, it is, by (15), ee . Hence gng Sin Po m=gn,sin do; or, approximately, m=g sing. Substituting in (14) the observed values of o at various heights, and the value w=*2271, we find the values of g which are contained in Table III. TABLE III. om VE hj. g. O’king pee — Sakthinesee ace al tas) Li i Gaemcne atk GOGOL: COE ak ae eee 1260 RR ene 28 ‘1661 TORE aay nek -1218 So Nene ea Ss | Ml (Oe eae 1179 AeA, OP Ae dee 1554 — 1 De tie'=)) hacd ene 1134 5 ANTE wah 1479 13h Oe ee ‘1101 Gina oat ca _ 1420 Tf Ae EME 1063 Fen de Ree BGO Ell Ged eae eee ‘1020 The values of m will thus vary within the approximate limits 18 and-10. In any particular survey there will be a mean value of m, more or less applicable to the totality of measurements made, and the values should lie, as they do, within these limits. 5. Curvature of the Light Rays. Consider that portion, PQ (fig. 3), of a ray which lies between the heights /;_,; and h;=h;-1+1, where we suppose ho=t km: @=1, 2).22..8).. let. the Jensth of thisiiiaes of the ray be 1;; let it subtend an angle y; at the centre of the earth; and let @;-; and ¢; be its zenith distances at Terrestrial Refraction. Doe P and Q respectively. If there were no refraction, we should have ve) Lea iniee Ay 5 “n= SM REE sin $i-1) But owing to refraction, we must add the bending of the tangent along the ray PQ. This is found by t taking first differences of the refrachions in aloe JUL, Writing the first differences in the form A?d;, we have, very nearly : Op = fi sin bia) + IN 8 (NN) i giamsin' (Gos th sin $;_-1 ) Se ccune (awe) u+ ’ For the length J; we have, approximately, 1? = 4(¢R+4;_1)(R+ A,) sin? (7) +1, SMe Wey oe dU Oy) Cae mend fe oan ine tras (4) If we assume that PQ is an are of a circle of radius R/m;, we have RA*¢; ae a oy Sua, mare nd; = = 558 Mall ee ielicodten By means of these four formulee, we are enabled to get the values of @; and m; for the eight parts of a ray corresponding to 2=1,2,....8. The values of m; for the rays already considered are given in Table IV. From this table it is seen that there is a considerable change in the curvature when @p and h; vary. There is a slight irregularity in the curvature of rays which pass very close to the horizon. The errors due to the approximate character of (30)—(33), which are most important here, do not account for this. Other slight irregularities, especially for small values of #9, are due to errors in the calculation caused chiefly by errors in the last figures of the logarithms used in determining the refractions Ady. Tashe TV. (Values of m,.) Do: My, - M5. Ms. ae Ms. MN. Me. Mss SONS Ear as Rear "1820 149 | 142 > 134. 7126 “118. ene SI) OY aoe 73> “tbh 1469 2-187 > «128°: “19S ee ro) mae SENG 169" SST 145" 2-1388> .-129°.. 120) SO ti eee A660 o6y “447 ec138 7129 - 71205 poli Si el ease 164° 156° “147 138 129. 120, ee SOx. Cees 164. 15d 146. 1387) 128. 119) ae Sse heey 164, lab) 7147 9 AST 128° » “1S Sea Sota) hit ees "162" -"1as 145 71836.. -127.. 118) Se SO) venga eee AGL . 1538. 144-135 = 126° | IS Oana MD) steve ences “160 149 *142° 133 «+224 115: 2 eee MON ie ces "15d 145. 138) 2-180° 7120 12) eae Opry ge wake "143 -13t 7125. 7120 -11] "1025 (OSS aes DOU cee: "126 °° :120 “sll 104. -098 » -091. ){0Sa ee AOI saee ‘10% 100,092 -090. 081 077 20a FLA Yt PRS ak ‘056° 053 ~~ 050 “046 - 046-039. {0siaiae OTe aan: 000 §=-000 - -000 -000 @6-000 6-000 | “000 Fane The dips corresponding to a given value of gp are given by the formula p= | — (i=1,2,....8).>).. een ~ The values of D; for the rays previously dealt with are given in Table V. The advantage in specifying a particular ray by do rather than by D; lies in the fact that po is fixed for the ray, while D; varies with the height. Terrestrial Refraction. 5oy Tasue VV. (Dips.) Do ID. ID Dee D, ay )5:. 0° 49’ 59”) Teva 26" 12887 15. 1°, 487 48% B00380% 62. 1h 120 99 32 1 Bey BO) 1 54 98 0). ae iL eae ale 38,19 5846 2, .6 20 ao ie 2 19..9 2 293 25 Ne 301 0) MN) Bieeir.. Baers I) 3 16) 28 314 20) BG) 37 ee A 6 y OT MD 6 4 18 46 A) 9452 eet... Bek ug 5 10 10 RG 13 SO) 1! 22 off ee Pg uaa Fi ae hate kate ksh 7 Midas 0) A LO eo OL eh eo GO ue y, W460. OOM oe 5 an ee Ve Alva eS 2 MD woe Lat Ge 48 70) /) hee Ty) A TEL @ 80) se OT eI iy LOT) is 0) CO! ae 30520) AV es SOU lS SOE 202 S0e Saeko 50) oa AOE Soe AO Ln: HO ene sONe en 398 e40n. O18 Ae... FO KOS HO KON Aon es iS) 4 50) lh 63 10 ee 70 OO 70) WO.20, 70, O80 (250s eOr AD Oa: OO OO OO nd YOO OO 8 G0" O10 $0 De. D, De. D, SO Sia Om a ennai) 2°96) 56 DOT? HO}! BONSO) os. Dees D)il(8) “Bk DESH) 3 2 41 44 BOM eo. Oy TS) ay 2 99. 43 PAN 1g 2 50 22 Boke heii, O) i Bil e388 By She Dis 3 19 54 Se yee 39 30) 28 3 46 45 3 58 58 AOD Soper eeu A 30158 4 36 49 4 42 49 4 48 30 Sie tinge i Oo) 1! Bey) al Domo im a) al Some oe NE Del 7 LORD i OR ih 90) 34 BORIS. LOM 125 OOM nignea ei MLO eile 10nn20). 57 7 a Lo NS woe mel MMO OW elo, Lome rz ye 15. 1a. 56 7s omen 20.06 One Onan DO smo ne 20) elon 14 GOmIG eed! BOS oO ee ONE aT Ase 30.0 n'88) 80 60 28 5) ine AO C2 LT AOR POL kd On Ser 56 | S40) ) AST Oh ee BOM oles ae OM len a0 VO vain! 50-815 9 DD cama TOM OM OOU TOR Ohi GTO. eatenka TO’ Ip 2S Ona eae. GO OM Ol ey OOMEON.03 2) 906) 2010: 90> ORG 6. Refraction Correction for Instruments. The values of A¢; given in Table IL, and calculated by means of (25), give the total turning of the ray of light (wave-length 5000 A.U.) when it proc eeds from a height zero (sea-level) toa height A; km. Accordingly, the values of Ad¢; 560 Mr. A. R. McLeod on should tend to the corresponding astronomical refractions as- limits, when h; increases indefinitely. The refraction correction for an observer’s instrument is the angle between the tangent to the ray at his position, and the chord of the ray joining its extremities. The value of this angle lies between 4A¢ and Ag, and should tend to the corresponding astronomical refraction as a limit when /; increases indefinitely. Denote its value by e;Aq¢; for a ray whose zenith distance, ¢o, is given, and whose upper extremity is at height h;. The values of e; may be calculated in the following manner. Consider a ray ABCD oe 4), the observer being at A and the points B, C, i). . being at Fig. 4 heights 1,2,3,....km.above him. The observer is supposed to be on the proud (sea-level). Let AT,, BT,, CTs, .... be the tangents to the ray at A,B, C,.... The lengths, ;, of the successive ares are given by (32). We suppose each are to be part of a circle, so that the angles made by the tangents. at the extremities with the chord are equal. Let 6; denote this angle for the 7th are, which lies between the heights h;-y=(i1—1) km. and h;=2 con. Let L; be the length of that chord of the ray which is drawn from A to the upper ~extr emity Terrestrial Refraction. 5D61 of the 7th are ; and let,’ be the angle between the successive chords whose lengths are L;_; and L, («=1, 2,....8). at re | the angles @ and ai’ are all small 1en we have, since the angles @ and a’ are all small, Mca, —O;, Be MG (Or, ee [; L; i or, 1L & 4 2 @W; = Oe . e e (35) fel I l; We have also where A’; represents the first differences of the refractions Adi, as already considered in §5. The approximate values of e; are then given by By means of these formule we find the following values for €;:— Taste VI, (Values for ¢;.) Dos €\0 Bie 3. Ching > E50 Ey: €,. E,. QOS os OOM Pot Mea2 I wad oie sa42y 6547 | soo! (ois) CKO aaa SOOM Ola Vole eb Q4s |) eH 2QON 535 O41 O47 OAS) el eater 00M COON Mole a2 Ih Sw o2G aS N ane. | 645 85 Decccec OOOO Ole Cols oo4 bale aS a4 SUR gies cs OOD GUS © Gilley | SPSS ape eT at (ak ge | (oO yr Me eerie U0 REO ol oly oes) ool. “ob60 | ods The refraction correction for an instrumental observation made on the ground is thus given by the product of the appropriate value of e; and the corresponding value of Ad; in Table IT. 262 Mr. A. R. Mcleod on 7. Examination of the ordinary Formula for frange. Let AB (fig. 5) be a ray of light subtending an angle vy at the centre, C, of the earth. Let the zenith distances at its extremities be dy and ¢, the suffix 7 being dropped for con- a aTence. ) ANic cupeoce mie lower: extremity, A, to be on the surface of the earth (sea-level), while the upper extremity, B, is at height h. Let w,) be the refraction correction at the point A; then Ad—@p 18 the refraction correction at B, A®¢ being the refraction of the ray. Finally, let s be the range, measured on the surface of the earth (the projection of the light ray). and let / be the length of the ray AB. Then (cf. Winkelmann’s Handbuch, Band vi. p. 535) cos ($,4+ 0-3) 9 sin (by) + @,—Y) The assumptions ordinarily made in simplifying (39) are as follows :— (i.) We replace 2R sin y/2 by s=yR. di.) We assume that the ray is an are of a circle of constant radius, so that w = Ad—w= my/2 where m is a numerical constant. (iii.) y? 1s negligible in comparison with unity. (iv.) y cot gy 1s negligible in comparison with unity. These assumptions lead to the well-known formula : e(1l— hi sicon De wo oy oe (39) == Roi Terrestrial Refraction. 563 The equivalent formula for h, involving the dip D=7/2—¢, is sin (D+Ag—s 7) : RTE ey G = oF pine, Al i 2K sin 5 Sa Adon (41) In dealing with (41) the assumptions ordinarily made are the same as before, except that (iv.) is replaced by (v.) mry/2 tan D is negligible in cemparison with unity. The resulting formula has the well-known form 2h é 2h(1 — m) tan D+ ny / tan? iD= aa) The preceding work enables us to examine these assump- tions numerically. Assumption (i.) is true to ni 1 part in 8000 for h<8 km., and gy < 90". Assumption Gii.) holds to within 1 in 400 for h< 8 km. and dy < 90°; or to 1 in 800 for h< 4 km. Assumption (iv.) is accurate to within 1 in 800 for. poo km: and 40° < od, < 90°. In assumption (v.) we may write, except when D is small, mry/2 cot instead of my/2tanD; and so (y.) holds whenever (iv.) is true. Only Gi.) is left. This we now consider. Writing @o=e/d, so that Ad—a,=(1—e) Ad, we have ‘ l 0 Ss 5) Writing ees eml 2 ml vy dae aoa and 2(1—e) ml Ad —® = | Ss : s we have the following formule instead of (A )) and (42): 2 ') Se (im Zem she Gene ; § Dla / ud A = stan D-S, (1 0 wis ). wnnen (44) s h = scot do+ 064 Mr. A. R. McLeod on In applications of these formule the quantities in brackets are taken as constant. Consider the case of (44) when it is a question of determining s as a function of h and D. Writing (44) in the form : Ss h = stan D— sp - e ° e . (45) we have for the range error, ds, due to a small variation dR’, in the value of R’: Ss _— = pe: OF ps . ° . ° . e (46) : Also Y21-", ... ) ee where 1—e)ml Q= = (48) In any numerical formula, we must assume a value for R’. This requires the assumption of a mean value for Q, which ‘quantity varies from one ray to another, as well as along any . particular ray. To take a particular case, let us examine a formula in practical use. In Chauvenet’s ‘ Manual of Spherical and Practical Astronomy,’ p. 180, the following formula occurs : D = 22:14d+39-072/d, where D is the dip in seconds, 2 the height in feet, and d the range in statute miles. Changing the units to radians and kilometres, we get the equivalent formula : h s nee DiS hy 7199: . . «9 Writing R/ =7497, we have for this formula : R 6370 : Ro 7497 ~ 293 and therefore O = 478. Now if we take for ml, in (48), the expression xml; = RAd, Terrestrial Refraction. 565 where A@ is the total refraction, we have the following value for Q for the ray considered : q = Sane De) 2 59 a OES OU a ea es we have, on writing °850 for R/R’ in (47), Ts Oe Oa a Naa a ey ds alk Cer a D) ; (ey y Whence This last formula gives us the fractional error in range, as ealeulated with the equation (45). The error results when we replace the value of R’ in (45), which is given by (47) and (50), by the value R'? =7497. Simneien (percentage) values for 100ds/s are given in Table VII. and the corre- sponding curves are plotted in fig. 6. The values for go=90° are only rough indications, since ds is not small. ihe curves’ for h=l km. and hao km. are practically coincident. Tasue VII. (Values of 100ds/s.) Opree sts. 90°. 89° 30’. 89°. 88°. h Q. 100 ds/s. Q. 100 ds/s. Q. i00 ds/s. Q. 100 ds/s. ] 577 —20°4 550) ~— 1°31 538 = —'388 929 —-096 wa 521 —183 Oi 1520 511 —-388 d05 =—:0%8 5 495 —l14 494 — 635 491 —-210 487 —-O47 4 479 — 98 476 + -090 471 +:138 468 +:068 dD 464 +161 460 +1:04 455 +:°548 450 +:264 6 448 +4402 443 12°26 438 +1-08 431 +:445 a 434 +648 427 =+3°62 422 +1:68 415 +°681 8 419 +95 412 +5:°09 405 +2°40 396 +:985 Ono ea OU eg 86°. Soo. A. Q. 100 ds/s. Q. 100 ds/s. Q. 100 ds/s. 1 525 — O41 524 —'028 524 —:015 2 502 —-041 502. —:024 500 —:014 3 485 — O17 485 —-010 485 —-007 4 465 +:045 465 +:025 465 +°016 5 446 +129 446 +:076 444 +-049 6 429-4253 427 =+:'142 425 +:093 7 410 +°:368 408 +°226 408 +149 8 3938 +:520. 390 +:321 390 +:212 iniwMag. &,6. Voli 3s8sNo. 227. Nov. 1919, 2R 566 Mr. A. R. McLeod on From these figures it is seen that the formula h s 3 tan D = ee + aRO : . : 5 . (54) gives the range to within 1 part in 400, when h=1 km., for values of a not exceeding 88° 48’; w hile for h==25s a 4km. the results are better still. The errors are least for h=4 km. They are negative up to 3 km., and positive above 4 km. — Fic. 6. 100. ds /s PERCENTAGE RANGE ERROR An inspection of the values of @ in Table VII. shows that the values of @ contained in Table VIII. are more suitable RANGE ERROR =ds/s +05 pom 58 ro) @ Y © nN +°O! } Terrestrial Reyraction. 567 than the value Q =478, for the corresponding heights and small dips. Ifa more accurate formula than (54) is desired for a particular height, the value of R’ given in Table VIII., or one calculated in a similar way to suit the case, should be taken and substituted for R'‘?=7497 in (54). These formule, so obtained, will be accurate to 1 in 400 for zenith distances in the approximate range 40°< dq) < 89° 30'.. Probably the lower limit can be much decreased in many cases. Masia, WORE h. Q. R’. Peart ts cals alae oer 530 7642 Ea ies ga er 505 T7570 Sete Th hen Minera 487 7520 Ae by Sieg co A 468 7467 Ea SLE sini aaa Hea 450 7418 Gh tease vee 431 7367 (sagh take ane tame eae 415 7324 Sabre dy aossale ire tat a 396 7275 igs 7 ERROR DUE TO REPLACING TAN D BY D is DIP =D The error caused by replacing tan D by D in (54), thus obtaining Chauvenet’s formula (49), is plotted in fig. 7 2 ie ‘ols fs 268 Prof. C. V. Raman on the Scattering of for h=0 and h=8, using the equation 2 He (Deane) ee The error is always positive and limits the use of the formula (49) to small dips. Values of @) and D are interchangeable by means of Table V. 8. Lifect of a Change of Density. Suppose the values of the observed density given in Table I. are increased by 2 per cent. Then since the same increase occurs in 7%, p,g, k, Wa, Vb, Vc, the values of Ad and A?d are increased by 2 per cent. Actual calculation shows that the percentage changes in y; and /; are small, except for heights less than 2 km., when they may amount to an increase of +1 per cent. This is so for ¢@)=88°, while for dy =90° the changes do not exceed ‘5 per cent. Hence the values of m; will not differ from the values in Table IV. by more than 2 per cent. The formule of § 6 show that the changes in ¢; are inappreciable. The values of Q given by (48) will increase in very much the same way as the values of m, i.e. by not more than 2 per cent. This increase will therefore be about +10 in the value of Q. But the values of dQ obtained with the aid of (51), and used in the calculation of Table VII., varv from —88 to +100. Hence the refraction errors in Table VII. overshadow changes which are due to an increase of 2 per cent. in the density. Note that for h;<4km. the density increase makes the errer in Chauvenet’s formula less. LVI. The Scattering of Light in the Refractive Media of the Eye. By C.V. Raman, MA., Palit Professor of Physics in the Calcutta University®. 1. Introduction. ie his treatise on Physiological Opticst, Helmholtz has discussed the explanation of the interesting phenomena observed when a very small and intensely luminous source of light is viewed directly by the eye against a dark background. * Communicated by the Author. + Page 180, 1896 (German) Edition. Light in the, Refractive Media of the Lye. 569 An enormous number of luminous streamers appear to emerge from the source, and stretch out from it in more or less exactly radial directions. Helmholtz was of opinion that these streamers were due to the diffraction of light at the irregular margin of the pupil of the eye; and this view is snpported in his treatise by a description of the phenomena observed when the source of light is seen through a small hole in a metal plate placed in front of the eye so as partially or wholly to screen the margin of the pupil. Some doubt as to the adequacy of the explanation advanced b Helmholtz having been felt by me, a careful study of the effects was undertaken, the results of which are described in the present paper. The phenomena being of a subjective character, the assistance of a number of independent ob- servers with normal vision was obtained in order to confirm my persona] observations. This appeared all the more necessary, as the conclusions arrived at as to the origin of the phenomena differ from those of Helmholtz. 2. Description of the Phenomena. The character of the luminous phenomena seen is widely different in the two cases in which the source emits white and monochromatic light respectively. Tor observing the phenomena with white light, the most suitable arrangement is to condense the light of an electric arc upon a pin- Thales ina large dark screen, and to view the issuing light with the eye placed ata distance of about three or four “yards from the sereen. The luminous pin-hole appears surrounded in the first instance by a circular patch full of luminous streaks starting out more or less radially from it, and occasionally crossing each other. These streamers are generally white, but appear here and there tipped with streaks of colour. The circular patch is surrounded by a relatively dark ring, outside which again the streamers reappear passing radially through a luminous coloured halo* surrounding the dark ring. The halo is, in fact, made up of short sections of the streamers which, here, are strongly coloured. Outside the halo, the streamers emerge again, “but are much fainter, and they form a broad and a een ill-defined ring of lumi- nosity extending to a considerable angular width from the source. The inner margin of this luminous rin @ is greenish- blue, and the outermost visible periphery is of an orange-red colour, but some fainter fluctuations of luminosity and colour may be observed within it. For observations with monochromatic light, a Westinghouse * Of angular radius a little less than 2 degrees of are. 310 Prof. C. V. Raman on the Scattering of 3000 c.p. silica mercury-vapour lamp with glass dome fur- nishes a suitable source. The lamp is suspended immediately behind a small aperture in a dark screen and is viewed from a distance of two or three yards. A green-ray filter may be put in front of the aperture, but is not essential. An alternative arrangement for obtaining a small and powerful source of homogeneous light is to load the carbon rods of an electric are with plenty of common salt, and to condense the light from the luminous yellow mantle of the are upon the pin-hole in the screen. With either arrangement, the appearance of radial streamers issuing from the source as seen in white light is not obtained. We see, instead, circular area round the source filled with granular patches of light, and outside this, a relatively dark ring, followed by a well-defined circular halo, and some faint outer rings of luminosity. A remarkable feature which is worthy of mention is the peculiar circulatory or irregular movement which is best seen in the radial’ sepeatnene surrounding a white source of light, and less clearly in the granular patches surrounding a mmomogiironen te source of light. These movements disappear gradually when the eye is held with a fixed gaze towards the source, but start again immediately whenever any movement of the eyeball or of the eyelids occurs. 3. Discussion of Observations. The effects described above obviously present a striking resemblance to the phenomena observed when light 1s diffracted by a large number of irregularly placed apertures or particles of uniform size, e. g. lycopodium dust strewn ona glass plate. Recently, De Haas * has published an elaborate study of such diffraction phenomena, and has shown that the formation of the radial streamers in the coronas surrounding a source of white light, and their replacement by a granular field in monockromatic light, may be explained by considering the interference of light ditfracted by individua! particles which are assumed to be irregularly distributed over the aperture. The resemblance is not merely qualitative, as some careful visual estimates made by me seem to show that the relative intensity of the streamers in the area immediately sur- rounding the source and in the first circular halo is about the same as that observed when a source is viewed through a glass plate dusted with lycopodium. ‘There thus seems little * Proc. Roy. Sos. Amsterdam, 1918, p. 1278. It may be remarked here that the phenomenon discussed in the present paper is different from the coloured haloes seen in certain pathological states of the eye and described by Tyndall in his lectures on Light, and by other writers. Light in the Refractive Media of the Eye. Dill doubt that the phenomena described above have to be referred to the diffraction of light by a large number of particles of more or less uniform size included in the structure of the refractive media of the eye. The peculiar movements mentioned above would then ‘naturally be as- cribed to the movements of these particles. These may be imitated by observing a source of light through a plate of glass on which a little dilute milk has been flowed. The movements of the diffracted streamers of light can then be easily seen. Observations show that the angular diameters of the circular patch containing the streamers and of the circular haloes surrounding it are “entirely independent of the aperture of the pupil of the eye. This is readily proved by altering the intensity of the source of light under observation, with the result that the aperture of the pupil automatically adjusts itself. An ordinary candle-flame at three metres distance, and the light of an electric arc at the same distance or even nearer the eye, give identical measurements for the angular widths, though the aperture of the pupil must have been greatly different in the two cases. This is exactly what we might expect if the effects are due to particles contained in the structure of the eye, but it is very difficult to reconcile with the view of Helmholtz that the phenomena are due to diffraction at the margin of the pupil. ‘The observed os cannot he explained if we merely postulate any arbitra irregularities 1 in the circular shape of the margin of the =i It would be necessary also to assume a regular corrugation or periodicity in the margin of the pupil*, and even such an assumption, apart from its being purely hypothetical, fails to explain the observed effects. For, with any change in the aperture of the pupil, the distance between successive corrugations should also alter and influence the observed phenomena. This is inconsistent with the observed inde- pendence of the aperture and the observed effects. We are thus led to reject the view that the diffraction at the margin of the pupil determines the phenomena seen. It is useful in this connexion to note that the intensity of * A useful analogue is furnished by the milling on the circular edge ofa coin. Ina paper on diffraction which is in course of publication, S. Kk. Mitra has shown that a cirenlar disk with corrugated edges pro- duces a coloured halo surrounding the usual Fresnel-: \rago central bright spot, and the angular radius of this halo is equal to A/ 27a, where ais the radius of the disk and 2 is the number of corrugations, A similar phenomenon is shown in convergent light by a circular aperture with a corrugated edge, but in a less striking manner. Thenumber of corrugations remaining the same, the angular radius of the halo would vary inversely as the radius of the aperture. 572 Scattering of Light in Refractive Media of the Eye. the light diffracted by the pupil (supposed perfectly circular) ina direc ion making 30’ of are with the source would be only Tawvoe of that seen in the direction of the source. (This is calculated from the formula Ie[J,(<)/z|?, the radius of the pupil being taken to be 2mm. and A=5600 A.U.) Actually, the streamers surrounding the source can be seen in directions making an angle of 200’ and even more with its direction, and it seems safe to say that their intensity half a degree away from the source is a much greater fraction of its apparent intensity than i ONO A further test of the view that the effects are principally due to the structure of the eye and not to diffraction at the margin of the pupil, is furnished by ee with very thin metal screens containing apertures placed in front of the pupil of the eye. Using a circular hole with smooth edges smaller than the pupil and placed in front of it, the intensity of the streamers surrounding the source of light i is reduced, but does not vanish. When the screen is turned about an axis normal to its plane, the hole being continually kept in front of the pupil, the streamers of light seen in the field remain visible and fixed in position, showing that they are due to the structures of the eye through which the light passes, and not to the margin of the pupil, or the edges. of the hole. Another and probably more convincing demon- stration is obtained by using a square aperture smaller than the pupil of the eye and placed i in front of it. (This may easily be contrived with the aid of four Gillette blades forming the four sides of a very small aperture.) In this case, the effect due to diffraction at the boundaries of the aperture is very clear and marked, but can be shown (on rotating the aperture in its own plane) to be entirely distinct and separate from the phenomena now under discussion. The angular diametersand intensities of the haloes are such as to suggest that we are dealing not with one but with two sets of structures contained in the refractive media of the eye, averaging in size about 13 ~ and 7 respectively. The structures of the latter (smaller) size are indicated hy the outermost halo, which appears to ke composite in character and due to the superposed effect of the two sets of particles. These structures in the living eye are presumably to be localised in the cornea and in the vitreous humour, as bisto- logical evidence of the existence of cellular structures in these bodies is available. Upon this question, however, the author does not venture to express any opinion. Calcutta, April 22nd, 1919. Lame | LVII. On the Partial Tones of Bowed Stringed Instruments. By ©. V. Raman, M.A., Palit Professor of Physics in the Calcutta University*. 1. Jntroduction. ee of the outstanding questions in the acoustics of the violin family of instruments which has not as yet been fully cleared up,is the manner in which the tones elic ited by bowing depend on the position of the bowed “ point” on the string. ‘The problem is to some extent complicated by the existence of other variabie factors influencing the character of the vibrations excited, e. g. the bowing pressure and speed, and the width of the region of contact between the bow and string. Ina monograph of which the first part has been recently publisheat, | have attempted a systematic treatment of the mechanical theory of bowed stringed instruments, and have dealt with various problems relating HO) Md © Lin eS} jorKO= posed in the present paper to apply some of the results contained in the monograph in order to discuss the variation of the amplitudes and phases of the partial vibrations with the position of the bowed point within the musical range of powing. We may here assume that in the cases of musical interest, the steady vibrations of the string excited by the bow have approximately the character of the simple Helmholtzian type in which the vibration-curve of every point on the string is a perfect two-step zig-zag; and the problem is to find the nature and extent of the small deviations from this form of vibration depending on the position of the bowed point and other variable factors. If the vibrations were always ewvactly of the simplest Helmholtzian type irrespective of the position of the bowed point, it can readily be shown that for a given speed of the bow, the amplitudes of all the components of the vibration would be inversely proportional to the distance of the bowed point from the bridge, and would thus increase indefinitely according to a hyperbolic law as the bow is brought nearer the bridge : ; the ratios of the amplitudes, and the relative phases of ‘the partials, would be independent of the position of the bowed point. In practice, however, we know that the foregoing statement does not correctly represent the facts. For, when the bow is applied at a point of aliquot division of the string, e. g.at a point distant 1/5 or 1/6 or 1/7 &c. of the length of the string from the * Communicated by the Author. t+ Bulletin No. 15 of the Indian Association for the Cultivation of Science, Calcutta, 1918, pages 1-158. ae Prot. C. V. Raman on the Partial bridge, the partials having a node at such point cannot be excited by the bow, and must thus be absent in the motion maintained by it. The graphs representing the relation between the amplitude of the partials and the position of the bowed * point ” cannot thus be of the simple hyperbolic form mentioned above but must deviate from it, especially in the neighbourhood of the nodes of the respective partials. What, then, are the actual forms of these graphs? Then, again, in the simple Helmholtzian type, the phases of the partial vibrations are such that at two epochs in each vibration the displacements are everywhere zero. To what extent are these phase-relations modified in actual practice? It is proposed in the present paper to furnish an answer to these two queries. 2. Kinematics of Motion under the Action of the Bow. It is not intended here to enter into any detailed discussion of the mechanical theory of the action of the bow and of the manner in which the pressure and speed of bowing inflnence the character of the motion. For this, I would refer the reader to my monograph. It is sufficient for our present purpose to remark that, in general, when the pressure of the bow is sufficiently large in relation to its speed, the speed of the bowed * point” in the forward motion attains equality with that of the bow. But in the backward motion, the speed of the bowed point is generally non-uniform. In certain special cases, however, that is when the bow is applied with sufficient pressure exactly at one of the nodes distant 1/5 or 1/6 or 1/7 &c. of the length of the string from one end, the speed in the backward motion closely approaches or attains uniformity, the partials having a node at such point completely dropping out. It is obvious that the bow has te be removed to some distance from a node, before the corresponding partials can be fully restored in the motion maintained by the bow, and in the intermediate cases the character of the motion at the bowed point becomes slightly modified. The speed in the forward motion remains equal to that of the bow, but the speed in the return motion is non-uniform. The kinematies of these intermediate or ‘‘transitional modes ” of vibration have been fully discussed by graphical methods in my monograph. My object here is to show how the harmonic analysis of these transitional modes enables us to trace the variation in the amplitudes and the phases of the partial vibrations with the position of the bowed point. In the cases of musical interest, we are only concerned Tones of Bowed Stringed Instruments. 575 with relatively slight modifications of the principal (Helm- holtzian) type; and the general character of the vibration is determined by the movement, to and fro, on the string, of one large discontinuous change of velocity, and of a number of mznor discontinuous fluctuations of velocity. The positions and magnitudes of these minor fluctuations must be sucli that in the vibration-curve at the bowed point, we have a uniform gradual rise followed by a steep and generally non- uniform fall, the ratio of the time-intervals occupied by the two stages being the same asin the simple Helmholtzian type. In my monograph I have shown by several examples, that the same dispositions give for the vibration-curve of a point close to the end of the string, a “fluttering ”’ or irregular rise, followed by a steep and uniform fall. 3. Analysis of the Transitional Modes. The general expression for a discontinuous vibration is ao . iG 9) 33 | Se NTL 5 inert 7 zien) = S es ® Sees ej CS ¥ = 51D i | sin . +0, COS T cp where 1 | hh OF nT 9 ay = — <5, | dy cos —— +d, cos —-,-~ + Ke. }> n ey | 1 L 2 L and LL oe eto © . naC, Ss | a Sin Se Seal in = + &e. |> Nie rat L L 4 L dy, dy, &c. being the magnitudes of the discontinuous changes of velocity, and Cy), ('s, &c. their positions measured from the origin at time ¢=0. Ifa discontinuity is moving with the positive wave, its position is given by its a-coordinate ; but if it is moving with the negative wave, its position is given by (2/—z). From this expression, the amplitudes (a,+0;)® and the phases tan7' b,[d»n o£ the harmonics may be readily calculated, when the magnitudes and positions of the discontinuities are known. In finding the phases, it is convenient to choose the origin of time in such a manner that the phase anele of the fundamental component of the vibration is zero. When we have only one discontinuity, we may write d,=d;=d, &e. =0; and taking the origin of time such that the discontinuity d, is initially at the end of the string, C)=J, we have ; = (a ae _ nie . 2nrt y=d, >-—— a SID ~—— SIN : fT eae L i This is the simplest Helmholtzian type of vibration including 576 Prof. C. V. Raman on the Partial the complete series of harmonics. When the bow is applied at one of the points of sanju division, e. g. 1/5, or 1/6 &e., the subordinate series of harmonics havi ing a node at that point drop out, and it can easily be shown that the resulting ceils type of motion is determined by one large positive discontinuity, and a number of small equal negative discontinuities. For example, with the bow applied at //5, we have one large discontinuity equal to 4V, and four small discontinuities each equal to —V, where V is the velocity of the bow. Similarly, with the Lew a: 1/6, we have one dis- continuity equal to 5V, and five econiiaci fe: each equal to —V, and so on. In the intermediate or “transitional modes” also, the motion is of a generally similar character ; the positions and magnitudes of “the discontinuities must Me such that in the forward motion at the bowed point the speed is uniform and equal to that of the bow. The speed of the backward motion is necessarily non-uniform in greater or less degree except when the bow exactly coincides with one of the points of aliquot division. To illustrate the method of calculation of the amplitudes and phases of the harmonics from the formula given above, we may take a specific case in which the bow is applied at a point v=a, where 1/5 >a >1/6. We have then six dis- continuities of which five are necessarily equal and negative, oD and the other is larger and positive. The magnitude of the large discontinuity may be taken to be 5V, and of the small Ae Continuitics —V. We have then the following scheme:— d,=d,=d,;=—V, adg— DN, d;=d,=—V. ci— 0) C= Za. i C,=2l—6a, C= 2 , C,= 2l1—2a. When a=// 6, the initial position of the discontinuities is identical with that in the Helmholtzian type obtained by bowing at //6. It will also be seen that when a=//5, Cz; and C, become identical, and the discontinuities d; and d, therefore merge into each other, reducing the number of discontinuities to five, and the magnitude of the large dis- continuity to 4V. Thus, both when a=//6 and when a=//5, the transitional modes become identical with the Helmholtzian types. The cases in which the bow is applied between 1/7 and 1/6, or beween //8 and 1/7, may be similarly worked out. Tones of Bowed Stringed Instruments. ont A, Results. The amplitudes and phases of the first eight partials excited by applying the bow at various points lying between //9 and £/5 have been calculated and tabulated in the manner ex- plained above. To illustrate the general character of the results obtained, the amplitudes and phases of the fourth, fifth, sixth, seventh, and eighth partials within the range of bowing have been represented graphically and shown in figs. 1 to 5. Fig. 1 Amplitude x /6 NC) IS Gh OQ SC) © VAs) Y8 V7 V6 YS Position of the Bowed Point. Amplitudes and Phases of the Fourth Harmonic. The first three partials do not show any specially noteworthy feature within this range, their amplitudes increasing gra- dually as the bow is brought nearer the bridge, and their 578 Prof. C. V. Raman on the Partial phases remaining practically identical, and the same as in the simple Helmholtzian type. As regards the fourth and higher partials, the following are the principal features exhibited by the graphs, which embody the results of the investigation: — (1) At the points of aliquot division, that is, when the bow is yg YE V7 V6 VS Position of the Bowed Font. Amplitudes and Phases of the Fifth Harmonie. applied at 7/5 or 1/6 or J/7 &., the graphs give the Helmholtzian values for the amplitudes. (2) Elsewhere, they ‘show marked deviations from the Helmholtzian values, both in regard to amplitude and phase. (3) The deviations are greatest. when the bow is applied not far from a node of the partial under consideration, and decrease as the bow is Tones of Bes Stringed Instruments. 579 brought nearer and nearer the bridge, till ultimately the phase- differences vanish, and the amplitude increases re- eularly, instead of in a fluctuating manner, in inverse proportion to the distance of the bow from the bridge. (4) As the bowed point approaches the node of the partial Amplitude x 36 ——> Phase —~> yg 1/8 V7 6 Ws Position of the Bowed Forint, Amplitudes and Phases of the Sixth Harmonic. I under consideration, the amplitude of this partial at first rises above the value given by the Helmholtzian type, and then gradually falls to zero at the node itself. (5) In the same circumstances, the phase of the partial alters, being in advance or in rear of the Helmholtz a phase, acco ding as the bowed point is one side or the other of the node. The 580 Prof. C. V. Raman on the Partial maximum alteration of phase is +7/2. As the result of these alterations of phase, the transitional modes of oscillation are generally of an unsymmetrical character. Thisis perhaps the most novel and interesting result indicated by the investigation. Amplitude 4 4Q—-> yg ys V7 6 YS Position of the Bowed Fo/nt. Amplitudes and Phases of the Seventh Harmonie. It must be noted that the values of the amplitudes and phases shown in the graphs are valid only for the particular positions and magnitudes of the discontinuities assumed in working out the “analy sis, and are therefore to be regarded “as only of a representative character. In actual practice, they may be appreciably modified to an extent depending on Tones of Bowed Stringed Instruments. D81 the pressure, speed, and other mechanical conditions of bowing. It is not proposed here to enter into a discussion of these details. : Amplitude x 64 —> Vg Vg V7 V6 VS Position of the Bowed Point. —> Amplitudes and Phases of the Eighth Harmonie. I have to thank Mr. Durgadas Banerji, M.Sc., for useful assistance in carrying out the numerical work. Calcutta, May 2nd, 1919. ini. Mag. 5. 6. Vol. 38. No. 227. Nov. 1919. 28 LVI. An Absolute Determination of the Coefficients of Viscosity of Hydrogen, Nitrogen, and Oxyg ygen. By Kra-Lox Yen, PA.D., Research Assistant im Physies, University of Chicago, “recently appointed Pr pee of Physics, Government University, Peking, China * ‘Plate VIII] LTHOUGH historically several methods have been ve employed in the determination of the viscosity of gases, practically all of the most careful measurements fea heretolors been accomplished with the transpiration method. But, since it is an impossibility to obtain ecapil- laries of uniform bore and next to impossible to accurately determine the size and shape of the bore when it is not uniform, and since there is always a possibility of the fone of eddies at the ends of the eapularies and a consequent distortion in the lines of flow of the transpiring gas, the transpiration method is neither mathematically rigorous nor experimentally exact, and so even the most eareful and accurate determination is but relative. It was not until the development of that most accurate method for the determination of the elementary electrical charge, the ‘‘oil drop”? method, the accuracy of which is eee only by the accuracy in the mea surement of the absolute ralue of the coefficient of viscosity of the gas in which the droplets fall, that there was created a demand for the most accurate determination possible of the absolute value of the coefficient of viscosity of gases. In order to meet this demand there has been developed, by Professor Millikan and his pupils, in the Ryerson Laboratory, University of Chicago, a constant deflexion apparatus the behaviour of which has proved most con- sistent and the results of which can lay claim to accuracy of a much higher degree than has ever been achieved by any other merhoden As the ae of air determined by means of this apparatus came out somewhat lower than the values usually assigned to that coefficient {, the desirability of the redeter- mination of the viscosities of other gases was at once ap) arent ; and the work herein described is but a fulfilment of a small part of a general plan to revise the whole of the * Communicated by Prof. R. A. Millikan. + See Gilchrist, Phys. Review, n. s. vol. 1. No. 2, Feb. 1913; pp. 124— ah also Harrington, ibd. vol. viii. No. 6, Dee: 1916, pp. 738-751, if See Gilchrist and Harrington, oc. cit. i er ask ks Viscosity of Hydrogen, Nitrogen, and Oxygen. 583 heretofore accepted tables of viscosities contained in the existing physico-chemical tables. The Theory. The method employed here consists in causing one cylinder to revolve with a constant angular velocity about another which pee =. is suspended by a torsion thread. If o a be the radius of the cylinder which \ Dorf a\\_\ is revolving with an angular velocity ia <--_]' | w, and } the radius of the suspended \ a / j cylinder which has turned through SS (“fan angle @ away from its position a. when the outer cylinder is at rest, the equation for the viscosity can be derived in the following manner * :— The tangential force Ff acting on any cylindrical surface at a distance r is BF = 2nrlndvfdr, where / is the length of the suspended cylinder, 7 tbe coefficient of viscosity, and dv/dr the velocity gradient at 1 due to the slipping of one cylinder over another. Since YP duldr = rdw/dr + wdr[dr. As the term wdr/dr enters only when we have a rigid body, we have here simply dvu/dr = r dw/dr. BF = 2rr7ly diwidr, or the moment of shear Therefore Kr = 2ar*ln dwldr or Fr dry? = 2nlndw. and for summation we have malend Ooi Tw Hr \ dr/r? = 2aln dw, b @ Cy 0 since Hr does not depend upon »*. We then have Rr = Anln| a?b?/(a*—b’) * This particular form of derivation is taken from unpublished lecture notes of Professor Millikan. 2312 UW. ~ } } = d84 Prof. Kia-Lok Yen on the Coefficients of But if the suspended system has a moment of inertia I and a period ¢, we also have the relation Fr = An’l6/t?, and therefore An J@/t? = Aaln [ a7b?/(a? — b”) |. Therefore ; w19(a?— 6b?) la?b*t?w ?) = Thus we have an expression for the viscosity 7 in terms each of which may be made of such a magnitude that it may be measured with the highest degree of precision. The Apparatus. The apparatus is the same one used by Harrington with the suspension he designated as F in his report: this suspension, by the way, has been in use for a few years now and hae, never heen found to alter its zero position.’ Fig. 1 shows the elevation of the apparatus. The outer cylinder O is geared to the chronograph drum K by means of a shaft running through the a hole length of the steel pipe Q, and thus the chronograph is employed as the driving apparatus which causes the outer cylinder to rotate and also as the recording device which registers the speed of rotation. The inner cylinder I is suspended from the top of the tube T by means of an elastic steel suspension 8, so that the drag due to the viscosity of the surrounding gas will cause it to deflect from its position of rest when the outer cylinder is in rotation. A pair ot stationary cylinders G, G are provided as guards for the inner cylinder in orden to eliminate the end effects. The cylinders are supported by the heavy brass frame F, which is mounted upon the thick steel plate P. The lower end of the tube Q, together with the shaft and transmission gears, is immersed in a cup C filled with mercury in order to render the apparatus absolutely air-tight. The whole of the apparatus is covered by a glass jar J and sealed with mercury. A hole H is provided for letting in or pumping out the gas the viscosity of which is to be determined. The temperature inside of the apparatus is registered by a Beckmann thermometer, calibrated with a standard Baudin thermometer, and reading directly to 0:005 degree centigrade. The room in Hen the determinations — are eee is itself a constant temperature room. Viscosity of Hydrogen, Nitrogen, and Oxygen. 585 The time for the rotation is measured by a trace, on a sheet of waxed paper covering the chronograph drum K, iow. p= ile ARAL JU 3 PoRenSt Sra BESO ES ‘a (fp S = ERS eee ee - — Af f NANNANNY: made by a stylus which is electromagnetically connected to a standard clock in the Laboratory and which travels from one to the other end of the drum during one observation. The deflexion of the inner cylinder is indicated by a very sharply defined spot of light reflected from a mirror, 586 Prof. Kia-Lok Yen on the Coefficients of attached to the suspension, upon a transparent scale situated at a distance of two metres from the centre of the mirror. This scale is divided into millimetres and is bent to form a part of the circle the radius of which is two metres, in quedlere that the angle of deflexion may be accurately replaced by the quantity df: 2D, where D is the distance of the centre of the mirror to the scale, and d the distance, or rather the are, through which the spot of light is deflected : all aaememeee in centimetres. This is the only alteration of the arrangement as left by Mr. Harrington. As Mr. Harrington has given in his paper a much more detailed description of the construction and the adjustment of the apparatus, and also a full discussion of the deter- mination of the various quant tities that enter into the final calculation of results, it only remains here to reiterate his assertion that ihe probable error in the final result is not more than one part in one thousand. The Observation. When the apparatus is filled with the gas the viscosity of which is to be determined, the. temperature inside of the chamber is recorded ; then the chronograph i is set in motion and this motion is transmitted to the outer cylinder through the gears and shaft by means of a small clutch provided for that purpose. The inner cylinder will at once begin to swing in the direction of rotation of the outer, and if not checked, will continue to swing beyond the proper angle @ on account of its inertia, and will continue to oscillate for some time. ‘This oscillation is eliminated in the followin manner :—, while the bulb B contained distilled water into which a small amount of potassium permanganate had been dissolved in order to serve as catalyser. After the apparatus Fig. 4. was thoroughly evacuated the water was caused to drop slowly through the stopcock S upon the chemical, and reaction was thus started and the evolution of gas begun. The gas thus generated passed through a wash-bottle containing concentrated sulphuric acid, then through a tube containing lumps of KOH, and hen through the coil T, into the pyrex tubes containing Ca and CuO. These tubes were heated up to about 500 degrees centi- grade. Thereafter the gas passed through a battery of Py O; tubes and then entered the coil iy where it was liquefied. This was accomplished by immersing the coil Ts in a vessel of liqnid air. When a reasonable amount of gas was thus liquefied the evolution of the gas was stopped by the stopeock 8. The stopcock S, was then closed. Then the liquid-air vessel was removed from under T, and placed under T,, and thus the gas liquefied in T; was allowed to boil and return, in its gaseous state, through the purifying agents into the coil Tj, where it Viscosity of Hydrogen, Nitrogen, and Oxygen. 595 was again liquefied. This operation was repeated over and over until the gas was found by analysis to be pure. Then the gas was introduced into the compression-suction apparatus C (in fig. 2) through the stopcock 8,. The stopcocks S, to 8; controlled the shunt passages, as in the other two apparatus. The analysing apparatus consisted of a pyrex tube D connected through a ground joint to a volumetric bulb EH, which was in turn connected to a MacLeod gauge. D was filled to two-thirds of its length with granulated copper and surrounded by an electric heater (net shown in fig.). For analysis S; was closed, and both the bulb E and the tube D were “odemeriee through So. When the pressure inside was reduced to somewhat ie than 1 mm. mercury, the tube D was heated up in order to drive out whatever moisture there might have been inthe copper. The evacvation was continued shal discharge could hardly pass between the terminals of the vacuum tube V. ‘Then the pressure-gauge was read and the stopcocks Sg8_ were closed. Then a certain amount ok the gas to be analysed was introduced into E through §,. Then S; was closed and 8, opened and the gas in Ptwas allowed to go into D, chiens it was absorbed by the copper which had in the meantime been heated up to about 900 degrees centigrade. When the gas was pure, both the vacuum tube V and the pressure- gauge showed tlie same indications as before when the gas had not been introduced into E TaBLE ITI.—Viscosity of Oxygen. No. of Temperature ‘Temperature Mean observation. at beginning. at end. Temperature. n x10". o ©: AG ©, ee cri ee 22-990 23-000 22°995 2041-75 hove deletes 23030 23°050 23040 2045-21. 5 aA 22995 23°000 222998 2042-14 Ae Dae ae 23010 23°C00 23005 2044-24 Dee oe cs 23°010 23°010 23°010 2041-95 OEMs as. 22-999 22°980 22/985 204321 Ef aNaRs eae 23000 23000 23-000 2041-21 Sie es: 22995 23°C00 22-998 2042-01 Derek RE. 22980 22:970 22-975 2043-15 10) ere 22-980 22980 a2: 980 2045-07 SU ear 23:005 23-010 23008 2041-25 We rac ig 23010 23-020 23016 2042-12 Mg Se tees 22990 22980 22985 2043-04, ley gdawssens 23010 23010 23010 2041-05 OMe ee, 22°995 23°000 23:998 2041-05 HOR Mees oe 235°000 23000 23-000 20-41-00 ieee 23°010 23:020 23°015 2042-50 MSR eed. | 23-005 23-010 23-008 2041-41 1S) AA Re oe 22-990 22-980 22-985 204142 SAAN en eee 2-995 23010 23°008 2042°36 IM Gath cs aentensansitstcars 23° 0 01 2042: 35 596 Viscosity of Hydrogen, Nitrogen, and Oxygen. It should be pointed out here that even this analysis was only a qualitative one. Since the degree of purity required was much higher than that which could be reached by this method of analysis, and since no accurate spectral analysis of oxygen is possible, it had to be assumed that the gas was pure ; “and with the method and procedure followed in the production and purification such assumption was not by any means unjustifiable. Table III. contains the results of the individual obser- vations and their mean. As in the case of hydrogen and nitrogen, the observations were made with three changes of the content of the chamber. Summary. 1. The viscosities of hydrogen, nitrogen, and oxygen were determined by means of the constant deflexion apparatus which was designed and developed by Professor Millikan. - These determinations were made at a temperature of 23°-00 C. and under a pressure of 76 cm. mercury. 3. “The results obtained for these gases were :— Hydrogeny. 2.2). 7X10 = | sozaie Oxay emus alow aot. 3. «= 204025 INiirocetesem Gus 7 ee = 1764-30 each with a probable error of 0°15 of one per cent. In conclusion the writer wishes to say that it was under Professor Millikan’s direction that these determinations were made, and that it was the University of Chicago that defrayed all the expenses involved in the experiment. Addendum.-—Since there is in existence an enormous amount of literature concerning the subject of viscosities 7. was thought advisable to give ne references only those work that are closely connected with this particular method. The several methods for the production and purification of gases were found in Gmelin, Kraut und Friedheim, Handbuch Fee Anorganischen Chemie. The. employment of metallic calcium for the absorption of nitrogen and oxide of copper for the removal of hydrogen was found in Dennis’s ‘Gas Analysis,’ The particular form "of arrangement and order of procedure were results of what is known as “‘learning by trial and error.” Ryerson Physical Laboratory, University of Chicago, Chicago, IIL., US July ord, 1919! bacon || LIX. The Mass carried forward by a Vortea. By Wo Meviions, 7 AiS HEN a vortex aggregate is moving steadily through an irrotational liquid we can in general distinguish three definite regions of fluid motion, (1) that of the ring or aggregate itself which is in rotational motion and which keeps its identity and constituents throughout however the energy may alter, (2) the portion in irrotational cyclic motion surrounding the first, which also keeps its identity and volume so long as the energy is constant, and which travels uniformly through the liquid like a solid, (3) the irrotational acyclic motion, outside the second region which remains at rest at infinity and no portion of which is ever displaced by more than a small amount. The distinction between the rotational region (1) and the irrotational (2), (3) is funda- mental and well known. Less attention than it deserves, however, appears to have been devoted to the discussion of the relationships between the 2nd and 3rd regions. In this regard the following note may be interesting. When any such agoregate is travelling uniformly through an unlimited fluid with translatory velocity U relative to the fluid at a great distance, we may in order to study the relative motion suppose the ring brought to rest by imposing everywhere a velocity equal and opposite to U. In this case the boundaries between the three regions appear as fixed. The first is always a ring surface, even when the aggregate closes up, or the aperture diminishes to zero. The boundary between (2) and (3) may be a ring surface, or it may appear as a singly-connected surface, past which the outer liquid streams. This is what an observer travelling with the ring sees, and we can in this way determine the boundary mass and energy of the portion which goes bodily through the liquid, and the energy of the external part or region (3). Take as an instance a circular unicyclict vortex ring. When the ratio of the cross section of ring to aperture is very small, it is well known that the velocity at the centre is less than A of translation. Consequently when the system is brought to rest the flow at the centre is in the opposite direction to that of the original translation. In this case the boundary (2), (3) must cut the equatorial plane somewhere * Communicated by the Author. + I.e. without bicyclic or gyrostatic quality. Piul. Mag. 8. 6. Vol. 38. No. 227. Nov. 1919. > id 598 Prof. W. M. Hicks on the between the centre and the filament, and it will be ring- shaped. As the energy diminishes, the aperture becomes smaller, and the ratio of cross section to aperture larger. The velocity of translation increases, but the velocity at “the centre increases at a greater rate, until a state is reached at which the two become equal. In this case the acyclic boundary just loses its ring form and its section has a lemni- scate form. As the energy still further increases this boun- dary cuts across the straight axis of the vortex and the volume of region (2) w ill ultimately diminish, until the vortex itself closes up into the spherical aggregate, when it entirely dis- appears. Thereafter it is the actual rotational portion alone which is propagated through the surrounding fiuid. In determining the energies of the three portions it is most convenient first to find their values for the first and second regions when in the stationary state. In this case the stream functions y along the boundaries are constant. If E, E’ denote energies of the actual and stationary states respectively KH’ =7u(yY%2—%X1) + 2ar \\ wx dxdy, where y2, x; are the constant stream functions along the inner and outer boundaries of a region. When the vorticities are constant throughout the rotational portions, w =const. for two-dimensional motion and =)~# for three. For two dimensions @ X area of section =4y, =4wly—x) + AM yaeedy, For three dimensions oa=Axv, odA=tdy=rArdA= ~ dV, _ dm 277 K’=7E(x2—%¥1) + ve \\axae dy. Passing now to the case where the system is moving with velocity U thr ough tlie fluid at rest at infinity, let p, q denote the component Seichen cn fee stationary state, so that ptt, q are the actual ones. Then E=3\{ (p+ U)?+¢?}drdy= E' +3) U*dedy+ U\ pdedy, \ pdedy = momentum in stationary case = 0. E=E'+4(vol. of region) U?. Mass carried forward. by a Vortex. 599 But we might have applied the general theory direct to the actual motion. In this case, omitting the rotational region for the moment, H= — m\ypo'ds, in which the stream function wv is y+4Uz2’, and v’, the velocity along the boundary, is y : Paes o+U a where v is the corresponding velocity in the ds j stationary case. Hence |= —t \( v¥+4U2’ )eds—nU\(x +3 4U x’) dy. Here \xdy =x) dy=0 along a closed curve. - il \ edy along the two boundaries = — — (vol. of region). a Q Th oe > vol. orneciom) U2 — San U \a7vds. Comparing with the previous result it follows that Ja?vds = S=())) @ye anaes \z2vds round one boundary is equal to that round the other. Clearly this theorem is general, and states * that in a stationary condition \ePeds is the same along any two stream lines, provided no rotational region intervenes between them. For the core, the term ee wirdudy becomes Qa ? Tb ee | xydaedy + — ——— rele dxdy. m J Now Qar\.x" Cid yi \ xe. 27 udady is the moment of inertia of : d 2 Cc Cc (é — ii the whole ring Panna the translation axis =mk?, where k is the corres Sania radius of gyration, and is dieaotl ex- co) = 5 pressible in terms of the shape of the section. Thus Be imU? + — oe ir Ul pvds = E,'+4mvU?. * The following direct proof of this, I owe to the lindness of Mr. G. H. Livens. Let p,qdenote the component velocities at a point on a stream line. Then integrating alone two boundaries (5 5 re t ie i 's } |) v®vds= \ai(pdetgdy)= \\ : ~ (a°g) — es (=p) da dy =2\\. vg de dy e 1 ‘ ques : X quantity carried forward in the stationary state = 3 ah 9) TT -) 600 Prof. W. M. Hicks on the Therefore \a?vds taken round the stationary core boundary i) The corresponding theorem in two dimensions is fav M5 Mba. where w is the distance of the centre of gravity of the section from the translation axis. For region (3), the energy in the translation state is. Hh; = H3+ E,—H, == TX, — 4 (vol. core) U?+ SU a" vds— H,, the integral being taken over the core section =x, + 4(total vol. carried forward) U? + 5 BUR. When the portion carried forward is singly-connected, y,=0, K3= 5 HUF —4(translated mass) U?. This can be verified at once in the case of the spherical vortex in which U=yp/5c, k?=2c?, This gives H;=4mU? or energy of translation of half mass of fluid displaced by the sphere, and is correct. The corresponding theorem for two dimensions is on one side of axis of wv H;=3wUx—3 (translated mass) U’, or for the whole motion H;=Ue—4 (translated mass) U?, H.=pvy2+4 (mass of region 2) U®. Of this general theory the present note discusses in more detail the two special cases of (a) two parallel straight vortices, (6) a single ring vortes of uniform vortucity. (a) Two parallel straight vortices. Take first the case where they are so far apart that their sections can be regarded as circles of radius c, with centres at a distance 2a. If w denote the rotational constant and p the circulation, w=27¢.wc=27c’w. The velocity at any point due to one vortex, at a distance r from it, is w¢e?/r= p|(27r). In the figure A, A’ are the centres of the vortices, Mass carried forward by a Vortex, 601 P, P’ any symmetric points on opposite sides of the axis Oy. Then the stream function y at P= flow through PN due to Fig. 1. | | A, A’= flow through PP’ due to A alone = flow across P’Q where AQ=AP. Hence AP hey 1 Diy Ge JB ae x eg) ? age le PIES» AQ Also velocity of translation is velocity at A’ due to A, or ge ‘ Ara Hence when hrought to rest the stream function is sf (e+ta)y+y7? T Dy (@+ayr+y’ a en : a dy * aa "8 (a — a)? + 2" He B X= Urt 7 lo The velocity at the centre relative to the surrounding liquid is p/wa or 4U. Consequently in the case of parallel straight vortices the cyclic boundary (2, 3) is singly-connected. The stream function along it has the same value as for the axis and is zero. The equation to the boundary is therefore, taking the unit of length =a, get ty? Os (e—1)?+y? ae v 602 Prof. W. M. Hicks on the As this equation is independent of c, the shape of the trans- lated mass remains unaltered, and its area simply varies as a’, so long as the filaments are not so close that the shape of the cross sections deviate considerably from circles. So long as they can be treated as such, the shape remains the same however the energy alters. It may therefore be drawn once for all. For this purpose the equation may be written in the form suitable for logarithmic calculation with tables of hyperbolic functions * y=v/ { (coth #/4—«a)(#— tanh a/4)}. This cuts the plane of the filaments at a distance given by eoth #/4=x, De oe pel or 5) = O2e ny, 9 whence e=?:087288 =aa. It also cuts the axis of y aty=,/3. =a. The calculation is easily carried out by the aid of the Smithsonian tables. The curve when drawn is seen to be very close to a true ellipse. Indeed, if an ellipse be graphi- cally constructed with the same axes it is difficult to dis- tinguish any difference. For #=1, near which we should expect a maximum deviation, the ordinates of the two are 1:5203 for the ellipse and 1°5257 for the boundary, corre- sponding to a distance between the two curves of -0035a. For practical purposes we may therefore take the area to be aX 2;0872 x ,/3a°=3'6lma’, or that of a circle of raame EO OMa: The energies are H,=2 x 4$x2+4 (amass of region 2) U* =Ef2log%* 1-4} + 3(mapa*— ae?) 7. * Eg.“ Hyperbolic Functions,’’ Smithsonian Mathematical Tables. Mass carried forward by a Vortea. 603: Now aB=2:0872 /3=3°6151, f 2a ¢ & ey ea a yee, 2740 1 BE, =Ua—2 x 36151 U? i EK, ee oo =1x 4:38497a2U? =1~x 1:2130 x translated mass. U? = peck. Tis The last result can be verified from the fact, that since the form of the translated mass differs only inappreciably from that of an elliptic cylinder, the energy of the external motion is aa/Ba of the liquid displaced by the cylinder. Now a/8=1°205. ‘The difference from 1:2130 is probably due more to the velocity changes due to the slight change in form, than to the difference in area of the two cross sections. The curious fact emerges, that so long as the filaments are not so close as to appreciably affect the shape of their cross sections, the energy of the external fluid is constant and is independent of the velocity of propagation. As the velocity increases, the quantity carried forward diminishes so that this result follows. This fact might have been foreseen from a consideration of dimensions, remembering that with the proviso above, there is only one length a at disposal to define the system. The most interesting portion of the motion, when the two filaments close in to be almost in contact, is unfortunately not at present capable of being treated. It is easy to show, however, that if the shape of the pore conten referred to its centre of gravity be given by r=a(1+5,& cos 26+ ...) where =c/a, the change in U a on &. The above values are therefore approximately good even when a con- siderable amount of deformation is present. (b) Lhe circular ring. When with diminishing energy the aperture just closes up, the aggregate, as is well known, takes the form of a sphere. 604 Prof. W. M. Hicks on the It ce denote its radius mass=m= 3 TC, 7 = = =— 13), 4 On TO Thus the external energy is less than one quarter the internal. The only other state in which an approximate solution is attainable with the present state of theory is the ring state when the ratio b/a is not large, where } is the radius of the eross section, and a that of the circular axis. Here ma=2raX rh? =2r7ab? = 410 or ab?= — °°, For this case : vw—a)yr+y? where i —= , i= ( i uh (e2+a)*+7¥ (Ga a) a eee il : Sea flow through disk paralle! to the plane ‘TT = of the ring whose rim passes through the point z,y. When brought to rest, theretore, the stream function in the stationary condition is be by and denotes Meas: Par), Se ? Sa x out er , < - where now Ve Te = (L- i) LL denoting log. , In general the velocity at the centre of the ring is less than that of translation. In this case region (2) is also Mass carmed forward by a Vortex. 605 ring shaped. To find the condition that it may be singly- connected it must be possible to find a point on the aXI1S where the flow is zero. Write de eer: he” OTR Y— 7 SX. Near the axis, that is k very small, it is easy to show that es RTD x= 35 8, Ata point of bifurcation dy/dx=0, il aX a ee Caen oo also ee Sl ae near the axis, a whence it easily follows that ¢ 2/3 e+y=a ( ) = el: lj Hence L—i<2n7, =< 625338) b s Se OO a: a For this, and considerably larger, values of b/a, the ex- pressions for x, U will still hold with great approximation, and we shall be able to obtain some intormation as to the singly connected states. For b/a=°0116, y=0, and the boundary has a node at the centre. The two configurations require separate treatment. Singly-connected conjiguration.—Here b/a>‘0116. The equation to the boundary i is given by y=0. It cuts the axis at a point given by (G y 2/3 u| n=y/ ea) —1}. With given b/a, U is caleulated. The equation to the boundary can then be written X=iVa. R ye Or “= a) = ii) Pe ae 3 aV L—2 SSA Oo (Saye) 606 _ Prof. W. M. Hicks on the Now X is a function of k alone. The curve is then traced by giving a series of values to £, and calculating # from this equation. The corresponding ordinate is found from the circle k=const. or (e+a)?+7?= = . Ifa complete set of these circles (corresponding to bipolar co-ordinates) be drawn the points corresponding to given x van be found graphically, although the method is not susceptible of great exactness when the points are in the neighbourhood of the equatorial plane. If the boundary be ales for a given value of A, ‘that for any other suitable value can be drawn at once in the following way. Suppose the new value of A is fo, I thie boundary for A cuts one of the circles in P, draw PN perpendicular to the axis and on it take P'N=f. PN. The perpendicular from P’ to the axis of w (or equatorial plane) will cut the same circle at Q, which is a corresponding point on the new boundary. It will therefore be sufficient to draw the curve ave wtabely for a paraioulas value of A only. The curve J in fig. 2 is drawn for A= A=1 corresponds to b/a=:002 and is ios small to give a tae -connected space. A=2 corresponds to 6/a=-368, - which is far too lar ge for our approximate formule to hold. But this is immaterial for a standard curve from which to draw others for which the approximation holds. For the limiting case A=1:1741 or j= 987. In general 8 Te FICE | Tey a oem These give b/a when f is given, cr f when b/a is given. The table * at the end gives the values of X, log X XS ior a series of values of k feat; in the calculations of pe paper. Curves II, III, V, give the singly-connected boundaries for the three cases of bja= ‘1, :05, (0116—the last being the limiting case. They were drawn graphically from I by the method indicated above. The areas and volumes found graphically from the curves are : Area, * Vol. Tala ae 122008 5% 2°18 a? AB Bapecetaes Pats "478 a? 1°685 a? pu Giugrehae tos Spa "268 a? 1°362 a®. * The values of F, E are taken from the values given in Bertrand’s. Calcul Intégral. Mass carried forward by a. Vortex. 607 In connexion with the construction circles :=const., they cut the axis of x at points 4) Wee 1+H! Di Ue 9 eee so that oe) 1) FAG a , distance of centre from axis = : ue — re —1. Toroidal configuration.—Passing now to the consideration: of the case where the second region is ring-shaped, the boundary is given by the loop of that stream line which cuts itself in the equatorial plane, at a point where the velocity is zero. The value of the stream function for this is clearly negative, since in the aperture between the centre and this: point the velocity is everywhere negative. As before the stream function is given by TX = (aw)X —4V 2". be Under given conditions, i.e. V given, we have to find the value of # which makes dx dx = -O0 when y=0. Substituting this value of # and y=0 in vy then gives the constant value (say x1) of the bounding stream line, and the equation of the latter is y=. The finding of tie root of dy/d#w can be carried out by ordinary approximation when numerical ex- amples are required. Our present purpose, however, is not so much to get the result for a given state (value of b/a) as to follow the changes as the states vary. For this purpose it is best to choose positions for sets of nodes, and calculate the values of U required to satisfy dy/dz=0. "The corresponding value of b/a is then found from U. For instance choose kf, where #/a=(1—k'\/(1+4%’) since now w 1-4, _| | é 2:6 BEC eee 3 4 ! eee & : 2:4 | aT ie : 22 Somme Ee Sa ee —s SSA TY aiiease 1 a he ain joj yeenmene| [|| | PING PAY es OA 7O .CcO. °O§ FLATINUM. the rise starts corresponds to combustion to CO, not to COg. Petrol also has a flat base and a slow approach te the lower limit. ‘The last two are more sensitive to hot-wire ignition than hydrogen. | Ignition of Gases by Hot Wires. 627 Methyl and Ethyl alcohol, fig. 7, take the same mean current, 1:'2 amperes, but the former is more difficult to 7A JN eee Noy LS \ A tm oa a | oe Bs iy SUN, SARTRE ADEE IA j nS ALCOHOL ABSEISGAE POUBLED. p in Seas oe oO ie eel Bee SS Pa SR DWE UN | As PER CENT OF EAS IN AI ignite in the weaker mixtures, the curve rising from 12:0 per cent., the mixture for perfect combustion. The base of methyl alcohol is flat, resembling methane, that of ethyl alcohol slopes rather more than the curves of fig. 5. Ethyl ether is slightly more inflammable, with sharp limits and a flat base. Benzene compares with ethane in maguitude of igniting current and has a practically flat base. There is therefore strong evidence that ignition of most gases by hot wires is within a small range of current inde- pendent of the proportion of combustible gas present. This had been observed by Couriot and Meunier in the case of methane, and is characteristic of certain forms of spark ignition, for exainple of hydrogen and carbon monoxide by break sparks*, methane by impulsive discharge f. All cases of stepped ignition by sparks are examples ¢f the same phenomenon ¢, the difficulty of ignition changing suddenly at each step, where the proportion of oxygen to combustible gas passes through integral values, or as in Paterson and Campbell’s work at critical electrical con- ditions. Since oxygen is the only active gas common to all mixtures and since the igniting current is the same in so * Roy. Soc. Proc. A. vol. xc. pp. 284-285 (1914). + Roy. Soc. Proce. A. vol. xcii. pp. 881-401 (1916). } Roy. Soc. Proc. A. vol. xci. pp. 17-22 (1914). 628 Prof. W. M. Thornton on the many and varied cases, the suggestion is again strong that ignition begins by some Achiont from the fae wire that is dominated by oxygen. The probability is rather against the emission of positive oxygen lions at low temperatures since oxygen is electronegative. Hydrogen is so intimately re- lated to platinum by the phenomena of absorption that its sensitiveness to ignition by platinum wires can be readily understood on the eae that the low temperature emission is of hydrogen ions The action of methane on hot wires is unknown, except that in certain gas-aetectors in which a heated platinum wire forms one an m of a resistance bridge, a cooling effect is co) observed when the wire is placed in seals. mixtures of methane and air. This requires further investigation, but in the present work it has been observed fr equently that wires of all metals o glow mere br ightly at the same current in air than in an inflammable mixture. It is highly probabie that any double layer or surface charge would be removed by the flame. 11. Influence of Gas Pressure. (1) Below Atmospheric. A. further illustration of the indifference of ignition to external physical conditions is given by reducing ‘the pres- sure of the mixture. The curves of fig. 8 are “practically mm: SROUEN /SelPeRlcent LATHE an8 <2 ceo Hail EaNkzosnasnana= | METHAN 1E "$0 Pe. By pau QS Ten DINATES To BES COVBLED|. 0 6S5—_ 6) 63506 620: 2S0« 630 35 («40 «45 «2«450 5S «60 «C65 CFO FS Figc.8 PRESSURE. CM. MERCURY. identical ; the pressure at which ignition begins to fail is between 12°5 and 20 centimetres ‘of mercury, and below 10 em. it was impossible. From this point to atmospheric pressure there is no variation of igniting current. Ignition of Gases by Hot Wires. 629 The rates of absorption and adsorption of gases by metals both vary with pressure, the former as the square root, the latter as the cube root, and it might be expected that any effect depending upon either would vary with pressure, unless the rate of thermionic emission of gases once absorbed were independent of it. One property of a gas that is independent of pressure is its viscosity, and on Lodge and Clark’s view of the origin of the dust-free space around wires which are strongly illuminated* it is at least possible that heat removed by convection should be controlled by viscosity. On the other hand, gases of such widely differing molecular properties as hydrogen, ethylene, and carbon monoxide have the same igniting currents. (2) Above Atmospheric. For the higher ioe glass apparatus \ was replaced by the construction of fig. 9. A wrought-iron tube 32 inch bore and 18 inches long, had at the upper end a cross-piece fitted with a sparking-plug to which the wire was soldered, a quartz window through which it could be observed, and an attach- ment for a pressure-gauge and gas connexions. At the lower end a T-piece had 0 or dinary water-taps fitted, one, B, joined eas to the water main, the other, A, to drain the pipe. At the start the three-way tap C is opened to the atmosphere, A opened to drain the tube. The latter is then closed and B opened, sweeping air or products of com- bustion through C. The appearance of water here indicates that the tube is full. Bis then closed and C turned to the gas supply, A opened and a measured quantity of water drawn off, corresponding to the pressure required, the object being to have at least 30 cubic centimetres of eas to explode. A is then closed and B opened, admitting water and com- pressing the gas. The pressure could be readily controlled up to 100 pounds to the square inch, the highest pressure of the mains. Explosions are heard asa faint click or can be seen through the window. The cock on the gauge must be closed before explosion. . The results with hy drogen and coal gas, the only gases to be had at the time in large volumes, using platinum wire, were as follows (Table Or * “On the Phenomena exhibited by Dusty Air in the neighbourhood of stronely Hluminated Bodies,” O. J. Lodge and J. W. Clark. Phil. Mag. March 1884, p. 214. Phil. Mag. 8. 6. Vol. 38. No. 227. Nov.1919. 2X 630 Prof. W. M. Thornton on the TABLE V. Pressures above Hydrogen Coal Gas atmosphere. 30 per cent. 15 per cent. 0 lb. per sq. inch. 5:45 amperes. 6°35 20 5°50 6:40 40 5°50 6°40 60 5°60 6:45 80 5°80 6°50 100 6:0 70 sree 45 =|; G Y Y ZL REN HU ENSSSSSS SLLILLLELIS Z QUARTZ WIN DOW. WATER MAIN The slight rise is no doubt owing to the cooling effect of the more rapid gas bombardment at higher pressures. When Ignition of Gases by Hot Wires. | 631 the pressure is raised tol-several hundred atmospheres it is impossible to ignite the gas by any current short of fusion of the wire * We have then the remarkable result that from pressures of 20 to 600 centimetres of mercury the same current causes dgnition. This again points to ignition being a trigger effect. 12. Absence of Influence of Electric or Magnetic Fields on Hot Wire Lgnition. The connexions for obtaining a high-tension alternating electrostatic field are shown in fig. 1. Vi oom hy beep Onn senens Lo ralOy Alien 1b 17 40K Wie. eee Calcites os) | 22 Rg 12O A QA 11° 49! 467-0 a tnt ea ee er Aalee OLOG 11 49 450% Ree Sale. cdevuy teeteete a echoes NS Sho LI) 12 44 46-0 fe IPI Se eee BY apo ONG 12 44 59 -7 Ly. Roekssal tac lee meaee TES aS ees o PIEZ Sipeee HAS oles Ls Pier ean. Lass, Olees Pleo Ole se pe Td) Loe ehSees Mean ene ee Meo 13290 645 When the lattice-constants for rock-salt log 2d=°7503541 {at 18°) and for calcite log 2d ="7823347 as deduced in the earlier paper are used for evaluating these measurements, the following wave lengths are obtained :— TABLE VI. Precision-measurements in the X-Ray Spectra. Xe XG). Mean. IP Os Biante erie Mes dscts 1484-52 PES 1 Vit PAP Payee ee a 1484-51 i? — 1484:°52 RIG Tite ae atanelecs hee 1473°44 Pile, @ Thi steateee ns sa: 1473°52 ; = 1473°4 Ply 4: MUA KES nested ums 1279°1 Je Tel.) 13 rigid DY 6) ey aguary cen as 1279-19 ett DAT ity an | joy ee ab JG/ESs pAb eee eee 1241-92 se bs ole MOIGHS Aa ies ae bs oko 1241-90 ee A Oe eos (Wa yy ns leeeen forace ae 1095°53 The good agreement between the values for the wave- lengths “obtained from calcite and from rock-salt strongly supports the accuracy of the fattice-constant of calcite in reference to that of rock-salt. As is well known, the L-series consists of three groups of lines a, 8, and y. The fainter lines of these groups have been measured with. reference to one of the above stronger lines in the same croup. Some very faint lines were not Peco ble in the comparator but were only estimated to 5 mm. In the following Table VII. these wave-lengths are given to five figures only. TABLE VII. — 1 r pine ASU 1B, AEE Ciaten E. Dershem. Gl Sar ane 1675-05 944-02 Ca ed Lo iccinsins L4S4'52 613°85 1484-6 1485 Cae) ee eee 1473-48 618°45 1473°6 1472 ify TOR Cae 1417°7 642-78 a= 1416 Uae ee. «ntsc IAS ISP ee 701-66 1298°7 1298 — [1287-1] 708-08 — 1287 3, ae 1279-17 712-39 1279-2 1278 wo kee. 1260-00 723-23 1260-2 1259 Veal Op maaeel arene 1241 91 733°76 1242°1 1244 —- (1239-5) 735°29 — — s [1220-5] 746-63 1218-7 1220 i (12118) 752-00 am 1210 eee Det icine 2 sists 1203°1 757-43 ~ Bs: [1128-4] 807°57 ae 1129 DEVAN Girccrereieaic L095°58 85181 1096°5 1095 3.) 1065°84 854-98 1065°3 1065 4 ate ae 1059-65 859:97 1058-4 1059 el syircsi's 102647 887-77 1028-1 1025 Wale Mag. 5.0, Vol. 38, No. 227. Now. 1979. 2 Y 646 Precision-measurements in the X-Ray Spectra. In the 4th column are given the values of A. H. Compton* calculated from the reflexion-angles with the above lattice- constant instead of that used by Compton. The agreement is surprisingly good, as Compton only estimates his angles to be exact within 05 (for the stronger lines). A com- parison of our values shows that the probable errors in the measurements of Compton are only about O“1l. It may be remarked that Compton uses a registrating spectro- graph on the ionization method, whereas our- measurements are based on a photographical method. In the next column are tabulated the wave-leneths of Kh. Dershemt. These measurements are not of the same exactitude as the former, but in compensation therefor some fainter lines have been found. Two lines (l, 5) which certainly belong to the L-group of oe are missing. Some of the other fainter lines have also been reftound on my spectrograms. Whether they belong to the tungsten spectrum is still open. Besides ihe values in Table VIL, Dershem grves the wave-lengths 1177, 1071, 1043. Of these, the first probably is the Pb-l line. In some interesting papers Sommerfeld t has given evidence to show that the L-series can be separated into. two groups with constant frequency-differences between the lines. The present measurements show that there is a rather good agreement between the values of the first group, but that there is only approximately the same frequeney- ae In the 3rd column of Table VIL. are given the values of ~ PR where R stands for the Rydberg constant. From these values we find the following frequency-ditterences :-— TaBLE VIII. 6H = a a =, ISG Bal = Ba, — 98-54 OS Y= Yas Bb 98°05 @-% = ¥,--B. = 97-55 It seems from this table that the difference decreases with decreasing wave-lengths. Tund, Physical Laboratory, June 1919. * A. H. Compton, Phys. Rev. June 1916. EK. Dershem, Phys. Rev. 1918. A. Sommerfeld, Ann. d. Phys. li. pp. 1, 125 (1916). ++ -+ ee a LXAIV. Precision-measurements in the X-Ray Spectra. Part III. By Dr. Manne SrecBaan and Mr. A. B. Leipe*. An A-ray Spectrograph for medium wave-lengths. N the two former parts of this paper X-ray spectrographs have been described for long waves in vacuo and for very short waves, both allowing a sufficient degree of accuracy in the measurement of wave-lengths. It is not impossible te let these two apparatus complete one another, and with them to photograph the whole X-ray spectrum-range as yet known. But in this way there will be a region in which the measure- ments are not as reliable as may be wished. For the case of the vacuum-spectrograph this arises from the fact that the X-ray reflexion ata crystal-face is not a surface effect. On the contrary, when shorter waves come into play the pencil may reach a considerable depth in the erystal before being wholly reflected. This, as shown by Blake and Duanef, is very remarkable, sole an ionization method is used for registering the reflected ray. When a photographic spectrogram is taken, the breadth of the spectral lines in comparison with the slit-breadth gives a representation of the thickness of the reflecting layer. But there are in spite of this some reasons which make it desirable not to use a spectrograph of the type described in the first part for. the study of somewhat shorter waves, say between 0°5- iA (Gy An essential condition for the spectrograph giving sharp lines is that the reflecting layer shall coincide with the rotation axis of the crystal- ‘table e, a condition which naturally can only be fulfilled when the thickness of the reflecting layer is small. Farthermore, at the small reflexion- angles corresponding to short waves a little fault in the adjustment may have a great influence. To illustrate these effects it may be sufficient to take a spectrogram of e. g. the tungsten ‘K-series with a rotating crystal spectrograph. With rock-salt as analysing erystal it is hardly possible in the first order to separate the «doublet even with the smallest slit. A spectrograph intended for more penetrating waves may therefore be constructea on the principle already used for the spectrograph described in the second part, that is with the slit behind the crystal. A spectrograph where this is * Communicated by the Authors. + Blake and Duane, Phys. Rey. 1917. te Oe 648 Dr. Manné Siegbalin and Mr. A. B. l.eide on Fig. 1. Precision-measurements in the X-Ray Spectra. 649 realized, and which allows a considerable degree of accuracy in measuring the reflexion-angles, is shown in figs. 1 and 2. As is seen from the figure, the crystal-table is in fixed con- nexion with a high-precision circle-scale which can be turned in relation to the slit- and plate-holder. Fig. 2. To fix the crystal in the required position and for micro- metrical adjustments, the circle is provided with a tangent- screw attached to the arm for the plate-holder. Crystal- table and plate-holder as a whole can finally be turned about the same axis. A circle-scale and nonius on the back of the spectrograph serve to give an approximate adjustment of the crystal to receive the X-ray pencil under the reflexion- angle desired. The plate-holder and slit-stand are mounted on the same bed-plate, and a thick lead cover laid over to prevent blackening through diffused rays. The incoming pencil passes to broad slits of lead which have in a later construction been supplied with micrometer- screws to facilitate the necessary adjustments of spectrograph and X-ray source. As a spectrograph of this type seemed to require a rather 650 Dr. Manne Siegbahn and Mr. A. B. Leide on good specimen of crystal hitherto only calcite has been used as lattice. The measurements of the reflexion-angles were carried out in the following way:—The fine slit (with gold edges) was adjusted so that the normal from the rotating axis to the photographic plate passed through it. The crystal was placed on the table with its reflecting plane in the rotation axis. After this adjustment the spectrograph is ready for use. The crystal is then turned so that the normal slit-plate forms an angle with the erystal-face approximately equal to the reflexion-angle for the spectral line sought. This can be done in two ways: in one case the plate-holder may be turned to the right side in order that the incoming pencil shall be reflected ; in the other case a turning of the plate- holder to the left is necessary. Tt will be seen at once that the place of the line on the photographic plate depends solely on the situation of the crystal in relation to the fine slit. When the crystal after exposure of the desired spectral line on the first side is turned exactly 26+180° and a new exposure made, the two lines will coincide. In general the angle is not exactly 26+180°, so there will be two lines on the spectrogram. The little correction which must be added to the known angle of rotation is readily determined by measuring the distance between the two line-images and the distance from slit to plate. A disadvantage of this spectrograph is that it ranges over only arather small wave-length region for a definite position of crystal and plate-holder. : This spectrograph is suitable for wave-lengths between 2 A.U. to about 0:5 A.U., and so bridges over the range from the vacuum-spectrograph to the spectrograph for very short waves described in the second part. But as shown in the iater paper, good results may be obtained with as short waves as the K group of tungsten (0°2 A.U.). Asa first proof the new spectrograph has been used for a measurement of the Ke line of copper. The essential reason for this was a desired control of the measurements with the vacuum-spectrograph with waves of these lengths. It may be remembered that the Cu lines were the shortest of the long waves for which the vacuum-spectrograph had been used. The result for this wave-length, Ka, of Cu, obtained in the first part of this paper was X= 153736 XU. Precision-measurements in the \-Ray Spectra. 651 As tothe estimated accuracy of this value, it has been foand after continued working with the vacuum-spectrograph, that the method formerly used for measuring the distance rotation axis to plate was not as reliable as desired. This may answer for some units in the last decimal. ", according to Dr. Chapman. Near the sun r cannot be greater than 7.10’, the sun’s radius, so that vy must be about 4°4.10-*. Since n=2, e=4:78.107?9 and m=4.1°'65.. 10- 4 oramme, the aeaellanse momnisn lO. 10!? vemi/sec:”| , Lt is Clean that no beam of anything like the required density could persist for the time required to reach the earth, which must exceed 500 seconds. The only alternative would be to assume that the & particles carried an equivalent number of electrons with them. This would approach the theory propounded below, but reasons will be given to show why it is improbable. Suppose, however, that these a rays could be produced and dispatched from the sun in the required beam. Assume 3A 2 672 Prof. F. A. Lindemann on the their velocity to be that of ThO’ rays, the fastest known, viz.. 2°22.10° em./sec. This corresponds to an energy 1°53.107° erg, so that the a ray could only strike the earth even if 2 aimed at it in the best possible way if a = or HK, <5:2.10", E, being the earth’s charge and a its effective us The value calculated bypy Die: ‘Chapman 1s. oe . e. 6000 times greater. It need scarcely be added that these objections hold with additional force if 8 rays are assumed on account of their smaller mass. It would appear, therefore, that existing stream theories of magnetic storms are untenable and must be replaced by some modified form. Since the storms are undoubtedly due to solar ohencee there are apparently only two alternatives, namely, that the influence is electromagnetic, or that some corpuscular radia- tion intervenes. Ordinary induction effects are obviously impossible. Electromagnetic radiations can onl lonize a gas and separate the charges a distance of the order of the free path of an ion in the gas. This would increase the. electrical conductivity, but Dr. Chapman has given reasons for believing that this would not explain ‘the observed magnetic phenomena. One must therefore consider how the « energy of a magnetic storm, of the order U=10* ergs, could be transmitted by corpuscular radiations. It is clear that charges of one sign are not sufficient, a me mv for since the charge at any one time E,< >, > ClbheEs ZN Z must be enormous, which could only be explained by ussuming a radioactive substance of very short life, or E, must be alle oe E, is small the total charge EH ee be 4 HA HH . large, for U= — = i, which cannot exceed ——. JE Was a YAO) large the number of particles emitted per second must be large, and the mutual repulsion would render anything of the nature of a beam impossible. Therefore approximately an equal number of positive and negative charges must strike the earth. If U=10” ergs is the energy of the storm it is clear that N, the number of particles ie aoe | : striking the earth, is cit If v is so great that the time of transit from the sun is short compared with the duration of the storm, the radius of the beam must be of the order Theory of Magnetic Storms. 673 mE *Ro=2°2.10 em. and the total number emitted therefore TAG 2 oT TaN j : of the order (3 ‘ °) 4, T, being the duration of the To ah a storm, Ry the distance of the sun, and 7, the time of one revolution of the sun. Since the only way to account for a large value of v is to assume the presence of a radioactive substance, and since even in this case it cannot be much greater than 2.10° without leading to an inordinately short life, this number must be large and involve an impossible concentration of radioactive substance in the part of the sun involved. Therefore it seems unlikely that radioactive pro- cesses are involved, and v must be so small that the time of the particles in transit is not short compared with the duration of the storm. In this case of course the number is large, but since there is no need to postulate radioactive origin for the particles this does not matter. Further, as was stated above, there is no undue tendency for the particles to spread since their resultant electric charge is small. The hypothesis to be examined therefore is that an approximately equal number of positive and negative ions are projected from the sun in something of the form of a cloud and that these are the cause of magnetic storms and aurore. It is known that such clouds are emitted in the neighbourhood of sun-spots, as is observed in the protuber- ances, and it will be shown that the observed velocities might well be acquired by a neutral gas under the influence of light pressure. It will further be shown that the gas would become almost completely ionized as the pressure diminished, that it would spread out by approximately the right amount without recombining on its passage from the sun to the earth, that its density on reaching the earth would be of the right order of magnitude, and that the observed magnetic and auroral effects would be accounted for bv the difference in range of the positive and negative particles. ‘The connexion of magnetic storms with the sun-spot period and the tendency of storms to recur after periods corre- sponding to one solar revolution, would on this theory be obvious consequences of the tendency of protuberances to occur above sun-spots. It is beyond the scope of this paper to attempt to explain why masses of gas are ejected by the sun at various spots, and it is sufficient to accept this phenomenon as an observed fact. Strictly speaking it might be considered enough also to accept the fact that the radial velocity of these masses ‘of gas is comparatively great, often exceeding 8.10* em./see. \ 674 Prof. F. A. Lindemann on the Since it has sometimes been suggested, however, that these high velocities can only be caused by electrostatic forces, this point is worth considering, for obviously electrostatic forces could only act on a cloud with a definite charge. It is easy to show that appreciable electrostatic forces cannot exist on the sun. The outer layers of the chromo- sphere must certainly be highly ionized as will be shown below, so that any charges ‘of the sun as a whole would rapidly be neutralized by the emission of ions. This would continue until the electrostatic force E,en aoe the gravitational attraction Mymj7, EH, being the sun’s charge, M, its mass, € and m the charge and mass of the ion i valency n, and f the universal ‘constant of gravity. Since hydrogen ions must certainly always be present ne/m cannot Mof be less than 2°9.10'! and E,=—” cannot be greater than nes mM 4°6.10". Since the capacity of the sun as 6°96. 10! the potential therefore cannot exceed 6°6 E.S.U. or 1980 volts. | It might be imagined that large differences of potential might be caused by a or 8 rays coming from deeper layers. It may be shown that they can only produce a vanishingly small field on account of the high conductivity of the hot ionized gases in the chromosphere. If Xo is the range of an a par ticle in a substance of density 1, its range at density Po. is Ap/py. The fraction escaping from a depth h is therefore 27° sin (1 —cos d) Aarr? =}sin ¢ (1—cos d) h ee ee where cos = If the radioactive substance is distributed evenly therefore the total radiation per unit surface will be 0 alt (1—h/rd) V1 —12)? dh=X(r/8 —1/6) rv times the number emitted per cm.? If x, is the number of particles emitted per gramme per second, the current is. therefore 0°0594 7X9 . Ze per cm.? The current due to @ particles may be estimated in a similar way. Since § particles are absorbed exponentially the total fraction to escape is 2 _ph i i dh \ eo «dw=ip. eo eo Putting w the absorption coefficient equal to pou) therefore Theory of Magnetic Storms. 675 9: ? . 9 . the current per cm.” is a e. Since 1/py is about 100 times larger than 2» for most. substances the current may be ING assumed to be —”— tpg Ne: The electromotive force necessary to balance this is negli- : 24 mu gible. Since plenty of electrons are present X= — —, myer t 2 being the current, n; the number of electrons per cm.’, m the mass and e the charge, v the velocity due to thermal agitation, / the free path, and X the potential gradient. If Ni i is the number Oh atoms per cm.® and a their diame eter, therefore /=1/N,7o*, so that if v, is the fraction of atoms ionized, p, the fraction of the total mass consisting of radio- active matter of average life 7, and a the atomic weight of Bion 1/1830, ome finds X=—1e22 2 2 Ae a Vp a where A is the atomic weight of the radioactive substance. On the earth, and presumably on the sun, since a tempera- ture of 6000° is far too low to influence radioactivity, Als of the order 200 and py about 5. p, must be less than | and 7 greater than 107°, since the sun does not change funda- 26 Na mentally in 30 years, so that X< Lee bae ey uit vi, em. figure is almost certainly still far too large, but even were it accepted and vy, put as low as 107°, the potential gradient in the chromosphere could not exceed ‘1-2 volts per kilometre. It seems certain, therefore, that no intense electrostatic force can obtain in the solar atmosphere, and the high velocity of the protuberances cannot be attributed to electro- static repulsion. There is therefore no reason to assume that they are charged electrically. Prof. Strutt has shown that such velocities cannot be due to ordinar y gaseous expansion, and one is therefore reduced to considering the possibility of their beiwg caused by radiation pressure. Without going into details it is clear that an atom which hy scatters the energy fv subtracts the momentum — from the Cc incident radiation. If the incident radiation is due to an incandescent sphere it therefore acquires a radial velocity hv me? atom will be ionized, 7. e. the vibrating electron removed from it altogether if it absorbs the radiation hyp=eV, e being the charge on the electron. It will then of course no longer , m being its mass. If V is its ionization potential the 676 Prof. F. A. Lindemann on the be able to scatter light unless it recombines. It seems natural to suppose that the chance of scattering a quantum hy is proportional to the intensity of radiation of frequency v present in the incident radiation, 7. e. proportional to 3 a ek —1 suming some definite atomic model it is impossible to form any accurate idea of what happens in a many-line spectrum. The fundamental fact remains, however, thaé the intensity of, in the case of complete radiation. Without as- say. the blue hydrogen line of frequeney v= eee a Bay, J s | Viglen je 4° LG is about 2.10° as great as the intensity of the last hydrogen 4 27 me ie Lee which corresponds to ioniza- line of frequency vy= ~- 73 tion. One might reasonably expect, therefore, that some- thing like 10° quanta would be scattered before the atom was Dh vas git! om and m=1°65. 10-*, the velocity sh increment will be 82 em./see. each time a quantum is scattered by a hydrogen atom. It does not seem unlikely, therefore, that Sieh atoms will on the average acquire a velocity of the order of 8.10’ cm./sec. before being ionized. From this one may conclude that light-pressure is of the right order to explain the velocities observed in protuber- ances, and that there is reason to believe these to consist of an equal quantity of positive ions and electrons. That such clouds of gas will be highly ionized may be shown in a more satisfactory way which takes account of recombination. Taking the case of hydrogen, which is the only substance about which enough is Soper to enable quantitative results to be obtained, it is not difficult to form an estimate of the ionization in terms of the temperature and pressure. As is well known, the equilibrium constant ionized. Takingv= K,= EOP , Where 7, 2, and p’ are the partial pressures of fie ions, electrons, and atoms respectively, is given by Ni (eva E ("s i = aaa Ty Ty yrns — A > e log K, Lp al RE), C,dT + i 2m*me* . , ee Here eV=hy= — ae ees the work necessary to ionize the atom, k is Boltzmann’s constant Ht C, is the atomic heat, N? . ©. 4 oie - o - A ce ne ae Theory of Magnetic Storms. 677 . e ‘ 1 a 4 2V ‘ and 2 the chemical constant. If ''=6000, 2 965-She alii ho atomic heat of the atoms is presumably hyo. 222 (i ae 5/2R—3R-,——-. = 5/2R, hvo (ee? —1)? he since => 26°8. That of the ions may probably be put kV mai oe equal to 5/2R, so that the term ( Rp ( >C,dT reduces to 5/2logT. The chemical constant / has been shown to be 2-303 = 1, 6-+3/2 log A), A being the atomic weight, in all monatomic snhstances examined. Whether tig) may be extended to electrons, however, appears extremely doubtful, for if it were permissible the chemical constant of electrons would be —15, and the thermionic emission would be re- Mneeduoy aytactor of the order 3 10e% (Ti the term 3/2 log A is neglected in the case of electrons S7= —3°7 and age K= ~ 6000° 9 a ; : ; : ,P where wis the fraction dissociated and — wv P the total pressure, whence if 7 is the number of atoms per unit volume, 2 ie Ko nk Obviously therefore ionization becomes important when nkT is of the same order as or smalier than K, ; in the present case therefore where T'=6000°, ionization would be almost complete when n= 10%. Though this estimate is extremely rough on account of the uncertainty about the chemical comets it is clear that the cloud of gas would become ionized and therefore fade out of sight when the pressure became sufficiently low. The next question to be examined is what modification in constitution and shape these clouds of ionized gas are likely to experience on their way from the sun to the earth. Taking the speed as 8.10’ em./sec., a velocity often ob- served, the time in transit would be about 1°9.10° seconds. This is in good accord with the value found by Ricco, who arrived at a time 1°5.10* seconds by comparing the passage of sun-spots with the occurrence of magnetic storms. The most important question as regards “constitution is the amount of recombination to be expected. Taking hydrogen Now ke — 678 Prof. F. A. Lindemann on the ions and electrons, 7. e. positive and negative particles, and taking account of electrostatic attraction, the free path ar 1 eo oy, ma? 1+ —— mua 26. mie : o being the radius of the electron 3! and—5- = b/ 2k | me? 2 the equipartition energy. Putting T=6000° this reduces to- Pel —-——, » being the number of ions per unit, volumm: 2 Since 7 is small and continually decreasing as the cloud of gas expands, it would seem that collisions and recombination may therefore be neglected. ‘The shape of the cloud will consequently only be modified by the movement of the particles under their own initial velocities, for there are no appreciable macroscopic electric charges present and therefore the electrostatic forces are small. A few electrons will probably escape from the cloud, but as soon as the potential of the cloud becomes so high that the work done in escaping is of the order of the thermal energy, this will cease. The resulting electrostatic forces can only cause an expansion of the same order as that due to. the velocity of the ions. Otherwise the electrons will naturally be forced to conform to the velocity of the more massive hydrogen ions. One may therefore estimate the rate at which the cloud expands laterally as somewhat. greater than ie =5.10° em./sec., if T=6000° and A as in hydrogen. If the radial velocity of the cloud is taken as 8.10’ therefore it will expand to a radius somewhat greater than 10!'-cm. by the time it reaches the earth’s. orbit. The length of the cloud will be very much greater, since it is not to be assumed that the radial velocities of the particles in the cloud are exactly the same but rather that the velocities will be distributed about the mean velocity. Suppose, for instance, the velocities to vary from 6.10° em./sec. to 10.10’ cm. Isec., the cloud would take 10° seconds to cross the earth’s orbit, as its length when the particles of le 8.10° cm./sec. reached the earth’s orbit would be 7°5.10! em. Since the earth’s velocity in its orbit is about 3.10° cm./sec., it would on the average take a time of the order of 7. 10 seconds to pass through such a cloud. Dr. Chapman’s Theory of Magnetic Storms. 679 estimate of the duration of an average magnetic storm is 20 hours or 7-2.104 seconds. In general also the less time it takes a cloud to reach the earth, the denser it will be and the less time wili the earth take to pass through it. Thus one would expect the more intense storms to last a shorter time than the milder storms, as is borne out by experience It appears certain therefore that such clouds of ionized gas can exist, and that they would be projected radially from the sun at such a speed that they would naturally spread out enough by the time they reached the earth to account for the observed duration of magnetic storms. The next point to be examined is whether they would produce the phenomena on the earth associated with such storms. The question of the influence of the earth’s magnetic field is difficult to treat. A slight separation of the ee charges would take place owing to the different ratio-of charge to mass, but any large effect would be prevented by electro- patic attraction. ‘Farther, as was pointed out above, a slight excess positive charge of the whole cloud is probable owing to the escape of the more mobile electrons. This would be very small and the ratio of e/m would be of the order 2°5.10° E.S.U. Whether this would give rise to sufficient deflexion to cause the particles to predominate in polar regions is difficult to say, since in the interior of such an ionized cloud displacements of charge might be compensated by induced displacements at the edges. A more serious difficulty appears to be formed by the fact that auroree occur on the average at about 10’ cm. height above sea-level. The range of a hydrogen ion moving at about 8.10% em./sec. would be of the order 3°7.10-4 em. in air at standard temperature and pressure. This means that it would be stopped after passing through a column of air containing 101° molecules per em.? It is usually assumed in working out the composition of the upper air that the isothermal layer extends to an in- definite height. Making this assumption the number of molecules at height Ah above the bottom of the stratosphere is obviously - ae Ah =e Ah NEY + No€ Spe SS Ny, Ng, etc. being the numbers where isothermal equilibrium commences. The total number per em.? through which an ion must pass, therefore, when approaching from the sun would be gA —gAs RT ny — po 4 Ns en 4 oie pee ee ot te) —68C Prof. F. A. Lindemann on the Using Jeans’s value of Sn=10** at 10° em. and assuming isothermal equilibrium (‘7=220°) from there upwards, one finds that particles penetrating to 10’ em. should have passed through the equivalent of 13:1 cm. of Hy», 0°6 em. of He, ()'25 cm. of No, and 0:01 em. of Os, at standard temperature and pressure. This corresponds to a range in air of about 3°6 cm. or an @ particle of velocity 1°7.10° eim./sec., a mode- rately slow ray. Itis clear that the hydrogen ions should be stopped very much higher up if the above method of calcu- Jation is correct. There seem, however, to be some very doubtful assumptions involved : firstly that the air contains hydregen to the extent of one part in 10° at ground-level, and secondly that the isothermal region extends indefinitely from 10° em. on upwards. A number of estimates of the proportion of H, in air are given by various observers ranging from nothing at all to one part in 10*. Now the hydrogen molecules in the outer layers. of the atmosphere would certainly predominate if there were more than one part in 4.10° at sea-level, and it seems strange if this is so that no hydrogen lines have been identified in spectra of the aurora. The hyde ogen lines are some of the easiest to produce spectroscopically, and their total absence leads one to infer that there is very little hydrogen present above 10’ cm., and therefore say less than one part in 10° at sea-level. If this is admitted, helium is the predominant gas in auroral regions with a small ad- mixture of nitrogen, and this is apparently in good accord with spectroscopic results. The second assumption is more important and probably more doubtful. If «x is the absorption coefficient of unit mass of gas of molecuJar weight A, then the energy absorbed by a gramme-molecule of gas at temperature 'T is of the form AF (v, T), which is made up of the difference between the energy absorbed from the layers at a higher temperature T’ and the energy emitted at temperature T. If a gramme- molecule rises at velocity si = V, therefore AF (ae, T)dh C,dT — vdp = -—— arecahe or since po= hE and = =— UN dh, aa eDires al eal BG: a dh Y 16, gV j Theory of Magnetic Storms. 681 It is clear that the constant adiabatic temperature gradient. iat = obtains as long as the energy of convection gV is large compared with the energy interchanged by radiation HB (2; T), When T becomes much smaller than ‘Il’ however, then, eee the gas absorbs, 2. ¢. if x 1s appreciable, F(x, T) must become large and the term t ae becomes important. J The absorption of radiation in the atmosphere is mainly due to CO,, H,O and possibly U3, CO, especially being at any rate par ily responsible for fle isothermal Jayer. But the percentage of CO, in the atmosphere decreases extremely rapidly as the height increases and must diminish, for example, to 1/100 of its original value in 5:6. 10° em. It cannot therefore reasonably be supposed that the tempera- ture of the stratosphere, which is partly conditioned by the CQO,, remains constant up to heights of this order. The same holds good, though at a greater height, for H,O, and it may be doubted whether the feeble absorption of the other gases is capable of overcoming the up and down currents, which must tend to form in an atmosphere in which pressure differences and winds of considerable magnitude are known to exist. These considerations would seem to show that it is impossible a priora to predict anything about the tempera- ture and constitution of the outer atmosphere. The fact that auroree occur at heights of the order of 10% em. cannot therefore be brought in evidence that the particles producing electrification must have a normal range of 3°6 em. in air and a velocity of the order of 1°7.10° cm./see. The last point to be considered is the total number of particles involved, and the conditions they would produce in the upper atmosphere. Since the range is approximately proportional to the mass the electrons hose energy corre- sponds to a potential drop of some 1°8 oie. will be stoy yped at their first encounter. The positive particles will unin until their kinetic energy is used up in encounters or electro- static potential relative to the outermost layer, which is predominantly negatively charged. They will then spread out under their serene repulsion and a radial current will be formed by the inflow of negative and outflow of positive particles on their way to recombine. Dr. Chapman gives the total energy of an average magnetic storm as 10” ergs. If this is equated to the neni energy of the tneident stream, and if one assumes the 682 Prof. F. A. Lindemann on the average speed v to be 8.10‘ cm. as above, one finds the total mu ’ mass M=~ 5 =3'1.10' grammes. In reality, of course, the uv" mass will be greater than this, but how much greater depends upon the efficiency /, of the stream, 7. e. how much of the slowing up of the ions is due to electrostatic attrac- tion by the electrons which have been stopped in their early encounters and how much is due to loss of energy by collisions. If more were known about the outer layers this could probably be estimated; a rough attempt to evaluate the order of magnitude is given below. $1. 107 1 . 10 grammes of air, so that there is no reason to expect if spectr: al lines to show. It does not seem impossible, however, that the green line might be due to the anne element projected from the sun. Since the gas recombines comparatively quickly however, the dilution is in reality far greater than that given above, so that the appearance of the spectr um of the injected gas does not seem very probable. It the substance leaving the sun is a gas of atomic weight A and carries n charges, ‘the total ae EK of either sign is oe ni, F being Haraday’s constant 9°654.10° coulombs or 2°896.10" H.8.U. Therefore This mass grammes will be diluted by at least which is projected onto the earth is 2\Uirold 7 eee 3 Oe I SUNGee in yes if the gas is assumed to be hydrogen. U , The potential difference P cannot exceed hay Vues ie Me a1 in electrostatic units. At first sight this may appear unexpectedly small. It must be borne in mind, however, that the conductivity of a gas increases enormously when the potential drop on a free path becomes large enough for the ion to form new ions by collision. It is clear that such a potential difference could not be materially exceeded. As has been pointed out above, a hydrogen ion moving at 8.10 em./sec. could, at its best, only penetrate a column of air containing some 10!° molecules per em.?. The free path of course varies with the density of the gas. Assuming isothermal equilibrium it may be shown that the distance Ah between the first collision, at which the electrons are presumably stopped, and the vth collision r n= is Ae Disee Se vy? where V is the ionization perentia Theory of Magnetic Storms. 683 the ions would gain enough energy trom the field to ionize by collision after one free path. Since the main constituent above a height of 10’ cm. must be He, A=4 and V=20°5, so that if Ah is of the order 10’ and T greater than 100°, P must less than 2300 volts or 7°7 E.S.U. The above value of 11:1 &* may therefore he considered quite reasonable, in fact one may deduce from this that 4, is probably of the order of magnitude Oknly2. In this case the mass striking the earth is cone 6.10% grammes. The total mass in the cloud, whose radius was found above to be of the order 10! em., \ srowlld therefore be 1-4.10! grammes, by no means an excessive quantity. It corresponds, near the sun's surface at the density estimated above tor complete jonization, toa diameter of about 2.10° em. or nearly 1’ which is of the right order of magnitude. It has been assumed throu ehout Uae IONE, Chapman’ s con- clusion.is correct, namely that radial motion of the char ge on the earth is aniicemneto account for the magnetic phenomena. If one assumes, as he does, that the charges rapidly spread all over the surface of the ear an as seems probable on account o£ the high mobility at such low pressures, there would seem no reason to expect other phenomena than those he accounts for. Changes in the electrostatic potential gradient ob- viously cannot be caused by the two layers of opposite sign. On the other hand, the magnetic effect of the radial flow which occurs as they recombine must be the same as that found by Dr. Chapman when his layer of « particles expands. In both cases there may be a certain difficulty in explaining the sudden inception of a storm, though this point could only be studied with the help of actual records. Apart from this, no real objections appear to exist in the theory set forth above. Though the premises throughout are too uncertain to allow one to claim that the quantitative results confirn the theory, they do not seem in any way inconsistent either with one another or with the accepted laws of physics. Summary. It is shown that existing stream theories of magnetic storms, more especially that worked out bya: Chapman, are untenable, because :— They require incredible radioactivity on the sun. Even if this be granted, the particles could not remain together on: their way from sun to earth on account of electrostatic repulsion. Bven if such a beam of charged particles could exist, it could not impinge on the earth after the first few seconds on account of the earth’s charge. 684 On the Theory of Magnetic Storms. It is suggested that the storms are due to the recombina- tion of electrons and ions in the upper atmosphere which arise from ionized clouds of gas ejected from the sun, such as are observed in the protuberances. It is shown that no appreciable electrostatic forces can exist in the chromosphere, and it is shown that the observed velocities are therefore pr obably due to light-pressure acting on the neutralatoms. It is shown that the gas will become completely ionized and there- fore invisible when the density has become sufficiently low. It is shown that recombination will not take place on the way from sun to earth and that the normal expansion of the gas will increase the size of the cloud sufficiently to account for the observed duration of magnetic storms. It is shown that the usual assu: mption that only pa of relatively high speed could penetrate to the levels at which auroree are observed is probably erroneous, since it implies the existence of hydrogen which does not appear in the spectrum and since it requires the isothermal region to continue to an indefinite height. This is improbable since the percentage concentration of the heavy absorbing gases CO,, H,0, and possibly O3, would’ diminish exponentially with ‘lie height. It is show that the total mass of the particles striking the earth need not exceed 6.10’ on this theory, the iota charge being of the order 2.107) E.S.U. The potential difference which would be formed between the electrons which would be stopped first and the positive ions which would penetrate deeper would be of the order of 1000 volts. It is shown that this is of the order to be expected since ionization by coilision would be important if it were appreciably exceeded and the charges would recombine until some such value were reached. It is shown that the total mass in one of these assumed clouds need not exceed 1'4.10!? grammes and that its. apparent diameter before it becomes completely ionized would be of the order of 1’. It is shown that this theory does not lead to any change in the surface atmospheric potential gradient, and it is pointed out that the magnetic phenomena to which the recombina- tion of these particles would give rise would coincide with those explained by Dr. Chapman’s theory. It appears therefore that the theory leads to results which are of the right order of magnitude when the quantities are examined and that it is free from the inconsistencies inherent in the stream theories promulgated hitherto. Ciarendon Laboratory, Oxford. July 14, 1919. [ag LXXL. The Kinematics of the Kye. By Horace Lams, £’.R.S.* HIS subject should be of interest to mathematicians, for it is almost the only field where the theory of Finite Rotations, so elegant geometrically, finds a practical application. The theory of the movements of the eye was investigated very completely by Helmholtz t, and forms a considerable chapter of his Phystologische Optik}. ‘The mathematical part of the exposition is rather elaborate, and has I think somewhat obscured the essential simplicity of the matter. There is a decided gain in this respect, as I wish to show, if we adopt a purely geometrical treatment, based mainly on theorems of Donkin and Hamilton, which were indeed anterior in date to Helmholtz’s work but could hardly have been known to him. : A brief recapitulation of the physiological principles involved may be convenient. Fortunately these are simple, and (I believe) well established. The movements considered are of course relative to the head, which for the present purpose is to be regarded as fixed. The precise attitude of the head is unimportant, but for tacility of statement it is supposed to be erect. The eye- ball rotates very approximately about a fixed point O §, and so far as the muscular equipment is concerned it has the nsual three degrees of freedom. But in its normal operation the movements are so coordinated that there is vir tually freedom only of the second order, the position of the eye as a whole being completely determined by the direction of the “visual axis ||,’ 7. e. of the line drawn from O to that point of the external field which is the object of direct vision. This s ‘Donders’ Law’ (1847). The limitation to freedom of movement which it asserts is essential in order that the same object, seen in the same position relative to the head, should affect the same elements of the retina whenever the gaze is directed to the same point of it. If this were not the case * Communicated by the Author. + “Ueber die normalen Bewegurpgen des menschlichen Auges (1863), reprinted 1 in Wissenschaftliche Abhandlungen, vol. 11. p. 860. t Ist ed. 1866; 2nd ed. 1896. § Here and elsewhere in the present subject the conditions are slightly idealized. || German ‘ Blicklinie. For reasons connected with the structure of the eye this is not exactly coincident with the line drawn to the object from the anterior nodal point. The distinction is hardly important except in the case of very near objects. hel. Mag. 8: 6. Vol. 38. No, 228. Dec, 1919. 3B 9 686 Prof. Horace Lamb on the the interpretation of visual sensations would be enurmously complicated. The law which defines the position of the eyeball in terms of the direction of the visual axis is attributed to Listing (1857). It has been tested by various observers and may be regarded as established, aut all events for normal eyes. There is a certain ‘primary’ direction OA of the visual axis, or line of sight, to which all others are referred. Roughly, this may be described as the position assumed when with head erect we look tewards a distant point of the horizon, straight in front. For the present purpose a more precise definition is unnecessary. Listing’s statement is equivalent to this, that when the visual axis takes any other direction OP the position finally assumed by the eyeball is. that which would be derived from the primary position by a rotation about an axis perpendicular both to OA and OP, through an angle AOP. In the actual. transition from Ox to OP, the gaze may wander about in any manner, but the final position of the eyeball must always be the same, in virtue of Donders’ law. The various directions of the visual axis may be distin- guished by their intersections with a spherical surface of arbitrary radius described about O as centre This will be referred to as the ‘spherical field.’ If ne represent the primary, and P any secondary position, the above rotation is. conveniently indicated, after the manner of Donkin and Hamilton, by the ereat-circle are AP. Helmholtz investigates in the first instance the relation between any two secondary positions, represented (say) by P and Q on the spherical field. This can be found very simply. It is known that, if ABC be any spherical triangle, successive rotations of a rigid body represented by 2BC, 2CA, 2AB will restore the “body to its initial position, and accordingly that successive rotations 2BU, 2CA are equi- valent to 2BA*. It follows, in the present application, that the transition from one secondary position (P) to Tee (Q) is equivalent to a rotation 2X Y, where X, Y are the middle points of the ares AP, AQ respective ely. For the transition may be supposed made, first from P to A, and then from A to Q. * The theorems, in this form, are due to Donkin, Phil. Mag. (3) vol. xxxvi. p. 428 (1850), and (4) vol. 1. p. 187 (1851). They were given independently by Hamilton, ‘Lectures on Quaternions,’ pp. 328- 330 (1853) as direet interpretations of quaternion formule. Donkin’s simple proof will be familiar to readers of Nouth’s § Advanced Rigid Dynamics.’ > el a Kinematics of the ye. 687 Hence the transition from P to any other position what- ever, such as Q or R in the figure, is represented by some great-circle are through X; 2 e. it is equivalent to a rotation about some axis at right angles to OX. This line Joe dle a.) asi ae R OX is therefore called by Helmholtz the ‘atropic line’ for the position P. He remarks, further, that the existence of an atropic line in the case of infinitely small displacements from P is an immediate consequence of the limitation to two degrees of freedom, without restriction to any particular law, such as Listing’s. For if OI, OJ be the instantaneous axes for two such displacements, any other displacement from P will be compounded of rotations about these lines, and will therefore consist of a rotation about some axis in their plane. We have thus an ‘atropic line,’ viz. the normal to this plane. A straight line in the external space is represented in the spherical field by a great circle. If this circle passes through the primary point A, then as the fixation point travels along the line the successive portions are imaged on identically the same elements of the central region of the retina *, in virtue of Listing’s law. In other words, the various parts of the line appear to be, as they actually are, exactly superposable. It is obvious, moreover, that Listing’s is the only law which fulfils this condition. The same statement does not, however, hold with regard to straight lines which do not meet the primary position of the visual axis. Exact superposition in the case of all straight lines is in fact intrinsically impossible, * It is not necessary for the purposes of the argument to assume that the retina has any special form, spherical or other, 3B2 088 Prof. Horace Lamb on the apart altogether from the validity of Listing’s law. For consider any triangle PQ of the spherical field, and imagine the fixation point to travel round itin the order of the letters. In order to satisfy the above condition of exact superposition the movements of the eye must consist of successive rotations represented by the arcs PQ, QR, RP. By a beautiful theorem due to Hamilton *, the result would be a rotation about OP through an angle equal to the spherical excess of the triangle PQR. This is incompatible with Donders’ fundamental principle that the position of the eyeball depends only on the direction of the visual axis. It is therefore a matter of interest to ascertain what lines, if any, in the external field satisfy the requirement of super- position as tested by the retinal image. Now if the eye can fig, 2. A ce) rotate continuously about a fixed axis (IJ) through QO, the visual axis will trace out a small circle on the spherical field, and the projection of this from O on any plane (or other surface) will evidently give a line having the desired property. * ‘Lectures on Quaternions, p. 335. The theorem is most easily proved by examining the successive positions of the great-circle are which is initially coincident with PQ. Kinematics of the Hye. 689 But in order that this type of motion may be possible, the atropic line, being at right angles always to the axis of rotation, must deseribe the parallel great circle (MN). It P be any point on the small circle, and FOX the corresponding position of the atropic line, we have seen that AX=XP- As a particular case AM=MC, in the figure, and it follows that the small circle must pass through the ‘occipital point’ @ of the spherical field, 7. e. the point diametrically opposite to A. Conversely it appears that as the fixation point describes any circle through © the eye rotates about a fixed axis, and the required condition is fulalled. The various circles through Q are called by Helmholtz ‘direction circles,’ since they correspond to lines in the external field which have apparently a constant direction. Moreover it appears that circles which have a common tangent line at Q will correspond to lines having the same direction. It is of course only the portions of the circles within a moderate distance (say 40° at most) from A which have any real sienificance. An interesting alternative proof of the preceding result may be given. Let the spherical field be projected from © on the tangent plane at A. In this projection let PP’ represent an element of a line of constant direction, and let AA’ be the line whose image falls on the same linear element of the retina when the gaze is directed to A. By hypothesis AA’ has the same direction whatever the position of P on the line considered. The position of the eyeball hig. 3. (S) when the visual axis points to P is derived from the primary position by a rotation now represented by the straight line AP, by Listing’s law. Since angles are unaltered in stereo- graphic projection, it follows that. the angles P/PQ, A'AQ in the figure are equal. Hence PP’ is parallel to AAl; the locus of P is a straight line; and the corresponding locus in the spherical field is a cirele through Q. The lines in the external field which are apparently straight are accordingly those which project from O into direction 690 Prof. Horace Lamb on the circles of the spherical field. In particular, if we consider a plane perpendicular to OA, the lines in question which lie in this plane are a doubly infinite system of hyperbolic ares *. The annexed diagram (fig. 4), which is similar to the one given by Helmholtz, shows two sets of such curves, the boundary corresponding to an angular distance of 45° from OA. Ifa strong after-image be formed of a short horizontal line at A, the directions assumed by it as the eye is directed to various points of the plane are indicated by one set of these curves. The after-image of a short vertical line at A is in its various positions tangential in like manner to the other set fF. The fact that lines which are really straight, but do not meet OA, appear to be concave towards A is an immediate consequence. This completes the mathematical theory. The physio- logical basis of Listing’s law is a more abstruse question. * Their equations are of the form av ty*— (la+my)?+ 2a(lv+my) =0. + We have here (in principle) one method of testing Listing’s law. Kinematics of the Lye. 691 There is mabnine in the arrangement of the muscular attach- ments of the eyeball to sugyest this particular law as natural or advantageous. An explanation must therefore be sought elsewhere. The question is discussed at length by Helmholtz from the empirical standpoint *. The arg ument is long and intricate. The following investigation retains the central idea, which is interesting on mathematical as well as other grounds, but is simplified by the introduction of an auxiliary assumption which has (I think) something to be said for it. The assumption is that in the primary position of the eyeball the atropic line, which necessarily exists for infini- tesimal displacements, coincides with OA; i.e. that the instantaneous axis for any small displacement is peupendicut ir to OA. In the first place, a horizontal displacement of the visual axis requires only the action of the lateral muscles (rectus internus and externus), and invoives therefore ouly rotation about a vertical axis. The atropic line is accordingly assumed to lie in the horizontal plane through OA. Its posi- tions in the two eyes must moreover obviously beymmetrical, Next suppose the visual axis to be slightly displaced vertically from OA. If the atropic line did not coincide with OA, this displacement would involve rotations of the two eyes whose components about OA would have opposite senses. This would be disastrous for binocular vision in the most important part of the field. Let us next suppose that the gaze is directed to a point P of an object of small angular dimensions; let Q, R,S, ... be adjacent points of the object; and let », g, 7, s, ... be the points of the retina respectively affected. If the gaze be shifted from P to Q, this means that a certain impression is now transferred from Y to p, whilst others ere transferred from 7 to 7’ (say), s to s’, and so on. For the correct inter- pretation of visual sensations it would ie desirable that exactly the same transfers from 7 to *’, s to s', &c. should be consequent on the transfer from ¢ ‘to i whatever the initial direction of the visual axis. When the object is near the centre (A) of the field this condition is sensibly fulfilled, but in excentric positions a complication is introduced by the component rotation about OP which attends the transition trom P to Q f. Hellowing Helmholtz we may regard any such component * Loc. cit. The mathematical calculation appeared in the first edition of the Phystologische Optik. A revised version is given in J %ss, bh. yol. 11. p. 896 (1883). tf An attempt is here made to condense the rather lengthy argument of Helmholtz, which should be studied in the original. 692 Prof. Horace Lamb on the rotation as an ‘error.’ Such errors are unavoidable, but we may (still following Helmholtz) inquire under what law regulating the ocular positions the mean square of the errors involved, under similar conditions in different parts of the field, is a minimum. Any position of the eyeball may be defined on the usual plan by three angular coordinates. We denote by @ theangle which the visual axis (OP) | makes with its primary position OA, by W the inclination of the plane AOP to the horizontal plane ‘through OA, and by ¢ the angle which some eee (OPQ) thr ough OP, fixed in the ey eball, makes with the plane AOP. For definiteness this plane OPQ may be taken to be that which in the primary position is horizontal. On Listing’s law we should then have ¢6= —, but this is not at present assumed. Since the position of the eyeball is determined by the direction of the visual axis, we regard } as a function of the independent variables @ and W, to be ascertained, By a known kinematical formula any small displacement involves a rotation whose component about OP is dot cosGdy. ©. 3) 2) ae If ds be the displacement of P on the spherical field, making an angle e with the plane of 0, we have d0=coseds, sinOdy=sineds. - Sey Hence (1) becomes 1 Sf eoset st 1 al S% #2088) sine | ds. : ee The mean square of this for all directions of ds is 1}/0¢ 1 ! 2 5 (35) +5 & +0088) ds. Sa We have to take the mean value of this in the field over which the visual axis can range, for a given standard of ds. Kinematics of the Eye. 693. Denoting the mean value by Ids”, we have = all sin (55) a ee +cos é) A dO dy. (5) We take the field to be circular, of angular radius a, so that the limits of 0 areQand«. The limits of w may be taken as 0 and 27, but it is to be noticed thatif @ be supposed to vary continuously its values at these limits will not be the same. Thus for a given infinitely small value of 6 will be small for yw=0; it will change to —7 (nearly) as ar increases from 0 to 7; and again from —7 to —27 (nearly) as w further increases from 7 to 27. The two sides of the line y~=0 are therefore to be included as part of the boundary of the region considered. Taking the variation of the integral in the usual manner, we find [sin SS es 5g] 1 =0 [= fe a\ | (ap tooele Be a? oO a9 w=o Sin @ ee ) 1 oy { \ | vp (si 9) an = tae ie | ob dé dw. (6) Hence we must oe / sin 02 = g Sues 9 9? oe + bene Wea a( to with the boundary conditions ($6) =0 for aimee en a (8) and ES ov the values of 6@ for W=0 and w=27 being necessarily equal. An obvious solution of (7) is GaN ert EMEN ON cc YOR hush ion tek! tae CEA) which also satisfies the bound: ary conditions. On reference to (3) we see that in order that the rotation may vanish for 6->0, in accordance with our assumption, we must have W=2n + cos ah —=Or anh values! Of Gs hi) .(9)> OE eye A ay Nl! «4s heel Ct) 694 Prof. Horace Lamb on the Hence A= —1, and o= Sp. e s e e e s (239) regard being had to our convention as to the zero of reckoning of @. As already stated, this is the analytical form of Listing’s law. If we put (loo amet ec ae aes the equation (7) takes the form ag 0: se re The function +7 is single-valued and_ periodic with respect tor. The general solution, subject to the condition of finiteness for 0- Sn is therefore d+ w=3(A, cossp+B,sinsw)t-§, . . (15) where s=0,1,2,3,..... The boundary condition (8) requires Ace= 0.5 a= Osi acy a le rior for s>0, whilst Ay=0O in virtue of the convention as to the origin of @. We are thus restricted to the solution (12). ‘he formula (8) for the rotation about OP now reduces to Bec —— dam SO SIMNE;dS.. 2) iis The mean square of this over the region considered is 2 ean Ae tan? 3@sin 9d0= eee : 5) as (18) 4 sin? de J, sin? D For «=40° this ='0316ds". The error of mean square is therefore '178ds. The maximum error, corresponding to oe e= im, is tandeds, or ‘364ds if 2<=40°. For comparison I have caleulated what would be the error of mean square on the hypothesis that the eye works like an aliazimuth instrument. If we measure @ from the vertical OZ, and denote the azimuth by >, we have 6=0, everywhere. Fora displacement ds of the visual axis, ina deren ‘making an angle e with the vertical circle, the rotation about that axis Is cot@.dssine. * 4) 0 eee The mean square of the errors is therefore Ids’, where cos” 2 _—— 4a (eee dre omen Ge eT ee a ee Kinematics of the Eye. 695 We must first integrate with respect to 0 between the limits $7 + y, where COS a COs DU A OG MI aunt 22 ).") cos wv arowe suppose y=0 in the primary position. The result is te ane: flog tan Gar+4y)—siny}dw. (22) we hi SIN 144 0 Fig. C. Putting «=40°, I find by a quadrature [=-0692, and the error of mean square is accordingly °263ds, as against “178ds in the former case. ‘the errors near the centre of the field are about twice as great as on Listing’s law (but eae distributed), and the maximum error is taneds, "839 ds if a=40°. The preceding argument of Helmholtz has been repro- duced (in a simplified form) on account of its mathematical interest. The question presents itself, however, whether the sardinal advantage of the fact that the eye ‘conforms to Listing’s law does not consist simply in this, that the test of superposition is fulfilled for straight lines which pass through the centre of the field of view, in the primary position. Tf this view be adopted the ‘above inyvestiga- tion still retains its value as showing a further consequent advantage. [ 696 ] LXXIT. On the Molecular Theory of Solution. By SAMUEL CLEMENT BRADFORD *. ae the proposition, in 1887, of the dissociation theory of solution by Arrhenius, little attention has been paid to the attractions of the solute and solvent particles both for themselves and for one another. As such attractions. undoubtedly exist, they must be taken into account in a complete theory. In the present paper an endeavour has been made to treat the subject from this point of view and to show that the method indicates immediately the order in which substances arrange themselves with regard to any given property of motion The idea of molecular attra ictions being involved in the pro- cess of solution appears originally to have occurred to Newton. But Handl [1872] seems first to have suggested that the solubility of a substance is conditioned by the establishinent of a state of equilibrium between the number of particles. leaving the solid bodyeand the number returning. Nicol’s views [1883, 1884, 1886] are particularly worthy of mention. He considered that since the solution of a salt in water, in the form in which it crystallizes at the ordinary temperature, is usually attended by absorption of heat, the energy involved in the liquefaction of the solid must be greater than that evolved by union with water. He used the fact, that the specific gravity of a solid is affected by the temperature at which it crystallizes, as a criterion of the mutual attraction of the solid molecules at that temperature; and showed that, where solubility increases with tempera-— ture, the cchesion of the solid molecules is weakened by heating, and conversely, as in the case of sodium sulphate, decreasing solubility with rise of temperature is aceccm- panied by increasing attraction of the solid molecules for one another. Nicol also thought that loose packing of the molecules, or large molecular volume in the solid state, is. due to weak attraction between the molecules, and concluded that the greater the molecular cohesion of a salt the less its. solubility, and vice versa. Recently Traube [1909] has also approached the subject. from the molecular point of view. Unforiunately he considers merely the attraction of solute and solvent, which he calls their “cohesion pressure.” This cohesion pressure is indicated by the effect of a solute on the surface tension of a liquid. The more a solute diminishes, or increases, the * Communicated by the Author. On the Molecular Theory of Solution. 697 surface tension of a solvent, so much the smaller, or the greater, is the force with which it is attracted by the liquid. Traube shows that when salts are arranged in order of their cohesion pressures, as indicated by their effect on the surface tension of water, practically the same order is retained in the case of almost every property of solutions. I. Cohesion pressure. H K > Na. Il. Capacity for hydration. Cs Br> Cl> OH > S80, > CO. H>NH,> Li(hyd.) >K>Na. V. Reduction of vapour-pressure, ete. NO,ClO;>NO,>Br>Cl>OH>F, etc. ete. In the case of organic liquids, Traube showed that their solubility follows the order cf their cohesion pressures without exception. It will be seen later that this is because the molecular cohesion of organic liquids is smaller than that of water. Unfortunately for Traube’s theory, the solubility of solid substances follows the reverse order of their cohesion pressures. 698 Mr. 8S. C. Bradford on the The medern electrical theory of atomic structure appears to allow more insight into the nature of solution than bas hitherto been possible. Atoms should attract one another in the same way as electric doublets. Usually there would be a residual field of force surrounding a molecule, with the result that molecules should attract one another, whether alike cr unlike, with a foree varying inversely as the fourth power of the distance. A theory of molecular attraction of this kind was treated at some length by Sutherland in a series of papers in this journal. The idea has recently been developed by Sir J. J. Thomson [1914]. Representing the force between two atoms A and B as Now saN oe where Cy and Cg are the moments of the corre- sponding electric doublets and r the distance between their centres, the force between two molecules AB and CD eile ee Ce 2) ez atoms and radicles will be additive. The forces arising from electrons and positive charges will depend on the orientation of the atoms and will not always be radial. It is only by taking the mean value that they can be treated as radial and determined by distance. In the position of comparatively close packing, as in liquids, the attraction of a pair of unlike molecules may be influenced by their configuration, as affecting the distance of the electric charges, and may differ considerably from that deduced from their mutual attractions or from their behaviour in regard to other molecules. This effect must be taken into account in attempting to calculate solubility. It is the more likely to come into play in adsorption by solid substances, in which the orientation of the attracting molecules is more or less fixed, and may account, at least in part, for selective adsorption. Thus, as has usually been supposed hitherto, the electrical theory implies attraction within a narrow radius as a universal property of the molecules of matter. It follows, as Laplace supposed, that solids and liquids alike must have an enormous surface energy due to the unbalanced molecular attractions at their surfaces, while internally the forces are in equilibrium on account of the equal molecular attractions en all sides. Since, according , which shows that the effect cf g to the kinetic theory, all molecules of matter, above the absolute zero of temperature, are in constant motion, while their momentum changes continually in direction sad amount, molecules at ine surface of a Molecular Theory of Solution. 699 body which have a momentum sufficient to overcome the sur- face forces will escape as vapour. But, as the molecular cohesion cof solid substances must usually be greater than that of liquids, the vapour tension of solids must be less in the same degree. Certain solids such as camphor and iodine have appreciable volatility. Usually solids are non-volatile. However, this appears to be due in part also to the aggregation of the molecules in the solid state. Any cause which tends to balance the molecular forces at the surface will increase the number of molecules which are able to escape. The surface energy of two solid bodies can be considerably reduced by bringing them into such close contact that the particles at the two surfaces come within the range of molecular attraction. Under such circum- stances the compensation of the surface forces allows the migration of particles from one substance to the other. A familiar instance is where pieces of lead and gold are pressed together. | | In exactly the same way the surface tension of a solid must be reduced by introduction into a liquid. Although the adhesion of the liquid and solid molecules will usually be less than in the case of two solids pressed together, the contact at the boundary surface is the more complete, and particles which cross the surface are able to move away, so that solution in a liquid is more rapid than in a solid. Thus it is evident that solution must be of the same nature as vaporization. The presence of a solvent causes a diminution in the surface forces, so that many more particles have suflicient energy to cross the surface layer. Representing the force between two molecules as M,M naa uo ce ae where M, and Mz are the resultants of the moments of the atomic doublets in each molecule, it follows that the adhesion of a molecule of solid to a molecule of liquid will usually be less than the cohesicn of the solid molecules themselves and greater than the cohesion of the liquid mole- cules, Therefore, the particles at the surface of a solid body in contact with a liquid will still experience a resultant force acting towards the solid, although this force will be less than in the absence of the liquid. Only such particles at the surface of the solid will be able to escape into the liquid as acquire a velocity normal to the surface which is equal to or greater than, a given value s, such that their momentum 4 — 700 Mr. S. ©. Bradford on the in this direction is sufficient to overcome the unbaianced surface forces. And, of those that so escape, any which come again within the range of attraction of tke solid particles will be reclaimed, so that solution will continue until the number of particles leaving and returning to the solid is equal. As in the case of the vaporization of a liquid, the kinetic theory shows, in so far as it is applicable, that in this state the number of molecules in unit volume of the liquid 7g, will be equal to the number of those in tnit s2 volume of the solid, np, multiplied by e © where « is the most probable speed of the molecules, that is : Na=npe 8 LS 2) a The value of s is given by the relation that the momentum normal to the surface, 4ms’, of a particle of the solute having this velocity is Just sufficient to balance the forces acting against it. The force tending to prevent the solution of a solid particle, with greater cohesion than that of the solvent, is the difference between the attractions for it of the solid particles and of the particles of the solvent. It will therefore be M52 MSM, proportional to aa ;-, Where the suffixes s and w Ss Vsw correspond to the solute and solvent respectively. The average distance, 7sx, separating particles of solute and solvent will usually differ from that between the particles of the solid, partly on account of the natural difference in molecular volumes and partly because of the change in volume due to the different attracting forces. Solution of a particle of solid will, however, be favoured by the difference in the attractions of the particles of the solvent for the solute particle and for themselves. This may be repre- M >M w Whe sented as proportional to age a Thus the resultant SW Ww force hindering solution is proportional to gw Me ie. 4 i 2 gMsMee Mu? @ Tf the molecular cohesion of the solute be less than that of the solvent, the adhesive forces will be greater than the cohesion of the solute but less than that of the solvent. The solution of a particle of solute will be favoured by the difference between its attractions to other particles of the a a 4 LES V sew ps Molecular Theory of Solution. 701 ‘solute and to those of the solvent. Solution will be hindered by the difference between the attraction of the solvent particles for themselves and for the particle of solute. Thus the resultant force opposing solution will be proportional to M,?_, MM. | M,? fe — my Feats any Gen bp Vw Psw I's which is of the same form as before. This is the initial force when solution commences. As it proceeds, it would be necessary to take into account the attractions exerted by the dissolved particles of solute. In a somewhat similar way Bingham [1907] deduceda rather general expression for the surface forces in the special case of the miscibility of two liquids. Having obtained a complete expression for the force opposing solution, and knowing the numerical values of the terms, it would be possible to calculate the work done by a particle in crossing the boundary layer into the solution. By equating this to the momentum 34 ms’, the resulting value of s could be substituted in (2) giving a numerical expression for the solubility. In default of this desirable result, the approximate formula (3) for the force opposing solution, in conjunction with (2), allows certain deductions of practical utility to be made. Evidently solubility will be the greater, the smaller the value of f, though not directly in inverse proportion. Solu- bility willincrease with temperature proportionately with the increasing momenta of the solute particles. Hxceptions must be due to changes in the molecular forces, or in the distance through which they act. With solutes having greater mole- 2 cular cohesion than that of the solvent, the term = will be Ss the greatest, and the force opposing solution will be the greater, the greater the cohesion of the solute particles. But, from the second term in (3), the greater the cohesion of the solute, the. greater its adhesion to the solvent. And since this must be reflected by the increased surface tension of the solution, it follows that the more a solute increases the surface tension of its solvent the less its solubility. The series (I.) and (II.) quoted by Traube show that this is the ease for salts. Conversely, as in the case of organic liquids and gases, dissolved in water, where the cohesion of the solute is less than that of the solvent, the middle, or adhesion, term in (3) will ‘be greater than that representing the cohesion of the solute iaie Mag. 5.6. Vol. 38. No. 228. Dec. 1919. 3 C 702 Mr. S. ©. Bradford on the particles, and the force opposing solution will decrease as the cohesion of the solute particles and their adhesion to: the solvent, or ‘cohesion pressure,” increases. This is the rule already found by Traube. The capacity for hydration, regarded as a function of the molecular forces, must be related to the adhesion of solute to solvent. This property, therefore, should follow the order of Traube’s cohesion pressure and the opposite order of the solubilities of substances which increase the surface tension of the solvent, as in series (III.). Similarly solutes which increase the surface tension of the solvent must increase the internal pressure, and since their com- pressibility decreases with increase of internal pressure, the order of the compressibilities of sclutions -(1V.) will be the inverse of that of the cohesion pressures, or directly as that of the solubilities of the solutes. As Gilbault [1897] has pointed out, the decrease in compressibility of solutions of salts does not suggest dissociation of ions, which ought to be accompanied by great expansion owing to the release of chemical combination. If the diminution of vapour pressures. and depression of freezing-points of solutions are related to increase in surface tension, the effects should be in the same order as in (V.) and (VL). The last series quoted by Traube gives the order of the molecular volumes of salts. As has previously been noted, it was first suggested by Nicol that molecular volume is related inversely to molecular cohesion. The apparent molecular volume of a substance may be supposed to include the actual volume occupied by a molecule, together with the space in which it is free to move, which corresponds to Traube’s co-volume. It is natural to regard this as inversely proportional to the forces of molecular attraction. Jt follows that those saits which have the larger molecular volume will be the more soluble, as is found to be the case. However, the apparent molecular volume of a substance in an condition will depend on the internal pressure to which it is subjected in that condition, consequently changes in molecular volume will occur with change of state, whether of the pure substance or by solution. Therefore substances in solution will generally be subject to a different intrinsic pressure than in the pure state. And surrounding each particle of solute, whether in true or colloidal solution, there will be a thin skin in which the intrinsic pressure will be intermediate between that of solute and solvent. The relations between compressibility, molecular volume, surface tension, and other properties have been investigated by a number of Molecular Theory of Solution. 703 workers. Since, however, the intrinsic pressures of organic liquids are not very different, the small change in internal pressure on mixing will not greatly affect the molecular volumes, and, as Holmes and Sageman [| 1906, 1909, 1913 | assumed, such liquids must mix together in the same way as inelastic spheres. From this assumption they were able to deduce the molecular complexity of a number of liquids. Although molecular association may indicate strong mole- cular attraction, this extra affinity will have been satisfied in the aggreyated molecules and they will have a smaller resultant field of force. Moreover, in any case, the mole- cular volume is the resultant of molecular forces. It seems, therefore, that molecular volume may be taken as indicating the weakness of the surrounding field of force whether the molecule is simple or complex. No cases are known of solid substances which have an infinite solubility without previous melting. With liquids the condition of complete miscibility is of importance. From (2) and (3)it is obvious that liquids will be completely mixcible when the average energy of their particles is able to overcome the force opposing sulution. It follows that, at low temperatures, liquids will be completely miscible if their relative molecular volumes fall within certain narrow limits, which will increase as the temperature is raised. If liquids are arranged in order of their molecular volumes, having regard to their degree of association, they should bein order of their solubilities. Liquids which are adjacent should be mutually miscible. Liquids remote should be immiscible. Holmes and Sageman found this to be the case, without exception, for a large number of different liquids belonging to all classes of organic compounds. This suggests that the closed solubility curves found by Rothmund for certain substances may be due to a change in the relative degree of association of the two liquids, or to some intramolecular rearrangement causing diminution of molecular volume. If the aggregated molecules of one of the liquids were to break up with increasing temperature more rapidly than those of the cther, or if, in any way, the difference in molecular volumes increased, the smaller molecules would become immiscible until their energy had increased sufficiently to overcome their increased mutual attraction. Coming to the solubility of gases, the first two columns in the following table are usually quoted in text-books from Bunsen’s fioures for the solubility of different gases in water and alcohol, accompanied by a remark to the effect that, 3 C2 704 On the Molecular Theory of Solution. although the order of solubility in each liquid is the same, no general law is evident by which the actual solubilities are determined. Consideration of molecular cohesion and adhesion immediately throws much light on the relations. Bingham has advanced evidence in confirmation of the - hypothesis that the critical pressure is a measure of molecular attraction at the corresponding temperature, since this attraction is just balanced by the repellant forces due to kinetie energy. The critical pressure in each gas is given in the table as well as the corresponding temperature. | Solubility. y ie Critical | Critical Gas. | Pressure. Temperature. Water. Alcohol. | | a Ng Ne AW, SE OS 2 S| ee pans Sulphur dioxide......... 43-56 14455 | 79 | 155-4 ‘Sulphuretted hydrogen) 3233 | 954 | 887 — 100 [Nitrous oxide .....2....-. 0778 | 3268 | 750 | 354 'Carbon dioxide ......... 1-002 Oe Ts | a Sie iCarbon monoxide ...... 0:0243 0-2044 35°9 |\— 141°] Oxyeongl ober eee 0:03 0284 || 500 — 118-0 INiitRogeMn\-.ha.4- acre ace 00145 0'1214 30'0 —1460 NELy dozen’ pies. eene- C0193 0-0673 14-0 — 240°8 Since the molecular cohesion of gases is less than that of either solvent, their solubility should be the greater, the greater their molecular cohesion, and the less the cohesion of the solvent. Therefore gases should be more soluble in aleohol than in water. Moreover, since the value of the factor My, corresponding to the solvent, in the second term of (3), is much greater than that corresponding to the attraction of the solute, the difference in the solubilities in the two liquids should increase as the molecular cohesion of the gas grows less. ‘These deductions are well borne out by the figures in the table. It is fair to assume that the molecular attractions of the gases are affected by the temperature of observation. That the solubility of a gas increases as the cohesion of solvent diminishes is substantiated by Christoff’s experi- ments on carbon monoxide [1906]. He determined the solubility of this gas in water, aniline, nitrobenzene, benzene, toluene, chloroform, ethyl alcohol and acetic acid, and found that the solubility of the gas increased as the surface tension of the solvent diminished. Thermal Conductivity of Solid Insulators. 705: References. Binewam (1907). Amer. Chem. J. xxxviii. p. 91. CuRISTOFF (1906). Zezt. physik. Chem. lv. p. 622. GILBAULT (1897). Zeit. physik. Chem. xxiv. p. 384. Hanpu (1872). Wien. Akad. Anz. 1872, p. 125. Hormes and Saceman (1906, 1909, 1913). J.C. S. Ixxxix. p. 1774 ;. xev. p. 1919; exiil. p. 2147. Nicon (1883, 1884, 1886). Phil. Mag. (5) xv. p. 91; (5) xvii. p. 537 ; (8) xxi. p. 70. Tuomson (1914). Phil. Mag. (6) xxvii. p. 757. TRAvBE (1906). Ber. deut. chem. Ges. xlii. p. 86. PXXMTL. Lhe Thermal Conductivity of Solid Insulators. By W. M. TuHornton, D.Sc., D.Eng., Professor of Electrical Engineering in Armstrong College, Newcastle-on-Tyne™. HE mode of conduction of heat in substances such as quartz or paraffin-wax is shown, by their deviation from the Wiedemann-Franz law, to be different from that in metals. Free electrons are few, and heat energy consists of elastic vibrations of atoms about fixed positions. From the following experimental results it is probable that the forces that control the elastic movement of atoms in solids are those by which sound is transmitted. In the latter case all movement in the wave-front is in the same phase; in the transmission of heat there is complete confusion of motion, and its rate of propagation is by comparison extremely slow. Both movements are, however, quite definite in type, and there should therefore be some relation between the con- ductivity of heat and the velocity of sound in a solid. insulating medium. From inspection of the recorded values of these constants it was observed that while the square of the velocity of sound V is equal tothe ratio of the elasticity to the density p, the coefficient of conductivity of heat & is equal numerically to their product, so that t=Hp=V°p’. This simple relation holds with surprising accuracy over a wide range of solids, as shown by the following table. There are few materials for which all the coefficients are known and, with the exception of certain glasses, practically none in which the measurements have been made on materials of known composition or in the same state. The figures given are the most probable values taken from the tables of Landolt and Bornstein, Kaye and Laby, Everett, and Kempe; and for convenience reference is made to these rather than the * Communicated by the Author. 706 Thermal Conductivity of Solid Insulators. original memoirs. The elasticity of pure graphite was deter- mined specially by the author, and the velocity of sound where not known has been calculated from E and p. The conductivities of crystals though functions of elasticity are too complex to satisfy the above relations. Thermal Velocity of ‘ ; Conductivity, 4, : Density, Elasticity, Sound, V. me hae in calories. Me tee a: po we {Oe ee coe 10-8. 10 10 a S b. 25 ea aaa woe oo 2:66, K.1is 63, Kei: 3°0, cal. 181 ery ee fused Flint Glass ...... 29, K.L. 48, LB. AA ACT, 14-1 14°1 143348: rowiie. sk 2:5. KK Miss disse eae 5:4, KL. 18:3 183 18°3, L.B. Steet eae eee D8 Kelly. — co uelaaise Dioseacals 2, 92-0) 992, L.B. Graphite ......... Die Se 4 is 4°65, cal. 4 ~) Pass 12°0, K.L. Marble 25.0 826252 OF Kila ee Sie Sod Ocal: 7:0 7-0 7 gl eed TEN Mahogany, Hond. 62, K. SSG kes oily. cal. 0°54 0-52 0:5, K.L. eal ee. ae ne oh saline woe e by 2 wee Os ys Eee 0:45 0-44 0-4, K.L. HCO rece ce ae “Oi (Kelas, PEAS eK SE: 1-74, cal. 0°25 0-25 0:22. KE iParafin Wax; )2-2) °91,b:B. elo EB: 1:30) LB. 0137 O14 0°14), L.B. V. Indiarubber... 1:02, K.L. -:05, K.L. me nee 0:05 ee 0-45, KL. CDE ieee gt 0°24, K.L. 0055, cal. 48, DB: 0-0013 00013 013, K.L. | It was not to be expected that a quasi-solid such as indiarubber should conform to the rule. Its velocity of sound calculated from E and - is 0°7.10° centimetres a second; the cbserved value (K.L.) is 0°07.10°. In either case the values of Ep or V’p? are of a lower order than the conductivity observed by Lees, though the numeral is the same. The elasticity of cork calculated from the velocity of sound is 0:0055.10". Its value for steady stress is about one half of this; Ep=-00132 . 10", or exactly one hundredth of Lees’ value 0°13. The cellular structure of cork makes it difficult to say how far the vibrations are transmitted by the walls or contained air. Baking the cork to a dark brown softens it and improves its value as a heat insulator. The cell-walls are then seen under the microscope to be slack and collapsed, so that they undoubtedly have some part in the transmission. On the other hand, pumice-stone has a con- ductivity much the same as cork and has a more rigid structure. The agreement between the values of k, Ep, and V’p’, for substances of such different composition as quartz and wax, indicates that there is some fundamental process, a conse- quence possibly of the electrical structure of matter, certainly independent of molecular structure, beneath these phenomena. On the electron theory of matter mechanical forces are A Positive Ray Spectrograph. 107 eventually expressed in terms of electric charge or the move- ment of charge. Resistance to compression is a repulsion between the structural elements of matter possibly of the nature of an electrostatic repulsion between electrons which form the outer frame of atoms. The sudden compression of a sound-wave then gives rise to electrostatic forces of recoil and the velocity of sound is the rate of uniform: transmission of these through a loaded ether. On the other hand, the heat energy of irregular motion may be considered as that of alternating atomic currents attracting and repelling at random and so transmitting the motion electromagnetically. It is possible to extend this to a general consideration of thermal conductivity in. insulators as an electromagnetic phenomenon. The object of the present note is to direct attention to the above relations between /, KH, p, and V which must be explained by any theory of conduction in non- metallic solids. The chief practical qualification of a material as a heat insulator is that it should be light and inelastic. LXXIV. A Positive Ray Spectrograph. By F. W. Asron, M.A., D.Sc., Clerk Maxwell Student of the University of Cambridge™. [| Plate LX. | | ae analysis of positive rays by electric and magnetic fields giving deflexions at right angles to each other has been very completely worked out by Prof. Sir J. J. Thomson7. Alternative methods have been suggested by Dempster{ and others. Dempster’s arrangement depends on the knowledge of the potential through which the rays have fallen in the discharge-tube, and is therefore only practicable in the case of low velocity rays. Positive rays obtained from an ordinary discharge-bulb vary both in mass and velocity. An electric field will spread them into an ‘electric spectrum’ with deflexions proportional é c e e . o —53;a magnetic field will spread them into a ‘magnetic vw e spectrum ’ with deflexions proportional to —.. In Thomson’s Mv method of crossed deflexions, in which both fields are applied e e é . . td simultaneously, rays having constant — will lie on parabolas$§ m * Communicated by the Author. , + ‘ Rays of Positive Electricity,’ p. 7 et seq. t Phys. Review, vol. xi. p. 816 (1918). § Luce pe l2: 708 Dr. F. W. Aston on a so that masses can be compared by measuring the ordinates.. This method, though almost ‘ideal for a general survey of masses and velocities, has objections as a method of precision, many rays are lost by collision in the narrow canal-ray tube,. the mean pressure in which must be at least half that in the discharge-bulb; very fine tubes silt np* by disintegration under bombardment; the total energy available for photo- graphy falls off as the fourth power of the diameter of the- canal-ray tube. The first two can be overcome, as will be described below, by replacing the brass or copper tube by fine apertures made in aluminium, a metal which appears to suffer no appreciable disintegration, and by exhausting the space between these apertures to the hichest degree” by means of a subsidiary charcoal tube or pump. The falling off in intensity of the parabolas as one attempts to make them finer is a very serious difficulty, as the accuracy and resolving power depend on the ratio of the thickness to the total magnetic deflexion ; and if we increase the latter the electric deflexion must be increased to correspond and the parabolas are drawn out, resulting again in loss of intensity. Methods of increasing the intensity of the spot. The concentration of the stream of positive rays down the axis of the discharge-bulb is very marked, but there is good evidence for assuming that the intense part of the stream occupies a fairly considerable solid angle. This suggests the: possibility of an increase of intensity by means of a device which should select the rays aimed at a particular spot on the plate whatever direction they come from. For example, a thin gap between two coaxial equiangular cones would allow the rays to be concentrated at the vertex. The dimensions of the patch formed would be roughly those of one given by a cylindrical canal-ray tube of diameter equal to the width of the gap. ‘The increase of intensity would therefore be considerable; but the method is not easy to put into practice, and, in tlie case of deflexions through large angles, would necessitate a curved photographic surface. Clearly the simplest way of increasing the intensity of the spot without increasing its dimensions, at any rate in one direction, is to use two parallel straight slits. In the case of the method of crossed deflexions this device would only be of use in a special case such as the resolution of a close. double, as the parabolas will only be sharp at points where. they are parallel to the slit. - ep. = Positive Ray Spectrograph. 709 Possibilities of focussing positive rays. The very great accuracy attained in the spectrometry of light depends largely on the fuct that a considerable solid angle of divergent rays from a point source can be brought to a point image by means of a lens. It is of importance to. inquire if any such convergence can be applied to rays of charged particles by any electric or magnetic device. As regards the ordinary lens the problem appears rather hopeless, but electric or magnetic analogues of the cylindrical lens can be made theoretically in several ways. Thus magnetically homogeneous rays diverging from a point or slit source S (fig. 1) will be brought to a first-order Fie. 1. focus Fif the integrated intensity of the field traversed by any one ray is proportional to its angular distance from the line SF, e. g. by the use of parallel pole-pieces of wedge-shaped section. Such a ‘magnetic lens’ is not of much immediate value as the magnetic spectrum of positive rays is very complex. The electric spectrum, on the other hand, possesses one very important simplicity, namely, that the distribution of intensity in itisa property of the discharge rather than of the particles carrying it, and is to a great extent the same for all particles: in other words, the value of mv? giving the brightest result for particles of one mass will in general give the brightest result for all. Fie. 2. A variety of electrical analogues of the cylindrical lens are possible. In fig. 2 divergent rays of constant mv? passing through the field between two charged plates whose sections are two concentric coaxial rectangular hyperbolas will be brought to a focus on the line passing through the source 710 Dr. F. W. Aston ona and the centre of the hyperbolic system. For the field between plates of this form is such that its intensity varies directly as the distance from this line. Again in fig. 3 a beam of such rays generated by the use of a point source and an annular slit, if passed between two charged concentric spherical surfaces. will travel in great aircles and come to a foeus on the axis of the system. Principle of the Spectrograph. The above devices will give: bright spectra of the rays in : De me : | : cog terms either of —~ or — ; what is actually required is an Neece e : ? ; anode m arrangement which will give a spectrum in terms of — only e -and, at least over a small range, independent of rv. This is done in the present apparatus as diagrammatically indicated —_onT we owen oe 8 1 ; t i] t tx Ss, 5, Po+ oat ; Lge os epee, Ciena ae oo) < a = ZS Roo in fig. 4. The rays after arriving at the cathode face pass through two very narrow parallel slits of special construction S, S., and the resulting thin ribbon is spread out into an Positive Ray Spectrograph. 711 electric spectrum by means of the parallel plates Pj, Pe. After emerging from the electric field the rays may be taken, to a first order of approximation, as radiating from a virtual source Z% halfway through the field on the line 8,8). A group of these rays is now selected by means of the stop or dia- phragm D, and allowed to pass between the parallel peles of amagnet. For simplicity the poles are taken as circular, the field between them uniform and of such sign as to bend the rays in the opposite direction to the foregoing electric field. If Gand @ be the angles (taken algebraically) through which the selected beam of rays is bent by passing through fields of strength X and H, then é Ov? = IX — @), ‘and- ¢o=LH— (2), where /, Lare the lengths of the paths of the rays in the fields. Equation (1) is only true for small angles, but exact enough for practice. It follows that over the small range of § selected by the diaphragm @v? and gv are constant for all rays of given e/m, therefore eu peg and et ROE ih -so that 07 _ 26g when the velocity varies in a group of rays of given e/m. In order to illustrate in the simplest possible way how this relation may be used to obtain focussing, let us suppose the angles (exaggerated in the diagram) small and the magnetic field acting as if concentrated at the centre O of the pole- pieces. If the length ZO=), the group selected will be spread out to a breadth 656 at O, and at a further distance r the breadth will be 180 + (80 +86) or 50[0-+r(1+ 39) |. re) Now as the electric and magnetic deflexions are in opposite directions, @ 1s a negative angle. Nay @6=—6’. Then if b> 26’, the quantity (3) will vanish at a value of r given by r(p—20')=b . 26’, which equation appears correct within practical limits for large circular pole-pieces. LZ Dr. F. W. Aston on a Referred to axes OX, OY the focus is at rcos (p—20’),. rsin(d—26'), or 7, b. 26’: so that to a first-order approximation, whatever the fields, so long as the position of the diaphragm is fixed, the foci will all lie on the straight line ZF drawn through Z parallel to OX. For purposes of construction G the image of Z in OY isa convenient reference point, é being here equal to 40’. It is clear that a photographic plate, indi-. _ cated by the thick line, will be in fair focus for values of e/m over a range large enough for accurate comparison of masses. | The arrangement, which has a distinct resemblance to the. ordinary quartz spectrograph, gives very complete control. The field between the plates can be adjusted to allow the brightest part of the electric spectrum to be used which, as has been shown, is in general the same for all normal rays. under steady discharge, and the values of e/m can be com-- pared very accurately from the positions of their lines relative to those of standard elements which can be brought to any desired position on the plate by varying the magnetic field strength. Preliminary results. In order to test the method before making the somewhat elaborate camera, a temporary apparatus was set up using an, existing camera, the plate being in the position indicated by the dotted line AB. Under these conditions the focus can only be good at or near the point B. The results so far obtained are exceedingly promising, and show that as far as intensity and sharpness of the lines are concerned no serious difficulty need be apprehended. Plate IX. fig. 1 is a photo-. graph taken with an electric field but with no current | assing through the magnet. It shows the undeflected spot as a sharp bright line—with a patch of fog above it due to some internally reflected light—and the electric spectrum of the positive rays spread out below. In fig. 2 the magnet has been turned on, other conditions being identical with fig. 1. It will be seen that although there is no diaphragm in this apparatus and therefore practically the whole of the electric spectrum is in use, yet the rays corresponding to the hydrogen molecule are concentrated as a sharp bright line 1-4 cm. above the undeflected spot and displaced a little to the right as the maynetic pole-pieces are not set truly vertical. Fig. 3 was taken with a smaller field, showing the hydrogen atom near the focal point and the molecule below it very much out of focus. Plate 1X. fig. 4 was taken with a much larger magnetic: Positive Ray Spectrograph. 713 field, the bright line with its four fainter companions is due to carbon and its compounds CH, CH., CH3, CH,. Lines corresponding to still heavier particles are seen as in- distinct patches which would come up in turn to the focal point if the magnetic field were still further increased. These results were obtained with residual gas from charcoal, the slits were ‘(05 mm. wide, and the current in the main discharge-tube roughly of the order of one milliampere at 30,000 volts. The duration of exposure was 2 minutes in the first three cases and 8 minutes at a rather lower current in the fourth. These remarkably short exposures indicate clearly the great -advantage obtained by the combined use of a slit system and a focussing arrangement. The other parts necessary to complete an apparatus suitable for a general investigation of the relative masses of positive rays are now being constructed, with which it is hoped to obtain results comparable in accuracy with those determined chemically. For masses not too great or widely separated an accuracy of one tenth per cent. is by no means impossible. If anything like this order is obtained, the composition of atmospheric Neon—element or isotopic mixture—will be settled beyond dispute (one of the prime reasons for this work) and several other problems laid open to direct attack. It is as well to point out in view of future developments, that second-order corrections in focus are clearly possible by varying the section or even the figure of the pole-pieces and electric plates; but it is not proposed to employ these refinements until such incidental difficulties as small stray fields due to electrification by the rays themselves have been successfully overcome. In conclusion the author wishes to acknowledge his in- debtedness to many friends for their kind advice and help, ‘in particular as regards the mathematical analysis of some of ‘the more complex systems considered which led up to the one finally adopted. Also to the Government Grant Committee -of the Royal Society for defraying the cost of parts of the apparatus used in the practical investigation. Summary. Precision methods of positive ray analysis are discussed and means of improving the brightness of the beam analysed are suggested. Theoretical electrical aud magnetic analogues of optical -cylindrical lenses for focussing positive rays are shown to be possible. 714 | A Positive Ray Spectrograph. A form of positive ray spectrograph giving a focussed spectrum depending solely on ratio of mass to charge is described. Actual photographic results obtained with a preliminary apparatus are submitted showing the great accuracy possible. by the method with which it is hoped to compare masses to one tenth per cent. Cavendish Laboratory. August 1919. APPENDIX. The Construction of the Shit System. The very fine slits used in this apparatus were made with comparative ease as follows:—A cylinder of pure aluminium about 10 mm. long by 5 mm. wide is carefully bored with a hole 1 mm. diameter. The resulting thick-walled tube is then cleaned and crushed with a hammer on an anvil until the circular hole becomes a slit about °3 mm. wide. Contin- uation of this treatment would result in a slit as fine as required giving the maximum resistance to the passage of gas, but its great depth would make the lining up of a pair a matter of extreme difficulty. The crushed tube is therefore now placed between two V-shaped pieces of steel and turther crushed between the points of the V’s at about its middle point until the required fineness is attained. Practice shows that the best way of doing this is to crush until the walls just touch, and then to open the slit to the required width by judicious tapping at right angles to that previously employed. With a little care it is possible to make slits with beautifully parallel sides to almost any degree of fineness, ‘01 mm. being easily attainable. At this stage the irregularly shaped piece of alumininm is not suited to accurate gas-tight fitting ; itis therefore filled with hard paraffin to protect it from small particles of metal &c., which if entering cannot be dislodyed owing to itsshape, and turned up taper to fit the standard mountings. These in the present apparatus are taper-holes in the back of the cathode and in a corresponding brass plug at the ends of a wide tube 10 cm. long. When turned, the paraffin is easily removed by heat and solvents. ) 174 OE) \de. " eu = } 2ar(1—r) § The probability that € does not lie between +e is then 9 / 1—Erte(saq—)) f ae O 2 2e7 22/271 —7) (o> ant) z (7) 7 Now, no matter how small e and 7 may be, it is always possible to choose n large enough to make this less than 7 ; accordingly, by making n great enough we can make the probability that m/n differs from r by more than any quantity assigned beforehand as small as we please. This is Bernoulli’s theorem. This does not, however, give the probability that m/n will tend to a limit as n tends to infinity. Hor if e be a small quantity fixed beforehand, the necessary and sufficient con- dition that m/n tend to a 7 as a limit is that a value of mp ean always be tound such that for all values of n greater than no, mjn—~r shall be less numerically than e. Now if «x be great enough to make e~~ small, we have the relation 1-Eriz= “— {14 0(4)}. _ ee & A a and the probability that m/n does not lie between + e€ is therefore, when n is great enough, cee | (A (ir) ')§ seo a, (HS? eh. € nT Aspects of the Theory of Probability. 719 Phe probability that any value of m/n for n>np lies outside these limits is therefore not greater than the sum of these expressions for all values of n from Ng + 1 to infinity. We see that this is less than 2. Pen aa 21) sa nge2/200 41+ 0S z)} € No {1 4e-@? W191) 1 go 262/2r Gees ae (10) The sum of the series is finite and independent of ny; hence we see that ny can always be chosen so as to make the probability that, for all values of n greater than 7, the value of m/n will lie between rte, differ from unity by as small a quantity as we like. The proposition required for the validity of Venn’s theory is : 2 can always be chosen so as to make the probability that, for all values of n greater than 7, the value of m/n will lie between + +e, exactly equal to unity. These two propositions bear a close resemblance to each other, but they are not equivalent. In fact, in consequence of the existence of modes of selection for which m/n does not tend to 7 as a limit, we know that the second proposition must be false. The first, on the other hand, has just been proved true; butit does not even establish a high probability for the proposition that m/n tends to r as a limit in any particular case. For it has been shown only thata certain result will have a very high probability when a single value of é has been assigned ; but there is no reason to infer from this that the probability i is high that it will hold for all values of ¢ whatever, which would Tape ie tie ete m/n were to tend toa limit. The dificulty is somewhat similar to that in the theory of infinite series, with regard to series that “ con- verge with infinite slowness.” Il. The Mathematical Theory of Probability. An essential assumption in order that analytical methods may be applicable to the theory of probability must now be stated, namely, that a correspondence can be established between positive real numbers and the propositions to which the fundamental notion of probability is applicable (relative im,each case to the appropriate data) which shall have the Following properties. 3 D2 720 Miss Wrinch and Dr. H. Jeffreys on some 1. To each combination of proposition and data corr esponds. one and only one number. 2. If in one combination the proposition is more probable relative to the data than in another, the number corresponding to the first is greater than that corre- sponding to the second. . If two propositions referred to the same data are mutually exclusive, the number corresponding to the proposition that one of them is true is the sum of those corresponding to the two original propositions. 4. The greatest and least numbers correspond to those combinations and only those in which the data imply that the proposition is true or untrue respectively. Several writers have defined probability as * quantity of belief,” or somewhat better, “ quantity of knowledge”; these are somewhat vague terms, and the transition from thesé expressions to the number series has usually been carried out without any explanation. Yet the use of numbers for the comparison of probabilities at all was perhaps the greatest advance ever made in the theory. The above assumptions are independent ; they are involved implicitly in every theory of probability yet introduced ; and. with their aid it is possible to make some progress with a logical theory. In the first place, we can show that the number corresponding to a proposition incompatible with the data is zero. For let a datum be that «=1; then on this datum the propositions e=2 and #=3 are both false, and each corresponds to the number a, where a is the least possible number of those involved in the correspondence. Further, e=2 is incompatible with «=38; hence by axiom 3 the. probability that one of them is true is 2a. But the proposi-_ tion “ #=2 or e=3”’ is incompatible with the datum “ #=1,” and therefore corresponds to a. Thus 2a is equal to a, and. a is therefore zero. If, then, probabilities are to be repre- sented by numbers, zero ‘must be the least number involved ; but adjustments could be made in our assumptions which would allow any other number to represent the minimum on the scale. If we divide all numbers of the series by that corresponding to a proposition implied by the data, all the above axioms wil apply equally to the numbers of the new series. We shall henceforth use the notation P (p:q) to denote the number of this series corresponding to the proposition p on the data g; we have P(q:¢)=1, and P(not-q: Di 0. P(p: g) may be read “the probability of p given q.’ . Consider two propositions p and g which are not mutually eS) Aspects of the Theory of Probability. 721 exclusive, referred to data f. Then the following four propositions are mutually exclusive, namely BiG ei PY se Pd Then by axiom 3 ECO) CP gry) (Plog <7) pee ge) pees Rpg i) es PWG) — peg Ih) tee (pis 9 77) HE CH pig: )- _ By addition we find EO eae BY Oat E Cpl. gar) sen Ge) which is regarded by Jevons and de Morgan as axiomatic. The second axiom yields as an obvious corollary the famous “principle of sufficient reason”; according to this, equal probabilities are assigned to propositions relative to data when the data give no reason for expecting any one rather than any other. In discussing the problems of probability t Poincaré, after disposing of the view of Laplace and Boole, seems in- clined to consider this principle as the only possible basis of the theory. Substantially the same view is held by Jevons. There is, however, an objection to basing the whole theory on the principle of sufficientreason. Tor the only way of passing from the notion of ‘‘ more probable” to the numerical esti- maté of probability in any particular case is to discover some set of mutually exclnsive and exhaustive alternatives, from which we can pick out some by our judgment as more probable than others; the most probable on the data then receives the greatest numerical estimate. But if we restrict ourselves to cases where we can obtain a set of alternatives that shall be all equally probable, we are arbitrarily limiting the field to which the theory can be applied. We could, indeed, only deal with those cases where some proposition that is certain on the data can be expressed as the disjunction of a number of equally probable and mutually exclusive propositions; the probability of any proposition that can be expressed as, or is implied by, the disjunction of any sub-class of these could then be assessed by means of the principle of sufficient reason and axioms 3and 4. Now there is no reason to believe that the notion of probability is applicable to no * ~~p denotes the proposition that p is false, and p.g denotes the proposition that p and g are both true. Thus -~p.~gq denotes the proposition that p and g are both false. The proposition that at least one of p and q is true is denoted by p v q, or the disjunction of p and q, + La Science et U’ Hypothése, 1904, 213-245. 122 Miss Wrinch and Dr. H. Jeffreys on some propositions other than those expressible in this way ; and it is habitually employed in scientific practice and every day life in cases where it seems likely that such expression is im- possible. Most of the problems of inverse probability, for instance, seem to introduce propositions not so decomposable. If then we wish to retain the customary applications of the theory (and this seems desirable at any cost), we must assume that axiom 2 is correct. The assumption that information can be obtained from the notion of ‘‘ equally probable ” alone without that of “‘ more probable,” which seems as intelligible a priort anyhow, demands that propositions can be decomposed - in this way in all these cases, whether there is any warrant for assuming this possibility or not. Thus axiom 2 is prefer- able to the principle of sufficient reason as a primitive proposition, since it covers as much ground and involves fewer assumptions. The use of the principle of sufficient reason in the cases where it is applicable leads to a proof of another proposition which is an axiom in Jevons’s theory. Suppose we have a class of n propositions, of which we know that one and only one is true, and any one is as likely to be true as any other. Then if any m of them are selected, the probability that one of these mis true is m/n. Let g then denote the proposition that one of these mis true. Consider another class of the original propositions, and let » denote the probability that some member of this class is true. The probability that p and g are both true is then the probability that seme member of the common part of the two sub-classes is true. Let the number of propositions in this common part be /. Thenif h denote the data we have at the beginning, we have P(p.q:h)=l/n lm mn == igang) i) eas he We see that all cases where the probabilities of propositions can be determined by decomposing a certain proposition into a finite number of equally probable alternatives can he treated in this way, so that the relation P(p.¢:h=Ppt gh) PG) ee is always true when the principle of sufficient reason is, applicable. But is there any reason to suppose that this relation still - Aspects of the Theory of Probability. 723. holds when the principle is not applicable? Some further assumption is necessary before it can be proved in these cases, and various suggestions could be offered that would bridge the gap without making it necessary to suppose that the relation is known @ priori in these cases. There seems, however, to be little or no ground for deciding between them, and the proposition may as well be assumed to hold in general without further discussion. From the propositions so far assumed or proved, with judgments of greater, equal, or less probability in particular cases, the mathematical theory of probability can be developed. Another point in connexion with the theory may be briefly mentioned. All that is strictly necessary in order that the notions of probability may be capable of logical treatment is that combinations of propositions and data can be arranged in a series so that whenever a combination A is not more probable than another, B, B shall not precede A in the series. With suitable assumptions regarding the position in the series of a combination, such as the disjunction of two contra- dictory propositions referred to the same data, a theory could be constructed. There is no reason save convenience why the number series should be the one employed for this purpose. So long as we confine ourselves to those cases where a proposition certain on the data can be decomposed into a finite number of equally probable alternatives, and the proposition whose probability is to be estimated is expressible as or equivalent to the disjunction of a class of these, the number series is obviously adequate ; in fact the series of all rational proper fractions in ascending order of magnitude would be adequate. This latter series is, however, at once found to be insufficient when we attempt to deal with cases where the number of equally probable alternatives required to cover the case considered is infinite. This difficulty was thought to be removed by using the series of all the real numbers less than unity instead of that of the rational numbers. But the question that arises now is, whether the series of all the real numbers is itself adequate for the purpose, and the answer seems to be in the negative, for there are evidently cases where the use of infinitesimals is necessary to a complete theory, and the discovery of others, necessitating the introduction of infinitesimals of different orders, is practi- cally certain. For instance, suppose we are given that 2 is a whole number, and that all whole numbers are equally probable values of w What is the probability of any particular value of «, say 1053? Clearly it is not finitely different from 0 ; for if it were X say, we could find a whole 724 Miss Wrinch and Dr. H. Jefireys on some number M whose reciprocal would be less than A; but 1/M is the probability of M being 1053 when there are only M possible alternatives, and the probability cannot be increased by increasing the number of alternatives. Hence the pro- bability that # is 1053 is less than any finite number, contrary to what was assumed. It is nevertheless different from zero, for then there would be no means of distinguishing between the probability of this, which is a perfectly possible pro- position on the data, and that of a proposition known to be impossible on the data. Hence this probability is less than any finite number, and yet is different from zero; in other words, itis an infinitesimal, in the original sense of the term*. Again, we can see that the probability of a particular real number chosen at random being rational is infinitesimal ; so is the probability that a function is analytic, given that all functions are equally probable. Nowa complete theory of probability must cover all these cases; but so long as we are confined to the series of the real numbers that is impossible ; for if this has C members, the number of possible functions whose values are real numbers is C°, which is greater; hence problems arising in connexion with the probability of func- tions demand the use of a series for comparison whose members are more numerous than the real numbers. Such series are known ; and perhaps one suitable for the purpose may be constructed which will include among its members the real numbers themselves. III. On Probability Inference. The characteristic feature of the type of inference with which classical logic is primarily concerned is that given the premises it is possible to establish the conclusions with absolute certainty from them. In many cases, however, such a result is unobtainable when it is nevertheless possible to * M. E. Borel remarks (Legons sur la Théorie des Fonctions, 1914, p. 184) that “there is a true discontinuity between an infinitely small probability, z.e., a variable probability tending towards zero, and a probability equal to zero. However small be the probability of the favourable case, this is possible; whereas it is impossible if the pro- bability be zero..... The same is not true of continuous probabilities ; the probability that a number taken at random may be rafional is 0; this must not be considered as equivalent to impossibility.” This use of zero to denote the probability of both an impossible alternative and a possible alternative with no finite probability seems likely to lead to confusion. The introduction of the conception of a limit does not help matters, for in making a single trial the probability of success is quite definite, and involves no notion of a limit. Aspects of the Theory of Probability. 125 show that the conclusion has a certain probability relative to the premises; an inference of this kind may be called a ‘“ probability inference.” The establishment by this means of a high probability in favour of the conclusion relative to the premises is often as useful as the inference that does not involve the notion of probability. The course thus indicated is always followed in empirical generalization, for in such a generalization it is never possible to establish the certainty of the conclusion from the data. The principles employed in such inference are therefore of extreme importance ; but as yet they are not well understood. Detailed treatment is most applicable to the type of probability inference known as sampling induction, and numerous discussions of this have been given, but even here various errors seem to have survived. ‘T'be problems capable of solution by this method are analogous to the following. Suppose that a bag contains m balls, an unknown number of which are white. Of these p+q have been drawn and not replaced; p of them have been white and g not white. What is the probability that the number of white balls in the bag is n? ) It is assumed that the balls are indistinguishable before being drawn, so that at any stage any individual ball is as likely to be drawn as any other. Let f(n) be the prior probability of any particular number of white balls. If n were the true number of white balls in the bag the probability that p white balls and q others would be picked in p+q trials would be ae . It follows that the prior Pra probability of a particular pair of values of p and g for a n(i m— given n is fom Hence, by the law of inverse probability, which follows easily from the proposition pg k= Eps alia Cg: hy: the probabilities on the data of particular numbers of white balls are in the raiio of the probabilities of the actual values of » and gq for these numbers of white balls; thus we find that the probability that any particular value of » is the true number of white balls in the bag, given the composition of the sample, is f(n) Os De "CoH Sn f(r) "Cp 220 — BGs } : (1) where the summation is to be extended to all possible values of nN. 726 Miss Wrinch and Dr. H. Jeffreys on some This gives the solution of the problem in the most general case, but in most cases more concise information, even though it may be only approximate, is desirable. The case ex- clusively considered in the discussions hitherto given is the very simple one where f(m) is the same for all values of n. The ground for this evaluation may be either complete ignorance of the relative number of white balls among the balls in the world, or knowledge that white and other balls have occurred equally frequently in all the ratios possible in this problem. On this hypothesis it can be shown that the probability of a white ball at the (y+q+1)th drawing is (p+1)/(p+q+2); and if qg is 0, the probability that all the balls in the bag are white is (p+1)/(m+1). It is however very rarely, if ever, possible to assume, on the data available before the sample is taken, that 7() is inde- pendent of n, and cases where it has other values are much more interesting. For instance, we kvow that there is a strong tendency for similar individuals to be associated, so that the greatest and least values of n are more probable on the initial data than the intermediate ones. Or suppose we are considering balls of another colour, say green. It would be absurd to suggest that a bag is as likely to contain green balls alone as to contain no green balls, for we know that in fact green balls are not nearly so common as balls of all other colours together. On the other hand in these cases it is not usually possible to decompose the propositions, whose probabilities we wish to assess in order to find f(n), into equally probable alternatives, so that the principle of sufficient reason cannot be applied; thus though we may be confident that /(7) lies within certain limits, we cannot say that it has’ any partieular value. It will, however, be shown that unless the form of this function is something very remarkable the probabilities to be assigned to particular values of » are practically independent of the prior probabilities, depending almost wholly on the composition of the sample taken, pro- vided this is large enough. To show how this comes about we need an approximation to”C,™-"C, when p and ¢ are fairly large. This is best obtained by a method analogous to that adopted by Dr. Bromwich*. If r and s are both large and r is large compared with s, formula (2) of Dr. Bromwich’s paper yields the approximation 2 log {(r-+s)!}=(r-+s+4) logr—r+} log 20 +3— 3 + terms of order a &e. (2) Lf * Loe. cit. Aspects of the Theory of Probability. 127 Put fo ae CON AO ae UTM EVIpE Ne CC, Then log ("C,"—"C,) =log { (am +ma)!}+ log {(m—2m— mv) I} — log p! — log gq! — log {(m—p+me)!}— log {((m—ny—g—ma)!}, (4) which gives on substituting the above approximation, pro- vided mz is not so great as to invalidate it, an expression that ee to m »—g +1 [Og aur amp m2 ee q) / | ) —p—q PRS) (p (p+g)im —log 24 —4a? ——_—+_... ._ (9) React Pa) ©) Hence the function considered is 2 maximum for n=7p, and its values for other values of » are distributed about this according to the Gauss law. The given sample is said to be a fair one if p/(p+-q) is equal to the ratio of the true number of white balls to the whole number of balls in the bag. The deviation from fairness is therefore represented by x. Substituting the approximation (5) in the formula (1), we find by summation that the probability that « lies between +e is (p+q) log h (p+ q)?me* SOU ONY a3 snrecnmarsan rs (6) a (ptg)ema® as * pgm—p—q) where in the denominator the summation covers a values of S f(n) exp — nm and in the numerator all values between m a Oe Pe) and 4 oe +e). Now the coefficient of a* in a exponent a 7 is always numerically greater than 2(p+q). If then 22(p+q)%e is greater than h, the exponential is less than e7”*, which is very small even when h is not remarkably small. Outside of the range the exponential factor is even smaller, and unless f(”) is so great that its greatness can counteract the aes of the exponential factor, the contribution to the denominator from the values of z not between +e is small. Thus the numerator and denominator are nearly equal and the probability that « lies between +e is nearly 1. 728 Miss Wrinch and Dr. H. Jeffreys on some We therefore have the theorem: if a selection of p+g mem- bers from a class of m «’s consists of p §’s and q not-§’s, and f(n) the prior probability of there being n 6’s and m—n not-£'s in the class 2 is such that when # is numerically greater than e, f(n) is never so great that /(n) exp—2(p+q)(a?—&) is comparable with f{mp/(p+q)}, then the probability that n/m lies within e of p/(p+q) differs from unity by a quantity of order not greater than exp—2(p+q)e*. Thus, unless the distribution of prior probability among various values of x is very remarkable, its precise form does not produce much effect on the probability that the true value lies within a certain range determined wholly by the constitution of the sample itself. It is worthy of note that the range within which it is probable that n/m must lie is of length proportional to (p+q) "2; it does not depend on m to any great extent, but if p and qg are very different the range may “be much shorter than this. This leads to the result that’ there is a strong presumption that a large sample is approximately a fair one even if it is small compared with the whole of the class; and that the range within which the fractional composition is as likely as not to le is much the same however great the whole number of individuals may be. ‘The fact that the error likely to be committed in sampling is, except in extreme cases, limited by the size of the sample itself, may be of some importance in electoral and economic questions. It is also easy to infer from the results obtained that the probability of drawing a 6 at the next trial is not likely to be far from pl(p+q), agreeing with sufficient accuracy with the result of the ordinary theory. In a recent paper * Mr.C. D. Broad has given a suggestive discussion of the problem of inductive inference, in which he adopts the erdinary theory, according to which when g is 0 the probability that all the members are @’s is (p +1)/(m +1). This is not necessarily true, for the reasons given above, but this does not affect Mr. Broad’s main point, which is ‘that in all ordinary cases the number of observed instances is so small compared with the total number of instances that it is impossible to arrive by this means at any noteworthy prob- ability for a general law. General laws are, however, of various kinds. The type to which Mr. Broad devotes most attention is the statement that ‘‘all crows are black,’ based on the fact that all observed crows have been black. Nowa crow is an object defined by the conjunction of a number of properties, which may or may not include blackness. In the * ¢Mind,’ October 1918, pp. 389-404. Aspects of the Theory of Probabihty. 129 former case the inference becomes tautologous, and we are concerned only with the latter. Butit has been shown above that if 2 be the number of black crows in the world, n/m is not likely to deviate from pi(p+q) by more than a quantity of the order of (p+q)~?; and in this case, as p is great and gis zero, we are justified in inferring that the number of crows that are not black is a small fraction of thé whole, which is all that is inferred in pri nee ; for the possibility in Ee tisnal cases of sport, albinos, and so on is well known. The other type of general law is one that is held to be true in every instance of the entities to which it is held to apply. Such a law cannot be derived by means of probability in- ference, for it deals only with certainties. Here Mr. Broad’s argument is valid, and no such law can derive a reasonable probability from experience alone ; some further datum is required. One way of arriving at such laws may be sug- gested here. Suppose we have an a priort belief that either every c has the property ¢ or every « has the property w. If then a single «, say c, is found to satisfy @ but not yr, we ean infer deductively the universal proposition that all 2’s satisfy @. Such cases are fairly frequent: if for instance we consider that either Hinstein’s or Silberstein’s form of the principle of general relativity is true, a single fact contra- dictory to one would amount to a proof of the other in every cease. Before leaving the important question of induction, we propose to consider it in relation to the Venn view. If Venn’s definition of probability be adopted the existence of a numerical estimate of probability depends on the possibility (at least imagined) of indefinite repetition of the data, the truth or falsehood of the proposition whose probability relative to the data is to be estimated being recorded at each repetition. The probability is then the limit of the ratio of the number of favourable cases to the number of all cases. Now on this basis it is never possible, by what has been said already, to prove that in any given case such a limit will exist. All the axioms of the “ undefined concept” theory are therefore indemonstrable, and must be assumed @ prior? in’ the same way. [ven in the simple case of picking indis- tinguishable balls out of a bag the probability of picking any particular individual cannot be assessed without some hypothesis about the limit of the results obtained by making an indefinite number of selections, each bail being replaced after being drawn. In the problem of s sampling induction we can therefore by making enough assumptions of this character, which there seems ‘little or no reason to believe, 730 Miss Wrinch and Dr. H. Jeffreys on some obtain a proof of a proposition superficially the same as the chief theorem of this section ; but let us. consider what this result means on the Venn view. It would mean that if we had a large number of classes, each of m members, and from each we had picked out p+q ‘members, of which p were §’s and g not-§’s, then when the number of such classes is in- definitely increased the fraction of them in which the actual number of 6's does not lie within certain limits would tend to zero as a limit. Thus the already hopeless task of pro- ceeding to the limit of an infinite number of observations becomes in this case the still more complex one of repeating similar classes indefinitely. Such indefinite repetition of znjinite classes is called by Venn the construction of ‘‘cross-series” and forms an essential part of his theory of inference. It is necessary, for instance, in giving a meaning to the proposition connecting the probabilities of a proposition referred to different data P(p.g:h)=P(p:q.-h).P(g:h). For an infinite series is needed to give an account of Por: q-h), which is the limit derived from the frequency of the truth of p among entities for which gand hare true. Such entities, however, are only a part of those for which h holds. Thus to establish a meaning for the number P(p.g:h) we must consider all entities satisfying h, ate ed they satisfy g or not. Thus further series must be constructed which will show how often q is actually true, and this requires, according to Venn, an infinite number of series of entities all satisfying h, so that we can examine in one direction to find the frequency vf p given q¢ and h and in the other to find that of g given h. Thus the difficulty of obtaining enough terms, acute in the simple case, is here intensified ; further, there is no more reason to believe in the existence of limits in this case than there was in the other. The difficulties are merely complicated and not re- moved by the use of cross-series. There is no evidence that Venn ever attempted to meet these difficulties. Indeed, we may conclude from some passages of his work that they had never suggested them- selves. The following passage, for example, occurs in the third edition of his ‘ Logic of Chance,’ page 208. ‘The opinion according to which certain inductive formule are regarded as composing a portion of probability cannot, I think, be maintained. It would be more correct to say... . that induction is quite distinct from probability, yet co- operates in almost all its inferenees. By induction we determine for example whether and how far we can safely generalise the proposition that 4 men in 10 live to be 56: Aspects of the Theory of Probability. 731 supposing such a proposition to be safely generalised we hand it over to Probability to say what sort of inferences can be deduced from it.” The “undefined concept” view of probability can be developed so as to yield a theory of induction adequate for scientific purposes. There are difficulties in the way of obtaining such a theory from the frequency view, and we conclude that the balance is in favour of the “ undefined concept” view. Summary. It is shown that the attempt to give a definition of probability in terms of frequency is unsuccessful. Laplace’s definition, apparently in these terms, really involves im- plicitly the concept of probability and is therefore circular in character. Venn’s definition in terms of the limit of a series is unsatisfactory because there is no reason to believe that his series do in fact usually tend to a limit; it is shown that there are many cases where they do not; and as his process is incapable of being carried out, the existence of such a limit can in any case only be known @ prior if at all, so that his method offers no advantage over that of regarding probability as an entity known to exist independently of definition, intelligible without such definition, and perhaps indefinable. A set of axioms on which a mathematical theory of probability can be based is then given, which seems to offer certain advantages over the current ones. In particular itis capable of covering cases where the principle of sufficient reason cannot be applied to assess probability. It is also shown that a complete theory of probability must allow for the use of infinitesimals. A discussion of sampling induction is given, in which it is shown that when the sample is large enough the prior probabilities of different constitutions of the whole do not usually affect appreciably the probabilities inferred after the samples have been taken. Also the range within which the fractional constitution is as likely as not to he includes the fractional constitution of the sample, and its extent is inversely proportional to the number of the sample itself, whatever be the number of the whole. Bae LXXVI. On the Form of the Trailing Aerial. By H. C. PLumMer *. 1% fae problem of the form assumed by an aerial trailing from an aeroplane, which has been discussed by Captain Hollingworth (p. 452), is in essence scarcely so new as he seems to imagine. For in principle it resembles the problem of Bernoulli +, which is concerned with the shape of a narrow sail held between two parallel yards and filled with wind. The solution of this latter problem, when the wind is supposed to find an immediate issue and the weight of the sail is neglected in comparison with the pressure of the wind, is known to take the form of the commen catenary with its axis horizontal. Captain Hollingworth attempts to take the weight of the wire into account, to a first approxi- mation, but an unfortunate slip makes it appear that the apparent agreement of his results with observation must be largely illusory. It seems probable that for practical purposes the common catenary will of itself give a sufficient approximation. 2. The equation to be solved, as given by Captain Hollingworth, is d’s ds pt siné — de ~dd--” a ‘cos? @—p sin 8’ where p=w/Kv”, w is the weight of the wire per unit length, v is the constant horizontal velocity of the aeroplane, and K is a constant depending chiefly on the altitude and the diameter of the wire. This may be written ib d e=)= pcosé pcos@+2 sin cos 8 wa 5 0 co’’O—psin@ cos? —p sind or p cos @ 1—psin 6— sin?@" “log [en (cos? @—p sin Me Let | 1—psin @— sin? 6=(tana+ sin @)(cota— sin 8), so that p= tana— cola=—2cot2a. _ : 3. Thus, p being small, « is an angle slightly in excess of 45°. * Communicated by the Author. Tt Routh’s Statics, 1. p. 334. On the Form of the Traaiing Aerial. 733 Hence Ne E (tan «+ sin @) (cot a— sin 6) | dé | __ tana— cota ( cos cos @ ) ~ tane+ cota \tane+ sin@ cota— sind eee ad ee tana+ sin@ — — € SO Soreres y 7 ado) cot a— sim Or and therefore tan a+ sin @\7— ©9824 cot a— sin 6 (tan a+ sin €)(cota— siné)=C ( Hence the complete integral is : do ais =c 9 9 23in?2 a ° 2 a (tan e+ sin @)°°**(cot a— sin Oyen a 2) The tension T' and the radius of curvature p at any point of the wire are connected by the relation T= Kv’p(tan a+ sin @) (cot a— sin @) ' = — cos 2a =CKe? (= a+ sin 5) ee) cota— sind 3. The integral in (2) seems by no means too complicated to use directly for the construction of a curve by tangents according to the method employed by Captain Hollingworth. Since « is near 45°, tana, cota, 2 cos? a, and 2 sin’ are all near 1. Instead of using the accurate first integral, how- ever, he substitutes an approximation (X.) by the partial neglect of powers of p above the first, and in doing so commits an error. To the first order in p we have p= —2cot2a, a=l1(7r+ p), tana=2sin?a=1+p, cota=2cos*a=1—4Lp. Hence (tan a+ sin 0)?°* =(1+ip+ sind)’ ®, =(1+ sind+4p) exp. [—$p log (1+ sin @)] =(1+ sin @)[1+43p(1+ sin @) ~'—4p log (14 sin 4)], (cot #— sin 6)" ante =(1— sin 0)[1—4p(1— sin 0) -!+ 4p log (1— sin @)], (tan «+ sin Gyn FF 8 cot ec — sin Aya k 1+ sin @ a) ST: UE ects = sec” @ E + sec? @ sin @+ 2p logs — aa 4 : Phil. Mag. S. 6. Vol. 38. No. 228. Dec. 1919. 3 E 734 Prof. H. C. Plummer on the This is the first approximation to the integrand in (2) when the first order in p is retained, and the result of integration is $—Syp=C[tan 6+ p{4 sec? O@— sec? + tan @logtan(47+4@)}]. . . (A) 4. Captain Hollingworth’s equation XII., which corre- sponds with (4), isa very mixed approximation. Tt contains a further error in the eduction from X., which can be amended by substituting (2—3p)/,/p for the factor (1— 3p) in the second line. Then ie developing the oe form of XII. to the first power in p, or much more simply by applying the same process to X. and atterwar ds integrating, it is found that §— sy =Cl[ tan 6+ p(dsec® A+ sec 6) ], the discrepancy from (4) being due to the error in X. 5. The constant C employed here is not at all the same as Captain Hollingworth’s constant C. In accordance with (3), if W is the suspended weight and the air resistance on it be neglected (which can never be strictly right), so that @=0 at the end of the wire, W=CKe3(tan a)? There are two compensating errors in the deduction of XI. from X. Apart from this, W/Kze? must be a constant slightly in excess of the true one. Since the tables on pp. 460-461 and the corresponding fig. 1 depend on the equation X., they cannot be correct. Int tig. 2 the description of the full and broken curves is at variance with that given in the text and is presumably inverted. ‘The resemblance between the results of calculation and observation is probably due to the predominance of the common catenary, which is the lowest approximation. If so, 1t suggests that the total neglect of the weight of the wire may be justified, at least in the application to practical circumstances in which the condition of uniform horizontal flying will seldom be accurately maintained. ‘The effect is obtained by making p=0, a=45°, and the tension constant. 6. According to (3) above, the radius of curvature and the tension both become infinite when sin@= cote. But the tension supports the attached weight and only part of the weight of the wire, the latter being partly supported by Form of the Trailing Aerial. Te) the normal pressure. Hence the tension cannot become infinite without the weight, and therefore the length, of the wire being infinite too. This is a physical proof that the integral in (2) tends to infinity as sin@ tends to cote. Hence it is clear that there is no point of inflexion at a finite distance along the wire, but the manner of approach to infinity, whether by a rectilinear asymptote or by a curvi- linear branch, does not seem obvious*. But this scarcely affects the practical problen. When the wire is unweighted, the constant C=0, the wire hangs straight at the angle given by sin@=cota, the above equations become illusory, ‘and it is necessary to return to the stage of the problem anterior to the formation of equation IV. 7. When, as in $3 above, the first power of p only is retained, it is possible to express the rectangular coordinates of a point on the curve in terms of the single variable 6. Let the axis of x be taken horizontal in the direction of flight, and the axis of y vertical. Then e = sin 7] = (C1 +psec? @sin @+p log tan (47 +40) ] sec 0 tan 0 dé = sec) + p(t tan? @— tan @+ sec @ log tan (477 +48) | and a =\t cos 0
  • —a) > cosh, a+bcos x da 21 2b asimer fees 2 603) Jatbsina at+bsine = — (b?—a’) =f eogh a se a+6 sin & recommend themselves best for the purpose. They can be retained by a single act of memory, being immediately deducible from one another, and once remembered they can be readily changed to any other desired form. The limiting cases, when b= -+a, hardly require special mention. But when a=1 they all follow the single rule dv 1 dy sin aa: yH=it eos”? the most general function of this type being y= sin’ c(a@+a)/2c’. Enero LXXVII. Connexion between Light and Gravitation. By Sir Otiver J. Lover *. INSTEIN’S great achievement is to begin the association of Gravitation with the electro-magnetic phenomenon called Light, as it has long been associated with the other presumably electrical phenomenon called Matter. The inertia of Matter is affected by motion, and in the case ot Light seems to be generated by motion. Assuming that the mass of Light is proportional to its kinetic energy, while the weight of Lightis proportional to its whole energy, Hinstein’s result obviously follows. (Phil. Mag., August 1917, page 93.) Another mode of regarding the result would attribute to the mere neighbourhood of Matter a small influence on the specific inductive capacity of otherwise free space, analogous to the influence familiarly exerted in itsinterior, In other words, to extend to a gravitational field a trace of the effect known in a cohesional field; a sort of loading of the Aither by Matter at a distance. A spherical field whose re- fractivity was proportional to the gravitational potential would then bend a ray in accordance with \uds stationary, and would give a deflexion inversely as the shortest distance between ray and centre of attraction, as fact requires. ‘The -—1 required at each point to give the observed deflexion, quantitatively, is the ratio between the energy of free fall from infinity and the energy of motion with the velocity of light. The former method seems the simpler, but both require a detailed theory of weight beyond what has yet been attained. A particle falling towards the sun would be accelerated longitudinally and also deflected transversely. Light cannot be accelerated longitudinally: all the accelerative effect of gravity must go (gyrostatically) into transverse deflexion. Consequently when it has reached the shortest distance all the deflexion to be expected from weight has been produced. The other half deflexion is generated during recession. * Communicated by the Author. p98) i LXXVIILI. Periodic Precipitates. By the late Lord Rayirien, O.IL, YRS. [Norr.-—Thigs paper was found in my father’s writing-table. drawer. It is not dated, but I believe it was written in 1917. It was no doubt withheld in the hope of making additions. It is published exactly as found.— RaYLzieu. | | OW my knowledge of this subject, as well as beautiful specimens, to Prof. 8. Leduc of Nantes. His work on the Mechanism of Life * gives an account of the history of the discovery and a fairly detailed description of the modus operandi. ‘According to Prof. Quincke of Heidelberg, the first mention of the periodic formation of chemical precipitates must be attributed to Runge in 1885. Since that time these precipitates have been studied by a number of authors, and particularly by R. Liesegang of Diisseldorf, who in 1907 published a work on the subject, entitled ‘On Stratification by Diffusion’.” In 1901 and again in 1907 Leduc exhibited preparations showing concentric rings, alternately transparent and opaque, obtained by diffusion of various solutions in a layer of gelatine. “The following is the best method of demonstrating the phenomenon. A glass lantern slide is carefully cleaned and placed absolutely level. We then take 5 c.c. of a 10 per cent. solution of gelatine and add to it one drop. of a concentrated solution of sodium arsenate. This is poured over the glass plate while hot, and as soon as it is quite set, but before it can dry, we allow a drop of silver nitrate solution containing a trace of nitric acid to fall on it from a pipette. The drop slowly spreads in the gelatine, and we thus obtain magnificent rings of periodic precipitates of arsenate of silver. . . . The distance between the rings depends on the concentration of the diffusing solution. The greater the concentration, the less. is the interval between the rings.” In considering an explanation, the first question which presents itself is why should the precipitate be intermittent at all? I suppose the answer is to be found in the difficulty of precipitation without a nucleus. At a place where the second material (silver nitrate) has only just penetrated, there may be indeed a chemical interchange, but the resultant (silver arsenate) still remains in a kind of solution. Only when further concentration has ensued, can a pre- cipitate in the usual sense be formed, and a visible line * Translated by W. Deane Butcher, Rebman Limited, Shaftesbnry- Avenue, London. Periodic Precipitates. 139 of silver arsenate constituted. But this line will not thicken itself far outwards, since the silver arsenate forming a little beyond, as the diffusion progresses, will preter to diffuse back and deposit itself upon the nucleus already in existence. In this way the space just outside the nucleus becomes denuded of the weaker ingredient (sodium arsenate). This process goes on for a time, but ultimately when the stronger solution has penetrated to a place where a sufficiency of the weaker still remains, a condition of things arises where a new precipitation becomes possible. But between these lines of precipitation there is a clear space. The process then recurs and, as it appears, with much regularity. This view harmonizes with the observed diminution of the linear period as the concentration increases. We may perhaps carry the matter a little further, con- sidering for simplicity the case where the original boundary is a straight line, the strong solution occupying the whole of the region on one side where w (say) is negative. For each line of precipitation « is constant, and the linear period may be called dx. According to the view taken, the data of the problem involve three concentrations—the two con- centrations of the original solutions and that of arsenate of silver at which precipitation occurs without a nucleus. The three concentrations may be reckoned chemically. There are also three corresponding coefficients of diffusion. Let us inquire how the period dw may be expected to depend on these quantities and on the distance # from the boundary at which it occurs. Now dw, being a purely linear quantity, can involve the concentrations only as ratios; otherwise the element of mass would enter into the result uncompensated. In like manner the diffusibilities can be involved only as ratios, or the element of time would enter. And since these ratios are all pure numbers, dx must be proportional to a. In words, the linear period at any place is proportional, ceteris paribus, to the distance from the original boundary. Jn this argument the thickness of the film—another linear quantity—is omitted, as is probably for the most part legitimate. In imagination we may suppose the film to be infinitely thin or, if it be of finite thickness, that the diffusion takes place strictly in one dimension. The specimens that I have prepared, though inferior to M. Ledue’s, show the leading features sufficiently well. I have used the arsenate of silver procedure, and the broadening of the intervals in passing outwards is very evident when the plate is viewed through a Coddington lens. 740 Geological Society :— Hookham’s Crystals. Another remarkable example cf fine periodic structure was brought to my notice by Mr. George Hookham. In this case double refraction plays an important part and a careful study of the crystals requires the use of a polarizing microscope. I have had the advantage not only of receiving interesting specimens and a sample of one of the solutions employed, but also of witnessing for myself Mr. Hookham’s procedure. The active ingredient is copper sulphate ; but, as it is desired to obtain a film which is initially amorphous, other ingredients must be added. In the solution given me there is both salycine and sugar. Mr. Hookham describes it as consisting of a solution saturated (in the cold) with copper sulphate and salycine, to which is added 3 per cent. of strong syrup. A few drops are placed upon a strip of glass, such as are ordinarily used for microscopic slides, and are spread with the finger. The slide is then warmed over a spirit lamp, when any excess of liquid may be thrown off. By a further application of heat the whole is then dried somewhat rapidly. There is usually immediate formation of erystals at the edges, but throughout a space in the interior the film should be amorphous and nearly invisible. At this stage the amorphous film shows nothing in the polariscope, but in a short time after cooling deve- lopments set in and proceed with rapidity. There is much here to excite admiration and perplexity, as in other similar phenomena of crystallization, but the feature in which | am specially interested, viz. the formation of a structure periodic several thousand times in the incb, does not appear to present itself unless the plate is kept warm until crystal- lization has set in. Mr. Hookham mentions a temperature about 30° F. above that of the room. I have usually placed the slides over hot water pipes or on the mantelpiece. LAXIX. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 668. | May 7th, 1919.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. HE Presiprnt said :—Major R. W. Brock, formerly Director of the Geological Survey of Canada, was called upon last year to undertake on behalf of the War Office an arduous journey in Palestine, during which he had to devote particular attention to On the Geology of Palestine. TA the Dead Sea region. At the request of the Officers of the Society, Major Brock has kindly undertaken to tell us something of his observations in the country. It is needless to say that the region is of surpassing interest to geologists, and I am sure that the Fellows will appreciate the opportunity of hearing how its remarkable features have impressed so acute and experienced a field-geologist. Major Reeinatp W. Brocx, M.A., F.G.S., then proceeded to deliver his lecture on the Geology of Palestine, his observations being summarized as follows :— The following formations are recognized : QUATERNARY. Alluvium. Dunes; Valley and Plains clay, | and Silt; Desert Crust. Heavy Diluvium. ‘Terrestrial. lLisan Formation volcanic (Jordan-lake-beds). | flows, Marine. Upper Calcareous Sand- ( aes stone & Limestone. ashes, Lower Calcareous Sand- | tuffs, ete. stone. TERTIARY. Pliocene. Lacustrine. Eocene. Nummulitic Limestone. Danian : ' volcanics, (Senonian 4 Campanian basalts | Santonian ; | Upper Turonian MEsozotc. Cretaceous. + | Cenomanian. | | | Lower Nubian Sandstone. Jebel-Usdum formation (?). Jurassic. On Lebanon & Hermon only. PALZOZOIC. Carboniferous. Possibly south-east of the Dead Sea. Cambrian. Dolomite and sandstone. PRE-CAMBRIAN. Volcanics and arkose. Red granites and porphyries. Grey granites, gneiss, and crystalline schists. The structure was shown to be that of a tableland bisected by a great rift-valley (graben), and flanked by a coastal plain. A section was exhibited illustrating Hast Jordanland acting asa horst; the boundary-faults of the Jordan Trench; the unequal sinking of the contained blocks; the western section of the tableland sunken with relation to the eastern, and thrown into an asymmetric anticline the limbs of which rise in steps through monoclinal flexures or faults. Lantern-slides were used to illustrate the character of the country and outstanding features in its geology, more particularly the following :—the dependence of the topography upon geological structure, slopes depending on the attitude of the rocks and eleva- tion upon the raising or depressing of fault-blocks or on lava-flows ; basins and sunken areas in the tableland; the scarps bounding the 742 Geological Society :— Jordan Trench, especially the western fault and fault-blocks in the Trench or against its walls that have not sunk equally with others; the upturned block of Jebel Usdum which, with the Dead Sea bottom and the block north of Jericho, indicates a median fault between the boundary-faults ; the interbanding of the Jebel- Usdum salt with sandstone and shales that resemble Nubian Sand- stone; the unconformity of the Jebel-Usdum formation with the Jordan lake-beds (Lisan formation) which with the chemical com- position of the salt (lack of bromine, ete.) shows that it is not a Jordan lake-deposit; high lake-terraces in the centre of the Trench, with corresponding ones north of Tiberias and south in the Araba Valley, showing that the Jordan lake stood 1400 feet above the present Dead Sea, and that there has been no marked warping since their formation; cld gravel-filled cafions of the Arnon and Terka. Main which prove that the level of the Dead Sea before Jordan-lake: days stood at about its present level, and that climatic conditions must have been about the same, also that it did not long precede the Jordan lake; the isan formation of the Jordan lake-beds, thin layers of mechanical and chemical sediments veneered along the Jordan river with fluviatile clays; bad-land topography near the wadis and in the Lisan formation: narrow box canons of the wadis in the Jordan Trench and the more open valleys above producing a sort of ‘ hanging valley.’ In the main, Blanckenhorn’s recent work was confirmed, in particular the fault forming the western border of the Trench and disturbances and sinking in the tableland; but new points were mentioned, such as the evidence of a median fault within the Trench; the sea-cliffs of Lisan and Jebel-Usdum; the wave-cut shelf and the salt of Jebel Usdum; the tilting of the Jebel-Usdum block and its independence from and unconformity with the old lacustrine beds; lack of disturbance and of warping since their deposition ; the age and former level of the Old Dead Sea and the recent rise in the present sea, the latter indicating an increase in moisture and not drier conditions as generally supposed. May 2:st.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. The following communications were read :— 1. ‘The Silurian Rocks of May Hill.’ By Charles Irving Gardiner, M.A., F.G.S. With an Appendix by Frederick Richard Cowper Reed, Sc.D., F.G:S. The district of May Hill comprises a small area of ashy grits, which Dr. Callaway in 1900 considered to be of Pre-Cambrian age. The evidence now available does not seem to warrant any definite opinion as regards the age of these beds. Llandovery sandstones. are extensively developed, and are of Upper Llandoveryage. They consist of a lower division of coarse sandstones and conglomerates, and an upper one of fine sandstones. No beds of Tarannon age occur. The Silurian Rocks of May fill. TAB: The Woolhope Limestone is never thick, and fossils in it are very few. The Wenlock Shales and Limestone show a normal deveilop- ment. The latter is very fossiliferous, and shows coral-masses in the position of growth. | The Ludlow Beds are, in the main, of a brown sandy nature. No Aymestry Limestone is present, and the Ludlow Beds cannot be separated into an upper and a lower division. A bone-bed is seen at the top of the Ludlow Beds by the side of the road near Blaisdon. This was described by H. E. Strickland in 1863, who saw it in the railway-cutting close by. | Downton Sandstone oceurs in the north of the district, where it is about 300 feet thick; but it is only some 11 feet thick near Blaisdon on the south. It is conformably overlain by Old Red Sandstone. The Silurian rocks are arranged in an anticline in the part of the district where May Hill is, but elsewhere show no such arrange- ment. On the north they are much broken bv faults. Near Flaxley, in the extreme south, rocks from the Wenlock Shale to the Old Red Sandstone inclusive are overfolded. Dr. F. R. C. Reed describes a new species of Lichas from the. Wenlock Limestone and a new variety of Calymene papillata. 2. ‘The Petrography of the Millstone Grit Series of Yorkshire.” By Albert Gilligan, D.Sc., B.Sc., F.G.S. Since the pioneer work of Sorby on this subject, published in 1859, the clastic deposits of the Carboniferous System have been unaccountably neglected by petrologists. The author has followed the usual methods of investigation, and has collected a large number of pebbles and specimens from widely - separated areas which have been examined microscopically. Numerous separations of the heavy minerals have also been made from all types of rock, varying from coarse conglomerates to shales, which occur in the series. Quartz-pebbles are dominant, and vary much, both in size and in colour. The largest are found in the coarse-grained beds at the bottom and top of the series. They often show double-sphenoid forms suggestive of derivation from mechanically-deformed rocks, which inference is shown to be correct by the undulose extinction, the crenulate and mylonized structure seen when sections of them are examined in polarized light. Blue and opalescent quartz is very common, containing inclusions often of indeterminable character arranged in streams or rows; others contain liquid with movable bubbles, while needles and hair- like inclusions are also usually present. The quartz of the finer material is similar in character, and the inclusions in the grains suggest that it has been originally derived for the greater part from such rocks as gneisses and schists. Felspar-pebbles are abundant in all the coarse beds. They are dominantly microcline or microcline-microperthite, and, when 744 Geological Society :— broken, are found to be perfectly fresh, the lustre of the cleavage- faces being most remarkable. Blebs of quartz are frequently present in these felspars. In many of the rock-sections grains of microcline and oligoclase, quite fresh and unaltered, are common. Fragments showing the intergrowth of blue or opalescent quartz -and microcline are fairly abundant. Chert-pebbles are plentiful in the coarse beds at the base of the series ; they are also sporadically distributed throughout the upper beds, and in some of these oolitie structure has been observed. One pebble of silicified oolite shows a microscopic structure strongly resembling a structure found in the Torridon Sandstone. A few fragments containing microscopic organisms have also been obtained. Mica is not plentiful in the coarser beds, but increases in amount with decrease in grade of the material. From the Middle Grits of Airedale a remarkable assemblage of pebbles has been obtained, including the following types :—gneisses, granites, schists, quartz- and felspar-porphyries, quartzites, grits, sandstones, and mudstones. One of these pebbles has been recog- nized as the black schist associated with the Blair Athol-a-Nain Limestone of Scotland. Another pebble is doubtfully referred to the rhomb-porphyry of the Christiania region. The results of the investigations into the heavy mineral contents may be summarized as follows, dividing for this purpose the Millstone-Grit Series into three more or less well-defined groups :-— (a) Lower Division—Base of the Ingleborough Grit to the base of the Leathley Sandstone. (b) Middie Division—Leathley Sandstone to the base of the Flags below the Rough Rock. (c) Upper Division—Flags and Rough Rock. The minerals are tabulated in decreasing order of relative abundance :— (a) Coarse beds contain garnet, ilmenite and leucoxene, zircon, tourmaline, rutile, monazite and magnetite. Fine beds contain zircon, rutile, garnet, and tourmaline. (6) Coarse beds contain zircon, rutile, garnet, tourmaline, ilmenite and leuco- xene, magnetite and monazite. Fine beds contain zircon, rutile, tourmaline, and garnet. Some of the separations from the shales of Otley Chevin were almost entirely zircons, only a few grains of other minerals being present. (c) Coarse beds contain garnet, ilmenite and leucoxene, zircon, rutile, tour- maline, monazite and magnetite. The Flags at the base of the Rough Rock contain zircon, rutile, garnets, and tourmaline. The monazite has been determined by spectroscopical and chemical tests. In view of the similar work which is being done among the younger sedimentary rocks, it is important to record that, although the author has not yet discovered staurolite in the Millstone Grit, he has found it to be common in some of the sandstones near the top of the Coal Measures in Yorkshire :—namely Ackworth Rock, Pontefract Rock, and the Red Rock of Rotherham, and also in -basement Permian at Conisborough. Petrography of Millstone Grit Series of Yorkshire. 745. In Yorkshire alone, to which area for the greater part the researches have been limited, the Millstone Grit forms the surface of 840 square miles; while, if that which lies beneath the newer rocks and that represented by outlers on the Pennine Fells were taken into account, it must have extended over at least 2000 square miles. If 1000 feet be taken as its average thickness, the York- shire Millstone Grit would represent a volume of 400 cubic miles, the equivalent of a range of mountains 800 miles long, 1 mile: high, and 1 mile wide at the base. The beds attenuate southwards, and the only possible conclusion from their stratigraphy, reached by Sorby, and later confirmed by Edward Hull and A. H. Green, is that the material was derived from a northern source. The evidence which the author has. obtained corroborates this view. The ancient land-mass of the Midlands must be excluded as a. possible source for more than a small fraction of the material, both on account of the inadequacy of the area and on account of its lithological constitution. The Lake District was probably submerged in Viséan times, and for that reason could not have supped material to the Millstone Grit. Further, the abundance of monazite in these beds and its. absence from the granites of the Lake District, as shown by R. H. Rastall & W. H. Wilcockson, definitely exclude that area. Southern Scotland may have contributed to the homotaxial deposits. farther north than Yorkshire, but inadequacy of area is again pointed out. Thus, by elimination of other areas for one reason or another, the author shows that the most probable source of the material lay still farther north in a land-mass of continental extent, of which Scandinavia and the North of Scotland represent the remaining fragments. In these areas alone can the mineralogical demands of the Millstone Grit be satisfied, and the author institutes a comparison between the Torridon Sandstone and the Millstone Grit, which shows that their similarity of constitution is altogether too great to be merely fortuitous. He infers that, despite their disparity im age, they had a common source in that northern continent. , . That continent had probably been base-levelled in pre-Millstone Grit times, and the advent of this period was brought about by renewed uplift rejuvenating the rivers, which removed the old rotted soil-mantle and exposed fresh unleached rock. The extension of the land-mass across the North Atlantic would produce a monsoon type of climate, and the rock-débris broken up under semi-arid conditions, as seems clear from the extreme freshness of the felspars in the grits, would be swept along rapidiy by floods to the deltas of the large rivers. The author concludes by postulating one such large trunk river flowing southwards from the northern continent, and receiving tributaries from what are now Northern Scotland and Seandinavia, debouching somewhere off the north-east coast of England, the deltaic material of which (now consolidated) forms the Millstone Grit. 4AG Geological Society :— June 4th.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. The following communications were read :— 1. ‘On the Dentition of the Petalodont Shark, Olimaaodus. By Arthur Smith Woodward, LL.D., F.R.S., P.L.S., F.G.S. 2. ‘A New Theory of Transportation by Ice: the Raised Marine Muds of South Victoria Land (Antarctica).’ By Frank Debenham, B.A., B.Sc., F.G.S. A series of deposits of marine muds are found on the surface of floating ‘land-ice’ in the deep bays of Ross Sea (Antarctica). Similar deposits are also found on land up to a height of 200 feet, in some cases on old ice, in other cases on moraine. The deposits are briefly described, and former theories concerning ‘them are discussed. A new theory is put forward, prefaced by an account of the nature of the typical ice-sheet which bears them. The upper surface of the sneet is known to suffer a net annual decrease, and evidence is given to show that the lower surface has a net increase by freezing from below. The theory is that the sheet will freeze to the bottom in severe ‘seasons, and enclose portions of the sea-floor. Owing to the method of growth of the sheet by increments from below, the enclosed portions will ultimately appear on the surface, thus being raised vertically as well as translated horizontally. The application of the theory to other localities is_ briefly sketched, with especial reference to the shelly moraines of Spits- bergen and the shelly drifts of the glacial deposits of Great Britain. The general results of such a method of transportation are shown vo be the raising of marine deposits above their initial level, the preservation of the organisms, the deposition of small patches of muds with ordinary supra-glacial moraine, and the col- lection of remains of fauna from different depths in one horizon. June 25th.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. The following communications were read :— 1. ‘Outlines of the Geology of Southern Nigeria (British West Africa) with especial reference to the Tertiary Deposits.’ By Albert Ernest Kitson, C.B.E., F.G.S., Director of the Geological Survey of the Gold Coast. The oldest rocks in Southern Nigeria comprise a series of quartzites, schists of various kinds, blue and white marble, grey limestones, altered tuffs and lavas, amphibolites and gneisses. ‘Their strike varies from west-north-west and east-south-east to north-east and south-west. They occur in the north-western portion of the country (Yorubaland), north of lat. 7° N., and in the Oban- Hills region in the east. They may be classed provisionally as Pre- Cambrian. Intruded into these are large masses of granites of various kinds, syenite and diorite, with pegmatite- and aplite-dykes. In some parts these rocks have shared in the dynamic alteration Geology of Southern Nigeria. 747 to which the oldest series has been subjected ; but usually they are practically unchanged. ‘There is no definite evidence to show to what period they belong, but they are certainly Pre-Cretaceous, probably Middle and Early ‘Paleozoic. So far as observed, there is a great hiatus between the Pre- Cambrian and the next known sediments, the Upper Cretaceous. Normally, these are slightly inclined rocks: they include (1) Marine fossiliferous shales, mudstones, limestones, and sand- ‘stones in the great valley between the Oban Hills and the Udi plateau. The fossils are principally ammonites and mollusca ; (2) Estuarine fossiliferous carbonaceous shales, mudstones, anal sandstones along the eastern foot of the Udi escarpment ; (3) Lacustrine sandstones, shales, and black coal-seams, with numerous plant-remains; and (4) Fluvio-lacustrine sands, shales, and pebble-bands in the lower and upper parts of the Udi plateau. Flanking this plateau on the south and south-east, and extending thence over the southern part of the great valley to the Cross River, is a series of Eocene estuarine shales, clays, and marls, with septarian nodules and pieces of coal and resin, and a rich fauna consisting principally of mollusca, but including fragmentary remains of whales, birds, fishes, and turtles. A thick series of sandstones, mudstones, shales, and seams of brown coal forms a large portion of the basin of the Niger, west of the Udi plateau. ‘These rocks appear to be of lacustrine origin, and are probably Eocene. ‘They contain numerous remains of un- determined plants, largely of dicotyledonous types. Their relation ‘to the Cretaceous and to the Hocene estuarine series 1s uncertain. In the Ijebu Jebu district are bituminiterous sands and clays with Plocene estuarine shells. Extending over practically the whole of the country south of lat. 7° 10' N., and west of the great valley of the marine Cretaceous is ‘a varying thickness of (usually unstratified) clayey sands, probably late Pliocene—the Benin Sands Series of Mr. J. Parkinson. Along the coast-line and extending for considerable distances up the Niger and Cross Rivers are fluviatile, deltaic, littoral, and swamp gravels, sands, and muds of Pleistocene and recent age. In the Cross-River basin, intruded into the marine Cretaceous, are voleanic necks of decomposed agglomerate, and sills (?) and dykes of olivine- ‘dolerite. These are probably Pre-Hocene. Faulting and local folding are visible in various portions of this district. Numerous silver-!ead-zine-iron lodes occur along these fault-lines, with brine-springs in several localities. The Yorubaland erystalline rocks contain magnetite in considerable quantities, while these and the crystalline rocks of the Oban Hills show smaller quantities of cassiterite, gold, monazite, and columbite. 2. ‘Notes on the Extraneous Minerals in the Coral- Limestones of Barbados.’ By John Burehmore Harrison, C.M.G., M.A., E.G.S., F.1.C., and C. B. W. Anderson. Characteristic representative specimens of the fossil reef-corals and of the beach-rock of the high-level and low-level limestone 748 Intelligence and Miscellaneous Articles. terraces of Barbados were examined chemically and microscopically, in order to ascertain the composition, nature, and origin of their extraneous mineral contents. A special method was used, whereby the extraneous mineral matters were separated, practically with- out alteration, from large quantities of the limestones. Chemical analyses of the residua were made, and the results of these and of the microscopical examinations are tabulated in the paper. The extraneous minerals present were found to be apparently-fresh and largely-unaltered fragments of wind-borne volcanic minerals and glass. It was found that the voleanic minerals enclosed in the reef-corals on which they fell have been protected from change ; those in the clastic limestone or bed-rock show signs of detrition and weathering prior to the consolidation of the limestone. Similar minerals separated from clay normally formed and accumulated in a pothole in the limestone supply evidence of weathering changes after being set free from the rock. It is shown that the composition of the sedentary residual soils on the higher limestone-terraces of Barbados corresponds in its essential: parts with the residua separated, either naturally or artificially, from the limestone. The proportions of magnesium carbonate present in the coral- rock are briefly discussed, and complete analyses of the high-level and the low-level limestones are given. A note on the proportions of titanium oxide in the Barbados Oceanic clays and in some of the Challenger and Buccaneer deep-sea dredgings is appended to: the paper. LXXX. Intelligence and Miscellaneous Articles. ON A NEW FORM OF CATENARY. To the Editors of the Philosophical Magazine. Dear Sirs,— M4* I be permitted through you toexpress to Mr. Mcleod my sincere apologies for having unintentionally published my paper “ On a New Form of Catenary ” when the subject had already been dealt with by him. I was requested to undertake the investigation during the summer of 1917, and a summary of it was incorporated in a monthly report of the R.N.A.S. Establishment, Cranwell, early in 1918. This was a highly confidential document and was never published ; and since then the paper has been lying by awaiting permission for open publication, which was only granted this summer. I may add that I was quite unaware of Mr. McLeod’s work until I read his letter in your November issue. Yours faithfully, College of Technology, J. HOLLINGWORTH. Manchester, Noy. 10th, 1919. ASTON. : Phil. Mag. Ser. 6, Vol. 38. Pl. [X. Ine, H Fig. 2. Fie. 3. Niele © INDEX to VOL. XXXVIII. ans ADENEY (Prof. W. E.) on the rate of solution of nitrogen and oxygen in water, 317. Aerial, on the form of the trailing, 452, 661, 732, 744. Aleebra, on a non-metrical vector, 115. Allen (Prof. F.) on the discovery of four transition points in the spec- trum and the primary colour sensa- tions, 55; on the persistence of vision of colours, 81. Alpha particles, on the recoil of, from light atoms, 533. Amsler planimeter, on the slip-curves of an, 489. Anderson (C. B. W.) on the coral- limestones of Barbados, 747. Antonofft (Prof. G. N.) on surface- tension and chemical interaction, 417. Aston (Dr. I’. W.) on a positive ray spectrograph, 707. Atomic orientation, on, 463. Atoms, on the structure of, 256; on the recoil of alpha particles from light, 533. Audition, on the resonance theory of, 164. Barrow (G.) on high-level sands and eravels at Little Heath, 662. Barton (Prof. FE. H.) on the resonance theory of audition, 164; on a syn- tonic hypothesis of colour vision, 338. Becker (H. G.) on the rate of solution of nitrogen and oxygen in water, 317. Bedson (11.) on a new magnet-testing instrument, 542. Bessel-Clifford function, on ths, 501. Books, new :— Lewis’s System of Physical Chemistry, 197; Findlay’s Osmotic Pressure, 198; Livens’s The Theory of Electricity, 199 ; Annuaire du Bureau des Longi- tudes, 200; Shaw’s Lectures cn the Philosophy of Mathematics, 495; French’s Applied Optics (Steinheil & Voit), 661. Bradford (S$. C.) on the molecular theory of solution, 696, Brock (Major R. W.) on the geology of Palestine, 741. Phil. Mag. 8. 6. Vol. 38. No. 22 Bromwich (Dr. T. J. I’a.) on electro- magnetic waves, 145 ; on approxi- mations in the theory of proba- bilities, 231. Browning (Miss EK. M.) on the reso- nance theory of audition, 164; on a syntonic hypothesis of colour vision, 338. Bursill (A.) on a new magnet-testing mstrument, 542. Campbell (Dr. N.) on time-lag in the spark discharge, 214; on the meas- urement of time, 652. Carslaw (Prof. H. 8.) on problems in conduction of heat, 200. Catenary, on a new form of, 452, 661, 732, 748. Chapman (Dr. 8S.) on the possibility of separating isotopes, 182. Chemical interaction, on surface ten- sion and, 417. Clifford function, on the, 501. Colour sensations, on four transition points in the spectrum and the primary, 55. vision, on a syntonic hypothesis of, 338; on a theory of, 402. Colours, on the persistence of vision of, 81. Conductivity of solid insulators, on the thermal, 705. Contrast, on the fundamental law for the true photographic rendering of, 187, 633, 637. Critical speeds of machinery, on, 395. Cyclone, on the travelling, 420. Cylinders, on the potentials of ellip- tic, 465, David (Major W. T.) on the origin of radiation in explosions, 492. Debenham (F’.) on the raised marine muds of South Victoria Land, 746. Diffraction, on, by apertures with curvilinear boundaries, 289; on the use of the Clifford function in problems of, 520, Du Toit (Dr. A. L.) on the geology of the marble delta of Natal, 665. Earth, on the use of the Clifford function in the theory of the figure of the, 513. ‘ Eason (A. B.) on eritiecal speeds of machinery as related to vibrations, 395. So Peo gO 0. oF 700 Electric conductivity along a glow discharge and in flames, on a method of measuring the, 352. field constants, on, 201. Electromagnetic waves, on, 90, 143; on the propagation of, round the earth, 569. Electroscopes, on a method of mea- suring the capacity of, 245. Elliptic cylinders, potentials of, 465. Everett (Miss A.) on proofs of elementary theorems ot oblique diffraction, 480. Extinction, on the molecular theory of, 269. Eye, on the kinematics of the, 685. Field constants, on electric and mag- netic, 20]. Films, on the optical properties of metallic, 98. Flames, on the conductivity in, 352. Fluids, on the spreading of, on glass, 49; on the flow of, past a rigid boundary, 433; on the motien of, in a vortex, 997. Force-function in the theory of rela- livity, on the, 118. Friction, on static, 32. Gardiner (C. I.) on the Silurian rocks of May Hill, 742. Gases, on the ignition of, by hot wires, 613. Geological Society, proceedings of the, 267, 425, 496, 662, 740. Gilbert (C. J.) on high-level sands and gravels at Little Heath, 662. Gilligan (Dr. A.) on sandstone dykes in the Cumberland coaltield, 499 ; on the millstone grit series of Yorkshire, 748. Glass, on thespreading of fluids on, 49. Glow discharge, on the conductivity along a, 302. Gold, on the series system in the spectzum of, 1. Gold-leaf electroscopes, on a method of measuring the capacity of, 246. Gravitation, on hght and, 737. Gray (Prof. A.) on electric and mag- netic field constants, 201. Greenhill (Sir G.) on the Bessel- Clifford function and its applica- tions, 501. Gwyther (R. F.) on the equations for material stresses, 255. Hammick (D. L.) on latent heat and surface energy, 240. LN Dae Hardy (J. K.) on statie friction and lubricating properties, 32. Hardy (W. B.) on static friction and lubricating properties, 32; on the spreading of fluids on glass, 49. Harrison (J. B.) on the coral-lime- stones of Barbados, 747. Heat, on the conduetion of, 200; on latent, and surface energy, 240, Hicks (Prof. W. M.) on the series system in the spectrum of gold, 1; on the value of the silver oun, 301; on the mass carried forward by a vortex, 597. Hollingworth (Capt. J.) on a new form of catenary, 452, 748. Houstoun (Dr. R. A.) on a theory of colour vision, 402. ; Hydrogen, on the coefficient of vis- cosity of, 582. Insulators, on the thermal conducti- vity of solid, 705. Iron rods, on the magnetic effects of vibration ir, 528. Isotopes, on the vapour pressure and affinity of, 173; on the possibility of separating, 182. Jackson (L. UC.) on the stability of atoms, 256. Jettreys (Dr. H.) on the ferce-function in the theory of relativity, 1138; on the derivation of the Lorentz- Einstein transformation, 349; on some aspects of the theory of probability, 715. Kendall (Prof. P. l’.) on wash-outs in coal-seams, 496. Kinematics of the eve, on the, 685. Kitson (A. E.) on the geology of Southern Nigeria, 746. Lamb (Prof. H.) on the kinematics of the eye, 685. Latent heat and suriace energy, cn, 240, Leide (A. B.) on precision-measure- ments in the X-ray spectra, 647. Light, on the molecular theory of refraction, reflexion, and extinction of, 268; on the diffraction of, by apertures with curvilinear bound- aries, 289 ; on proots of elementary theorems of oblique refraction of, 480; on terrestrial refraction of, 546; on the scattering of, in the refractive media of the eye, 568; and gravitation, on, 797. INDEX: {Lindemann (Prof. F. A.) on the vapour pressure and affinity of iso- topes, 173; on magnetic storms, 669. Lodge (Sir O.) on light and gravita- tion, 737. Loeb (Dr. L. B.) on the recoil of alpha particles from light atoms, 553. Lorentz-Kinstein transformation, on the derivation of the, 349. Lubricating properties of substances, on the, 32. Machinery, on critical speeds of, 395. McLeod (A. R.) on terrestrial re- fraction, 546; on a new form of catenary, 661. Magnet testing instrument, on a new, 542. Magnetic effects of vibration in iron rods, on the, 528. field constants, on, 201. storms, on the theory of, 669. Material stresses, on the equations for, 235. Merton (Dr. T. I.) on an experiment relating to atomic orientation, 463. Metals, ignition of gases by hot, 613. Mitra (3. K.) on the large-angle diffraction by apertures with curvilinear boundaries, 289. Molecular theory of refraction, re- fiexion, and extinction, on the, 269. of solution, on the, 696. Mukerjee (A. T.) on a method of measuring the capacity of gold- leaf electroscopes, 245. Natanson (Prof. L.) on the molecular theory of refraction, reflexion, and extinction, 269. Nitrogen, on the rate of solution of, by water, 317; on the coefficient of viscosity of, 582. Oblique refraction, on proofs of elementary theorems of, 480. un, on the value of the silver, 501. Owen (8S. P.) on a vapour pressure equation, 655. Oxygen, on the rate of solution of, py water, 317; on the coefficient of viscosity of, 582. Partial tones of bowed stringed in- struments, on the, 573. Pascoe (Lieut. 1. H.) on the early history of the Indus, Brahmaputra, and Ganges, 667. Periodic precipitates, on, 738. Photographic rendering of contrast, on the, 187, 633, 637. certain 7ol Planimeter, on the slip-curves of an Amsler, 489. Plummer (Prof. H. C.) on the form of the trailing aerial, 732. van der Pol (Dr. b.) on the pro- duction and measurement of short continuous electromagnetic waves, 90; on a method of measuring without electrodes the conduc- tivity at various points along a olow discharge and in flames, 3o2; on the propagation of electromagnetic waves round the earth, 365. Porter (Prof. A. W.) on the law for the true photographic rendering of contrast, 187, 637. ee ray spectrograph, on a, GOT Potassium, on the optical properties of films of, 98. Potentials of elliptic cylinders, on the, 465. Precipitates, on periodic, 738. Probability, on approximations in the theory of, 231; on some aspects of the theory of, 715. Radium and uranium, on the relation between, 483. ; Raman (Prof. C. V.) on the scattering of light in the refractive media of the eye, 568; on the partial tones of bowed. instruments, 573. Rankine (Dr. A. O.) on the magnetic effects of vibration, 528. Rayleigh (Lord) on the travelling cyclone, 420; on periodic precipi- tates, 738. , obituary notice of, 427. Reflexion, on the molecular theory of, 269. Refraction, on the molecular theory of, 269; on proofs of elementary theorems of oblique, 480; on terrestrial, 546, Refractive media of the eye, on the scattering of light in the, 568. Relativity, on the force-function in the theory of, 1138. Renwick (I. F.) on the law for the true photographic rendering of contrast, 633. Resonance theory of audition, on the, 164. Richardson (Lieut.-Col. A. R.) on stream-line flow from a disturbed area, 433, = J Le 4 : : 752 LN GDS Sen (N. R.) on the potentials of elliptic cylinders, 465. Siegbahn (Dr. M.) on precision-mea- surements in the X-ray spectra, 639, 647. Silberstein (Dr. L.) on non-metrical vector algebra, 115; on a time- scale independent of space mea- surement, 382. Silver oun, on the value of the, 301. Slade (Dr. 03) ) on the law for the true photographic rendering of contrast, 187, 657. Slip-curves of an Amsler plani- meter, on the, 489. Soddy (Prof. F.) on the relation between uranium and radium, 483. Sodium, on the optical properties of films of, 98. Solution, on the molecular theory of, 696. Sommerville (Dr. D. M. Y.) on the slip-curves of an Amsler plani- meter, 489. Spark discharge, time-lag in the, 214. : Spectrograph, on an X-ray, for medium wave-lengths, 647; on a positive ray, 707. Spectrum, on the series system in the, of gold, 245; on four tran- sition points in the, and the primary colour sensations, 5d. Stability, on the Clifford function in problems of, 501. Stream-line flow from a disturbed area, on, 455. Stresses, on the equations for material, 235. Stringed instruments, on the partial tones of bowed, 573. Surface energy, on latent heat and, 240, tension and chemical inter- action, on, 417. Terrestrial refraction, on, 546. Time-lag in the spark discharge, on, 214. -scale independent of space measurement, on a, 382, 652. Thermal conductivity, on the, of solid insulators, 705. Thornton’ (Prof. W. M.) on the ignition ot gases by hot wires, 613; on the thermal conductivi ity of solid insulators, 705. Todd (Prof. G..W.) on the Kundsen vacuum gauge, 381; on a vapour pressure equation, 655. Trechmann (Dr, C. T.) on interglacial loess in Durham, 425. Tungsten, on the A-ray spectrum of, 639. Urauium and radium, on the relation between, 483. Vacuum gauge, on the theory of the Knudsen, 381. Vapour pressure equation, on a, 695. Vector algebra, on a non-metrical, 115. Vibration of flcors and critical speeds of machinery, on, 395; on the magnetic effects of, 528. Violin family of instruments, on the partial tones of the, 573. Viscosity. coefficients of, of hydrogen, nitrogen, and oxygen, 482. Vision, on tke persistence of of colours, 81. Vortex, on the mass carried forward by a, 597. Water, on the rate of solution of nitrogen and oxygen by, 317. Waves, on electromagnetic, 90, 145; on the propagation of electro- magnetic, round the earth, 365. Whitaker (W.) on the section at Worms Heath, 668. Wires, on the ignition of gases by hot, 613. Wood (Prof. W.) on the optical properties of films of sodium and potassium, 98. Worth (R. H.) on the geology of the Meldon Valleys, 267. Wrinch (Miss D.) on some aspects of the theory of probability, 745. X-ray spectrum of tungsten, on the, 639. Yen (Prof. Kia-Lok) on the co- efficients of viscosity of Ps drogen, nitrogen, and oxygen, 582. Young (Dr. F, B.) on the magnetic effets of vibration, 528. END OF THE THIRTY-EIGHTH VOLUME. Printed by TayLor and Francis, Red Lion Court, Fleet Street. le n= Raa Yet a | | S S)5 6 7g il WANN 3 9088 ————